Passive VHF Radar Interferometer Implementation, Observations

Transcription

Passive VHF Radar Interferometer Implementation,
Observations, and Analysis
Melissa G. Meyer
A thesis submitted in partial fulfillment of
the requirements for the degree of
Master of Science in Electrical Engineering
University of Washington
2003
Program Authorized to Offer Degree: Electrical Engineering
Graduate School
This is to certify that I have examined this copy of a master’s thesis by
Melissa G. Meyer
and have found that it is complete and satisfactory in all respects,
and that any and all revisions required by the final
examining committee have been made.
Committee Members:
John D. Sahr
Donald B. Percival
Robert H. Holzworth
Robert S. Hiers
Rachel A. Yotter
Date:
In presenting this thesis in partial fulfillment of the requirements for a Master’s
degree at the University of Washington, I agree that the Library shall make its copies
freely available for inspection. I further agree that extensive copying of this thesis is
allowable only for scholarly purposes, consistent with “fair use” as prescribed in the
U.S. Copyright Law. Any other reproduction for any purpose or by any means shall
not be allowed without my written permission.
Signature
Date
Abstract
Passive VHF Radar Interferometer Implementation,
Observations, and Analysis
by Melissa G. Meyer
Chair of Supervisory Committee:
Professor John D. Sahr
Electrical Engineering
In this work, we describe a new method for remote sensing of ionospheric plasma
turbulence: an extension of the passive radar technique to include interferometry.
We discuss the implementation of a passive radar interferometer, and show many
observations of varied targets, including ground clutter, aircraft, and meteor trails, as
well as plasma density irregularities in the E-region ionosphere. Because of the very
fine resolution of our instrument (as fine as 0.06◦ in azimuth, or 1.5 km2 at a distance
of 1000 km), we can form two dimensional images of these targets, and we are able to
show that many E-region irregularities exist in compact scattering volumes (as narrow
as 30 km in the transverse dimension), and that significant structure exists on very fine
scales. We also describe in detail the passive radar interferometer cross-correlation
estimator and its statistical properties, and perform an analysis of the resolution
capability of the instrument. Finally, we demonstrate how the interferometer can be
used to measure geophysical information, such as electric fields and velocity shears
across scattering volumes in the ionosphere.
TABLE OF CONTENTS
List of Figures
v
List of Tables
ix
Glossary
x
Notation Conventions
Chapter 1:
xii
Introduction
1
1.1
The Manastash Ridge Radar . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2.1
The Manastash Ridge Radar Interferometer . . . . . . . . . .
6
What can be Learned . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.3
Chapter 2:
System Description
8
2.1
An Example of Target Detection with MRR . . . . . . . . . . . . . .
12
2.2
Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.3
Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
Chapter 3:
Interferometer Observations
18
3.1
Ground Clutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.2
Airplanes and Compact Targets . . . . . . . . . . . . . . . . . . . . .
23
3.3
Geophysical Targets
25
. . . . . . . . . . . . . . . . . . . . . . . . . . .
i
3.4
3.3.1
Meteor Trails . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
3.3.2
Auroral Observations . . . . . . . . . . . . . . . . . . . . . . .
31
Analysis of Data Products . . . . . . . . . . . . . . . . . . . . . . . .
33
3.4.1
Two-Dimensional Interferometer Images . . . . . . . . . . . .
37
3.4.2
Azimuth Aliasing . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.4.3
Other Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
Signal Processing
43
Chapter 4:
4.1
Preliminary Assumptions . . . . . . . . . . . . . . . . . . . . . . . . .
43
4.2
MRR Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.3
Motivation for Statistical Calculations . . . . . . . . . . . . . . . . .
49
4.4
The Interferometer Estimator . . . . . . . . . . . . . . . . . . . . . .
51
4.5
Expected Value of the Interferometer Estimator . . . . . . . . . . . .
53
4.5.1
A Model for the Received Scatter . . . . . . . . . . . . . . . .
53
4.5.2
A More Complicated Scatter Model . . . . . . . . . . . . . . .
59
4.5.3
The Expected Value after Coherent Integration . . . . . . . .
63
Variance of the Interferometer Estimator . . . . . . . . . . . . . . . .
64
4.6.1
Variance with the Simple Scattering Model . . . . . . . . . . .
64
4.6.2
Variance after Coherent Averaging . . . . . . . . . . . . . . .
69
4.7
Interferometer Implementation Issues . . . . . . . . . . . . . . . . . .
69
4.8
Statistics of Other Estimators and Additional Comments . . . . . . .
73
4.8.1
The Periodogram (FFT-Based Spectrum Estimation) . . . . .
73
4.8.2
Application to MRR Signal Processing . . . . . . . . . . . . .
74
4.8.3
Periodogram Variance . . . . . . . . . . . . . . . . . . . . . .
76
4.6
ii
Chapter 5:
Simulations
78
5.1
A Simulated Target . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
5.2
Empirical Estimation of Cross Spectrum Variance . . . . . . . . . . .
81
5.3
Evaluation of Interferometer Resolution . . . . . . . . . . . . . . . . .
87
5.4
Target Detection by Phase Compactness . . . . . . . . . . . . . . . .
91
Chapter 6:
Analysis of Ionospheric Events with Interferometry
94
6.1
Description of Irregularity Backscatter . . . . . . . . . . . . . . . . .
95
6.2
February 2, 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
6.2.1
99
6.3
6.4
Natural Progression to Two-Dimensional Images . . . . . . . .
March 24, 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.3.1
Velocity Vector Measurements . . . . . . . . . . . . . . . . . . 102
6.3.2
Electric Field Measurements . . . . . . . . . . . . . . . . . . . 108
March 30, 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Chapter 7:
Conclusions
112
7.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.2
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Bibliography
116
Appendix A: Variance Derivation
120
A.1 Isserlis’ Gaussian Moment Theorem Applied to Analytic Radar Signals 120
A.2 Variance Derivation, Continued . . . . . . . . . . . . . . . . . . . . . 122
A.3 Analysis of 8th -order Correlations . . . . . . . . . . . . . . . . . . . . 122
A.3.1 Term 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
A.3.2 Term 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
iii
A.4 The Full Variance Expression . . . . . . . . . . . . . . . . . . . . . . 128
Appendix B: Simulating Appropriate Passive Radar Signals
130
B.1 Simulation of an FM Signal . . . . . . . . . . . . . . . . . . . . . . . 130
B.2 Simulation of a Target Signal . . . . . . . . . . . . . . . . . . . . . . 134
B.3 Creating the Scattered Signal . . . . . . . . . . . . . . . . . . . . . . 136
iv
LIST OF FIGURES
1.1
2.1
2.2
2.3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
An illustration of radar interferometry. The radial and transverse dimensions are shown at bottom right. . . . . . . . . . . . . . . . . . .
An illustration of the MRR system. . . . . . . . . . . . . . . . . . . .
A range-Doppler diagram, the typical data product of MRR. Reflections from the Cascade mountain range can be seen at zero Doppler
near 100 km; at a range of 1000 km, echoes due to ionospheric plasma
turbulence are present. . . . . . . . . . . . . . . . . . . . . . . . . . .
A scatter plot of in-phase vs. quadrature components of a typical reference FM transmission. The station is 96.5 MHz; this is 1 second of data
taken at 200 kHz. Note that the samples are zero-mean, and the inphase and quadrature components are uncorrelated. The narrowband
time series can be considered an analytic signal. . . . . . . . . . . . .
A range-Doppler display from MRR. Examining the near-ranges reveals
ground clutter and airplanes. . . . . . . . . . . . . . . . . . . . . . . .
An interferometer image with the same ground clutter and airplanes
shown in figure 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Interferometer data from the Mt. Rainier echo. Both antennas show a
narrow, DC spike, and the correlation coefficient between them is well
above the 95% significance level at zero Doppler. . . . . . . . . . . . .
Interferometer data and power spectra for the airplane with Doppler
shift -50 m/s (also shown in figure 3.1). Again, we include the 95%
significance level for the coherence spectrum. . . . . . . . . . . . . . .
An illustration of the wavevectors involved in Bragg scatter. A backscatter radar can detect the scattered signal due to ks when θ = 0. . . . .
A range-Doppler diagram showing an echo from a meteor trail (790
km) during the 2002 Quadrantids meteor shower. . . . . . . . . . . .
Cross- and self-spectra for the range containing the meteor echo shown
in figure 3.6. The 95% significance level for the coherence is also shown.
v
5
10
13
15
20
20
21
23
26
28
29
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
A meteor trail image, produced with MRR interferometer data. . . .
Individual range spectra for the meteor trail shown in figure 3.8. (Observed during the Geminids meteor shower, December 14, 2001.) . . .
Selected meteor echoes observed by MRR during the year 2002 with
corresponding interferometer data. For each group of plots, the top
panel is a power spectrum, the middle panel is the coherence, and
the bottom panel shows the cross-spectrum phase. In each case, the
Doppler velocities range from -400 to 400 m/s. Some of the echoes are
very low SNR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An MRR range vs. Doppler diagram showing an echo from E-region
turbulence between 600 and 800 km. . . . . . . . . . . . . . . . . . .
A two-dimensional interferometer image of the same auroral echo shown
in figure 3.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Histogram and sorted samples of simulated coherence values. In the
bottom panel, dotted lines are shown for the 50th , 75th , 95th , and 99th
percentiles. These graphs represent a sample size of 500,000. . . . . .
An irregularity observed on 18 April, 2002, at UT 06:03. Power spectra
from both antennas as well as an interferometer image are shown. The
color scale applies to all panels. . . . . . . . . . . . . . . . . . . . . .
An irregularity observed on 24 March, 2002, at UT 05:07. Power spectra from both antennas as well as an interferometer image are shown.
The color scale applies to all panels. . . . . . . . . . . . . . . . . . . .
Normalized histograms showing the trend toward Gaussian distribution
after coherent integration. Gaussian probability density functions with
the same mean and variance as the data have been overlayed in the
right panels. Each figure represents the real part of 106 complex data
points, sampled at a rate of 100 kHz. . . . . . . . . . . . . . . . . . .
4.2 A typical autocorrelation function for an FM transmission (96.5 MHz).
The signal is mostly decorrelated after 10 µseconds, indicating that
samples taken at 100 kHz are independent. These samples were taken
at 200 kHz, roughly the bandwidth of the signal. . . . . . . . . . . . .
4.3 A comparison of 2-D image resolutions obtained by range averaging and
varying the number of phase bins used in the interferometer program
(essentially “phase averaging”). The data shown here is an auroral
echo observed on February 2, 2002, UT 05:11. . . . . . . . . . . . . .
30
31
32
34
35
36
40
41
4.1
vi
45
46
50
4.4
Cross-correlation magnitude of the meteor trail echo shown in figure
3.6. The noise, clutter, and interference contribution is contained in
lag zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
A comparison of bias removal techniques. The top plot shows the
uncorrected spectrum; the next two demonstrate different correction
methods discussed in the text. The data is from the meteor trail observed on January 3, 2002 at UT 19:23. . . . . . . . . . . . . . . . . .
72
5.1
A simulated target, with -13 dB SNR.
. . . . . . . . . . . . . . . . .
79
5.2
An illustration of the variance estimation method used here. . . . . .
82
5.3
The top three panels show different representations of cross-spectrum
variance versus SNR. The bottom panel shows the number of variance
estimates that were averaged together to form the final value. . . . .
84
5.4
Cross-spectrum phase variance plotted as the spectrum SNR increases.
87
5.5
A comparison of transverse resolutions in interferometer data. From
left to right, the phase is separated into consecutively fewer bins, resulting in degrading transverse resolution. The range resolution in each
figure is 1.5 km. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
Power spectra at 3 different ranges for the auroral echo we observed
on February 2, 2002, at UT 05:11. . . . . . . . . . . . . . . . . . . . .
91
An example of cross-spectrum phase compactness where a (simulated)
target exists. The standard deviation of phase estimates can be used
to detect targets, in some cases more reliably than power spectrum
magnitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
The MRR field of view, over southwestern Canada, shown with contours of constant range and aspect angle. The edges of the figure are
labeled with geographic latitude (N) and longitude (E). Credit for creating the figure goes to Dr. Frank Lind. . . . . . . . . . . . . . . . .
96
A range-Doppler display from MRR showing an auroral echo near 1000
km; an area of high intensity and large Doppler shift is flanked by a
larger, diffuse “type 2” echo, which we show to be spread across the
transverse dimension, indicating a shear. . . . . . . . . . . . . . . . .
97
4.5
5.6
5.7
6.1
6.2
vii
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
The cross-spectrum and self-spectra for one range in the auroral echo
from figure 6.2. Due to the highly organized phase, we observe that
the echo seems limited to one interferometer lobe; the phase width
implies a scattering volume extent of approximately 30 km. The 95%
significance level for the coherence is also shown in the middle panel.
Cross spectrum at range 984 km for the auroral echo on February 2,
2002, UT 05:11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2002, UT 05:11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2002, UT 05:11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2002, UT 05:11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Power spectra (from one antenna) and interferometer phase vs. range
for the auroral echo on February 2, 2002, UT 05:11. . . . . . . . . . .
The February 2, 2002 auroral echo, represented in a 2-D image of range
vs. transverse width. The resolution in this plot is 3 × 5 km (range ×
transverse dimension). . . . . . . . . . . . . . . . . . . . . . . . . . .
2D interferometer images from March 24, 2002, UT 05:07—05:15. . .
Measured velocity vectors for the drifting irregularity shown above in
figure 6.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An example of Doppler power spectrum versus range from irregularity
scatter obtained on March 30, 2002. . . . . . . . . . . . . . . . . . . .
A sequence of 5 consecutive interferometer images from the March 30
event, 4 minutes apart in time. The first three frames indicate a mean
transverse drift speed of 70 m/s. The final frame clearly shows that the
scattering volume has split, a feature unrecognizable from the rangeDoppler spectra in figure 6.12. . . . . . . . . . . . . . . . . . . . . . .
B.1 Scatter plots of actual (left) and simulated FM data. . . . . . . . . .
B.2 Power spectral densities of actual (left) and simulated FM data. . . .
B.3 Autocorrelation functions (real part) of actual (left) and simulated FM
data. Each sample is marked with a dot. . . . . . . . . . . . . . . . .
B.4 A comparison of histograms of actual FM samples and samples from a
simulated FM signal. . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
98
99
100
100
101
103
104
105
106
109
110
132
133
134
135
LIST OF TABLES
2.1
Latitude/longitude coordinates for some locations relevant to MRR. .
11
5.1
Interferometer transverse resolutions appropriate for a given SNR. . .
89
ix
GLOSSARY / FREQUENTLY USED ACRONYMS
Autoregressive. Describing a random time series which can be written in
terms of a linear combination of its previous values, plus a random component.
AR:
ARMA: An autoregressive moving average process (see MA).
CUPRI: The Cornell University Portable Radar Interferometer.
CW: Continuous wave. A continuous (non-pulsed) transmission.
DFT: Discrete Fourier transform.
ERP: Effective radiated power.
FFT: Fast Fourier transform.
Frequency modulation. A scheme for encoding information into the frequency of a carrier signal.
FM:
GPS: Global Positioning System.
IMF: Interplanetary magnetic field. The sun’s magnetic field, carried outwards by
the solar wind.
MA: Moving average. Describing a random time series which can be written in
terms of a mean value plus a series of perturbations based on a white noise
process.
MATLAB: A commercial numerical computation and visualization software tool.
MRO: Manastash Ridge Observatory, MRR’s namesake and where its scatter-
receiving antennas are located.
x
MRR: The Manastash Ridge Radar, designed and developed at the University of
Washington.
PLL: Phase-locked loop. A circuit with an oscillator that locks on to the frequency
of an input signal.
SCR: Signal-to-Clutter Ratio.
SHERPA: The Système HF d’Etudes Radar Polaires et Aurorales, an HF coherent
radar operated from Schefferville, Québec.
SNR: Signal-to-Noise Ratio. Typically measured in dB, this is the ratio of signal
power to noise power. In this document, we obtain SNR by a “peak to floor”
measurement.
STARE: The Scandinavian Twin Auroral Radar Experiment.
RF:
Radio frequency.
RX:
Receiver.
UW: University of Washington, where this work was conducted, and where one
of the MRR receivers is located.
xi
NOTATION CONVENTIONS
C(·): A magnitude-normalized cross spectrum (complex coherency).
C∞ :
Clutter contribution from all ranges.
CN :
Clutter contribution from a single range.
K̂(·): Cross correlation estimator for MRR.
RXY (·): A correlation function between signals x and y, defined as hx(t)y ∗ (t − τ )i.
S(·): A spectral density function.
S (ρ) (·): The periodogram, a spectral estimator.
h·i:
Denotes expected value of the argument.
xii
ACKNOWLEDGMENTS
First and foremost, I want to thank my advisor, mentor, and friend, John Sahr.
He has opened a whole world of opportunities to me, and it’s in no small part thanks
to him that I am having the time of my life in grad school. Hearty thanks also goes
to the other members of my committee, Don Percival, Bob Holzworth, and Robert
Hiers, and my many other excellent mentors over the years– Frank Vanzant, John
Hopkins, and Paul Crilly: you all have helped and inspired me more than you realize.
To Frankye Jones especially, as well as the rest of the EE, space science, and CS
communities here at UW: you have been like families to me since I came here. I do
so much enjoy working – and the talking that goes along with it – with you!
A huge hug goes to my bestest1 friend, Andrew, without whom I would certainly
have gone insane by this time; my wonderful housemates Lena, Adam, Matt, and
Shanaz, who are very dear friends despite the fact that I never see them due to being
on campus so much; and last but not least, my loving family, who gave me up so that
I could follow my dreams 3000 miles out here to Seattle.
1
It’s a technical term.
xiii
1
Chapter 1
INTRODUCTION
“Always beware when someone with a Ph.D. calls something
‘interesting.’ ”
— Dr. Steve Keeling
We begin with an appeal to the reader’s sense of curiosity. The primary motivation for this work is to develop ways to study the reasons and mechanisms behind
the aurora borealis, or northern lights. A fascinating phenomenon that has intrigued
Earth’s inhabitants for thousands of years, the aurora are ultimately caused by reactions in the Earth’s atmosphere to the sun’s energy in various forms.
The outermost layer of the atmosphere is called the ionosphere because it is composed partly of ionized particles (plasma) rather than completely of neutral particles.
At lower altitudes in the ionosphere (termed the D- and E- regions) the electrons and
ions are kept too energetic to recombine during daylight hours by the constant radiation from the sun; at night these layers disappear (become neutral). The outermost
layer, called the F-region, is much more sparsely populated and remains ionized all
the time. Thus, the Earth is surrounded by a sheath of plasma which absorbs the
most energetic radiation from the sun. Since these particles are charged, they respond
to electromagnetic forces, and are strongly influenced by both the Earth’s magnetic
field as well as any electric fields that may exist (due, for example, to the much
higher mobility of electrons as compared to ions). A very complicated and intricate
2
structure of currents results from the coupling between the ionosphere, the magnetosphere (the area within Earth’s “magnetic force-field”), and the interplanetary solar
wind (which carries the sun’s magnetic field “frozen” into it). In short, the northern
(and southern) lights are caused by energy pouring into the ionosphere from currents
traveling along magnetic field lines. These field-aligned currents excite particles in
the atmosphere, which emit photons (of different colors, depending on their chemical
makeup) as they decay back down to their previous lower energy levels.
The entire system can become very perturbed when the radiation from the sun
is highly energetic for some reason (for example, after a solar flare) or when the
interplanetary magnetic field (IMF) carried by the solar wind lines up so as to cancel
out part of the Earth’s magnetic field. During events such as these, the field-aligned
currents causing the aurora can be pushed southward into lower latitudes, where their
effects can be seen by, for example, plasma physicists at the University of Washington
in Seattle1 .
As interesting as the physics responsible for the aurora is, it is a challenge to
study experimentally due to the awkward altitude at which the interesting activity
takes place. Ionospheric heights (typically 90–180 km for the E-region) are too low
for satellite orbits, yet too high for balloons; rockets with scientific payloads can be
(and have been – the ERRRIS campaign, for example [24]) flown through the area of
interest, but these are expensive and short-lived experiments. Remote sensing makes
the most sense as a strategy for extended study of ionospheric processes, and much
work has been done toward developing robust and reliable radar techniques toward
this end. In this work, we present a new technique for passive radar remote sensing
of plasma turbulence in the ionosphere: passive radar interferometry.
1
Provided the sky is not overcast.
3
1.1
The Manastash Ridge Radar
The Manastash Ridge Radar (MRR) is a project at the University of Washington
[27] designed and developed as an affordable way to do remote sensing of ionospheric
E-region field-aligned irregularities (plasma turbulence usually correlated with the
visual aurora, and sometimes called the ‘radar aurora’). Located in the northwestern
United States, the MRR field of view covers a region over southwestern Canada in
the sub-auroral zone, corresponding approximately to the geomagnetic latitudes of
56 − 63◦ (50 − 57◦ geographic latitude). Therefore, it detects coherent scatter from
auroral irregularities only during disturbed ionospheric conditions. For example, in
the year 2002, we observed E-region irregularities with MRR on 27 separate days2 .
The most unique feature of MRR is that it does not have a dedicated transmitter.
Conventional active radars transmit successive pulses (or coded pulse sequences) and
measure the time of flight of the reflected pulses to detect and determine the range
of reflecting targets. MRR, however, takes advantage of existing commercial transmitters as a source of illumination; for this reason we refer to it as a passive radar.
The system, described more completely in chapter 2, consists of two receivers: one to
provide a reference copy of the transmitted signal, and one to collect the signal after
it has been scattered from various targets.
1.2
Interferometry
In coherent scatter studies of the ionosphere, interferometric techniques have frequently been used to resolve the transverse structure of scattering volumes and to
provide information about the location of scatterers in the field of view [5, 25, 28].
The interferometer we have created has very fine resolution, and this is useful for
2
This is still more often than we expected to detect auroral scatter, based on our sub-auroral
field of view.
4
determining transverse structures.
The basic idea behind interferometry is that the phase difference between the
signals generated by a wave arriving on two (or multiple, but we consider only the
two-antenna case here) closely spaced3 antennas provides information about the angle
of arrival of the wave. In particular, the phase difference between the signals on the
two antennas, φ, is related to the physical angle of arrival θ by
φ = kd cos θ =
2π
d cos θ
λ
(1.1)
where k is the wave number, λ is the wavelength, and d is the distance between the
antennas. Figure 1.1 illustrates this idea; it is convenient to think of φ as the phase
accumulated by the wave as it travels the extra distance d cos θ to the second antenna.
The interferometer will be able to uniquely map one wavelength of phase to a full
180◦ field of view (a front-to-back ambiguity remains) if the baseline is less than or
equal to half a wavelength;
d≤
λ
2
this will cause φ to vary between −π and π. If the baseline is any larger, the interferometer will have multiple lobes (one for each multiple of 2π radians of phase between
the antennas), and the physical angle information will be aliased.
We can write equation 1.1 for the general case by adding a few terms:
φ + φ0 + n2π = kd cos(θ + θ0 )
(1.2)
Here the n2π term represents any aliasing (n is an integer); φ0 is a net phase correction
term, necessary because of arbitrary phase delays in the system, such as transmission
lines between the antennas and receiver; and the θ0 term is a correction term for the
3
Here ‘closely spaced’ means that the evolution of a wave traveling between them can be completely described in terms of phase accumulation; i. e., the time of flight between the two antennas
is less than the correlation time of the wave.
5
Reflecting target
in far field
Plane wave with
Uniform phase front
Extra distance
to antenna 2
d cos θ
θ
strikes
antenna 1
first
d
2
1
transverse
dir.
radial
dir.
