EE 278: Statistical Signal Processing

EE 278: Statistical Signal Processing
Homework 8
Due: Friday, December 5 , 2014 at 3pm
1.
(a) For the symmetric 2-state Markov chain in Q. 2 of HW 7, compute its power spectral
density in terms of α (0 < α < 1). Plot the power spectral density for 4 representative
values of α. Qualitatively, how does the power spectral density depend on α?
(b) Compute the power spectral density of the process {V (t)} in Q. 3 in HW 7.
(c) For Q. 6 of HW 7, compute the power spectral densities of the processes {Xn }, {Yn }
ˆ n } in terms of α, |α| < 1 in the steady-state when n is large. Qualitatively, how
and {X
do the power spectral densities depend on α?
2. Given a random process {X(t)}, the autocorrelation function RX (t1 , t2 ) is defined to be
RX (t1 , t2 ) := E[X(t1 )X(t2 )].
(a) Express the autocorrelation function in terms of the covariance function and the mean
function of the process.
(b) What property does the autocorrelation function satisfy for a stationary process ?
(c) In class, we have defined the power spectral density SX (f ) of a zero-mean stationary
process to be the Fourier transform of the covariance function. For general stationary
process with arbitrary mean, define the power spectral density SX (f ) as the Fourier
transform of the autocorrelation function. Give a physical interpretation to SX (f ) and
justify it with some calculations.
3. In class, we defined the notion of a stationary process. In this question, we will explore
the notion of jointly stationary processes.
(a) Give a suitable definition for two processes {X(t)} and {Y (t)} to be jointly stationary.
The definition should be a generalization of stationarity for a single process.
(b) Given two processes {X(t)} and {Y (t)}, define the cross-covariance and cross correlation
function as:
¯ 1 ))(Y (t2 ) − Y¯ (t2 ))],
KXY (t1 , t2 ) := E[(X(t1 ) − X(t
RXY (t1 , t2 ) := E[X(t1 )Y (t2 )].
Argue that KXY (t1 , t2 ) and RXY (t1 , t2 ) depend only on the time difference t2 − t1 for
jointly stationary processes {X(t)} and {Y (t)} (using your definition in Part (a)).
(c) Are the cross-covariance and cross-correlation functions of two jointly stationary processes even? Explain.
(d) Let X(t) be a zero-mean stationary process with covariance function KX (τ ) and autocorrelation function RX (τ ). Let Y (t) = (h ∗ x)(t) where h is the impulse response of a
LTI system. Compute the cross-covariance and cross-correlation functions for {X(t)}
and {Y (t)} in terms of KX , RX and h.
(e) Argue that {X(t)} and {Y (t)} are jointly stationary. (Rigorous argument not required.)
4. Estimating the cross-correlation between two stationary random processes has several real
world signal processing applications. In this problem, we illustrate two such applications.
(a) Finding impulse response of LTI system. To find the impulse response h(t) of an LTI
system (e.g., a concert hall), i.e., to identify the system, white noise X(t), −∞ < t < ∞,
is applied to its input and the output Y (t) is measured. Given the input and output
sample functions, the cross-correlation RY X (τ ) is estimated. Show how RY X (τ ) can be
used to find h(t).
(b) Finding time of flight. The distance between a point and an object is often measured
by sending a signal and measuring the time of flight, the time it takes for the signal to
reflect off the object and return (assuming speed of signal, e.g., light, is known). Let
X(t) be the signal sent and Y (t) = X(t − δ) + Z(t) be the signal received, where δ is
the unknown time of flight and Z(t) the sensor noise. Assume that X(t) and Z(t) are
uncorrelated zero-mean stationary processes. The estimated cross-correlation function
of Y (t) and X(t), RY X (t) is shown in Figure 1. Find the time of flight δ.
RXY (t)
3
2
5
8
t
Figure 1: Cross-correlation function.
5. Suppose that {X(t)} is a sample path generated by a real-valued zero-mean continuoustime stationary process with covariance function KX (τ ). For any finite time window [0, T ],
one can compute the Fourier transform of the truncated signal
(
X(t),
if 0 ≤ t ≤ T,
XT (t) :=
0,
otherwise
2
as follows
Z
T
X(t)e−jωt dt.
Fω (XT ) :=
0
Note that Fω (XT ) is a random variable since it is a function of the sample path X(t).
In practice, the quantity T1 |Fω (XT )|2 , known as the periodogram, is frequently used to es
timate the power spectral density SX (f ) based on finite time series. Calculate E T1 |Fω (XT )|2
in terms of KX (τ ), and comment on why this is a good measure for estimating SX (f ) when
T is sufficient large.
6. The Madden Julian oscillation (MJO) is the largest element of the intraseasonal variability in the tropical atmosphere. It is a traveling pattern that propagates eastward, and
each cycle lasts approximately 30-60 days. The MJO is most obvious in the variation of
outgoing longwave radiation (OLR), a satellite-derived measure of tropical convection and
rainfall.
The data package at http://ee278.stanford.edu/restricted/hw8_data.mat consists
of one time series {Xn } of OLR in the Indian ocean that contains 1752 points. Each point in
the time series records a 5-day average of OLR over the region specified, from the beginning
of 1979 through the end of 2002. Suppose that the time series is generated by a stationary
process.
(a) Estimate the covariance function of the time series and plot it for time lags 0 through
30. Describe briefly on any cyclic characterstics that the covariance function might
present. What can you say about the periodicity of the MJO?
(b) Estimate the power spectral density (PSD) of this time series based on the covariance
function you estimate. Does it agree with your observation in Part (a) on the cycle of
the MJO?
(c) Now compute instead the Fourier transform of the sample path {Xn : 1 ≤ n ≤ N } (N =
1752), which is also a random variable. Plot the absolute square of the Fourier response,
which is the periodogram of the data (up to some normalization constant). Compare
and contrast it with the PSD plot you obtain in Part (b). Does it agree with your
derivation in Q. 5?
7. {Xn } is a stationary process with zero mean and covariance function KX (m) such that
KX (m) = 0 for |m| > L.
(a) Consider the random vector X = [X1 , X2 , . . . XN ]T by taking a block of length N of the
process. Let KX be the covariance matrix of X. Write down a specific example of KX
for your favorite choice of covariance function and N .
˜ X to make
(b) KX is Toeplitz but not circulant. Show how you can modify KX slightly to K
it circulant. You should only need to change a fixed number of rows of KX independent
of N . You can also assume that N > 2L. You should do this in general and not only
for your specific example.
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˜X.
(c) Find a set of complex eigenvectors and the corresponding eigenvalues for K
˜ X to see if their eigenvalues are close when
(d) Try out numerally an example of KX and K
N is large.
(e) (Bonus) Earlier in the course, we said that a symmetric real matrix always has real
˜ X real and symmetric? If so, why are the eigenvectors in part (c)
eigenvectors. Is K
complex? Can you find real ones instead?
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