Figure 1.1: An illustration of radar interferometry. The radial and transverse dimensions are shown at bottom right.
angle of arrival, and could be used to compensate for heterogeneous antenna patterns,
for instance.
If the antennas are oriented such that their gain patterns are directed broadside
to the baseline connecting them, the interferometer will emphasize azimuth angle
information, whereas a configuration with the antennas oriented in an “endfire” position relative to the baseline will respond better to the elevation angle of the target.
We will consider the azimuth-emphasizing orientation here, as this is the type of
interferometer we have implemented.
6
1.2.1
The Manastash Ridge Radar Interferometer
In order to perform interferometry at MRR, we need data from three antennas: two
separated by a known distance d to collect scattered signals, and one copy of the
reference FM transmission. The “two antennas” we refer to when describing the
MRR interferometer are the scatter-collecting antennas. The baseline between them
is approximately 47 meters long, or roughly sixteen wavelengths (16 λ) at MRR’s
carrier frequency of 100 MHz. Thus, the interferometer has 32 lobes, and the width
of a single lobe is approximately 5.6◦ in azimuth angle (100 km wide at a range
1000 km). The antenna configuration we have used is due mainly to the geometry of
the space available at the site where we keep our antennas. The large baseline was
available and convenient when we began this project, so we used it, and made plans
to set up additional antennas in the future.
The method we use to extract the phase difference between the antennas is a
complex cross-correlation (or cross-spectrum, in the Fourier domain) between the
target signals on the two antennas. We refer to the correlation coefficient between
the two antennas (or the cross-spectrum magnitude normalized by the power on
each individual antenna) as coherence [5]. The coherence provides information about
the transverse size of scatterers, while the phase provides information about their
locations. We describe in detail our passive radar interferometer cross-correlation
estimator and determine its statistical properties in chapters 4 and 5.
1.3
What can be Learned
If transverse structure information is available, we can learn many things about the
scattering volume. First, we can create two-dimensional images of the scatterer (where
the two dimensions are the radial direction away from the radar and the direction perpendicular to that, which is described by azimuth angle). Given the time evolution of
7
scatterers in these two dimensions, we can estimate their transverse velocities in addition to their Doppler shifts. Finally, by making approximations about the scatterer
motion based on known behaviors of plasmas, we can use interferometer information
to measure geophysical properties of the scattering volume, such as electric field.
Chapter 3 shows interferometer observations of many different types of targets,
the various data products we can create, and their interpretations. In chapter 6
we demonstrate the use of our interferometer on radar data containing scatter from
E-region irregularities, which is the primary focus of this work.
8
Chapter 2
SYSTEM DESCRIPTION
“The party line is that more data are a blessing — though bedraggled
grad students might disagree!”
— Dr. Don Percival
The Manastash Ridge Radar (MRR) is a unique instrument that utilizes ambient
radio illumination in the environment to detect and determine the characteristics
of sound waves and other density perturbations in the ionosphere. Because of its
passive nature, it does not require a dedicated transmitter, and this is an advantage
for many reasons. Most notably, radar transmitters are very expensive and powerful
instruments, and they therefore require a significant amount of maintenance and
caution. Furthermore, appropriate licenses must be obtained to operate high-powered
transmitters. In our case, we wished to operate a VHF radar in the 100 MHz frequency
range, but this band is allocated for FM radio broadcasts. Therefore, we designed
MRR to listen to the broadcasts themselves.
As it happens, FM radio provides extremely useful radar waveforms for studies of
the ionosphere. The transmitters are powerful and omnidirectional, and the effective
radiated power is high (typically 100 kW) since they are CW, not pulsed. The 100
MHz carrier frequency is nearly immune from ionospheric refraction and atmospheric
absorption effects, yet scatters readily from plasma turbulence. Most importantly,
the typical FM waveform has an excellent ambiguity function in the average sense
[8], often completely free of range and Doppler aliasing (of course, it changes with
9
time, so occasionally we experience “bad ambiguity”). The ambiguity function of a
complex-valued radar waveform envelope u(t) is defined as the output of its matched
filter,
Z
|χ(τ, ν)| = ∞
−∞
u(t)u (t − τ ) exp(j2πνt)
∗
(2.1)
and describes the radar signal’s ability to resolve targets in range (time lag, τ ) and
Doppler (frequency, ν) space [19]. Typical ambiguity functions from MRR data show
perfect range resolution (in the sense that there is no clutter due to range ambiguity)
at a scale of 1.5 km, corresponding to a 100 kHz receiver sampling rate, and Doppler
sidelobes that are quite low, especially when compared to other coded radar pulses,
such as the barker codes [8]. Essentially, the FM transmission acts like a stochastically
coded long pulse [11], which is very useful for overspread1 targets such as turbulent
events in the ionosphere. The bandwidth of the FM transmission is large compared to
the timescales of fluctuations in the ionospheric plasma, and this permits overspread
target pulse compression. Pulse compression is a signal processing technique for
achieving fine range resolution without sacrificing sensitivity. Narrow pulse widths
require a large receiver bandwidth, letting a lot of noise into the system and degrading
detectability. Also, the amount of power a transmitter can expel in an extremely short
pulse is limited, while a longer pulse can thoroughly illuminate a target. Therefore,
longer pulses are modulated (usually in a bipolar phase scheme) by pulse sequences
chosen specifically for their ambiguity function performance. After matched filtering,
these long pulses perform as though they had very narrow widths, but with the
sensitivity advantages of long pulses.
To extract useful information from our radar, we need both the original FM transmission (for matched filtering) as well as any radiation scattered from targets. How1
An overspread target is one which is simultaneously too distant and moving too fast to unambiguously resolve its range and frequency characteristics with a pulsed radar.
10
E-region Irregularities and Meteors
km
40
0
00
-1
10
40
0-
11
0k
gro
u
nd
Commercial
Transmitter
m
clu
Remote
Receivers
tter
Reference
Receiver
x(t)
y(t)
1
30 km
y(t)
2
150 km
Figure 2.1: An illustration of the MRR system.
ever, the transmitter can be over 100 dB louder than the faint scatter returning from
ionospheric targets, and this makes using a single receiver very difficult due to the
required dynamic range. Of existing passive radars, MRR solves this problem by
locating the transmitter recording and scatter collecting receivers in separate places,
approximately 150 km apart. This distance, as well as the Cascade mountain range
which lies between the two receivers, effectively prevents most, if not all, radiation
entering the scatter receivers directly from the transmitter. Therefore, MRR is a
bistatic (arguably “tristatic”) passive radar system. An illustration of MRR is shown
in figure 2.1; its field of view and the locations of the receivers and transmitter are
also shown in figure 6.1 (page 96). In table 2.1, we provide the exact locations, in
latitude/longitude coordinates, of the radar features indicated in these figures.
While separating the receivers solves the dynamic range problem, it causes problems of its own. For example, the receivers must be synchronized very precisely: an
11
Table 2.1: Latitude/longitude coordinates for some locations relevant to MRR.
Site
Latitude Longitude
Comments
UW
47.66◦ N
122.31◦ W
reference RX location
MRO
46.95◦ N
120.72◦ W
scatter RX location; elevation 3960’
KYPT
47.54◦ N
122.11◦ W
FM station at 96.5 MHz; 100 kW ERP
KWJZ
47.50◦ N
121.97◦ W
FM station at 98.9 MHz; 52 kW ERP
Mt. Rainier
46.85◦ N
121.76◦ W
prominent ground clutter
offset of 10 microseconds will cause an error in range greater than our current range
resolution. We synchronize the receivers with a time and frequency reference from
GPS receivers at each end (the references are accurate to 100 nanoseconds). Also, we
are now required to send large amounts2 of data over a sometimes tenuous internet
link. For a 10-second period every 4 minutes (currently the normal radar operation),
we sample the scatter receivers on four channels3 at 100 kHz, and the reference receiver
on two channels (a single antenna, but two frequencies). The scatter and reference
datastreams must be combined to extract useful information from the system. The
network load is not unreasonable, however, since in most cases we only need to send
a single file from the scatter receiver to a central location to be processed with the
reference data. The resulting internet traffic is 4 megabytes every 4 minutes, which
is sustainable. If this single antenna-frequency combination indicates an interesting
event in our data, however, we transfer the remaining data for processing and storage,
which often takes a long time and puts significant stress on the connecting network.
2
The default operation of MRR generates over 8 gigabytes in raw receiver samples per day. It
takes a full-time grad student to keep up with it all!
3
Two frequencies on each of two antennas. Soon there will be more channels, when we erect new
antennas for the interferometer.
12
Range information is obtained from the MRR system by correlating the original
FM transmission against successively delayed copies of the received scatter. Target
signals are boosted above the background noise floor when these two time series
align correctly. We wish to detect targets and determine their range as well as their
Doppler characteristics. Figure 2.2 shows a typical data product from MRR: a rangeDoppler diagram. These diagrams are composed of power spectra (of the target
signal), stacked up over multiple ranges. The vertical axis indicates Doppler shift, or
frequency information, while the horizontal axis shows the distance of the scatterer
from the radar. The greyscale is in dB (uncalibrated units). In this particular plot
we can see ground clutter features at near ranges (caused by the mountains which
lie between the transmitter and scatter receiver), as well as an example of scatter
from E-region turbulence at about 1000 km. The ground clutter is centered at zero
Doppler, as we expect, although ambiguity sidelobes in the Doppler dimension are
apparent. Occasionally we also detect aircraft (with appropriate Doppler shifts) at
ranges less than about 200 km. The E-region echoes are moving much faster, typically
with Doppler shifts near 450 m/s. This figure represents a 10-second time integration,
a typical length for MRR.
2.1
An Example of Target Detection with MRR
Using the latitude/longitude information in table 2.1, we can calculate the distances
between important features of the radar. For example, the total bounce distance from
the KYPT commercial FM transmitter to Mt. Rainier and then to the scatter receiver
at MRO is 160 km. At the speed of light, it takes 530 µsec to traverse this distance;
if our receivers take samples every 10 µsec (a 100 kHz sampling rate), then the echo
from Mt. Rainier should lag the transmitter waveform by 53 samples. However,
since the distance between the KYPT transmitter and our reference receiver at the
13
dB
April 20, 2002 UT 09:47
1500
232
230
1000
Velocity (m/s)
E−region turbulence
228
500
226
0
224
−500
−1000
−1500
0
222
Ground clutter, with
ambiguity sidelobes visible
220
200
400
600
Range (km)
800
1000
1200
218
Figure 2.2: A range-Doppler diagram, the typical data product of MRR. Reflections
from the Cascade mountain range can be seen at zero Doppler near 100 km; at a
range of 1000 km, echoes due to ionospheric plasma turbulence are present.
University of Washington is 20 km, the apparent time of flight for all scatter received
by the system is 7 lags fewer than the true time of flight. Because of the elapsed time
between a transmission and when our receivers start recording, the phrase “Warning:
Targets are further away than they appear” applies, and should be compensated for
in the radar signal processing. Thus, we expect the echo from Mt. Rainier to appear
around lag 46, as it does in figure 2.2 (we also show individual power spectra from a
ground clutter echo at this range in figure 3.3, page 21).
In the inverse problem, the number of lags can be converted into a physical distance. Backscatter radars determine the target range by halving the time-of-flight
range; however, bistatic distance is not quite as straightforwardly understood as the
circles of increasing radii one imagines for a monostatic radar. In our case, contours
14
of constant range are elliptically shaped (or rather, shells of constant range form ellipsoids), with the transmitter and scatter-receiver as the foci of the ellipsoids. Thus, for
close-range echoes seen by MRR, the “one half time-of-flight times the speed of light”
approximation is not useful. However, since the transmitter and scatter receivers are
still close together (150 km) with respect to many of the far-away echoes we detect
(450-1200 km), the approximation is sufficient for our purposes. Finally, we note
that the total distance traveled by the radar transmission is also affected by altitude
(constrained to 90-180 km for our E-region targets of interest). For this reason, radar
range estimates are often referred to as “slant ranges.”
2.2
Hardware
In the absence of a transmitter, the most prominent hardware components in the
system are its antennas, receivers, and the computing resources used to process the
data. The two antennas used to collect scatter from targets (with which we do
interferometry) are a Yagi (mounted on a 40 foot tower) and a log periodic, mounted
on the roof of a building. Both of these antennas are kept at the Manastash Ridge
Observatory (MRO), which is situated at a relatively high altitude (3960 feet). The
gain patterns of these two antennas are very different; the Yagi has a higher gain (14
dB) with a narrow field of view directed toward the northeast, but the log periodic
has a very wide field of view and a more modest gain (6 dB). The antenna used to
receive the transmitter signal is a simple half-wave dipole, kept at the University of
Washington in the Electrical Engineering Building.
The first generation MRR receivers were implemented with analog components
[20]. In late 2001, however, a second generation of MRR was implemented, making
use of the quickly developing digital receiver technology. We now use the Echotek
ECDRGC214/TS digital receiver for both the reference and scattered signals. These
15
x 10
In Phase / Quadrature Scatter Plot of FM Transmission
−2
−1.5
4
1.5
1
Imaginary Part
0.5
0
−0.5
−1
−1.5
−1
−0.5
0
Real Part
0.5
1
1.5
2
4
x 10
Figure 2.3: A scatter plot of in-phase vs. quadrature components of a typical reference
FM transmission. The station is 96.5 MHz; this is 1 second of data taken at 200 kHz.
Note that the samples are zero-mean, and the in-phase and quadrature components
are uncorrelated. The narrowband time series can be considered an analytic signal.
receivers sample the incoming RF signals directly at 56 MHz (thus, we require antialiasing bandpass filters) with 14-bit ADCs. They connect to PCI slots in desktop
computers and then implement further downconversion, mixing, and filtering digitally.
An example of the resulting in-phase and quadrature receiver samples is shown in
figure 2.3; the receivers produce very high-quality samples. Zhou [32] discusses this
receiver in more detail.
The computational capacity required to process the MRR receiver samples is
high, but easily handled by today’s “over the counter” computing resources. We
16
currently use a 750 MHz single processor desktop PC for all our computational needs.
Other hardware includes GPS receivers, which are needed at each digital receiver
to synchronize the nodes in the system; PLLs, which make use of the frequency
reference provided by the GPS; and passive filters and amplifiers for preconditioning
the received signal before digitization.
Finally, MRR is, inherently, a distributed system, an instrument composed of
widely separated components connected by computer networks. Therefore, in an
exhaustive list of hardware required for the system, we must include all the network
hardware involved in transferring our raw data files from their respective receiver
locations to the final storage area where the data is processed.
2.3
Software
With the exception of the antennas and digital receivers (the latter of which arguably
could be referred to as software), the majority of a passive radar such as MRR is
implemented in software. After both the reference and scatter receiver samples are
available (for a particular increment of time), the next task toward creating diagrams
like that shown in figure 2.2 is signal processing: we must deconvolve the transmitter
signal from the received scatter, and then perform spectrum estimation on the resulting time series. Currently the signal processing algorithms are such that the radar can
keep up with data processing in real-time (e. g., it takes approximately 10 seconds to
process a 10-second burst of data). However, this does not include the time required
to transfer raw data from remote receiver locations. The signal processing itself is
described in more detail in chapter 4; here we list the basic software components of
the system which accomplish the task.
The main signal processors that do the majority of the number crunching are written in the C and C++ languages. These processors are invoked and controlled on a
17
regular basis (during the “real-time operation of the radar”) by several scripts written
in bash, csh, perl, and python (linux shell commands and programming languages).
The diversity of software tools is mainly a weakness in the system caused by the
preferences of the different programmers developing the system. However, there are
many diverse tasks which must be accomplished in the day-to-day operation of the
radar, including: data acquisition, which involves both driving the digital receivers
to take data as well as determining when data needs to be taken; scripts for moving,
storing, and transferring data from computer to computer; processing the data to
create the range-Doppler diagrams (or other data products of interest); and finally,
creating images in a useful form from the data and publishing these images to the
internet.
Additionally, we have developed interferometer signal processing programs along
with an array of other programs and scripts for plotting and displaying this data. We
often find that the most difficult part of collecting and dealing with experimental data
is determining how best to display it. In the next chapter we show several examples
of our experimental results with the interferometer.
18
Chapter 3
INTERFEROMETER OBSERVATIONS
“There’s nothing new under the sun — but you could look under a rock.”
—Dr. John Hopkins
In this chapter we present several examples of data from the MRR interferometer
and provide general commentary on our results. Although we will be mostly interested
in upper-atmospheric targets (in particular, volumetric scatter from turbulence in
the E-region), we also investigate more “conventional” targets, such as aircraft, for
diagnostic purposes and so that our data may be readily compared with the products
of other radars. An obvious first target at which to direct our attention is ground
clutter from mountains in the area, a predictable feature present in all of our data.
Next, we consider aircraft; these are “point targets” and are useful because they have
simple, well-understood Doppler characteristics. We also show examples of scatter
from meteor trails, and demonstrate the use of our interferometer in detecting meteor
“tubes,” from which wind shears at mesosphere altitude can be inferred.
We fully describe the signal processing involved in producing these images later
in chapter 4; in chapter 6 we discuss in detail several observations of E-region irregularities, our primary purpose for remote sensing.
3.1
Ground Clutter
While ground clutter is generally undesirable and has a negative impact on target
detectability, it nevertheless provides a useful way to gauge the “health” of our system.
19
For example, in chapter 2, we determine the time lag after which we expect the echo
from Mt. Rainier to arrive at the scatter receiver. Indeed, in almost every rangeDoppler diagram MRR generates, the most noticeable feature is the ground clutter
due to Mt. Rainier. We use this strong echo as a crude system diagnostic: its shape
in range-Doppler space gives us an idea of the FM waveform ambiguity (at the time
the data was taken), and we know that the radar has become insensitive if we are
ever unable to identify Mt. Rainier. For this reason, the detection of ground clutter
is a convenient “sanity check,” and we briefly consider it with the interferometer.
First, to clarify our meaning, we use the term clutter to refer to echoes from
unwanted delays, which contribute energy at all ranges since the transmitter is CW.
However, we can look specifically at a “ground clutter” echo from, say, Mt. Rainier,
and determine its power spectrum. In the case of a mountain, we would of course
expect a narrow spike at zero Doppler, since the target is (hopefully!) not moving
or spewing rocks and ash into the air at different velocities. We would also expect
the cross-correlation between the two interferometer antennas to be large, provided
that the reflecting surface on the mountain is not larger than an interferometer lobe
(a limitation discussed in section 3.4.2).
Figure 3.1 shows a range-Doppler diagram of the first 300 km of the MRR field
of view. Several ranges show echoes on the zero-Doppler line, which we presume
to be ground clutter from the Cascade mountain range; a few aircraft are visible as
well (the most prominent ones have been marked with arrows). An image derived
from interferometer data is also shown in figure 3.2 for corresponding ranges; we are
able to compare targets in range-velocity space with their representation in a range
vs. transverse dimension image. The strongest ground clutter shows up in the black
pixels, while aircraft and weaker ground or mountain echos appear in dark grey. We
will discuss the generation of plots like figure 3.2 later in section 3.4.1. Other features
20
April 20, 2002 UT 04:23
dB
235
300
230
Velocity (m/s)
200
100
225
0
220
−100
−200
215
−300
0
50
100
150
Range (km)
200
250
210
300
Figure 3.1: A range-Doppler display from MRR. Examining the near-ranges reveals
ground clutter and airplanes.
Transverse
Position (km)
April 20, 2002 UT 04:23
−5
0
5
50
60
70
80
90
100
110
120
130
140
Range (km)
Figure 3.2: An interferometer image with the same ground clutter and airplanes
shown in figure 3.1.
21
Cross Spectrum Phase and Magnitude at range 69 km
Phase
2
0
−2
−400
−300
−200
−100
0
100
200
300
400
−300
−200
−100
0
100
200
300
400
Normalized
Magnitude
1
0.5
Self Spectra (dB)
0
−400
240
Log Periodic
Yagi
230
220
210
−400
−300
−200
−100
0
100
200
300
400
Doppler (m/s)
Figure 3.3: Interferometer data from the Mt. Rainier echo. Both antennas show a
narrow, DC spike, and the correlation coefficient between them is well above the 95%
significance level at zero Doppler.
visible due to the strong echoes in figure 3.1 are the ambiguity sidelobes of the FM
waveform. These are frequently seen with high SNR echoes, and are in general not
confusing and easily recognizable as ambiguity sidelobes.
In figure 3.3, we show interferometer data from the specific range (lag 46) at which
we expect the Mt. Rainier echo, plotted against Doppler velocity. As we expect, the
power spectra on both individual antennas (bottom panel) show a large, narrow spike
at zero Doppler. The middle panel shows the normalized cross-spectrum magnitude
(coherence), which grows to nearly a perfect correlation of 1, again at zero Doppler.
We also plot the 95% significance level for the coherence, which turns out to be 0.51 in
22
this case, obtained through the numeric simulation of 500,000 independent instances
of a random variable with the same distribution as our coherence estimates (discussed
further in section 3.4).
As with conventional interferometry, we find that the cross-spectrum phase (top
panel of figure 3.3) becomes organized in the same area that the coherence is large.
This is because the strongest signal on both antennas is arriving from the same direction. The cross-spectrum phase, then, is related to the angle of arrival of the
Mt. Rainier echo (see section 1.2), and it appears that the transverse size of the
reflecting surface of Mt. Rainier is significantly smaller than the width of one interferometer lobe at this range (roughly 10 km). However, even though the phase in the
top panel of figure 3.3 is compact, the average interferometer phase of Mt. Rainier
can change drastically1 on a minute-length time scale due to propagation paths and
channel effects. Therefore, we are unable to determine an actual angle of arrival for
the echo. Additionally, our data suffers from azimuth aliasing due to the multiple,
closely-spaced interferometer lobes. We would require antennas set closer together
(wider interferometer lobes) to deemphasize channel-related fluctuations and uniquely
determine the bearing of the target. Finally, we note that the phase remains concentrated over a Doppler extent much larger than that of the actual narrow-band echo.
This is due to the ambiguity sidelobes, which reach Doppler shifts far from the true
spectral peak. These Doppler features appear to arise from the same direction, and
generate phase similar to that of the mountain reflection itself.
1
Since the reflection from Mt. Rainier most likely enters both antennas through a sidelobe, we
believe the large phase variation in this case to be caused by antenna sidelobe effects, which also
confuse attempts at using averaging to obtain a stable value. For unmoving targets in the main
beam of both antennas, such large variation should not be a problem.
23
Phase
2
0
−2
−400
−300
−200
−100
0
100
200
300
400
−300
−200
−100
0
100
200
300
400
Normalized
Magnitude
1
0.5
Self Spectra (dB)
0
−400
240
Log Periodic
Yagi
230
220
210
−400
−300
−200
−100
0
100
200
300
400
Doppler (m/s)
Figure 3.4: Interferometer data and power spectra for the airplane with Doppler shift
-50 m/s (also shown in figure 3.1). Again, we include the 95% significance level for
the coherence spectrum.
3.2
Airplanes and Compact Targets
Next, we direct our attention back to figure 3.1, and consider the Doppler-shifted
echoes visible in the range-Doppler diagram. Spectral peaks, presumably from aircraft, exist at 180 m/s, +50 m/s, and -50 m/s. MRR routinely sees echoes like this,
and upon examining successive segments of radar time series, we find that the range
migrations of targets such as these are consistent with their Doppler velocities.
We show interferometer data for the range containing the plane with −50 m/s
Doppler shift in figure 3.4. Again, we see that the individual power spectra and the
24
cross-spectrum magnitude are consistent with a narrow-band target at −50 m/s; also,
the interferometer phase is concentrated in the same velocity bins where the spectral
peak exists.
In principle, it is possible to estimate the angular width of the scattering surface (or
volume) from the coherence [5]; however, it is also possible to use the phase estimates
within those Doppler bins where the coherence is significant to infer properties of
the target. For instance, we can interpret the range of phase values covered in the
concentrated area as a measure of the angular width of the target. In this case, the
range of the phase estimates (denoted ∆φ) over the area of concentration is about
0.6 radians (by inspection of the plot). We can approximate the azimuth extent with
∆θ ≈ ∆φ/(kd) if we assume the target is located broadside to the interferometer axis
(using φ = kd sin θ and sin θ ≈ θ for small θ). Then, with an antenna baseline of
47 m, kd = 47 × 2π/λ ≈ 98; thus, the azimuth extent ∆θ ≈ 0.6/98 ≈ 0.006◦ , and
the transverse extent of the reflecting surface is approximated by r∆θ ≈ 600 meters,
where we have used r ≈ 100 km from a-priori knowledge. Here, 600 meters is the
transverse distance covered by the airplane during the 10 second duration in which
data was taken. Thus, we know the total velocity vector of the airplane: 50 m/s in
the radial direction away from the radar, and 60 m/s perpendicular to the radar line
of sight2 , resulting in a speed of approximately 78 m/s, or 150 miles per hour.
From the multi-range interferometer data in figure 3.2, we can also detect the plane
at 132 km (with Doppler shift 180 m/s), although in most cases the mixture of ground
clutter and aircraft is difficult to interpret without additional Doppler information.
Again, even though the target phase is compact, we are unable to unambiguously
determine an angle of arrival due to the aliasing of the cross-spectrum phase.
2
Determining whether this transverse direction is eastward or westward is possible, and would
require examining the phase on each antenna individually.
25
3.3
Geophysical Targets
We now consider targets in the upper atmosphere that scatter radio waves by mechanisms other than reflection off a tangible surface3 . Some radars that observe volumetric scatter from “deep” targets in the atmosphere rely on a phenomenon called Bragg
resonance to produce echoes that can be detected by the radar receiver(s). MRR is
one of these; we wish to detect irregularities in the ionospheric plasma density distribution. Actually, we are interested in the plasma processes, space weather, and
magnetosphere-ionosphere coupling of which the irregularities are a symptom. We
are able to make measurements of various parameters related to these irregularities
remotely with MRR.
Without going into great detail or speculation about the means by which ionospheric density perturbations arise, we assume some mechanism exists that results in
a force on a portion of the plasma fluid [21]. The disturbance will typically propagate outwards as a wave4 with a certain periodic structure (wave number), causing
successive plasma density enhancements and rarefications, or, alternatively, successive areas of high and low electrical impedance. When a radar wave attempts to
propagate through such a periodically-structured area, each “layer interface” (in a
layered-media approximation) will reflect a small amount of energy. In order for
significant backscatter to occur, the plasma density structures must exist with an
appropriate spatial scale to produce phase delayed-reflections that will constructively
interfere; this process is known as Bragg scatter [29].
However, a perfectly periodic, wavelike plasma distribution is not necessary for
3
There are other scattering mechanisms which we do not discuss here. For instance, many weather
radars rely on Rayleigh scattering from volumes filled with many point targets of size much smaller
than the radar wavelength (i. e., raindrops).
4
For E-region plasmas at frequencies in which we are interested, this will usually be an ion
acoustic wave.
26
kscatter
ks
θ
kr
Figure 3.5: An illustration of the wavevectors involved in Bragg scatter. A backscatter
radar can detect the scattered signal due to ks when θ = 0.
Bragg scatter to occur. Arbitrary turbulent density fluctuations can be decomposed
into spatial Fourier components that, if matched appropriately with the illuminating waveform, will produce a reflection. The radar backscatter is thus due to the
component of the plasma spatial frequency spectrum which resonates with the radar
wavelength.
We can describe this relationship for arbitrary scattering angles (or, equivalently,
angles of incidence) by
λs =
λr
2 cos θ
or
ks = 2 sin θkr
(3.1)
(3.2)
(3.3)
Here λs (ks ) is the wavelength (wavevector) of the plasma spatial structure component; λr (kr ) is the radar wavelength (wavevector); and θ is the angle of incidence,
measured as shown in figure 3.5. In the case where there is backscatter (when θ = 0◦ ),
we have λs = 21 λr , or in terms of wavevectors, ks = 2kr . Thus, the radar wavelength
selects very specific spatial scales, and the nature of the scatter returned to the receivers is extremely dependent on the frequency of the illuminating wave. For targets
at far ranges, the bistatic geometry of MRR can be approximated by a monostatic
27
(backscattering) system. MRR operates at VHF frequencies (100 MHz), so λr = 3
meters; it therefore detects structures on a spatial scale of 1.5 meters.
3.3.1
Meteor Trails
Although the exact physical processes causing radio wave scatter associated with
meteors are not completely understood, it is widely believed that small meteoroids5
cause “tubes” of plasma to form in the upper atmosphere (near mesosphere heights)
by ionizing gas as they burn up during their entry. These plasma tubes can extend
over a large range of altitudes, and are subject to wind shears from different layers of
the atmosphere. Radars such as MRR can detect them via Bragg scattering processes.
The tubes dissipate and the plasma recombines quickly, so the majority of “meteor
echoes” are short-lived (they last approximately 1 second); however, they occur often
and are usually easily identified.
Radar observations of meteor trails are interesting for several reasons. For instance, their Doppler shifts reveal neutral wind velocities in the upper atmosphere,
and their spectral widths depend on the decay time constant of returned power, which
in turn is related to the relevant diffusion coefficient for the plasma [6]. Interferometer
observations of meteor trails can show their transverse extent, and when combined
with Doppler information, neutral wind shears at mesosphere altitude can be detected. By creating plots such as the one in figure 3.8, we are even able to form
rudimentary images of the plasma tubes caused by the meteors.
In figure 3.6, we show a typical range-Doppler display from MRR during the 2002
Quadrantids shower (January 3, 2002) when a meteor is present; the corresponding
two-element interferometer data for range 790 km (where the meteor appears) can be
seen in figure 3.7. The spectral peak is nearly centered at zero Doppler velocity; the
5
Of sizes on the order of a grain of sand.
28
dB
Velocity (m/s)
January 3, 2002 UT 19:23
230
300
228
200
226
224
100
222
0
220
−100
218
−200
216
−300
214
0
200
400
600
Range (km)
800
1000
1200
212
Figure 3.6: A range-Doppler diagram showing an echo from a meteor trail (790 km)
during the 2002 Quadrantids meteor shower.
trail is not moving quickly. However, the Doppler width covers 50 or 60 m/s, and
the interferometer phase shows a clear downward linear trend over the Doppler bins
with significant coherence. By examining the phase estimates in the concentrated
“linear” area, we find that ∆φ is approximately 1 radian. At a range of 790 km, this
indicates that the meteor trail is roughly 8 km wide in the radar field of view. Thus,
the scattering volume appears to contain a wind shear across at least 8 km in the
transverse direction.
We present another multiple-range interferometer image in figure 3.8; the total
backscattered power from a meteor trail has been plotted on a two-dimensional grid
(frequency information has been integrated out). MRR observed this echo during the
Geminids meteor shower on December 14, 2001. The transverse span of the plasma
29
Phase
2
0
−2
Normalized
Magnitude
−400
−200
−100
0
100
200
300
400
−300
−200
−100
0
100
200
300
400
1
0.5
0
−400
Self Spectra (dB)
−300
225
Log Periodic
Yagi
Doppler (m/s)
220
215
210
−400
−300
−200
−100
0
100
200
300
400
Doppler (m/s)
Figure 3.7: Cross- and self-spectra for the range containing the meteor echo shown
in figure 3.6. The 95% significance level for the coherence is also shown.
tube appears to be 20-30 km; it also extends through 2-4 range cells (3-6 km in slantaltitude). In figure 3.9 we show Doppler spectra from the range 585 km; we see that
this meteor trail has a net positive Doppler shift (it is blueshifted, or moving towards
the radar). It also has a large Doppler width: nearly 100 m/s. However, unlike the
meteor trail shown in figures 3.6 and 3.7, the phase estimates do not vary linearly
over the Doppler bins with large magnitude. They are still compact, but bunched
up, and not organized according to Doppler velocity. This indicates that the meteor
trail is present over a wide area, revealing different mesospheric wind velocities.
From the bottom panel of figure 3.9, we also see that the meteor trail spectrum
30
Meteor Trail: 14 December 2001, UT 04:39
One Interferometer Lobe (5−6o azimuth)
600
595
Range (km)
590
585
580
575
570
565
560
555
−20
−10
0
10
20
30
Transverse Position, Measured from Lobe Center (km)
Figure 3.8: A meteor trail image, produced with MRR interferometer data.
has roughly three peaks. We speculate that the scattering volume (area of ionization)
is due to a meteoroid which broke into three (or more) pieces during its entry into
the atmosphere. Each of these pieces had a different velocity relative to the radar
line of sight; thus, the resulting spectrum has three separate areas with significant
magnitude.
Finally, in figure 3.10 we show several power spectra from other meteor trails
observed by MRR, along with corresponding interferometer coherence and phase. In
many cases the coherence has a very high variance; this is due to the low SNR of the
meteor trail backscatter.
31
Phase (rad)
2
0
−2
−400
−300
−200
−100
0
100
200
300
400
−300
−200
−100
0
100
200
300
400
Self Spectra (dB)
Normalized
Magnitude
1
0.5
0
−400
230
Log Periodic
Yagi
220
210
200
−400
−300
−200
−100
0
100
200
300
400
Doppler (m/s)
Figure 3.9: Individual range spectra for the meteor trail shown in figure 3.8. (Observed during the Geminids meteor shower, December 14, 2001.)
3.3.2 Auroral Observations
The majority of our interest in geophysical targets lies in “auroral E-region irregularities,” or, as we mentioned at the beginning of this section, turbulence and waves
in the ionospheric plasma caused by various interactions with the magnetospheric
plasma and solar wind. Because of the location of MRR (in the northwest United
States), we focus on high latitude phenomena, and because of the structure of Earth’s
magnetic field and the aspect angle dependency of the plasma irregularities, we are
unable to detect F-region phenomena. Therefore, we concentrate on echoes from Eregion plasma turbulence in this work. In particular, we are interested in what the
Selected Meteor Echoes
32
Figure 3.10: Selected meteor echoes observed by MRR during the year 2002 with
corresponding interferometer data. For each group of plots, the top panel is a power
spectrum, the middle panel is the coherence, and the bottom panel shows the crossspectrum phase. In each case, the Doppler velocities range from -400 to 400 m/s.
Some of the echoes are very low SNR.
33
radar detects when the auroral oval expands into the radar field of view at lower
latitudes, sometimes called the “radar aurora” [28].
On April 20, 2002, the Kp index6 grew to above 7.0, indicating high geomagnetic
activity, and MRR recorded many examples of E-region turbulence. One such auroral
echo is shown in figures 3.11 (range vs. Doppler velocity) and 3.12 (two-dimensional
interferometer image). By examining these figures, we see that the scattering volume
extends for many range cells (over 100 km), and that its transverse width is approximately 20 km. The net Doppler shift of the entire echo is near 400 m/s (redshifted);
we might classify the echo as “type 2” because of its large Doppler width [28]. The
ranges with narrower spectra (650–700 km, 800–900 km) could possibly be “type 3.”
When the data is of high quality (e. g., it does not suffer from interference or
a momentarily-ambiguous transmitter waveform), much can be learned about these
irregularity echoes from the MRR interferometer. We will present other observations
and more extensive analyses in chapter 6.
3.4
Analysis of Data Products
We desire a complete description of the probability density function of the interferometer coherence estimates in order to determine what level of coherence is statistically
significant, indicating the presence of a target. We outline a possible approach for
analytically determining the probability density function of the coherence estimates,
and then describe a numeric experiment we conducted to estimate confidence levels
via simulated data.
The target signal, denoted s(t), is Gaussian, as is its Fourier transform S(f ), since
6
A measure of geomagnetic disturbance designed to assess solar particle radiation by its magnetic
effects as seen from the Earth at mid-latitudes [1]. The index value is the aggregate of 13 groundbased magnetometer measurements from around the globe and is scaled from 0 to 9. The name
Kp originates from ”planetarische Kennziffer” (planetary index).
34
April 20, 2002 UT 06:27
1500
dB
223
222
Velocity (m/s)
1000
221
220
500
219
0
218
217
−500
216
215
−1000
214
−1500
0
200
400
600
Range (km)
800
1000
1200
213
Figure 3.11: An MRR range vs. Doppler diagram showing an echo from E-region
turbulence between 600 and 800 km.
a linear combination of Gaussian-distributed random variables remains Gaussian. Its
coherence spectrum is formed by the complex conjugate multiplication of the Fourier
transforms of the signal on two antennas (p and q), normalized by the self power from
each antenna, as follows:
S ∗ (f )
Sp (f )
=q
×q q
Sp (f )Sp∗ (f )Sq (f )Sq∗ (f )
Sp (f )Sp∗ (f )
Sq (f )Sq∗ (f )
Sp (f )Sq∗ (f )
q
(3.4)
A large number of receiver samples allows us to form a final coherence spectrum from
the summation of M individual spectra. Therefore, a single coherence estimate (in a
single Doppler bin) is obtained by summing M K-distributed random variables, where
the K-distribution results from the multiplication of two unit variance Gaussians [28].
The functional form of the K-distribution is a Modified Bessel Function of the second
kind, denoted with a K, giving the distribution its name. A scheme is now needed
35
April 20, 2002 UT 06:27
900
850
Range (km)
800
750
700
650
600
−40 −20
0
20 40
Transverse Position (km)
Figure 3.12: A two-dimensional interferometer image of the same auroral echo shown
in figure 3.11.
for dealing with the large sum (up to 100) of random variables; a moment generating
function approach seems the most feasible.
However, now we turn to numeric simulation for an approximate result. Using
the MATLAB software (including its Gaussian random number generator), we simulated 500,000 instances of the sum of M complex, zero mean, unit variance Gaussian
random variables7 . The resulting histogram for M = 78 is shown in the top panel
of figure 3.13. We also sort the resulting coherences in order to determine their percentile, as shown in the bottom panel. We have plotted the 95% significance level on
7
M = 78, 39, and 19 for initial coherent integrations of 50, 100, and 200, respectively.
36
Number of Samples
Histogram of Sample Coherence Distribution
6000
4000
2000
0
0
0.2
0.4
0.6
0.8
Sorted Coherence Samples
Percentile
(samples below
coherence value)
95
75
50
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Coherence
Figure 3.13: Histogram and sorted samples of simulated coherence values. In the
bottom panel, dotted lines are shown for the 50th , 75th , 95th , and 99th percentiles.
These graphs represent a sample size of 500,000.
each of our experimental coherence spectra for reference.
Close observation of the cross spectra presented here will reveal occasional imperfections in the interferometer data and/or signal processing. For example, in the
middle panel of figure 3.7, the normalized magnitude exceeds unity, a physically impossible result which is caused by the removal of bias in the cross spectrum, a quantity
which must be estimated (see section 4.7). A second flaw remains in the phase (for
example, in the top panel of figure 3.7); the phase at Doppler shifts far from the
meteor peak is not distributed uniformly over [−π, π]. This is because the ambiguity
37
sidelobes of the echo reach to Doppler shifts far from the true spectral peak. These
Doppler features appear to arise from the same part of the sky, and generate phase
similar to that of the meteor peak. In figure 3.9, the coherence shows a false spike
(at least, a spectral feature that does not correspond with a high SNR feature in the
individual antenna spectra in the panel below). Since the interferometer coherence
is a correlation between the time series on two antennas, very low-SNR features can
show up prominently, and can potentially cause problems with interpretation.
3.4.1
Two-Dimensional Interferometer Images
While much can be learned from interferometer cross-spectra at individual ranges, we
find it useful to create plots like those shown in figures 3.2, 3.8, and 3.12 for easily
determining the large-scale structure of radar targets. These two-dimensional plots
of range vs. transverse position can be considered true images, since the pixels are
total power received (Doppler information is integrated out), organized on a grid by
physical position. To produce these images, we separate the total scattered power
from each range into a quantized set of positions across the transverse dimension.
Using interferometer phase information, we group each point in the cross-spectrum
into a certain (user specified) number of phase bins, then sum the (un-normalized)
spectrum magnitudes in each phase bin. Essentially, we create a phase histogram,
weighted by the cross-spectrum magnitude. We then stack these “transverse dimension spectra” according to range. In general, we try to keep the range resolution and
transverse resolution comparable, although this is not necessary. The transverse resolution is determined by the number of phase bins we specify for the interferometer
signal processing, and we consider this choice in great detail in chapters 4 and 5.
In creating the two-dimensional images, we have made some simplifying assumptions. First, we have taken the azimuthal beamwidth of the interferometer to be
38
5.6◦ ; this value assumes 32 equally-spaced lobes in a 180◦ field of view, which is not
entirely accurate. Also, while transverse size may be inferred using range information
together with the 5.6◦ beamwidth approximation above, the “transverse dimension”
in the plots remains, at best, phase information rather than a true physical dimension.
Second, we have taken contours of constant range to be circles, which is accurate for
a monostatic radar, but in our bistatic case this is another approximation. Angle
aliasing is also an issue, and we discuss this below.
3.4.2
Azimuth Aliasing
Since our interferometer baseline is several wavelengths long (16λ), the interferometer
has several lobes (32) within its field of view, and our data is strongly aliased in angle.
Therefore, we have no information about the absolute arrival angle of the radar
scatter. Much of this ambiguity may be removed by considering a-priori information
we have about the underlying physical processes causing the irregularities and various
elements of the system (such as the antenna beam patterns). As we will see, in many
cases the probable scattering volume location is constrained to an area only as large
as two interferometer lobes; this allows us to speculate about transverse structures of
irregularity echoes with some confidence.
For example, it is well known [29, 28, 17, 20] that these field-aligned irregularities
are strongly damped when traveling in directions parallel to Earth’s magnetic field
(or, at small aspect angles8 ), and this limits the possible location of any such echoes
detected by MRR, as shown in figure 6.1. Also, because of the geometry of the MRR
field of view and the magnetic field’s dip angle at high latitudes, we detect only
irregularities in the E-region of the ionosphere; thus, in our case, scattering volumes
are confined to exist at E-region heights. Finally, because our Yagi antenna has a much
8
We have defined 0◦ aspect angle as a line of sight parallel to Earth’s magnetic field.
39
narrower field of view (directed toward the northeast) than our log periodic antenna
does, we occasionally detect echoes with the log periodic that the Yagi does not
sense. In these cases, we speculate that the scattering volume lies in the westernmost
“solution” area (where the appropriate range and perpendicular aspect angle contours
intersect). Alternatively, we know the local time of the echo observation with respect
to magnetic midnight, which tells us whether to expect (in general) westward or
eastward convection, and thus the blue- or red-shiftedness of the echo can give us
information about whether the scattering volume lies to the east or the west of the
radar. Therefore, by considering the range, local time, and antenna gain pattern
associated with an irregularity observation as well as the constraints of E-region
altitude and perpendicular aspect angle, the location of scattering volumes can be
estimated to within a reasonable area with a good amount of confidence.
Another initial concern about our large baseline was that it might completely
prevent us from detecting auroral scatter with the interferometer. At first, we had
expected the scattering volume of most irregularities to exceed the 5.6◦ of angular
width available in each lobe, causing the phase to vary over an entire wavelength and
the coherence to remain low despite the existence of a strong echo. However, we did
not find this to be the case; in fact, in the year 2002 we collected interferometric
measurements from E-region phenomena on 27 separate days that show transverse
scattering sizes as narrow as 30 km (at a range of 1000 km).
Occasionally we observe irregularities at very close range, and in these cases the
interferometer lobes are extremely closely spaced, and too narrow to resolve even
moderately sized scattering volumes. For example, the right panels of figure 3.14
show individual antenna views of a close scatterer at 400 km; the echo also has an
uncommonly high SNR on both antennas. Yet in the corresponding interferometer
image on the left, the echo can barely be seen. We attribute this low detectability to
40
550
224
223
500
Velocity (m/s)
225
222
450
1000
500
0
−500
−1000
−1500
300
221
400
500
600
Yagi Antenna
220
400
219
218
350
217
300
−20
0
20
216
Velocity (m/s)
Range (km)
Log Periodic Antenna
226
600
1000
500
0
−500
−1000
−1500
300
400
500
Range (km)
600
Figure 3.14: An irregularity observed on 18 April, 2002, at UT 06:03. Power spectra
from both antennas as well as an interferometer image are shown. The color scale
applies to all panels.
a scattering volume which spans multiple interferometer lobes and thus shows little
coherence.
In contrast, figure 3.15 shows an echo that in many ways is more representative of
the type of irregularities MRR detects. First, the SNR of the echo on both antennas
is much lower than in the previous example (note that the greyscale is the same for all
panels in both figures 3.14 and 3.15). Next, the echo exists at a range of approximately
1000 km, much farther away than the one in figure 3.14. Thus, the interferometer
41
Log Periodic Antenna
1200
226
500
1150
224
223
1100
0
−500
−1000
222
1050
−1500
900
221
1000
1100
1200
Yagi Antenna
220
1000
1000
219
218
950
500
0
−500
217
900
Velocity (m/s)
Range (km)
1000
Velocity (m/s)
225
−50
0
50
216
−1000
−1500
900
1000
1100
Range (km)
1200
Figure 3.15: An irregularity observed on 24 March, 2002, at UT 05:07. Power spectra
from both antennas as well as an interferometer image are shown. The color scale
applies to all panels.
lobes are necessarily wider at this range, and are able to accommodate larger targets.
Though the received scatter in this case has a much lower SNR, the interferometer
still clearly detects a scatterer. This scatterer is almost certainly contained within a
single interferometer lobe, its transverse width spanning 20-30 km.
A consequence of phase compactness and the method by which we generate the
two-dimensional interferometer images can also be seen in the left panel of figure
3.15. Since all the phase estimates for the auroral echo are concentrated around the
appropriate value for the irregularity direction of arrival, the area adjacent to the
42
echo in the image is almost completely devoid of “noise.” This emptiness may be
regarded as a sign of local high SNR (at a particular range); nevertheless, we must
realize that the effect is a byproduct of our signal processing.
Currently, though we are often able to make useful measurements (of transverse
structure) with our interferometer despite its angle aliasing, we would still like to
determine scatterer locations. We have recently constructed two shorter baselines
(by adding a third antenna) to alleviate this problem; however, so far we have been
unable to make many measurements with the new antenna9 .
3.4.3
Other Issues
Other remaining nonideal elements of the system that will be characterized and either
changed in the future or calibrated and compensated for in the signal processing include the orientations and positions of the antennas, both vertically and horizontally,
and differing cable lengths between the antennas and the receiver. Finally, the antennas currently in use in our system are quite inhomogeneous; we obtained the data presented above from one Yagi and one log periodic antenna. The recently-constructed
(and even more recently deconstructed) third antenna is a half-wave dipole. While
these differences in gains and beam patterns are not necessarily a drawback (see discussion of irregularity echo location at the beginning of chapter 7), further analysis of
the field of view and other antenna-related properties of the system will be conducted
to ensure proper treatment.
9
The scatter-receiving antennas face many hardships that are difficult to simulate in MATLAB,
including, but not limited to, severe weather on Manastash Ridge and rural Washington natives
with shotguns.
43
Chapter 4
SIGNAL PROCESSING
“Their only advantage would seem to be that they would allow one to
whom the autocorrelation is sacred to apply it.”
— Drs. Bogert, Healy, and Tukey (In The Quefrency Alanysis of Time Series for Echoes: Cepstrum, Pseudo-Autocovariance, Cross-Cepstrum and
Saphe Cracking [2])
The basic signal processing algorithm employed at MRR is designed to efficiently
extract power spectral density measurements from the receiver samples we collect.
Our primary data product is a range vs. Doppler diagram (figure 2.2, for example).
However, we will be mainly concerned here with the slightly different problem of
estimating the cross-correlation (or cross-spectrum) between antennas, which is useful
in interferometry for determining angle of arrival and the transverse structure of
targets.
Before we begin, we state a few facts and make some assumptions about the nature
of the data produced by the passive radar system.
4.1
Preliminary Assumptions
We denote many signals using a functional notation with a continuous time argument:
f (t). However, to the extent that our analysis is meant to describe signal processing
implemented on a digital computer, it is convenient to think of t as quantized by the
associated sampling period 1/fs .
44
The most important assumption we have made is that all the stochastic signals
we are concerned with are zero-mean, Gaussian (in amplitude) processes. Although
the FM waveform in particular is definitely not Gaussian, we make this assumption
in order to simplify the discussion and make analytic analyses of the radar signal
processing tractable. When we consider the effect of a coherent integration on the
received time series, however, the Gaussian assumption is vastly more valid, thanks
to the Central Limit Theorem. To illustrate this, we compare histograms of radar
time series before and after a coherent integration of 50 consecutive points. Figure
4.1 shows (in the top two frames) relative-frequency histograms of the real part of 106
samples from an FM transmission. The right panel, after the coherent integration, is
clearly closer to a Gaussian distribution than the left; for longer integration times we
only expect this approximation to improve. The histograms have been normalized by
the number of data points and their bin widths so that they have unit area and may
be compared to probability density functions; Gaussian probability densities with
the same mean and variance as the data have been overlayed in the post-coherent
integration plots. The bottom two panels show the same information for the radar
“detected time series,” which is the result of matched filtering. The detected time
series (with no target present) appears to have an exponential distribution, but after
the coherent integration, it too appears Gaussian.
Also, odd moments of order 3 and above should tend towards zero as a distribution
function approaches that of a Gaussian random variable. We measured the skewness
(normalized third central moment) of the time series shown above both before and
after the integration. We found that, in both cases, the skewness of the decimated
time series was less than 10% of the skewness of the original time series.
The scattering signals themselves can be assumed Gaussian1 , and they are zero1
At VHF frequencies, the wavelength of the illuminating waveform is often shorter than the
45
Probability Densities of Radar Samples Before and After Coherent Integration
Transmitter Signal
8e−5
8e−5
4e−5
4e−5
Detected Time Series
0
−2e4
0
2e4
0
−5e5
0
5e5
0
2e9
1.5e−9
2e−8
0.75e−9
1e−8
0
−2e8
0
2e8
Before Coherent Integration
0
−2e9
With 50−Point Coherent Integration
Figure 4.1: Normalized histograms showing the trend toward Gaussian distribution
after coherent integration. Gaussian probability density functions with the same
mean and variance as the data have been overlayed in the right panels. Each figure
represents the real part of 106 complex data points, sampled at a rate of 100 kHz.
mean as long as the time average we use is long enough. Of course, we would like to
keep our time averages as short as possible, because we also assume that our radar
targets are statistically stationary over the time we observe them. Depending on the
time resolution used and the nature of the target, this assertion is questionable (see
discussion of E-region targets in chapter 6). However, to proceed with a tractable
theoretical analysis, it is a necessary assumption, so we make it.
Next, we consider the characteristics of the transmitter signal, an FM commercial
correlation length of the scatterer. For longer wavelengths, or shorter correlation lengths, the
received spectrum should be Lorentzian, according to collective wave scattering theory [10].
46
FM Transmission Autocorrelation Magnitude
1
0.9
Correlation Coefficient
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−50
−40
−30
−20
−10
0
10
Lag (microseconds)
20
30
40
50
Figure 4.2: A typical autocorrelation function for an FM transmission (96.5 MHz).
The signal is mostly decorrelated after 10 µseconds, indicating that samples taken
at 100 kHz are independent. These samples were taken at 200 kHz, roughly the
bandwidth of the signal.
radio broadcast. An FM process has a bandwidth around its VHF carrier of about
200 kHz. Frequency modulation does not affect the carrier amplitude, but it has a
nonlinear effect on the phase. Thus, the transmitted signal has constant modulus and,
from the point of view of an outside observer, essentially random phase. Properties
of FM broadcasts and their potential as radar waveforms are described in detail by
Hansen [8], Hall [7], and Lind [20]. Here we are mostly interested in the correlation
time, τx (shown in figure 4.2), which is typically 10µs.
Thus, it’s true that the FM signal autocorrelation Rx (τ ) is a sharply peaked
47
function on time scales of tens of microseconds. Sahr and Lind [27] use a Gaussian
model,
Rx (τ ) = e−πτ
2 /τ 2
x
(4.1)
where, in general, τ τx (τx is small compared to time lags in which we are interested). Given that our usual sampling frequency at MRR is 100 kHz, however,
the transmitter time series can be modeled as “white,” i.e., each sample in the time
series is independent from the next, sampled after the waveform decorrelates. Then
we can write Rx (τ ) = Rx (0)δ(τ ), where, to avoid problems with higher-order powers
of Rx (τ ), we use the Kronecker delta function so that δ(0) = 1 (which implies that
δ 2 (τ ) = δ(τ )). This model will vastly reduce the complexity in the derivations to
come. Armed with these tools, we proceed to a description and analysis of the signal
processing.
4.2
MRR Signal Processing
With our passive radar, we first perform a matched filter operation by correlating the
received scatter (denoted y) with the reference transmitted signal (the original FM
transmission, denoted x). Suppose we model the scattered signal purely as a delayed
copy of a perfect FM transmission modulated by a signal of interest s(t):
y(t) = s(t)x(t − r0 )
(4.2)
Then, since an FM process ideally has constant modulus, the matched filtering operation in the ideal case will recover s(t) exactly:
z1 (t, r0 ) = y(t)x∗ (t − r0 )
(4.3)
= s(t)x(t − r0 )x∗ (t − r0 )
(4.4)
= s(t)
(4.5)
48
From this point forward in this document we treat both x and s (and therefore
y also) as stochastic signals. We call the time series z1 (t, r0 ) emerging from the
matched filter the “detected signal.” It is a function of time, and is proportional to
the scattering amplitude of a target at range r0 . The range r0 is actually a sample
number with units of time quantized by the sampling period
1
,
fs
which we use as a
delay in the transmitter signal x(t), but in terms of the target, r0 corresponds to the
bistatic slant distance
r0
c
2
With a sampling frequency fs = 100 kHz, each sample is 10µs apart and each range
delay is 1.5 km.
During this “detection” step in the MRR signal processing, we also do a coherent
integration, typically of 50 points or greater:
z(t, r) =
D−1≥49
X
z1 (t + t0 , r) , t = 0, D, 2D, . . .
(4.6)
t0 =0
The time t now evolves on a much slower time scale than the samples t0 . We
are able to do this because the correlation time of the target is very large with
respect to the correlation time of the reference signal. Thus, this integration step
accomplishes a reduction in the data rate from the raw sample rate (100 kHz) to the
much lower typical target bandwidth (≤2 kHz). This eases the burden of computing
power spectra with the data, and puts appropriate limits on the range of Doppler
velocities computed. Furthermore, it reduces the power from clutter arriving at ranges
other than the range of interest, since x(t) becomes uncorrelated quickly during this
interval.
Also, the voltage integration gives us a processing gain (or, alternatively, the
variance of the target time series is reduced). Processing gain (for a lowpass operation
49
with real-valued coefficients) is defined by
f [k])2
GP ≡ P
(f [k])2
P
(
(4.7)
where f [k] is the window associated with the signal processor impulse response. In
the above integration and decimation step, we achieve a processing gain of D per
sample; the expression would become slightly more complicated if we used a data
taper or overlapped/sliding windows. When D = 50, our processing gain at this step
is 50, or approximately 17 dB.
Finally, the coherent integration step allows us to appeal to the Central Limit
Theorem. The sum of many independent samples of the transmitter waveform allows
us to approximate it as Gaussian distributed, even though it certainly is not. However,
if we can assume that x is Gaussian, then we can use Isserlis’ Gaussian moment
theorem [16], upon which we lean heavily in the estimator derivations below.
Thus, during typical MRR operation, we perform matched filtering and coherent
integration on the data, usually followed by FFT-based spectrum estimation. Below,
we consider the radar signal processing in great detail in order to learn more about
its effects on the statistics of the observed target.
4.3
Motivation for Statistical Calculations
A major motivation for these derivations is to determine an appropriate resolution
to expect from our instrument. In chapter 3, we showed several 2D interferometer
images which were obtained by separating the cross-spectrum phase into bins, and
plotting the total power in each bin over several ranges. The number of bins used in
this operation is variable, and we essentially specify the transverse resolution of the
interferometer by setting this parameter. Figure 4.3 shows a comparison of interferometer images with different range resolutions and transverse resolutions.
50
Range Resolution / Transverse Resolution Comparison
1.5 km / 1 km
3 km / 2 km
6 km / 5 km
12 km / 10 km
Range (km)
1100
1050
1000
950
900
−50
0
50
−50
0
50
−50
0
50
−50
0
50
Figure 4.3: A comparison of 2-D image resolutions obtained by range averaging and
varying the number of phase bins used in the interferometer program (essentially
“phase averaging”). The data shown here is an auroral echo observed on February 2,
2002, UT 05:11.
We claim that the range resolution value of 1.5 km (the finest resolution in the
grid, and the one used at MRR currently) is appropriate; with this range resolution,
each sample in the 100 kHz time series represents a separate range, and if the samples
from the transmitter are independent for each range, ideally there should be no range
ambiguity. Advantages in spectrum computation and SNR are available if rangeaveraging is employed, but from a theoretical point of view, we assert that the intrinsic
range resolution of a radar waveform with correlation time τx = 10µs is
c
2
× τx = 1.5
km.
We would like to get a similar idea for an appropriate transverse resolution to
51
use in our interferometer images. The idea of resolution is related to the uncertainty
associated with a measurement, which is in turn directly dependent on the variance
of the estimate being displayed as a data product. Error bars are a common way
to display information about the uncertainty of a measurement; often error bars are
displayed as ± one standard deviation of an estimate.
Tarantola [30] uses an instructive description of experimental data: while an estimated parameter can be completely described by its probability density function,
an estimator essentially makes a decision about the true data value based on the
distribution moments it estimates from experimental data. For example, in a grid of
data displayed as an image, each pixel represents the sample mean of the true value;
in the case of the interferometer, the location at which to plot each pixel is also an
estimate. Therefore, the image resolution should be based upon the variance in the
interferometer data products.
In the following sections we derive expressions for the expected value and variance
of our interferometer data product, the cross-correlation of a radar target between
two scatter-receiving antennas. Our intent is to quantify the statistical properties of
the estimate so that we can display our data in the most useful and least misleading
manner possible. Following Sahr and Lind [27] and Hall [7], who derived similar
expressions for the MRR estimator of the target autocorrelation, we identify sources
of bias and the factors contributing to the variance in our interferometer estimates.
We also describe the effects of signal degradation, such as interference and receiver
noise, on the estimator.
4.4
The Interferometer Estimator
To extract interferometric information from radar data, we estimate the complex
cross-correlation of the target signal as it arrives on multiple antennas in a known
52
spatial configuration. We consider only the two-antenna case here. Our goal is to
compute
D
E
pq
Rss
(τ ) = sp (t)s∗q (t − τ )
(4.8)
which is the complex cross-correlation of the target signal s as it arrives on antennas
p and q. With a conventional (pulsed) radar, the natural estimator takes the form
vp (t)vq∗ (t − τ )
where vp and vq are voltages on the scatter-receiving antennas. However, in the passive
radar case, we first need to deconvolve the reference signal from the received scatter,
as shown above in section 4.2. Therefore, we must use the detected signal z in place
of the antenna voltages for our estimator (which we have named K). The additional
z = yx∗ correlation step results in a much larger number of terms to consider when
evaluating the statistics of the passive radar interferometer estimator:
K̂pq (r0 , τ ) = zp (t, r0 )zq∗ (t − τ, r0 )
(4.9)
= yp (t)x∗ (t − r0 )(yq (t − τ )x∗ (t − r0 − τ ))∗
(4.10)
= yp (t)x∗ (t − r0 )yq∗ (t − τ )x(t − r0 − τ )
(4.11)
The estimate K̂pq (r0 , τ ) represents a single sample of the cross-correlation between
antennas p and q of the scatterer at range r0 . For convenience we will often drop the
clarifying subscripts pq and arguments r0 and τ , when these are clear from context.
In the next section we evaluate the expected value of this estimator. We show that
it is biased by clutter and interference, although at long lags (where the transmitter
signal becomes uncorrelated), it becomes unbiased.
53
4.5
Expected Value of the Interferometer Estimator
The expected value of this estimator is a 4th order correlation involving the reference
transmission and the scattered signal observed from two perspectives (each antenna
in the interferometer):
D
E
E
D
K̂ = yp (t)x∗ (t − r0 )yq∗ (t − τ )x(t − r0 − τ )
(4.12)
However, the scattered signal is itself formed from the interaction of the transmitter signal x and any target signals it encounters. Before proceeding, then, we
introduce a model for the voltages on the two scatter-receiving antennas.
4.5.1
A Model for the Received Scatter
In the ideal case, which we treat first, we call the scattered signal y(t) a superposition
of signals from all contributing ranges, modulated by successively delayed copies of
the transmitter waveform. Models for y(t) on two antennas p and q (assumed to have
the same gain) are given below.
yp (t) =
yq (t) =
R
X
i=1
R
X
si (t)x(t − i)
si (t)e−jkd cos θi x(t − i)
(4.13)
i=1
Here k is the wavenumber of the transmitter carrier frequency, d is the distance
between the antennas, and θi is the azimuth angle of the target (see figure 1.1 on
page 5). The signals si (t) are the target signals; we assume them in general to be
zero-mean, complex Gaussian processes (si (t) includes any Doppler information as
well as radar cross section). For simplicity, we assume all the scattering signals are
point targets in azimuth, so that θi is a single value with a probability distribution
that we model as Gaussian. However, superposition allows us to sum many targets
54
at one range in order to model targets that are distributed in angle, if we wish. R
is the number of contributing ranges, equal to approximately 800, or 1200 km in the
current implementation of MRR; after this distance, E-region targets fall below the
horizon, and we cannot detect them with our line-of-sight system. (At high latitudes
the magnetic field dip angle is such that VHF radars are unable to detect F-region
scatter.) We must include all R ranges, because the transmitter operates continuously.
As we will see, the continuous wave nature of the transmitter causes clutter from
all ranges to be present in every correlation (or power spectrum) estimate; this is
one striking difference between passive radars and conventional radars. There are
advantages associated with the 100% duty cycle, however: the transmitter effective
radiated power is much larger than that available with pulsed-mode radars. Finally,
we note that the possibility for direct illumination of the scatter receivers by the
transmitter has been included in the model, since we sum over all contributing ranges,
including range “zero” (direct illumination would be at range number 51, as this is the
amount of time it takes for light to travel directly from the transmitter to the scatter
receivers). We expect ranges 1 through 50 to be “empty” after matched filtering, as
the scatter and transmitter waveform should be completely uncorrelated.
55
Using the model for received scatter, then, we write the expected value of the
cross-correlation estimator for a target at range r0 as:
D
K̂
E
=
D
E
=
yp (t)x∗ (t − r0 )yq∗ (t − τ )x(t − r0 − τ )
x∗ (t − r0 )
R
X
(4.14)
!
sn (t)x(t − n) ×
n=1
R
X
s∗m (t
+jkd cos θm ∗
− τ )e
!
x (t − m − τ ) x(t − r0 − τ ) (4.15)
m=1
=
R X
R D
X
sn (t)s∗m (t − τ )e+jkd cos θm ×
n=1 m=1
E
(4.16)
E
(4.17)
x∗ (t − r0 )x(t − n)x(t − r0 − τ )x∗ (t − m − τ )
=
R D
R X
X
e+jkd cos θm
ED
E
sn (t)s∗m (t − τ ) ×
n=1 m=1
D
x∗ (t − r0 )x(t − n)x(t − r0 − τ )x∗ (t − m − τ )
where the last equality follows from the fact that the transmitter signal x and the
target signals s are zero mean and uncorrelated.
We now find ourselves with a 4th order correlation in x. As stated above, we
assume that x is Gaussian distributed, which allows us to apply Isserlis’ Theorem to
break higher-order even correlations into combinations of 2nd order correlations (see
discussion in section A.1 of appendix A). Under the Gaussian assumption, we have:
D
E
K̂ =
R X
R D
X
E
e+jkd cos θm Rs,n (τ )δnm ×
n=1 m=1
D
D
ED
x(t − n)x∗ (t − r0 )
E
x(t − r0 − τ )x∗ (t − m − τ ) +
ED
x(t − n)x∗ (t − m − τ )
E
x(t − r0 − τ )x∗ (t − r0 )
(4.18)
56
In writing hsn (t)s∗m (t − τ )i = Rs,n (τ )δnm , we have made the assumption that
target signals at different ranges are independent. Given our range resolution of 1.5
km, this assumption seems completely reasonable2 . This Kronecker delta function
will select m = n, and dispense with one of the sums:
D
K̂
E
=
=
R D
X
n=1
R D
X
E
h
E
h
i
e+jkd cos θn Rs,n (τ ) Rx (r0 − n)Rx∗ (r0 − n) + Rx (τ )Rx∗ (τ )
e+jkd cos θn Rs,n (τ ) |Rx (r0 − n)|2 + |Rx (τ )|2
i
(4.19)
(4.20)
n=1
In order to better interpret this expression, we make the simplifying assumption
that Rx (τ ) = Rx (0)δ(τ ). This is equivalent to saying that x is a white process
with power (spectral level) Rx (0). We justified this in section 4.1, and making the
substitution here leads to the following simplification:
D
E
K̂pq (r0 , τ )
=
R D
X
E
h
i
(4.21)
i
(4.22)
e+jkd cos θn Rs,n (τ ) Rx2 (0)δ(r0 − n) + Rx2 (0)δ(τ )
n=1
hD
E
= Rx2 (0) e+jkd cos θr0 Rs,r0 (τ ) +
δ(τ )
R D
X
E
e+jkd cos θn Rs,n (0)
n=1
We see that the cross-correlation estimator is unbiased for nonzero lags (or, for
a conservative estimate, lags greater than 20µs or so). At τ = 0, the delta function
(due to the assumed whiteness of the transmitter signal) turns on a large clutter term
containing the energy and interferometer phase from signals at all ranges. Thus,
this lag is essentially useless3 , but otherwise we have the values we seek: the target
2
E-region scattering structures typically have a lifetime (correlation time) of a few milliseconds,
which corresponds to less than a 10 meter extent when drifting at speeds near the ion acoustic
speed.
3
However, we make use of it as an estimate of the noise floor in section 4.7.
57
autocorrelation at the range of interest, r0 , and the interferometer phase, φ = kd cos θ.
The entire expression is multiplied by a constant factor (which we can estimate) of
the transmitter signal energy squared.
While identifying a condition under which the the correlation estimate is unbiased
is certainly a good thing, it is not particularly useful in the situation where the primary
radar data product is a power spectrum. This is currently the case with the normal
operation of the Manastash Ridge Radar. Although the correlation estimates are only
biased at short lags, the corresponding power spectra exhibit a nearly flat increase
in the noise/clutter/interference floor due to the clutter term. To show this, we take
the Fourier transform of equation 4.22:
D
E
Ŝpq (r0 , f ) =
Rx2 (0)
"
D
+jkd cos θr0
e
E
Ss,r0 (f ) +
R D
X
+jkd cos θn
e
E
#
Rs,n (0)
(4.23)
n=1
This result is the expected value of the cross-spectrum between antennas p and q
(not considering any effects of the cross-spectrum estimator itself, discussed in section
4.8.1). The desired target power spectrum is given by Ss,r0 (f ), and the interferometer
phase remains the same. The zero-lag bias acts like a flat spectrum in the frequency
domain, reducing the detectability of targets in which we are interested. Now, following Farley et al. [5], we normalize the magnitude by dividing by the individual power
spectra on both antennas, to obtain the complex coherency:
Cpq (r0 , f ) = rD
D
E
Ŝpq (r0 , f )
ED
Ŝpp (r0 , f )
(4.24)
E
Ŝqq (r0 , f )
The phase difference does not appear in the self-power terms Ŝpp and Ŝqq (since the
signals are multiplied by their own complex conjugates). Therefore, they are purely
real, and we have
D
E
D
E
Ŝpp = Ŝqq =
Rx2 (0)
"
Ss,r0 (f ) +
R
X
n=1
#
Rs,n (0)
(4.25)
58
so we can write
D
+jkd cos θr0
Cpq (r0 , f ) = e
E Ss,r0 (f ) +
P
n
ejkd(cos θn −cos θr0 ) Rs,n (0)
Ss,r0 (f ) +
P
n
Rs,n (0)
(4.26)
At frequencies where Sr0 is large (in the interesting case where there is a target with
significant amplitude), we can make the approximation
D
Cpq (r0 , f ) ≈ e+jkd cos θr0
E
(4.27)
which gives us a coherence of unity and the phase accumulated over the extra distance
traveled by the wave before it hits the second antenna. This is what we hope for and
expect. Otherwise, in the absence of a strong scatterer, the coherence is small4 , and
random fluctuations due to the clutter contribution in equation 4.26 dominate both
the phase and magnitude spectra.
The above approximation is useful because it demonstrates the ideal behavior
of the interferometer in the presence of a strong scatterer. However, in practice, the
coherence and phase estimates are severely affected by the random fluctuations caused
by clutter at other ranges. For this reason, we have devised methods for estimating
and subtracting the clutter-bias from our spectra. This will be addressed in section
4.7.
Finally, although we have arrived at equations 4.26 and 4.27 by modeling only
point targets in azimuth, the angular extent of the target also affects both the coherence and the interferometer phase [5]. In particular, if the angular extent is as large
or larger than one interferometer lobe (2π of interferometer phase), the coherence will
be low, the phase will appear unorganized (randomly distributed over the entire 2π of
interferometer phase, depending on the target characteristics), and the interferometer
will not detect a target, though the individual power spectra may clearly show one.
4
We briefly describe the statistical behavior of the coherence estimates in section 3.4.
59
In chapters 3 and 6 we show several experimental measurements of cross-spectrum
estimates, and attempt to offer interpretations of the data, drawing from these analytic results.
4.5.2
A More Complicated Scatter Model
In the previous section, we derived the expected value of an estimate of the crosscorrelation (and cross-spectrum) between two antennas in an interferometer. However, in our simple model for the received scatter (equations 4.13), we did not consider
any non-ideal effects, such as receiver noise and interference. Here we briefly consider
a new model for the received scatter which contains receiver noise and interference,
which we define as any signal appearing on both antennas p and q at the frequency of
interest (other than delayed copies of the reference FM signal). Interference, therefore,
includes cosmic noise and other ambient radio-frequency energy in the environment
which is either in the FM band or aliased into the FM band by our receiver5 .
With this new model, we will find that the cross-correlation estimate contains a
new bias term (at zero lag) consisting of the transmitter and interference signal energies. This type of analysis is useful in addressing larger questions such as whether the
radar sensitivity is noise-limited or clutter-limited. Of course, we are still neglecting
to model an endless number of nonideal elements in the system. Among these are
the transmitter and receiver filters; receiver nonlinearity; antenna gain patterns; the
propagation channel of the radio waves (weather, multipath, and fading effects); and
imperfections in the reference copy of the transmitter waveform. These last two are
discussed by Chucai Zhou [32] in his Ph.D. dissertation; we assume them all to be
ideal here.
5
The receivers sample the FM band at 56 MHz, which causes a lot of energy to be aliased into
our data. We use anti-aliasing filters; however, we still find interference from frequencies outside
the FM band to be a problem.
60
The new model for the received scatter on an antenna p is:
yp (t) = nrx, p (t) +
I h
X
i=1
i
nsky, i (t)ejkp̄·Ω̄i +
R h
X
i
sn (t)ejkp̄·Ω̄n x(t − n)
(4.28)
n=1
As before, the signals s(t) contain amplitude and Doppler information about the
targets at each range. We have written the phase term due to the angle of arrival
separately for each antenna to illustrate the action of the interferometer estimator.
The vector p̄ represents the location of antenna p and the unit vector Ω̄ represents
the direction of the target with respect to the origin (here, the midpoint between the
two antennas). Also present are a sum over all “interference noise” sources, nsky , and
a receiver noise term, nrx, p , which is specific to antenna p. By specific, we mean that
the receiver noise associated with the signal from one antenna is uncorrelated with
that associated with other antennas. With new digital radio technology, however,
the lines between receiver channels are becoming blurred. Our current system (see
chapter 2) samples RF signals directly, implementing most of the preliminary signal
processing (filtering, downconversion, amplification, resampling) digitally. This can
result in separate channels producing identical data, if they are tuned to the same
antenna at the same frequency and share other settings. This raises questions about
the independence of separate receiver channels. We still use, however, separate analog
amplifiers and passive filters on each antenna signal prior to digitization, so for now
our assumption of uncorrelated receiver noise is safe.
61
To find the expected value of the K estimator using the model in equation 4.28,
we again write the cross-correlation between the detected signal on antennas p and q:
D
E
K̂pq (r0 , τ )
=
D
E
=
*
yp (t)x∗ (t − r0 )yq∗ (t − τ )x(t − r0 − τ )
xx nrx, p +
nrx, q +
I
X
i=1
I
X
i=1
nsky, i ejkp̄·Ω̄i +
−jkq̄·Ω̄i
nsky, i e
R
X
(4.29)
!
sn xejkp̄·Ω̄n ×
n=1
R
X
+
sn xe
−jkq̄·Ω̄n
!+
(4.30)
n=1
On the second line we have dropped the complex conjugate signs and the time
arguments of the signals for compactness, mimicking the shorthand notation found
in Sahr and Lind’s description of the MRR target autocorrelation estimator [27].
This looks messy, but in fact, when we consider that the receiver noises nrx, p and
nrx, q are zero-mean and uncorrelated with each other as well as every other signal in
the expression above, the equation simplifies rapidly. Many of the cross-terms average
to zero and drop out of the expectation. This is an advantage we have since our goal
is the cross-correlation between antennas rather than the autocorrelation of a signal
received on only one antenna. As Paul Hall shows [7], the noise term survives in
the self-correlation and, in the end, contributes a value of noise power scaled by the
transmitter signal power to the zero-lag of the target autocorrelation estimate.
62
We also note that the interference sources nsky are uncorrelated with the transmitter signal x and the target signals s. This only leaves two terms:
D
E
K̂ =
*
xx
XX
i
=
*
=
+
n
jk(p̄·Ω̄i −q̄·Ω̄j )
xxnsky, i nsky, j e
+
sn sm xxe
jk(p̄·Ω̄n −q̄·Ω̄m )
+
(4.31)
+
(4.32)
E
(4.33)
m
XX
n
j
XX
i
nsky, i nsky, j e
XX
j
XX
i
jk(p̄·Ω̄i −q̄·Ω̄j )
sn sm xxxxe
jk(p̄·Ω̄n −q̄·Ω̄m )
m
E
D
hxxi hnsky, i nsky, j i ejk(p̄·Ω̄i −q̄·Ω̄j ) +
j
XX
n
D
hsn sm i hxxxxi ejk(p̄·Ω̄n −q̄·Ω̄m )
m
Finally, we reason that two separate sources of interference will be uncorrelated,
and repeat our assumption that targets at different ranges are uncorrelated. This
allows us to remove two of the sums:
D
K̂
E
=
XX
i
D
E
δij Rsky, i (τ )Rx∗ (τ ) ejk(p̄−q̄)·Ω̄i +
j
XX
n
=
X
D
m
E
Rsky, i (τ )Rx∗ (τ ) ejk(p̄−q̄)·Ω̄i +
i
D
E
(4.34)
D
E
(4.35)
δnm Rs,n (τ ) hxxxxi ejk(p̄−q̄)·Ω̄n
X
Rs,n (τ ) hxxxxi ejk(p̄−q̄)·Ω̄n
n
The second term in equation 4.35 is the same as the cross-correlation estimate with
the ideal model for y(t), and simplifies to the expression in equation 4.22 above. The
first term is a new bias contribution due to interference. If we use the approximation
Rx (τ ) = Rx (0)δ(τ ) again, we find that the new bias term is
δ(τ )Rx (0)
I
X
D
Rsky, i (0) ejk(p̄−q̄)·Ω̄i
E
(4.36)
i=1
which, like clutter, only contributes to the zero lag of the cross-correlation. Thus,
the noise floor in the cross-spectrum is raised by an additional amount equal to the
63
sum of the signal power of all interfering signals, times the power in the transmitter
signal.
4.5.3
The Expected Value after Coherent Integration
We have not yet considered a coherent integration of the radar time series in our
analysis, though we discussed this as part of the “default” MRR signal processing
in section 4.2. If we use a coherent integration of D points according to equation
4.6, our estimator of the cross-correlation (using the basic scatter model in equations
4.13) becomes
(d)
K̂pq
(r0 , τ )
=
=
D−1
X
zp (t
a=0
D−1
X
X D−1
!
+ a, r0 )
D−1
X
zq∗ (t
!
+ b − τ, r0 )
(4.37)
b=0
yp (t + a)x∗ (t + a − r0 )yq∗ (t + b − τ )x(t + b − r0 − τ ) (4.38)
a=0 b=0
(where the (d) superscript is for decimation, named for the decimation in the raw
data rate due to coherent integration). We note that successive time lags (τ ) are
now further apart by a factor of D, although the time (t) still advances at the raw
sampling rate.
The expected value of the interferometer cross-correlation using the estimator
given above is
D
E
(d)
K̂pq
(r0 , τ )
= Rx2 (0)
D−1
X D−1
X D
E
e+jkd cos θr0 Rs,r0 (τ + a − b) +
a=0 b=0
δ(τ + a − b)
R D
X
+jkd cos θn
e
E
Rs,n (0)
(4.39)
n=1
Essentially, the coherent integration causes each point of the target autocorrelation
(now sampled more sparsely) to be a weighted sum (a triangular window, actually)
over the adjacent closely-spaced lags. The equivalent frequency-domain operation
64
is the filtering of the power spectrum by a sinc2 (·) type smoothing window. Since
Rs,r0 (τ ) remains nearly constant over the interval ±D, the resulting correlation function of the target is much stronger with respect to any noise or interference that may
be present (these do not show up in our expected values since we assume they are
perfectly uncorrelated for nonzero lags, but in practice they do affect the radar data).
Rx (τ ), on the other hand, becomes decorrelated quickly within this same interval, so
there is no clutter price to pay. We will see that this effective “smoothing” will show
up as a reduction in variance in later sections.
4.6
Variance of the Interferometer Estimator
4.6.1
Variance with the Simple Scattering Model
We now work out the variance of the cross-correlation estimator using the model for
the scattered signal given in equations 4.13. The definition of variance is
D
Var K̂ =
D
=
D
K̂ − K̂
E
E D
K̂ K̂ ∗ − K̂
D
K̂ − K̂
ED
K̂ ∗
E∗ E
(4.40)
E
(4.41)
(which is just the second central-moment).
D
To ease the presentation, we first tackle the K̂
D
E
K̂
ED
ED
E
K̂ ∗ term, using the expression
for K̂ in equation 4.20:
D
K̂ ∗
E
=
R D
X
n=1
R D
X
=
E
h
2
2
e+jkd cos θn Rs,n (τ ) |Rx (r0 − n)| + |Rx (τ )|
E
h
2
2
∗
e−jkd cos θl Rs,l
(τ ) |Rx (r0 − l)| + |Rx (τ )|
l=1
R X
R D
X
i
i
!
×
!
(4.42)
E
∗
(τ ) ×
e+jkd(cos θn −cos θl ) Rs,n (τ )Rs,l
n=1 l=1
h
D
|Rx (r0 − n)|2 + |Rx (τ )|2
Note that in writing e+jkd cos θn
ED
E
D
ih
|Rx (r0 − l)|2 + |Rx (τ )|2
E
i
(4.43)
e−jkd cos θl = ejkd(cos θn −cos θl ) , we have assumed
65
that the scatterer angular distributions (θ) are uncorrelated at different ranges.
Next,
D
K̂ K̂ ∗
E
=
D
yp (t)x∗ (t − r0 )yq∗ (t − τ )x(t − r0 − τ ) ×
E
yp∗ (t)x(t − r0 )yq (t − τ )x∗ (t − r0 − τ )
=
x∗ (t − r0 )x(t − r0 )x(t − r0 − τ )x∗ (t − r0 − τ ) ×
R
X
sn (t)x(t − n)
s∗m (t)x∗ (t
!
+jkd cos θi ∗
− τ )e
x (t − i − τ ) ×
!
− m)
R
X
−jkd cos θl
sl (t − τ )e
!+
x(t − l − τ )
(4.45)
E
(4.46)
l=1
XXXXD
m
s∗i (t
i=1
m=1
n
R
X
!
n=1
R
X
=
(4.44)
*
ssss
ED
xxxxxxxx
ED
ejkd(cos θi −cos θl )
i
l
The last equality follows from the fact that s and x are zero-mean and uncorrelated
(and assumed to be Gaussian). Now we will handle the hssssi and hxxxxxxxxi terms
separately. As before, we rely on Isserlis’ Gaussian moment theorem to break up the
large correlations:
D
ssss
E
=
D
ED
sn (t)s∗m (t)
E
sl (t − τ )s∗i (t − τ ) +
D
=
|
Rs,n (0)δnm
{z
ED
sn (t)s∗i (t − τ )
}
|
{z
Rs,l (0)δli + Rs,n (τ )δni
term 1
E
(4.47)
(4.48)
s∗m (t)sl (t − τ )
∗
Rs,l
(τ )δlm
term 2
}
Again we have assumed that the targets are uncorrelated in range. The Kronecker
delta functions in term 1 will cause m = n and i = l, while in term 2 they force i = n
and m = l. Since these different conditions cause different correlations of the x terms,
we handle each piece separately:
66
D
K̂ K̂ ∗
E
XX
=
n
D
E
Rs,n (0)Rs,l (0) xxxxxxxx +
l
|
{z
}
term 1 D
ED
E
XX
∗
Rs,n (τ )Rs,l
(τ ) ejkd(cos θn −cos θl ) xxxxxxxx
n
(4.49)
l
|
{z
}
term 2
Next, we must deal with the 8th -order correlation in x. Using the Gaussian moment theorem, this will “reduce” to a total of 24 2nd -order terms. For details, please
refer to appendix A. For brevity, we skip to the final result after the substitution
Rx (τ ) = Rx (0)δ(τ ) has been applied:
Var K̂ =
"
Rx4 (0)
2
2
Rs,r
(0) + 6δ(τ )Rs,r
(0) + δ(τ )
0
0
X
2
(0)
Rs,n
n
+ Rs,r0 (0)
X
n
+
XX
n
Rs,n (0) 2 + 4δ(τ ) + 2δ(τ + r0 − n) + 2δ(τ + n − r0 )
Rs,n (0)Rs,l (0) 1 + δ(τ ) + δ(τ + r0 − n) + δ(τ + l − r0 )
l
+ δ(τ + l − n) + δ(τ + l − r0 )δ(τ + r0 − n)
+ 2 |Rs,r0 (τ )|2 + 8δ(τ ) |Rs,r0 (τ )|2
+ 3δ(τ )Rs,r0 (τ )
XD
∗
ejkd(cos θr0 −cos θn ) Rs,n
(τ )
E
∗
2δ(τ )Rs,r
(τ )
0
XD
ejkd(cos θn −cos θr0 ) Rs,n (τ )
n
+
E
n
+ δ(τ )
XXD
n
+
X
n
jkd(cos θn −cos θl )
e
E
∗
Rs,n (τ )Rs,l
(τ )
l
2
|Rs,n (τ )|
1 + δ(τ ) + δ(τ + r0 − n) + δ(τ + n − r0 )
#
(4.50)
To make this expression easier to interpret, we introduce the notation C∞ (τ ),
which we define to be the clutter contribution from all ranges (thus the ∞ subscript).
67
The argument τ indicates that the clutter term is the sum of all target autocorrelation
functions at lag τ . In particular,
R
X
C∞ (0) =
Rs,n (0)
(4.51)
n=1
is the sum of all reflected power from every range. Likewise, we define Cn (τ ) = Rs,n (τ )
to be the clutter contribution from only one range, n. Using this notation, we can
approximately 6 write equation 4.50 as
2
(0) + 2Cn2 (τ ) +
Var K̂ ≈ Rx4 (0) Cn2 (0) + 2Cn (0)C∞ (0) + C∞
R
X
Cn2 (τ )
n=1
+ δ(τ )Rx4 (0) 6Cn2 (0) +
R
X
Cn2 (0) + 4Cn (0)C∞ (0)
n=1
+
2Cn2 (0)
2
+ 2Cn2 (0) + C∞
(0) + Cn (0)C∞ (0)
+ Cn (0)C∞ (0) + Cn (0)C∞ (0) + Cn2 (0)
+ 8Cn2 (τ ) + 3Cn (τ )C∞ (τ ) + 2Cn (τ )C∞ (τ )
+ C∞ (τ ) +
R
X
Cn2 (τ )
+
Cn2 (τ )
+
Cn2 (τ )
(4.52)
n=1
≈
Rx4 (0)
2
C∞
(0)
+ 2Cn (0)C∞ (0) +
Cn2 (0)
+ (R +
2)Cn2 (τ )
2
+ δ(τ )Rx4 (0) C∞
(0) + 7Cn (0)C∞ (0) + (R + 11)Cn2 (0)
+
2
C∞
(τ )
+ 5Cn (τ )C∞ (τ ) + (R +
10)Cn2 (τ )
(4.53)
where we have also absorbed the autocorrelation function of the target of interest
(Rs,r0 ) into the “clutter at only one range” term, Cn .
We see that the variance of a single estimate of the cross-correlation at lag τ (the
particular range of interest is now indistinguishable) depends on the signal power,
6
We ignore phase terms and complex conjugates; we do not distinguish between individual clutter
ranges. We defend the approximation by claiming that the new notation is merely a crutch for
making the variance expression quickly meaningful; uniformly representing all clutter with the C
symbol and disregarding phase information can only cause us to overestimate the variance.
68
or radar cross section, of all scattering targets. It is also strongly affected by the
transmitter signal power. While part of the expression for the variance applies only
for short lags (the part multiplied by δ(τ )), the remaining portion is still quite large.
Finally, for a very over-simplified interpretation, if we assume that the clutter
arriving from only one range is much smaller than the clutter arriving from all ranges
(i.e., Cn (τ ) C∞ (τ )), and we avoid looking at very small correlation lags (τ > 20µs),
we obtain the following very simple approximation:
2
Var K̂ ≈ Rx4 (0)C∞
(0)
(4.54)
From the derivation of equation 4.50 in appendix A, we note that the fast decorrelation of the transmitter waveform helps immensely in reducing the variance due
to clutter, though it remains significant, and probably would prevent us from noticing all but the strongest echoes. However, we actually do much better than this.
Note that in equation 4.50 (or any subsequent statement of the variance), there is no
dependence on “N ” (a number of data points used in the estimation). This would
imply that we have an inconsistent estimator: one whose variance properties we cannot improve by providing more data. However, we have considered only a single
cross-correlation estimate formed from three voltage measurements at an instant in
time (one from the reference antenna and two from the scatter-collecting antennas).
In practice, averaging is used on many time series voltages before the correlation or
spectrum estimation is performed. Also, many individual cross-correlation estimates
are incoherently averaged to form a final result. The latter step serves to reduce the
√
variance from the expression in equation 4.50 by a factor of 1/ N , where N is the
number of estimates averaged together. We discuss both of these options in more
detail below.
69
4.6.2
Variance after Coherent Averaging
Again, we note that we have not yet considered any coherent integration of the target
time series in our analysis. If a coherent average is used, i.e.
z(t, r) =
X
1 D−1≥49
z1 (t + t0 , r) , t = 0, D, 2D, . . .
D t0 =0
(4.55)
we can achieve a result with smaller variance than what is indicated above. This is
because equation 4.55 is a lowpass-filtering operation, so the variance in the target
time series itself is reduced by a factor of 1/D, an improvement of 17 dB when D = 50.
Including the coherent average in the derivation of the variance immediately introduces a factor of 1/D4 ; however, it also introduces 4 additional sums and vastly
increases the complexity of the expression. Therefore, we do not work out an analytic
expression for the variance after coherent averaging here. We note that Hall [7] works
out a full expression for the single antenna case (autocorrelation estimator), and Sahr
and Lind [27] also show an expression, organized as a polynomial in transmitter correlation function (degree of clutter suppression). At this point, we lean on other
arguments to get an analytic idea of the variance, and we show numerical estimates
of variance in simulated data in chapter 5.
4.7
Interferometer Implementation Issues
As we mentioned earlier, the interferometer cross correlation estimator is unbiased for
lags greater than the correlation time of the FM waveform. This is a useful result, yet
if one prefers to work in the frequency domain (the preferred form of most MRR data
products), the energy contained in lag zero of the correlation function still appears
over all frequencies as a “noise and clutter floor.” Also, the interferometer phase is
significantly affected by the clutter at all ranges. Therefore, we require a method of
70
estimating the interferometer bias (mainly due to clutter), and removing it from the
cross-spectrum.
A preliminary technique for estimating the bias is to calculate the cross-spectrum
at a distant range7 assumed not to contain any signal of interest, but which will still
contain any interference as well as the clutter from all ranges. We then subtract
this bias estimate from the cross-spectrum at the range of interest. Point-by-point
subtraction allows us to distinguish between the bias that may be present in different
Doppler bins, which will be useful in a frequency-selective fading environment, or in
the presence of narrowband interference sources.
This technique is not perfect, and it can be shown that the “distant range” bias
estimate will miss errors that enter into the spectrum as cross-terms between a target
signal and clutter. To see this, consider the following:
(clutter + target)2 = clutter2 + 2(clutter)(target) + target2
(clutter + nothing)2 = clutter2
where the top line is the desired cross-spectrum and the bottom line is the bias
estimate.
Alternatively, we can use the zero lag of the cross-correlation function as a (Doppler
independent) estimate of the bias due to clutter, noise, and interference. We show
an example of a target cross-correlation function in figure 4.4; the data is from the
same meteor trail echo we showed in figure 3.6 on page 28. Comparing the zero-lag
value of the meteor correlation function with the spectrum floor value (bottom panel
of figure 3.7), we find that they are consistent. But this method, too, has the disadvantage that lag zero of the correlation function also contains energy from the target
7
We have also tried averaging spectra from many “empty ranges” together for a bias estimate; this
does not noticeably improve the correction-technique performance. It does, however, significantly
increase computational load.
71
16
Cross−Correlation Magnitude
2.5
x 10
Correlation Function of
Meteor Trail Time Series
Noise + Clutter
Level ~163 dB
2
1.5
1
0.5
0
−200
−100
0
Lag (ms)
100
200
Figure 4.4: Cross-correlation magnitude of the meteor trail echo shown in figure 3.6.
The noise, clutter, and interference contribution is contained in lag zero.
of interest itself.
In figure 4.5 we demonstrate the two techniques described above on the meteor
trail time series, and compare the resulting coherence and phase spectra to the uncorrected cross-spectrum. We conclude that some method of bias removal is needed,
based mostly on our experience viewing many examples of interferometer data. In
many cases, the target phase is indistinguishable from the background clutter unless
one of the above methods is used. Although in this particular example the zero lag
correlation method produces a more readable magnitude spectrum (the variance appears smaller), we have chosen to use the “alternate range” estimator for the bias
(mainly due to its success with the phase spectrum).
72
No Bias Removal
1
0.5
0
2
Normalized Cross Spectrum Magnitude and Phase
0
−2
With Alternate−Range Estimate of Bias
1
0.5
0
2
0
−2
With Zero−Lag Correlation Estimate of Bias
1
0.5
0
2
0
−2
−400
−300
−200
−100
0
Doppler (m/s)
100
200
300
400
Figure 4.5: A comparison of bias removal techniques. The top plot shows the uncorrected spectrum; the next two demonstrate different correction methods discussed in
the text. The data is from the meteor trail observed on January 3, 2002 at UT 19:23.
73
4.8
4.8.1
Statistics of Other Estimators and Additional Comments
The Periodogram (FFT-Based Spectrum Estimation)
Since we have worked out most of our formulas in the correlation (time lag) domain,
we have not yet considered one of the most widely used methods of spectral estimation
(indeed, the one that MRR currently implements): the periodogram. The unaltered
periodogram (in discrete frequency) is the magnitude squared of the DFT of the data,
Ŝ
(p)
1
[k] ≡
N
2
−1
NX
−j2πkn/N x[n]e
(4.56)
n=0
Using this method for estimating radar power spectra is computationally efficient,
since we are able to use the FFT algorithm. We have shown that clutter and interference cause bias in our power spectrum estimates by contributing to the noise
floor. By considering the periodogram as a power spectrum estimator, we discover
another type of bias due to finite data length. As shown in many texts which deal
with spectral estimation (Percival and Walden [23] and Oppenheim and Schafer [22],
for example), supplying the DFT with only a finite sequence of data is equivalent to
supplying it with (what must be assumed to be) an infinite time series, multiplied by
a rectangular window. In the frequency domain, the result is the desired spectrum
convolved with a sinc function, which, depending on the length of data supplied, will
have a mainlobe of a certain width (causing blurring of the spectrum) and sidelobes
of a certain height which persist into adjacent frequencies, causing bias and further
blurring at those frequencies. The problem might be better understood as one of frequency resolution; the function that is convolved with the true spectrum is called a
spectral window. As the length of data fed into the DFT becomes larger (the rectangle
becomes wider), the mainlobe width of the spectral window decreases, giving better
frequency resolution. Other techniques called data-tapering have been developed to
alleviate the “leakage” of power from one frequency to another due to the sidelobes of
74
the spectral window. These involve pre-multiplying the time series to be analyzed by
functions that taper the beginning and end of the times series more gradually toward
zero (forcing down the sidelobes in the frequency domain). There is a trade-off, however, between resolution and leakage (bias), since these data-taper spectral windows
also have increased mainlobe width. Much work has been done in exploring various
data-tapering and spectral estimation techniques and their associated trade-offs ([23]
and references therein, for example).
4.8.2
Application to MRR Signal Processing
Experimentation with data tapering at MRR has indicated that our power spectrum
estimates are not significantly impacted by sidelobe leakage effects, so we only consider
the above in the context of the trade-off between frequency (Doppler) resolution and
variance in our radar data products. Given a finite amount of data N from which
to estimate a power spectrum, if we use an M -point FFT, we are able to average
approximately N/M spectra together for a final estimate. The incoherent averaging
q
reduces the variance in our estimate by a factor of 1/ N/M =
q
M
,
N
which we would
like to be as small as possible; however, the larger M is, the better Doppler resolution
we achieve. Thus, we must balance the classical trade-off between frequency resolution
and variance. We now name some specific values for the parameters above that are
in use at MRR now.
In order to determine the amount of time for which we can take data (i.e., the timeresolution of our power spectra), we consider the statistical behavior of our intended
target. We would like to take data continuously for the longest length of time possible
while the target remains statistically stationary. A time resolution Tstationary could
be inferred for E-region irregularities by considering inherent time scales of plasma
processes, wave growth and decay mechanisms, etc. We do not discuss this particular
75
problem here, however. The current time resolution we use at MRR is 10 seconds8 .
Next, we need to determine a sampling frequency. There are two considerations
here: we want to capture the majority of the power in the illuminating signal, and
we would also like to not oversample so much that we have many samples during the
correlation time τx of the transmitter waveform. In our case, τx was shown above
(section 4.1) to be approximately 10µs, which corresponds to a receiver bandwidth of
100 kHz, which fortunately is also sufficient to capture about 90% of the FM signal’s
power. Therefore, we sample our receivers with an output data rate of 100 kHz.
Now we must consider the correlation time of the target. Since we’d like to end
up with a time series for analysis composed of independent samples, we decimate the
100 kHz time series, typically by 50 (see equation 4.6) to obtain a new time series
which evolves at the much lower bandwidth of the target (≤ 2 kHz). This gives
us an approximate Doppler range of ±1500 m/s; a longer decimation would give us
a smaller Doppler window. This decimation (actually a coherent integration of 50
samples, over which the target is correlated but the transmitter waveform is not) not
only achieves a reduction in the data rate and produces a time series of independent
target samples, but also suppresses clutter from other ranges and reduces the variance
in the radar data products (see section 4.6.2).
Finally, we are left with
(10 seconds) × (100 ksamples)
= 20, 000 samples
decimation rate of 50
with which to produce a spectrum. If we use a 256-point FFT, we get 78 individual
spectra which we can average for a final estimate, giving us a reduction in variance by
8
Long observation times (2 minutes of continuous data) have revealed field-aligned irregularities
that appear to be unchanging during this interval; however, we have also observed irregularities
that do not even last for the typical 10 second duration. The choice of 10 seconds is somewhat
arbitrary, but serves us well.
76
√
1/ 78, or approximately 9.5 dB. The 256-point FFT gives us a Doppler resolution of
fs λ
100 kHz × 3 meters
=
≈ 12 m/s
2 × FFT size × decimation
2 × 256 × 50
Fortunately, it is always possible to reprocess passive radar data, reducing its
variance at the expense of lower resolution in any of the dimensions of time, range, or
velocity9 . This result applies both to individual power spectra from single antennas
and cross-spectra between two antennas (for interferometry). In the latter case, rather
than squaring the magnitude of the DFT to obtain a power spectrum estimate, we
multiply the DFT of the time series on one antenna by the complex conjugate of the
DFT from the other antenna. This is the same operation in the frequency domain as
the interferometer estimator given in equation 4.11.
4.8.3
Periodogram Variance
We have been talking about variance in the radar power spectrum estimates; now we
cite some theoretical results describing second-moment properties of the periodogram.
Percival and Walden [23] show that, for a zero mean white Gaussian noise process,
the periodogram samples (with the exception of those at zero or Nyquist frequency)
can be described in distribution by a chi-square random variable with 2 degrees of
freedom (i.e., they are the sum of the square of two Gaussian random variables), times
one half the variance of the original process. They further show that
Var {Ŝ (p) [k]} =



S 2 [k], 0 < k < N/2

 2S 2 [k], k = 0 or N/2
(4.57)
for a white Gaussian process, where S[k] is the true spectrum of the time series, and for
any stationary process, asymptotically as the sample size N gets very large. Thus, the
periodogram is an inconsistent estimator: its variance does not decrease as the sample
9
This also gives us the interesting option of mimicking the resolutions of other radars.
77
size grows. Also, we see that the variance of the spectrum estimate scales as the square
of the value of the true power spectrum itself. This result is similar to the one we
show above in which the variance of the target auto- or cross-correlation estimator is
strongly dependent on the correlation functions of the transmitter waveform and the
targets themselves.
It would seem that stronger targets produce spectral estimates which are more
difficult to interpret, a counter-intuitive conclusion. Fortunately, we find that the
variance of the phase spectrum estimates decreases with increasing SNR, and this
is the quantity we are most interested in, at least for the purpose of determining
an appropriate transverse resolution for our interferometer. In the next chapter, we
describe this result and apply it to the resolution problem.
78
Chapter 5
SIMULATIONS
“In God we trust – everyone else has to bring data.”
—Dr. Jesse Poore
We have discussed causes of bias and variance in our interferometer data products
and derived analytic expressions for these for several cases. Now we turn to numeric
simulations to get an empirical estimate of the quality of our data. Simulations
are useful tools because they allow us to completely control the “environment” in
which the radar is operating, and we have perfect knowledge of each signal involved.
Also, true ensemble averages can be calculated, since we are able to use the same
illuminating and target signals over and over again, only changing the noise in the
system. However, there are many assumptions and approximations involved with
simulating real world processes; we describe in detail our techniques for simulating
passive radar data in appendix B.
5.1
A Simulated Target
For all of our simulations, we use MATLAB to generate reference and scatter data
files, then use our regular radar signal processors (in the C/C++ language) to do the
detection and spectrum estimation. An example of simulated interferometer data is
shown in figure 5.1. The target present at the displayed range was created with a
mean Doppler shift of 100 m/s and a Doppler width of 50 m/s; the echo originates
from a single azimuth angle of 30◦ , corresponding to an interferometer phase of -0.9
79
Cross Spectrum Phase and Magnitude for a Simulated Target
Phase
2
0
−2
−400
−300
−200
−100
0
100
200
300
400
−300
−200
−100
0
100
200
300
400
Normalized
Magnitude
1
0.5
Self Spectra (dB)
0
−400
antenna 1
antenna 2
190
186
182
−400
−300
−200
−100
0
Doppler (m/s)
100
200
300
400
Figure 5.1: A simulated target, with -13 dB SNR.
radians on an antenna baseline of 16λ; and the SNR of the target signal with respect
to the simulated noise is -13 dB. However, in the bottom panel of figure 5.1, where the
individual power spectra on the two antennas are shown, we see that the peak of the
target rises above the noise floor by approximately 6 dB. This increase in SNR is due
to signal processing gain: a matched filter is used on the target echo, and we also used
a coherent integration of 200 samples in this case, which provides a boost of about 23
dB to the target signal, which remains correlated over the interval. We are also able
to use 19 incoherent averages, with an FFT length of 256 (200 × 256 × 19 ≈ 106 ), and
√
this gives us a small processing gain of 10 log10 19 ≈ 6.5 dB.
We have purposefully simulated targets which only exist at a single azimuth angle
80
(no transverse width or azimuthal distribution). The reason for this is to prevent
the misinterpretation of angular spread as variance in the cross-spectrum. The effect
of angular spread on coherence and phase spectra is well understood [5]; we wish to
evaluate the variance of our estimator. Also, in the data shown in figure 5.1, as well
as in all the simulations we describe below, we have used a set number of “extra”
targets at other ranges. We do not vary these extra targets; this is our method for
simulating known, controlled clutter in the radar data. Each of the 3 clutter targets
we use has an SNR of 0 dB (before signal processing). Their range, Doppler, and
azimuth characteristics are unique.
From the discussion of interferometer bias in section 4.5.1, we expect the crossspectrum noise floor to be raised uniformly by an additional level equal to the transmitter signal energy squared times the sum of all the target signal energies:
Rx2 (0)
X
Rs (0)
This is the clutter contribution to the bias. Each signal in our simulation has an
energy of 1000 (the 2-norm of every signal, including the transmitter signal, noise,
and all targets, is approximately equal to 1000). Of course, after the -13 dB SNR is
applied to the “main” target, its energy becomes 0.05×1000 = 50. Next, we boost the
transmitter and scattered signal energies in our simulations by a factor of 1000 so that
when the simulated samples are truncated into the short data type, their information
will not be lost (without the gain of 1000, most samples vary between 0 and 1). This
adds an additional factor of 10002 to the noise plus clutter floor. Finally, we also take
into account the processing gains due to the coherent and incoherent integrations.
Then, we expect the noise floor in our simulations with clutter to be
81
noise floor
= 10 log10 (10002 × (3 × 1000 + 50)) (clutter contribution)
+ 10 log10 (10002 ) (boost of 1000)
+ 23 dB (coherent processing gain)
+ 7 dB (incoherent processing gain)
= 95 + 60 + 23 + 7 dB
= 185 dB
From figure 5.1, we see that the power spectrum noise floor is approximately 186
dB, which is consistent with this rough analysis.
5.2
Empirical Estimation of Cross Spectrum Variance
Now we use our simulator to generate many measurements of the variance in our crossspectrum estimates. For these simulations, as discussed above, we use three targets
at other ranges to represent clutter. These clutter targets, as well as the transmitter
signal and the target of interest, remain constant throughout all our simulations.
The only parameters we vary are the SNR associated with the target of interest, the
(complex, white Gaussian) noise itself, and the amount of coherent averaging1 used
in the signal processing before spectrum estimation. Also, to avoid complexity, we do
not consider normalization of the cross-spectrum magnitude here (we are interested
mainly in the statistical properties of the phase estimates), and we do not perform
any bias-removal techniques with our signal processor.
1
Instead of coherent integration, we use coherent averaging, which also divides by the number of
samples summed together. As we suggested in section 4.6.2, this acts to decrease the variance in
the target time series by a factor of 1/D, where D is the number of samples in the integration.
82
M spectral
estimates
.
.
.
.
variance of individual
spectrum estimates = v
average together
for "final" spectral
estimate
.
. . . . .
Doppler
variance of this
estimate = v/sqrt(M)
Repeat for many
instances to get
an estimate of the
variance of the
spectral estimates...
Figure 5.2: An illustration of the variance estimation method used here.
Figure 5.2 illustrates our variance estimation method: the simulated reference
and scatter time series are both 106 points long. Depending on the FFT length
and the initial coherent integration used, we obtain a certain number (denoted M
in figure 5.2) of spectral estimates (at one particular range containing a simulated
target of interest) from each simulation. Then, for each frequency bin, we obtain a
variance estimate using all the spectra. (Note that the variance of the “final” spectral
estimates, which are the result of averaging the M spectra together, is smaller than
√
the variance we estimate by an incoherent averaging factor of 1/ M . This is an
important distinction during our discussion of resolution based on variance in section
5.3.) We also determine the (post-processing) SNR2 at each Doppler bin, and organize
2
We first estimate the noise floor by taking the average value of the spectrum magnitude with
83
our variance results in this way. We repeat this process many times, and average
our variance estimates at each SNR. Figure 5.3 shows the resulting variance versus
SNR curves; we display the variance of the cross-spectrum in three different forms:
complex (“total”), phase, and magnitude. By “total variance,” we mean the complex
2nd central moment,
Var Ŝ =
=
D
D
D E Ŝ − Ŝ
E
D ED
Ŝ Ŝ ∗ − Ŝ
D E∗ E
Ŝ − Ŝ
Ŝ ∗
E
(5.1)
(5.2)
The bottom panel of figure 5.3, called “points averaged,” displays the number of
variance estimates (each computed from 19 spectral estimates) that were averaged
together to form the final variance estimate at each SNR. Since there are far more
frequencies with low SNR, many points went into the variance curves at low SNR,
while the number of points averaged drops off for the variance estimates at high SNR.
We also compare the variance in cross-spectra after different amounts of coherent
averaging of the target time series. Therefore, even though more spectral estimates are
available from the simulations using shorter coherent averages, for a fair comparison
we have limited the number of spectra used in each variance estimate to 19 (which
is the number available from 106 data points using a 256-point FFT and the largest
decimation we consider, 200). Thus, for every coherent average case, each variance
estimate has the same variance. As the top panel of figure 5.3 shows, the spectrum
variance decreases as larger coherent averages are used, which we expect (however,
the trade-off for large coherent averages/integrations is that we are unable to detect
faster-moving targets without additional signal processing tricks).
no target present (although the 3 clutter terms are still included), then estimate the SNR by
subtracting the noise floor from each value in our spectra. If this quantity is negative, we declare
the SNR to be zero. We note that the noise floor will be different depending on the amount of
coherent averaging used.
84
Cross−Spectrum Variance vs. SNR
Coherent Average Length
35
200
100
50
1
Complex
10
30
10
25
10
0
5
10
15
20
25
30
5
10
15
20
25
30
5
10
15
20
25
30
5
10
15
20
25
30
Phase
3
2
1
0
0
Magnitude
100
80
60
40
20
0
0
4
Points Averaged
10
2
10
0
10
0
SNR (dB)
Figure 5.3: The top three panels show different representations of cross-spectrum
variance versus SNR. The bottom panel shows the number of variance estimates that
were averaged together to form the final value.
85
Another observation about the “total” cross-spectrum variance in figure 5.3 is
that it tends to increase with SNR. We expect this increase in variance with SNR,
because when the SNR increases, the spectral values themselves become larger (by a
factor of a, for example), which will increase the variance by a factor of a2 :
D
E Var aŜ =
D
aŜ − aŜ
=
D
aŜ − a Ŝ
=
D
D E D
aŜ − aŜ
D E∗ E
aŜ − a Ŝ
D E |a|2 Ŝ − Ŝ
E∗ E
D E∗ E
Ŝ − Ŝ
= |a|2 Var Ŝ
(5.3)
Indeed, as we describe in section 4.8.3, the variance of the periodogram estimator
is proportional to the square of the value of the true spectrum. We find this approximately to be the case in the top panel of figure 5.3: the relationships between
variance and SNR are all roughly linear with a slope of 2 dB per dB of SNR. A linear
increase on a log-log plot indicates a power-law relationship, with the power equal to
the slope of the line. Thus, the variance is proportional to the SNR raised to the 2nd
power, and our estimator is performing as we expect.
Next, we compare the simulated variance to that predicted by the (first-order
approximation) analytic expression from equation 4.54,
2
Var K̂ ≈ Rx4 (0)C∞
(0)
(5.4)
Comparison with a more detailed expression for the variance (such as equation
4.50) is difficult due to the many specific clutter properties required in the formula.
The expression for “total variance” in equation 5.4 was derived for a single crosscorrelation estimate (i. e., no coherent averaging), so we expect it to agree roughly
with the dotted line (corresponding to a coherent average length of 1) in the top panel
of figure 5.3.
86
Proceeding in the same manner as in section 5.1, we evaluate the variance formula
with the 4 targets and transmitter power:
variance
≈ 4 × 10 log10 (1000) (transmitter power)
+ 4 × 10 log10 (1000) (boost of 1000)
+ 2 × 10 log10 (3 × 1000 + 50) (clutter)
≈ 120 + 120 + 70 dB
≈ 310 dB
≈ 1031
The value of 1031 is in the appropriate area of the top panel of figure 5.3, which
indicates that our very rough analysis is consistent with simulated data.
We are most interested, however, in the quality of the cross-spectrum phase and
magnitude estimates separately (especially the phase, since it determines our interferometer’s transverse resolution). The first thing we note from figure 5.3 about the
phase and magnitude variances is that they don’t seem to be affected by coherent
averaging (a counter-intuitive result). The magnitude variance still increases with
SNR, which is not surprising given the discussion above. The phase variance, though,
becomes smaller as the spectrum SNR grows larger. This is an excellent and intuitive
result for us, since it indicates that we will be able to use finer transverse resolution
with stronger targets. As a target, which exists in one particular region in the sky,
returns more power to the scatter receiver, the cross-spectrum phase becomes more
likely to take on the value corresponding to that particular direction than any random fluctuation. The range of possible phase values effectively shrinks. In the case
of the magnitude, when a target is present, the mean value of the magnitude shifts
upwards, but the range of possible values also grows, and no rule exists to constrain
those values.
87
Variance of Cross Spectrum Phase Estimates vs. SNR
Phase Variance (radians)
3
2.5
2
1.5
1
0.5
0
0
5
10
15
20
25
30
35
Signal to Noise Ratio (dB)
Figure 5.4: Cross-spectrum phase variance plotted as the spectrum SNR increases.
5.3
Evaluation of Interferometer Resolution
In figure 5.4 we show again a plot of variance in cross-spectrum phase estimates versus
SNR. To create this plot, we have used an initial coherent average length of 50, and
included the full 78 resulting spectra in each estimate of the variance. We ran the
simulation 30 times, varying the (pre-processing) target SNR between -5 and -15 dB.
The resulting plot is somewhat smoother than the one in figure 5.3, and we use it in
determining transverse resolutions for the interferometer.
We relate variance in the phase estimates to transverse resolution by the following
logic:
88
Variance = (Standard Deviation)2
2 × Standard Deviation = Resolution, in radians
2π/Resolution = Number of Phase Bins over 1 Interferometer Lobe
We are defining the transverse resolution (in radians) to be that phase range which
includes a phase estimate ± its standard deviation; this is a common way to display
error bars on data. To determine the variance implied by the use of, say, 100 phase
bins, we write
2π/100 = 2 × std = 0.063
var = std2 = 9.87 × 10−4
⇒ SNR > 35 dB
We see immediately that 100 phase bins is an unreasonably fine grid. Indeed, even
a very grainy image with only 5 bins across the 2π radians of interferometer phase
requires an SNR of 15 dB or greater, judging by our results here. It appears that our
interferometer has almost no resolution at all. However, we have not yet considered
the effect of incoherent averaging of the spectral estimates, and this reduces the
variance from that shown on the curve in figure 5.4 significantly. As we described in
section 4.8.2, our final spectrum estimate is actually the average of several spectra
computed independently from the available data (we assume the target is statistically
stationary over the time period of observation). Therefore, the final cross-spectrum
√
phase exhibits a variance reduced additionally by a factor of 1/ N , where N is
typically 78, for an initial decimation of 50 (in other words, the number of points
that went into our variance estimates shown above).
89
Table 5.1: Interferometer transverse resolutions appropriate for a given SNR.
SNR (dB)
Variance
Resolution
Resolution
# Phase Bins
(radians)
(radians) (km @ 1000 km)
26
0.004
50
π/25
2
17.5
0.025
20
π/10
5
11.1
0.099
10
π/5
10
any value
0.395
5
2π/5
20
We can now state appropriate transverse resolutions for our interferometer data,
given the signal-to-noise ratio of the target, in the particular case3 with a coherent
integration of 50 points and an FFT of length 256. These are summarized in table
5.1.
These values can be considered at best rules of thumb; we have made many
assumptions in coming to this conclusion and the simulations themselves are only
approximations. However, it is useful to have a general idea of the granularity of
data to expect out of our instrument. The method we recommend for determining
an appropriate transverse resolution is (unfortunately) on a case-by-case basis. The
post-processing signal-to-noise ratio of the corresponding power spectrum should be
estimated, and a decision about resolution made using that information. In figure 5.5,
we show as an example an auroral echo with 4 different transverse resolutions (the
range resolution remains 1.5 km throughout the figure). The leftmost panel, with 2
km transverse resolution, corresponds to an echo with SNR ≥ 26 dB, according to our
rule of thumb. The next two panels have resolutions accommodating SNR’s greater
3
The resolution limitations only relax for larger coherent integrations, so the values we state here
are lower (i.e., worst) bounds, in the sense that we rarely would want to use a decimation less
than 50.
90
50 bins / 2 km
20 bins / 5 km
10 bins / 10 km
5 bins / 20 km
Range (km)
1100
1050
1000
950
900
−50
0
50
−50
0
50
−50
0
50
−50
0
50
Figure 5.5: A comparison of transverse resolutions in interferometer data. From left
to right, the phase is separated into consecutively fewer bins, resulting in degrading
transverse resolution. The range resolution in each figure is 1.5 km.
than 17 dB and 11 dB, respectively. Finally, the rightmost panel, with a transverse
resolution of 20 km (at a range of 1000 km), can be considered accurate at any SNR.
In the case of the echo shown, which occurred at UT 05:11 on February 2, 2002,
we suggest that a resolution between 5 and 10 km may safely be used, according to
the power spectra from the same echo shown in figure 5.6. Determining the resolution
is a difficult problem, though, since the SNR varies with range (and Doppler velocity,
for that matter). We suggest a strategy of using finer resolutions when possible,
while keeping in mind the limitations of the instrument. The risk in using too fine
a resolution is that it becomes possible to misinterpret random fluctuations for true
91
Power Spectra at Different Ranges for an Auroral Echo
975 km
230
Max SNR: 18 dB
220
212
−1500
−1000
−500
0
500
1000
1500
225
989 km
Max SNR: 13 dB
218
212
−1500
−1000
−500
0
500
1000
1500
218
1005 km
Max SNR: 6 dB
215
212
−1500
−1000
−500
0
500
1000
1500
Doppler Velocity (m/s)
Figure 5.6: Power spectra at 3 different ranges for the auroral echo we observed on
February 2, 2002, at UT 05:11.
transverse structure in the lower SNR areas. This uncertainty associated with whether
the data is showing structure or coincidental patterns in noise is why we have spent
so much effort analyzing the variability of our data products.
5.4
Target Detection by Phase Compactness
As a final note, we describe another useful consequence of the decrease in phase variance with increasing SNR. As we have seen, the presence of a target causes a certain
“degree of organizedness” in the cross-spectrum phase; the decrease in variance is directly related to this phenomenon. Figure 5.7 shows a spectrum containing a target in
92
the top panel; in the bottom panel, the standard deviations of the phase estimates are
plotted versus Doppler velocity. We claim that it is possible to automatically detect
targets by setting a phase variance threshold and testing spectra for consecutive samples that fall below that threshold. This method would be no more computationally
intensive than search or test methods based on detecting peaks in spectrum magnitude: each variance (or standard deviation) estimate can be computed directly from
the cross-spectral estimates already available from the radar signal processing. We
have not implemented this technique, but we have noted on many occasions that while
magnitude spectra can be unreliable, the phase becomes compact quite predictably
when a target is present.
One caveat to note when applying this method is that one must consider the unnatural discontinuity introduced into the phase by unwrapping the circular spectrum
in the complex plane and plotting it as shown in figure 5.7. If the target phase exists
near −π or +π, small deviations in the phase estimates may result in much larger
standard deviation estimates, since the values may “roll over” (essentially, cross a
branch cut). There are many possible ways to deal with this problem; we do not
discuss them here.
std of Phase Estimates
Cross Spectrum Magnitude
93
140
Detection of Target by Variability in Phase Estimates
130
120
110
100
−400
−300
−200
−100
0
100
200
300
400
−300
−200
−100
0
100
200
300
400
2.5
2
1.5
1
0.5
0
−400
Figure 5.7: An example of cross-spectrum phase compactness where a (simulated)
target exists. The standard deviation of phase estimates can be used to detect targets,
in some cases more reliably than power spectrum magnitude.
94
Chapter 6
ANALYSIS OF IONOSPHERIC EVENTS WITH
INTERFEROMETRY
“Hey, that one’s cool! It looks like a jester’s hat! That’s a bat. Explosion.
Explosion. Funnel-cone ice cream thing. This is your brain. This is your
brain on drugs.”
— Dr. Mark Oskin
In this chapter we analyze several E-region field-aligned irregularities with the
interferometric techniques discussed earlier, and show what additional information
can be learned given knowledge of their transverse structure. We demonstrate that
our passive radar interferometer data products are consistent with what we might
expect from an active radar interferometer, such as CUPRI, and that we are able to
make the same types of measurements that have been done before with other active
radar interferometers [25, 26]. We further claim that the very fine resolution of the
MRR interferometer1 allows us to see for the first time that significant structure exists
in the irregularities at very fine scales. Finally, we note that the E-region irregularities
we detect must be inherently small, with scattering areas on the order of 10–50 km in
the transverse dimension, for our finely-spaced interferometer lobes to detect them at
all. Such narrow structure has hitherto been unconfirmed, to the author’s knowledge.
1
Typically, 1.5 × 2 km in the plane ⊥ to B. This is an extremely fine resolution relative to most
other ionospheric coherent scatter radars (cf. SHERPA, 30 × 60 km; STARE, 20 × 20 km).
95
6.1
Description of Irregularity Backscatter
As we discussed in section 3.3, MRR detects ionospheric phenomena via the Bragg
scattering mechanism, and thus it is sensitive to structures with scale sizes near 1.5
meters. We often find irregularity velocities to be near cs , the ion acoustic speed,
and a study of the Doppler statistics of echoes observed by the Manastash Ridge
Radar over a one-year period2 indicates that the majority of irregularities detected
by MRR are excited by the modified two-stream instability (otherwise known as the
Farley-Buneman instability [17]). These echoes might be classified as “type 1,” or fast
moving “type 2,” due to their large Doppler shifts and narrow- to moderate- spectral
widths.
Since the plasma wave modes believed to be causing the coherent backscatter are
very strongly damped in directions parallel to the magnetic field [28, 20], we can
expect scatterers detected by MRR to be localized to regions where the radar line of
sight is within 1–2◦ of perpendicular to B. Figure 6.1 shows the radar field of view,
over southwestern Canada, along with contours of constant bistatic range and aspect
angle. Even without direction-of-arrival information from the interferometer, we can
roughly determine the location of scatterers using the intersection(s) of the appropriate range contour and the 90◦ aspect angle contour. Furthermore, the altitude from
which irregularity scatter originates is limited to E-region heights, approximately
90–180 km.
2
In work not yet published, but presented at the IUGG General Assembly in Sapporo, Japan,
July 2003: Meyer, M. G., J. D. Sahr, D. M. Gidner, and C. Zhou, “A Year in Review: High
Latitude Ionospheric Irregularity Observations with Passive VHF Radar.”
96
Figure 6.1: The MRR field of view, over southwestern Canada, shown with contours of
constant range and aspect angle. The edges of the figure are labeled with geographic
latitude (N) and longitude (E). Credit for creating the figure goes to Dr. Frank Lind.
6.2
February 2, 2002
An example of an auroral irregularity observation from February 2, 2002 is shown in
figure 6.2. The range-Doppler diagram shows a scattering volume that extends from
approximately 970 to 1070 km; an area of high intensity exists which has a large
Doppler shift near 670 m/s, and a diffuse area with lower intensity and large Doppler
width exists alongside the strong scatterer.
Investigation of the interferometer data reveals that this auroral echo is most likely
97
dB
February 2, 2002 UT 05:11
1500
230
228
1000
Velocity (m/s)
226
500
224
222
0
220
−500
218
216
−1000
214
−1500
0
200
400
600
Range (km)
800
1000
1200
212
Figure 6.2: A range-Doppler display from MRR showing an auroral echo near 1000
km; an area of high intensity and large Doppler shift is flanked by a larger, diffuse
“type 2” echo, which we show to be spread across the transverse dimension, indicating
a shear.
limited to one interferometer lobe. The spectra shown in figure 6.3 were computed
at the range of 980 km, which includes part of both the bright spot and diffuse
scattering area. The coherence is large where the cross-spectrum phase is highly
organized, indicating a scatterer contained wholly within one lobe (approximately
100 km wide at this range). Again, we note that the self-spectra on both antennas
are consistent with each other and with the coherence.
Using the same methods for interpreting the phase information as we did in chapter
3, we estimate that the irregularity scattering volume is 30 km wide (at the particular
range of 980 km), and its longitudinal length is estimated to be 100 km from the
98
Cross Spectrum Phase and Magnitude for Range 980 km
Phase
2
0
−2
−1500
−1000
−500
0
500
1000
1500
−1000
−500
0
500
1000
1500
Normalized
Magnitude
1
0.5
Self Spectra (dB)
0
−1500
230
Log Periodic
Yagi
220
210
−1500
−1000
−500
0
500
1000
1500
Doppler (m/s)
Figure 6.3: The cross-spectrum and self-spectra for one range in the auroral echo
from figure 6.2. Due to the highly organized phase, we observe that the echo seems
limited to one interferometer lobe; the phase width implies a scattering volume extent
of approximately 30 km. The 95% significance level for the coherence is also shown
in the middle panel.
range information. Another feature to note in figure 6.3 is the upward trend in
interferometer phase across a Doppler width of about 500 m/s. The steady increase
in phase over these likewise increasing Doppler bins suggests a velocity shear across
the scattering volume in the transverse direction. In general, we expect some sort
of shear when large field-aligned currents are present, so this is consistent with the
disturbed conditions under which MRR detects irregularity scatter.
99
Phase
2
0
−2
−1500
−1000
−500
0
500
1000
1500
−1000
−500
0
500
1000
1500
Normalized
Magnitude
1
0.5
Self Spectra (dB)
0
−1500
225
Log Periodic
Yagi
220
215
210
−1500
−1000
−500
0
500
1000
1500
Doppler (m/s)
Figure 6.4: Cross spectrum at range 984 km for the auroral echo on February 2, 2002,
UT 05:11.
6.2.1
Natural Progression to Two-Dimensional Images
Examining the cross spectra from this echo for the next few ranges reveals the same
linear upward trend in phase across Doppler, but at more distant ranges, a discontinuity appears and the phase “steps” from one level to the next, indicating the separation
of the slower- and faster-moving parts of the scatterer. For example, the “step” can
be seen beginning to form in the interferometer phase at a range of 984 km (figure
6.4); at ranges 989 km (figure 6.5) and 992 km (figure 6.6), the two distinct levels
of phase (transverse position of scattering volumes) are clear; and by range 1005 km
(figure 6.7), the bright, narrow-Doppler feature is gone, and the type 2 feature exists
primarily at a single level of interferometer phase.
We find it useful to consider interferometer data such as this over the entire set of
ranges which contain irregularity scatter. For example, figure 6.8 shows magnitude
and phase-difference spectra from the same auroral echo, stacked up over several
100
Phase
2
0
−2
−1500
−1000
−500
0
500
1000
1500
−1000
−500
0
500
1000
1500
0
500
1000
1500
Normalized
Magnitude
1
0.5
Self Spectra (dB)
0
−1500
225
220
Log Periodic
Yagi
215
210
−1500
−1000
−500
Doppler (m/s)
UT 05:11.
Phase
2
0
−2
−1500
−1000
−500
0
500
1000
1500
−1000
−500
0
500
1000
1500
0
500
1000
1500
Normalized
Magnitude
1
0.5
Self Spectra (dB)
0
−1500
225
220
Log Periodic
Yagi
215
210
−1500
−1000
−500
Doppler (m/s)
UT 05:11.
101
Phase
2
0
−2
−1500
−1000
−500
0
500
1000
1500
−1000
−500
0
500
1000
1500
0
500
1000
1500
Normalized
Magnitude
1
0.5
Self Spectra (dB)
0
−1500
220
Log Periodic
Yagi
215
210
−1500
−1000
−500
Doppler (m/s)
Figure 6.7: Cross spectrum at range 1005 km for the auroral echo on February 2,
2002, UT 05:11.
adjacent ranges. A “footprint” of the echo can be seen in the stacks, both as a bump
in the magnitude spectra (self-power spectra from a single antenna) of the left panel
and as a compact area of phase in the right panel. This naturally led us to create twodimensional interferometer images (as discussed in section 3.4.1) by forming phase
histograms plotted versus range, expecting the “footprint” of organized phase to show
up as a bright area in two-dimensional colorscale plots. Additionally, we “weighted”
the phase histograms with the cross-spectrum magnitude by summing not only the
occurrence, but the corresponding magnitude, of each data point as it went into the
appropriate histogram bin. This method performed to our satisfaction3 , in that it
is a convenient and straightforward way to visualize the interferometer data. The
resulting two-dimensional interferometer image for the echo discussed above is shown
3
There are other methods for generating 2D interferometer images that we wish to try in the
future. For example, one which makes use of the normalized cross-spectrum magnitude estimate
of transverse width proposed by Farley et al. [5].
102
in figure 6.9; the larger-scale structure of the echo is now easily visible. We see
that the high-intensity, narrow-Doppler feature splits apart from the lower-amplitude
feature at a range of about 1000 km, as we predicted by examining the cross-spectra
at individual ranges.
Due to the aliasing issue, it is unclear whether these two features are part of the
same scattering volume, or whether they exist in different parts of the sky. Also, the
overall location (of either or both echoes) in the sky is unknown. However, recognizing
the highly field-aligned nature of these phenomena, it is reasonable to speculate that
the observed scatter exists in a single region of the sky. At a range of 1000 km, 2◦ of
aspect angle (90◦ ±1◦ ) spans about 200 km. At this same range, our interferometer
beams are approximately 100 km wide, indicating that any irregularity scatter we see
will most likely be contained within two lobes.
6.3
March 24, 2002
On March 24, 2002, MRR detected irregularities over a period of roughly 10 hours.
This extensive amount of data allowed us to make many successive interferometer
measurements, spaced at intervals of 4 minutes. Figure 6.10 shows 3 images from
adjacent time intervals during the 8 minute period from UT 05:07 to 05:15.
6.3.1
Velocity Vector Measurements
By determining the phase progression of the auroral echo from one panel to the next,
we can estimate the transverse drift velocity of the irregularity. For the timestep from
UT 05:07 to 05:11, at 5 different ranges (990, 1005, 1020, 1035, and 1050 km), we
find transverse velocities with both positive and negative values (both eastward and
westward drifts) on the order of 15 m/s (which accompany Doppler shifts of 800-900
m/s). For the timestep from UT 05:11 to 05:15, the transverse velocity at each of the
103
Interferometer Phase
Increasing Range
Increasing Range
Power Spectra
−1000
0
1000
−1000
0
1000
Figure 6.8: Power spectra (from one antenna) and interferometer phase vs. range for
the auroral echo on February 2, 2002, UT 05:11.
104
Auroral Echo: 2 February 2002, UT 05:11
1100
Range (km)
1050
1000
950
900
−50
0
50
Transverse Position, Measured from Lobe Center (km)
Figure 6.9: The February 2, 2002 auroral echo, represented in a 2-D image of range
vs. transverse width. The resolution in this plot is 3 × 5 km (range × transverse
dimension).
5 ranges is eastward, and slightly larger (30-60 m/s). The Doppler shifts in this case
are smaller, between 700 and 800 m/s.
With an unaliased interferometer, it is possible to determine whether a scatterer
lies to the east or the west of the radar receiver simply by observing the sign of the
phase difference between the east and west antennas. While the many interferometer
lobes in our system prevent us from determining a unique bearing to targets from our
scatter receivers, it is still possible to determine, by examining the phase progression
in time on each antenna, whether a moving target’s motion is in an eastward or
westward direction. Given this information along with the transverse drift speed
105
UT 05:11
Range (km)
UT 05:07
UT 05:15
1100
1100
1100
1050
1050
1050
1000
1000
1000
950
950
950
900
−50
0
50
900
−50
0
50
900
−50
0
50
Figure 6.10: 2D interferometer images from March 24, 2002, UT 05:07—05:15.
and Doppler shift, we know two perpendicular components of the irregularity drift
velocity: the Doppler shift, measured along the radar line of sight, and the transverse
velocity, in a direction normal to the radar line of sight. Assuming the radar is looking
perpendicular to B, and that all irregularity activity exists in the plane perpendicular
to B, this means we know the total velocity vector describing the irregularity motion.
We have analyzed the echo shown in figure 6.10 in this way. Figure 6.11 shows an
illustration of the velocity vectors for the two timesteps at 5 different ranges (15 km
apart). We see that the echo remains highly blueshifted over the entire time interval,
with a net drift toward the radar receiver. Also worth noting is the small flow angle.
106
Range: km
24 March 2002: Velocity Vectors vs. Range and Time
1000
900
Westward
Eastward
Toward
Radar
Radar Line
of Sight
UT 05:07 − 05:11
UT 05:11 − 05:15
Time
Figure 6.11: Measured velocity vectors for the drifting irregularity shown above in
figure 6.10.
107
We often find it to be the case that the Doppler shifts of irregularity echoes are much
larger than their transverse drifts.
It is possible that the reason behind the small transverse drifts is actually not
small flow angle, but rather aliasing in our measurements of interferometer phase
progression. Since the lobes are so finely spaced, a fast moving irregularity could jump
from one lobe to the next during the 4 minute time period between observations and
falsely appear to have very little transverse velocity. If this were the case, then echoes
at a range of 1000 km could be aliased in multiples of roughly 400 m/s. This is a rather
large error to incur (although the very large Doppler shifts in this particular example
still dominate the measured velocities even if the transverse error is 400 m/s). The
long 4-minute interval between observations is unfortunate. Experimentation with
splitting the 10-second datasets into multiple parts to get finer time resolution was
unsuccessful, as the variance of the phase estimates grew too large (from the necessary
decrease in the number of incoherent averages) to distinguish small changes in the
interferometer phase.
One possible way to alleviate this problem is to take 10-second bursts of data
much more often4 , opting to send only a few of the datasets over the internet link
to be processed, unless we find that they contain echoes we wish to analyze further.
Unfortunately, this technique was not in use during the March 24 event (or for any
event thus far). We do have 2-minute-long data bursts from recent irregularity events,
which would also serve to eliminate transverse drift aliasing, but these are plagued
by as-yet-unsolved interference problems.
4
Once every minute ought to be frequent enough for our purposes, as we do not often see irregularities with drift speeds greater than 1600 m/s.
108
6.3.2
Electric Field Measurements
If we assume that the plasma motion is due entirely to E × B drift (Hall drift), and
that the magnetic field is known and static (Earth’s dipole field), then we can use
the velocity vectors measured by MRR to roughly estimate the electric field in the
vicinity of the scatterer. The direction of E will be perpendicular to both B and the
irregularity drift velocity (v), such that
v=
E×B
B2
(6.1)
To determine the electric field strength, we take B to be 5.5 × 10−5 Tesla, a
typical value for E-region heights in the MRR field of view. In the ionosphere, the
ψ parameter also applies, which is related to the collision and cyclotron frequencies
of the ionized particles. A typical value for ψ in the E-region is 0.25. We use these
values with the equation
E = (1 + ψ)Bv
(6.2)
to obtain estimates of E. If we only have Doppler shift information (as opposed to a
total velocity measurement), we can include the flow angle θ:
E=
(1 + ψ)Bv
cos θ
(6.3)
In the example above, with velocities as shown in figure 6.11, we find the total
irregularity drift speeds |v| to be between 700 and 900 m/s. These speeds correspond
to electric field strengths of 48–62 mV/m, which are relatively large values and indicate the disturbed ionospheric conditions under which they were measured5 . We
can determine the direction of E by considering figure 6.11; B in this case is into the
page, which means the electric field is directed westward.
5
Kp during these observations was 6.0; an overflight of DMSP satellite F14 at the same time
showed disturbed magnetosphere conditions; GPS total electron content data showed depletions
in the MRR field of view, indicating low conductivity and a region able to support large electric
fields.
109
dB
March 30, 2002 UT 04:59
1500
236
234
1000
232
Velocity (m/s)
500
230
0
228
226
−500
224
−1000
−1500
0
222
200
400
600
Range (km)
800
1000
1200
220
Figure 6.12: An example of Doppler power spectrum versus range from irregularity
scatter obtained on March 30, 2002.
6.4
March 30, 2002
A geomagnetic storm lasting from 21:00 to 23:00 local time on March 30, 2002 caused
irregularities over western Canada which we were also able to observe with our interferometer. The range-Doppler spectra, one of which is shown in figure 6.12, reveal
echoes centered around 1050 km, ranging over approximately 200 km, with mean
Doppler shifts near 500 m/s and with Doppler widths from 100 m/s to 300 m/s.
All of these echoes appear to be “type 2,” with power spread uniformly across their
Doppler widths.
In figure 6.13 we show two-dimensional images from data taken at 4-minute intervals over a time window of 16 minutes. In the first three frames, the scattering volume
110
Figure 6.13: A sequence of 5 consecutive interferometer images from the March 30
event, 4 minutes apart in time. The first three frames indicate a mean transverse
drift speed of 70 m/s. The final frame clearly shows that the scattering volume has
split, a feature unrecognizable from the range-Doppler spectra in figure 6.12.
can be seen drifting in the transverse direction at roughly 70 m/s, which is consistent
with a possible guiding center E × B drift. Later, at UT 04:59, the scattering volume
separates into two main pieces (above and below 1050 km), not an apparent feature
when only the range-Doppler diagram in figure 6.12 is considered. Figure 6.13 also
illustrates the phase-wrapping inherent in the aliased interferometer measurements;
in the last three frames we can see that the echo is split between the two edges of the
frame, crossing an arbitrary phase boundary between interferometer lobes.
Using the transverse drift speed (70 m/s) inferred from the interferometer data
111
and the Doppler shift (500 m/s) from the power spectra, we find that the total
irregularity phase speed is approximately 505 m/s (again, the transverse drift makes
little impact when combined with the large Doppler shift). This speed corresponds to
an electric field strength of 34.7 mV/m, a reasonable value for a disturbed ionospheric
environment.
We note that if the Doppler velocity is “rotational,” as in the cases with velocity
shears, there is a nonzero curl in E, and Faraday’s Law states that the magnetic field
is time-varying in the vicinity. This also indicates that our view of velocity as E ×
B drift is only an approximation, although the majority of plasma motion can still
be described in this way, and the dominant component of B is still Earth’s magnetic
field. The current density J is also related to E by the conductivity σ, another
parameter characterizing the ionosphere. Thus, a wealth of geophysical information
may be gleaned from the radar data, particularly when interferometric measurements
are available. In future work we hope to explore all these possibilities, to extract as
much information from our instrument as possible.
112
Chapter 7
CONCLUSIONS
“Boy finishing that paper has made me feel goofy.”
— Future Dr. Rachel Yotter
We describe the first implementation of a passive radar interferometer for “deep,”
volumetric targets detected by Bragg scatter (as well as other targets, such as aircraft and meteor trails), and demonstrate its use in gathering information about the
transverse structure of these targets. Our interferometer is optimized for the study
of E-region ionospheric phenomena in that the baseline between its two antennas is
large (approximately 16 λ), giving us very fine transverse resolution (as fine as 1 km
at a range of 1000 km, or 0.06◦ in azimuth). While the long baseline prevents us
from uniquely determining the locations of scattering volumes we detect, this information is not critical in geophysical studies, and furthermore, we are able to roughly
localize the echoes we observe through a combination of a-priori information, such
as altitude limitation and aspect angle sensitivity, and information derived from the
heterogeneous nature of the antennas in our system.
7.1
Summary
In this work, we have described the passive radar technique (specifically, that of the
Manastash Ridge Radar) and discussed the implementation, in terms of both hardware and software, of a passive radar interferometer. We have shown many observations of varied targets, including ground clutter, aircraft, meteor trails, and plasma
113
density irregularities in the E-region ionosphere; we are able to form two-dimensional
images (in the plane perpendicular to B) of these targets and detect velocity shears
in both the radial and transverse directions. Because of our instrument’s very fine
resolution, we are able to show that many high-latitude coherent E-region irregularities exist in compact scattering volumes (as narrow as 30 km in the transverse
dimension), and that significant structure exists on scales as small as 1.5 square kilometers (1/1200th of one cell in typical data from SHERPA, for example [9]). Such
fine structure has previously been undetected due to the coarse resolution of available
instruments, and we are excited to offer our results to the scientific community.
We have also described in detail the passive radar interferometer cross-correlation
estimator and its statistical properties; we performed an extensive analysis of the
resolution capability of the instrument using both analytical arguments and empirical results, including simulations as well as experimental observations. Finally, we
have demonstrated how additional geophysical information, such as total velocity and
electric field, can be measured from the radar data using interferometer information.
Although we have focused on the observation and characterization of ionospheric
E-region irregularities in this work, the passive radar interferometer techniques we
have presented here are applicable in many other areas, including aerospace problems
(as Paul Howland has shown [12]), ionospheric effects on spacecraft communications
[3], and further understanding of plasma wave instabilities [4]. We consider our most
important contribution to be the development of radar techniques for remote sensing
of “difficult” targets (i. e., overspread, deep, volumetric targets).
114
7.2
Future Work
In the future, we hope to extend our interferometric capability by arranging two1
new antennas in such a way as to both relieve the azimuth aliasing problem as well
as provide a total of 6 baselines for use in imaging, perhaps with the radio astronomy
CLEAN algorithm or maximum entropy method (MEM), as Hysell has done [13,
14, 15]. We also hope to take advantage of the availability of receiver data from
other frequencies; differential changes in wavelength (or triangulation) could be used
to overcome azimuth ambiguity, and a study of scattering volumes illuminated from
many directions by different transmitters would be very interesting. Along these
same lines, the narrowband nature of the radar illumination is certainly a limiting
factor in the interpretation of MRR data (especially due to the Bragg backscattering
condition); including a sweep over many FM band frequencies during the normal
operation of the radar would be useful. We believe that advances in MRR signal
processing (in particular, methods for dealing with interference) will be necessary
before any of these multi-frequency possibilities will yield reliable results.
We plan to continue experimenting with ways of presenting interferometer data
so that it can be understood easily and intuitively; for example, alternatives to our
“phase histogram imaging” method include the abovementioned maximum entropy
method and estimators of transverse width that make use of the cross-spectrum magnitude [5].
An analysis of the received power in MRR data products and the radar cross
section of observed targets would be a very worthwhile study. We are also particularly
interested in multiple illumination frequency studies, as well as a study of ionospheric
targets illuminated from different directions (separate transmitters). Also, analyses
1
For a total of 4 scatter-collecting antennas, since that is currently the number of analog receiver
channels we have available.
115
of bias and variance in the MRR estimators for both interferometry (this work) and
power spectrum [7, 27] have indicated that the system is limited by clutter rather
than noise. Since we currently implement a matched filter (proven to maximize SNR
in receivers) in the MRR signal processing, a useful project for future work might be
to design an optimal detector based on signal-to-clutter ratio (SCR) rather than SNR,
and determine its effect on the detectability of scatterers. We make the preliminary
suggestion that such a technique might involve overresolution (i.e., sampling at a
higher rate than the bandwidth of the radar waveform). This would spread the
clutter power over a larger bandwidth [11], possibly increasing detectability; however,
it would also increase the computational (and network traffic) burden associated
with the radar, and it would mean we could no longer assume the noise plus clutter
background to be white, which is in general undesirable.
We are most interested in the use of the interferometer to study plasma irregularities, and to this end we hope to devise methods to glean as much geophysical
information from the radar as possible. Areas we hope to explore in the future include
experiments in time resolution, an investigation of the interferometer cross-spectrum
in the complex plane (where it belongs), and current density vector measurements.
Overall, we have found this project to be a fascinating combination of disciplines, and
we hope to continue in this direction in further research.
116
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120
Appendix A
VARIANCE DERIVATION
A.1
Isserlis’ Gaussian Moment Theorem Applied to Analytic Radar
Signals
In this thesis, we make extensive use of the Isserlis theorem [16, 18], which states
that 4th order moments of complex Gaussian random variables with zero mean can
be written as combinations of 2nd order moments as follows:
habcdi = habi hcdi + hadi hbci + haci hbdi
(A.1)
For our purposes, even-order moments always come in complex conjugate pairs,
so that we deal only with correlations such as
hab∗ cd∗ i = hab∗ i hcd∗ i + had∗ i hcb∗ i + haci hb∗ d∗ i
(A.2)
In the special case in which the random processes a, b, c, and d are analytic
signals, we are able to disregard all correlations between pairs of both-conjugated or
both-unconjugated random variables (such as haci and hb∗ d∗ i), as these correlations
will result in a value of zero. Then we can write
hab∗ cd∗ i = hab∗ i hcd∗ i + had∗ i hcb∗ i
(A.3)
which is the form of Isserlis’ theorem that have used in this text. The reason for this
is subtle, and we discuss it below.
The radar signals s and x involved in the analyses here can be assumed to be
analytic; they arise from real, causal processes, and are sufficiently narrowband (200
121
kHz at a center frequency of 100 MHz) that an in-phase and quadrature (IQ) receiver
produces real and imaginary samples that are, for all practical purposes, Hilbert
transforms of each other. Therefore, they exhibit the properties of an analytic signal,
which include the equality of the autocorrelation functions of their real and imaginary
parts and the odd symmetry of the cross-correlation function between their real and
imaginary parts [31]:
Ruu (τ ) = Rvv (τ )
(A.4)
Ruv (τ ) = −Ruv (−τ )
(A.5)
where we denote the real part of the analytic signal by the subscript u and the
imaginary part by the subscript v.
To show that “like conjugate” correlations such as haci go to zero, we define
analytic signals a and c to be partially correlated (otherwise their correlation would
certainly be zero) by introducing new random variables α, β, and γ and coefficients
A and B as follows:
a = Aα + β
(A.6)
c = Bα + γ
(A.7)
Then, the correlation haci is proportional to hααi. We can write hααi in terms of
real and imaginary parts (including a time lag τ for generality):
D
E
α(t)α(t − τ )
=
D
(αu (t) + jαv (t)) (αu (t − τ ) + jαv (t − τ ))
=
D
αu (t)αu (t − τ ) − αv (t)αv (t − τ )
E
(A.8)
E
D
E
+ j αu (t)αv (t − τ ) + αv (t)αu (t − τ )
(A.9)
α
α
α
α
(τ ) + Ruv
(−τ ))
= Ruu
(τ ) − Rvv
(τ ) + j (Ruv
(A.10)
α
α
(τ ))
(τ ) − Ruv
= 0 + j (Ruv
(A.11)
= 0
(A.12)
122
where the last two equalities follow from equations A.4 and A.5.
Thus, correlations between “like conjugate” pairs of analytic random signals have
expected value zero, and we only consider opposite-conjugate pairs of radar signal
correlations (such as equation A.3) in our analyses.
A.2
Variance Derivation, Continued
We now continue the derivation of the interferometer cross-correlation estimator in
section 4.6 here. From equation 4.49, we have
D
K̂ K̂ ∗
E
=
XX
n
D
E
Rs,n (0)Rs,l (0) xxxxxxxx +
l
|
{z
}
term 1 D
ED
E
XX
∗
Rs,n (τ )Rs,l
(τ ) ejkd(cos θn −cos θl ) xxxxxxxx
n
|
(A.13)
l
{z
}
term 2
where
D
xxxxxxxx
E
=
D
x(t − r0 )x∗ (t − r0 )x(t − r0 − τ )x∗ (t − r0 − τ ) ×
E
x(t − n)x∗ (t − n)x(t − l − τ )x∗ (t − l − τ ) (term 1) (A.14)
E
x(t − n)x∗ (t − l)x(t − l − τ )x∗ (t − n − τ ) (term 2) (A.15)
A.3
Analysis of 8th -order Correlations
To make the 8th -order correlations in x manageable, we again use Isserlis’ theorem to
decompose them into combinations of 2nd -order correlations. Due to the combinatorial
nature of this process and the numerous resulting terms, this is a tricky step. For
123
clarity (and generality) we write
haa∗ bb∗ cc∗ dd∗ i =



haa∗ i hbb∗ cc∗ dd∗ i +






∗
∗ ∗
∗

hab i hba cc dd i +



hac∗ i hca∗ bb∗ dd∗ i +





 had∗ i hda∗ bb∗ cc∗ i
(A.16)
The 6th -order correlations, in turn, break down as follows:
hbb∗ cc∗ dd∗ i =




hbb∗ i hcc∗ dd∗ i +



hbc∗ i hcb∗ dd∗ i +





 hbd∗ i hdb∗ cc∗ i
(A.17)
And finally, the 4th -order correlations can be handled in the straightforward manner that we have used previously:
hcc∗ dd∗ i = hcc∗ i hdd∗ i + hcd∗ i hdc∗ i
(A.18)
Thus, each 6th -order correlation yields 6 terms, which results in 24 separate terms
(each the product of 4 2nd -order correlations) for the entire 8th -order correlation.
Another way to see this is to write
haa∗ bb∗ cc∗ dd∗ i = ha♣i hb♦i hc♥i hd♠i
(A.19)
where it is clear that there are 4 possible choices for ♣ (any of a∗ , b∗ , c∗ , or d∗ ); 3
choices for ♦ after ♣ has been chosen; 2 possibilities for ♥; and ♠ is then the last of
the above choices. The number of separate terms, then, is 4! = 24.
Now we write equations A.14 and A.15 in terms of 2nd -order correlations according
to the above recipe.
A.3.1
Term 1
With the autocorrelation function defined as
Rx (τ ) = hx(t)x∗ (t − τ )i
(A.20)
124
equation A.14 becomes
D
E
xxxxxxxx =
Rx4 (0)
+
Rx2 (0)
|Rx (τ )|2 + |Rx (τ + r0 − n)|2 + |Rx (τ + l − r0 )|2
2
2
2
+ |Rx (τ + l − n)| + |Rx (r0 − n)| + |Rx (r0 − l)|
+ Rx (0) Rx∗ (τ + r0 − n)Rx (τ + l − n)Rx (r0 − l)
+ Rx∗ (r0 − l)Rx (τ + r0 − n)Rx∗ (τ + l − n)
+ Rx (τ )Rx∗ (r0 − l)Rx∗ (τ + l − r0 )
+ Rx (τ + l − r0 )Rx (r0 − l)Rx∗ (τ )
+ Rx (τ )Rx∗ (τ + r0 − n)Rx (r0 − n)
+ Rx∗ (r0 − n)Rx (τ + r0 − n)Rx∗ (τ )
+ Rx∗ (r0 − n)Rx (τ + l − n)Rx∗ (τ + l − r0 )
+ Rx (τ + l −
r0 )Rx∗ (τ
+ l − n)Rx (r0 − n)
+ |Rx (τ )|2 |Rx (τ + l − n)|2 + |Rx (r0 − n)|2 |Rx (r0 − l)|2
+ |Rx (τ + l − r0 )|2 |Rx (τ + r0 − n)|2
+ Rx (τ )Rx∗ (τ + r0 − n)Rx (τ + l − n)Rx∗ (τ + l − r0 )
+ Rx (τ )Rx∗ (r0 − l)Rx (r0 − n)Rx∗ (τ + l − n)
+ Rx∗ (r0 − n)Rx (τ + r0 − n)Rx∗ (r0 − l)Rx∗ (τ + l − r0 )
+ Rx∗ (r0 − n)Rx (τ + l − n)Rx∗ (τ )Rx (r0 − l)
+ Rx (τ + l − r0 )Rx (r0 − l)Rx∗ (τ + r0 − n)Rx (r0 − n)
+ Rx (τ + l − r0 )Rx∗ (τ + l − n)Rx (τ + r0 − n)Rx∗ (τ )
(A.21)
Now we use the simplification Rx (τ ) = Rx (0)δ(τ ); the validity of this assumption
is discussed in section 4.1. When we consolidate δ terms and enforce all the constraints
125
required for the δ’s to be nonzero1 , the above expression becomes
D
E
xxxxxxxx =
Rx4 (0) 1 + δ(τ ) + δ(τ + r0 − n) + δ(τ + l − r0 )
+ δ(τ + l − n) + δ(r0 − n) + δ(r0 − l)
+ 2δ(r0 − l)δ(τ + r0 − n) + 2δ(τ )δ(r0 − l)
+ 2δ(τ )δ(r0 − n) + 2δ(r0 − n)δ(τ + l − r0 )
+ δ(τ )δ(l − n) + δ(r0 − n)δ(r0 − l)
+ δ(τ + l − r0 )δ(τ + r0 − n)
+ 6δ(τ )δ(l − n)δ(r0 − l)
(A.22)
Using this result, term 1 in equation A.13 becomes
R X
R
X
D
E
Rs,n (0)Rs,l (0) xxxxxxxx =
n=1 l=1
"
Rx4 (0)
2
2
Rs,r
(0) + 6δ(τ )Rs,r
(0) + δ(τ )
0
0
X
2
Rs,n
(0)
n
+ Rs,r0 (0)
X
n
+
XX
n
Rs,n (0) 2 + 4δ(τ ) + 2δ(τ + r0 − n) + 2δ(τ + n − r0 )
l
+ δ(τ + l − n) + δ(τ + l − r0 )δ(τ + r0 − n)
1
#
For example, δ(τ )δ(τ + l − n) ⇒ (τ = 0 and l = n) ⇒ δ(τ )δ(l − n).
(A.23)
126
A.3.2
Term 2
Next, we consider the correlation of equation A.15. As before, we write it in terms of
the autocorrelation function of x:
D
E
xxxxxxxx =
|Rx (τ )|4 + Rx2 (0) |Rx (τ )|2 + Rx2 (0) |Rx (l − n)|2 + |Rx (τ )|2 |Rx (l − n)|2
+ Rx (0)Rx (l − n)Rx∗ (r0 − n)Rx (r0 − l)
+ Rx (τ )Rx (l − n)Rx∗ (r0 − n)Rx∗ (τ + l − r0 )
+ Rx (τ )Rx∗ (l − n)Rx (r0 − n)Rx∗ (τ + r0 − l)
+ Rx (0)Rx (τ )Rx (r0 − l)Rx∗ (τ + r0 − l)
+ Rx (τ )Rx (τ )Rx∗ (τ + r0 − l)Rx∗ (τ + l − r0 )
+ |Rx (τ )|2 |Rx (r0 − n)|2 + Rx (0)Rx∗ (τ )Rx∗ (r0 − n)Rx (τ + r0 − n)
+ Rx (0)Rx∗ (l − n)Rx∗ (τ + r0 − l)Rx (τ + r0 − n)
+
Rx∗ (r0
− l) Rx (0)Rx (r0 − n)Rx∗ (l − n) + Rx (r0 − l) |Rx (r0 − n)|2
+ Rx (0)Rx (τ )Rx∗ (τ + l − r0 ) + Rx∗ (τ )Rx∗ (l − n)Rx (τ + r0 − n)
+
Rx∗ (r0
− n)Rx (τ + r0 −
n)Rx∗ (τ
2
+ l − r0 ) + |Rx (τ )| Rx (r0 − n)
+ Rx (τ + n − r0 ) Rx (0)Rx (l − n)Rx∗ (τ + l − r0 ) + Rx (0)Rx∗ (τ )Rx (r0 − n)
+ Rx∗ (τ )Rx (l − n)Rx (r0 − l) + Rx∗ (τ )Rx∗ (τ )Rx (τ + r0 − n)
+ Rx (r0 − l)Rx (r0 − n)Rx∗ (τ + r0 − l)
+ Rx (τ + r0 − n)Rx∗ (τ + l − r0 )Rx∗ (τ + r0 − l)
(A.24)
127
Substituting delta functions for the transmitter autocorrelations, and grouping
delta functions with like-constraints, we find that expression A.15 simplifies to
D
E
xxxxxxxx =
Rx4 (0)
2δ(τ ) + δ(l − n) + δ(τ )δ(l − n)
+ 3δ(l − n)δ(r0 − l) + 8δ(τ )δ(l − n)δ(r0 − l)
+ 3δ(τ )δ(r0 − l) + 4δ(τ )δ(r0 − n)
+ δ(l − n)δ(τ + r0 − l) + δ(l − n)δ(τ + l − r0 )
(A.25)
Then, term 2 of equation A.13 becomes
R
R X
X
D
∗
Rs,n (τ )Rs,l
(τ ) ejkd(cos θn −cos θl )
ED
E
xxxxxxxx =
n=1 l=1
"
Rx4 (0)
3 |Rs,r0 (τ )|2 + 8δ(τ ) |Rs,r0 (τ )|2
+ 4δ(τ )Rs,r0 (τ )
XD
∗
(τ )
∗
3δ(τ )Rs,r
(τ )
0
XD
E
n
+
E
n
+ 2δ(τ )
XXD
n
+
X
n
E
∗
ejkd(cos θn −cos θl ) Rs,n (τ )Rs,l
(τ )
l
2
|Rs,n (τ )|
1 + δ(τ ) + δ(τ + r0 − n) + δ(τ + n − r0 )
#
(A.26)
128
A.4
The Full Variance Expression
D
Retrieving the result for K̂
ED
E
K̂ ∗ from equation 4.43 and applying the substitution
Rx (τ ) = Rx (0)δ(τ ) gives us
D
K̂
ED
K̂ ∗
E
= Rx4 (0)
R D
R X
X
E
∗
e+jkd(cos θn −cos θl ) Rs,n (τ )Rs,l
(τ ) ×
n=1 l=1
δ(r0 − n) + δ(τ ) δ(r0 − l) + δ(τ )
"
= Rx4 (0) |Rs,r0 (τ )|2 + δ(τ )
XXD
n
+ δ(τ )Rs,r0 (τ )
XD
∗
+ δ(τ )Rs,r
(τ )
0
XD
n
n
(A.27)
E
∗
(τ )
l
E
∗
(τ )
E
#
(A.28)
To finish off the full expression for the variance, we combine term 1 and term 2
D
E
of K̂ K̂ ∗ and subtract the above according to the definition of variance (given in
equation 4.41). We finally arrive at
129
Var K̂ =
"
Rx4 (0)
2
2
Rs,r
(0) + 6δ(τ )Rs,r
(0) + δ(τ )
0
0
X
2
(0)
Rs,n
n
+ Rs,r0 (0)
X
n
+
XX
n
Rs,n (0) 2 + 4δ(τ ) + 2δ(τ + r0 − n) + 2δ(τ + n − r0 )
l
+ δ(τ + l − n) + δ(τ + l − r0 )δ(τ + r0 − n)
+ 2 |Rs,r0 (τ )|2 + 8δ(τ ) |Rs,r0 (τ )|2
+ 3δ(τ )Rs,r0 (τ )
∗
+ 2δ(τ )Rs,r
(τ )
0
R D
X
n=1
R D
X
E
∗
(τ )
E
n=1
+ δ(τ )
R X
R D
X
E
∗
(τ )
n=1 l=1
+
R
X
2
|Rs,n (τ )|
1 + δ(τ ) + δ(τ + r0 − n) + δ(τ + n − r0 )
n=1
This result is repeated in chapter 4, where we discuss its meaning.
#
(A.29)
130
Appendix B
SIMULATING APPROPRIATE PASSIVE RADAR
SIGNALS
In chapter 5, we show numerical estimates of variance in simulated radar data
products. While simulations are useful for gaining empirical insight into analytically
difficult (or intractable) problems, care must be taken to ensure that the simulation
is an accurate and well-understood model for the problem at hand. As always, any
approximations and assumptions used in the simulation must be made available to
those who would interpret the results. In general, one must also be aware of numerical
and computational issues and the limitations of random number generators.
Some important considerations in our case are properly simulating the illuminating
FM signal and target signals. The distributions of the random numbers used in the
simulation must be accurate in order to provide a useful estimate of the variance
in our real data. Since the data is complex, we consider the distributions of its
magnitude, phase, and real and imaginary parts. Also, the simulated data must have
an appropriate bandwidth (power spectrum) and signal energy, at least with respect
to the energies of other signals in the simulation. In the development below, we
roughly follow the simulation work done by Hansen [8].
B.1
Simulation of an FM Signal
First, we generate the transmitter waveform. For our purposes here, we create an
ideal FM signal, just as we assumed that the reference copy of the transmitted signal
131
was perfect in the theoretical analysis above. Therefore, no noise has been added, and
the FM signal we generate has all the properties of an ideal FM signal: its envelope is
constant, the in-phase and quadrature parts of its complex samples are uncorrelated,
and its phase varies uniformly over [0, 2π]. Also, we do not simulate stereo channel
multiplexing or other similar features that are sometimes found in commercial FM
signals.
To create the FM signal, we first generate a baseband “audio signal” a[n] by
lowpass filtering white Gaussian noise (created with the function ’randn’ in MATLAB)
to a bandwidth of 8 kHz with a slow rolloff (we use a Chebyshev filter of order 3 for
the filtering). Next, we frequency modulate this signal according to
x[n] = exp j2πfc n + jκ
n
X
!
a[m]
(B.1)
m=0
However, we take the carrier frequency fc to be zero, since we would just immediately
mix the resulting signal down to baseband to create the reference time series for the
radar signal processing.
We conduct this entire process at the oversampled rate of 400 kHz in order to
generate a signal with the appropriate bandwidth (approximately 200 kHz). The
frequency deviation parameter κ, the amount of modulation in equation B.1, was
adjusted so that the “3 dB down” cutoff of x would be at 100 kHz (see figure B.2,
right panel). Finally, we downsample x[n] by a factor of 4 to create the 100 kHz time
series used in the simulations.
Figure B.1 shows scatter plots of the real and imaginary parts of complex FM
signals: the right panel is our simulated data, while the left panel shows samples
from an actual FM waveform (from a Seattle station at 94.9 MHz). Both signals were
sampled at a rate of 100 kHz. These plots show that the in-phase and quadrature
components are uncorrelated in both the experimental and simulated signals. The
simulated data clearly has the ideal constant modulus property, while the ring for the
132
Voltage Samples: In phase vs. Quadrature
4
x 10
1
1.5
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1.5
−1
−1
0
1
−1
−0.5
0
0.5
1
4
x 10
Figure B.1: Scatter plots of actual (left) and simulated FM data.
experimental data has a thickness, probably due to a combination of slight amplitude
modulation by the transmitter, noise, and other channel effects. Also, in the case
of the real FM data, a concentration of points exists with very small radius at the
center of the ring. This is most likely due to noise aliased into the signal because of
the insufficient sampling rate of 100 kHz. Finally, we note that, while the simulated
samples vary over the range [−1, +1], the magnitude of the experimental samples has
not been normalized, and thus the radius of the ring reflects the gain those samples
underwent, designed to utilize the full range of the analog-to-digital converters in the
receiver.
In figure B.2, we show power spectral densities of the same experimental and
simulated data. The simulated data spectrum is much smoother than that of the
measured FM data, which shows evidence of (perhaps) subcarriers or other additional
133
FM Signal Power Spectral Densities
4
120
100
2
dB
80
0
60
40
−2
20
−4
0
−20
−200
−100
0
100
200
−6
−200
−100
0
100
200
Frequency (kHz)
Figure B.2: Power spectral densities of actual (left) and simulated FM data.
structure in the FM signal. Also, the frequency response of the digital receiver filter
can be seen at the far edges of the spectrum, another feature that is not present in the
simulated data. Again, the receiver gain boosts the experimental data, so its power
spectrum shows a larger dynamic range.
Next, we note that each sample in either 100 kHz time series is independent from
the rest. This can be seen in the FM signal autocorrelations, shown in figure B.3,
and is a very important consideration in the simulations.
Finally, we compare histograms of the experimental and simulated data (106 data
points each) in figure B.4. The top plots show the magnitude distributions of the data;
again we see that the simulated signal is truly constant modulus, but in the measured
samples, there is some width to the distribution (with a heavier tail approaching zero,
because of the low-amplitude noise). Next, the phase distributions are shown, and
134
FM Signal Autocorrelations
9
2
x 10
20
1.5
15
1
10
0.5
5
0
0
−0.5
−1
−200
−5
−100
0
100
200
−200
−100
0
100
200
Lag (microseconds)
Figure B.3: Autocorrelation functions (real part) of actual (left) and simulated FM
data. Each sample is marked with a dot.
both appear to be uniform over [−π, π]. We expect this, since the FM signals are
zero-mean, yet constant modulus. Also, the unpredictable nature of the FM waveform
phase is the property that makes it useful as a radar waveform. In the bottom plots,
we show the distributions of the real part of the two FM signals; in the experimental
data, the central concentration can be seen in addition to the “smeared” edges of the
ring. As we would expect from the previously mentioned characteristics, the real and
imaginary part distributions of the two signals are very similar.
B.2
Simulation of a Target Signal
Next, we must generate a target signal for the transmitter waveform to scatter from.
Since our primary targets of interest are geophysical in nature (rather than “point
135
Experimental Samples
Simulated Data
5
Magnitude
2.5
5
x 10
10
2
8
1.5
6
1
4
0.5
2
0
0
0.5
1
1.5
2
x 10
0
0
0.5
1
1.5
−2
0
2
−0.5
0
0.5
2
4
x 10
4
Phase
2.5
4
x 10
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
−2
0
0
2
4
Real Part
5
4
x 10
10
4
8
3
6
2
4
1
2
0
−2
x 10
−1
0
1
2
x 10
0
−1
1
4
x 10
Figure B.4: A comparison of histograms of actual FM samples and samples from a
simulated FM signal.
136
targets”), we have chosen to use first-order autoregressive (AR) processes to represent
them. AR processes have a “memory,” or number of feedback elements, equal to the
order of the process (one, in our case). They decay smoothly in correlation; the
speed of decorrelation depends on the order of the process (higher orders decay more
slowly). An AR(1) process can be conveniently thought of as white noise that has
been filtered by a single-pole lowpass filter.
A useful way to generate target AR processes in our case is with the following,
which has been written in terms of the first and second moments of the power spectrum (fµ and f∆ , respectively):
s
1 − |β|2
w[t] + βs[t − 1]
s[t] =
2
where β = exp (−f∆ /fs ) exp (j2πfµ /fs )
(B.2)
The target s[t] has mean Doppler shift fµ and Doppler width f∆ (both in Hz). The
signal w[t] is the innovations process (complex, zero-mean, unit-variance, white Gaussian) and fs is the sampling frequency.
The theoretical power spectrum of this first-order AR process is
S(f ) =
1 − e−f∆ /fs
2 (1 − 2e−f∆ /fs cos(2π(f − fµ )) + e−2f∆ /fs )
(B.3)
where the roles of fµ and f∆ can be seen.
B.3
Creating the Scattered Signal
Once the transmitter and target signals have been generated, we construct the scattered signals on each antenna according to the simple model shown in equation 4.13.
This consists of adding together many delayed copies of the transmitted signal, each
modulated by one of the target signals. The delay corresponds to the target range;
an additional phase term is added on one antenna due to the target azimuth angle.
137
We have chosen to simulate targets only as concentrated at one particular azimuth
angle, and not distributed over a transverse width. This is to prevent the confusion of
variance in the estimator with variance in the cross-spectrum due to angular spread
(see section 4.5.1). Also, we do not consider interference or channel models in this
simulation; the received scatter is corrupted only by additive white Gaussian noise.
The addition of noise raises the important question of how to simulate specific
signal-to-noise ratios. The approach we have chosen is to generate all signals in
our simulation, including the noise, with the same energy (their 2-norms are approximately the same), and then to multiply each target by its own specified SNR
(typically less than 1; the signal is buried in noise, and must be detected by use of a
matched filter and other signal processing).
Power spectra and results from simulated data like we have described here are
shown in chapter 5.

Passive VHF Radar Interferometer Implementation, Observations

Transcription

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