Slides

DANISH GPS CENTER
Carrier Tracking Loop;
Loop Filter
GPS Signals And Receiver Technology MM12
Darius Plaušinaitis
[email protected]
Today’s Subjects
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• Demodulation of the GPS signal
• Tracking loop introduction
• Carrier tracking
– Phase Lock Loop (PLL)
– Frequency Lock Loop (FLL)
• Loop filters
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Demodulation
Or how to turn the radio waves back into the data
message that we are interested in
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Demodulation
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• The transmitted signal from satellite k is:
S k (t ) = 2 Pc (C k (t ) D k (t )) cos(2πf L1t )
+ 2 PPL1 ( P k (t ) D k (t )) sin( 2πf L1t )
+ 2 PPL 2 ( P k (t ) D k (t )) cos(2πf L 2t )
L1 C/A signal
L1 P(Y) signal
L2 P(Y) signal
D – data
C – C/A code
P – P(Y) code
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Demodulation
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• The transmitted signal from satellite k is:
S k (t ) = 2 Pc (C k (t ) D k (t )) cos(2πf L1t ) + 2 PPL1 ( P k (t ) D k (t )) sin( 2πf L1t )
+ 2 PPL 2 ( P k (t ) D k (t )) cos(2πf L 2t )
• After an RF front-end (L1 only):
S k (t ) = 2 Pc (C k (t ) D k (t )) cos(ω IF t ) + 2 PPL1 ( P k (t ) D k (t )) sin(ω IF t )
• The signal from one satellite after the ADC
(narrow filters and low sampling frequency):
S k (n) = C k (n) D k (n) cos(ωif n) + e(n)
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Demodulation
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• The signal from two satellites after the ADC:
S (n) = C 1 (n) D1 (n) cos(ωif 1n) + C 2 (n) D 2 (n) cos(ωif 2 n) + e(n)
• The code phase and the intermediate
frequency of the carrier wave should be
known parameters to demodulate the
navigation data from e.g. satellite 1.
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Demodulation
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• Convert the signal down to baseband:
S (n) cos(ωif 1n) = C 1 (n) D1 (n) cos(ωif 1n) cos(ωif 1n) +
C 2 (n) D 2 (n) cos(ωif 2 n) cos(ωif 1n) + e(n)
cos(a)*cos(b) = ½cos(a+b) + ½cos(a-b)
S (n) cos(ωif 1n) = 12 C 1 (n) D1 (n) + 12 C 1 (n) D1 (n) cos(2ωif ) +
C 2 (n) D 2 (n) cos(ωif 2 n) cos(ωif 1n) + e(n)
S (n) cos(ωif 1n) = 12 C 1 (n) D1 (n) + 12 C 1 (n) D1 (n) cos(2ωif )
+ 12 C 2 (n) D 2 (n) cos(ωif 2 − ωif 1n)
+ 12 C 2 (n) D 2 (n) cos(ωif 2 + ωif 1n) + e(n)
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Demodulation Visualized
IF
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f
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Demodulation
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• Code wipe off:
S (n) cos(ωif 1n)C 1 (n) = 12 D1 (n) + 12 D1 (n) cos(2ωif ) + e(n)
• After low-pass filtering (integration):
S (n) cos(ωif 1n)C 1 (n) = I = 12 D1 (n) + e(n)
• Received signal amplitude vs. tracking errors
sin(π∆f iT )
S
Ii =
2 R(τ i ) Di cos(∆φi ) + ei
N0
(π∆f iT )
• Conclusion: perfectly aligned code and carrier
replicas are required to do demodulation.
These replicas can be tracked using two
tracking loops.
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Tracking Loop
A way to generate exact copy of the received signal
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What does tracking do and
why?
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• The main goal is to receive the GNSS signal
”as clear and loud” as possible – the local
carrier and spreading code must be well
aligned with the ones in the signal
• GNSS adds one more requirement: to track
signal arrival (time) as precise as possible
• Advanced receivers can detect multipath to
some extent
• Additional task can be signal quality
monitoring
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The Tracking Loop Idea
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• Generate a local signal
• Correlate it with the received signal
• Measure (time/code-phase, frequency, phase)
error between the local and the received
signals
• Steer local signal generators to minimize the
error
• Pass demodulated data bit value stream to the
data processing task
• Repeat procedure
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Main Parts Of A Tracking Loop
Scaling of error – used to control
sensitivity to the tracking error
The error measurement
Tracked
signal
Θ1(s)
Error
measurement unit
(called error
detector or
discriminator)
Generated
signal
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Θ2(s)
Θe(s)
Filter
Kd
F(s)
VCO
K0/s
Signal generator generates
replica of the tracked signal
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Filter has two
main tasks:
•To filter noise in
the error
measurements
(due to noise in
the tracked signal)
•To shape the
tracking loop’s
response to the
tracking error
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Types Of Tracking Loops
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• There are 3 main types of the tracking loops
depending on the tracked property of the
tracked signal:
– Phase lock loop (PLL)
– Frequency lock loop (FLL)
– Delay lock loop (DLL)
• There are few error detectors for each type of
the tracking loop with different properties
• Variations of filter parameters will shape the
filter response and amount of noise filtering
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Plan For The Tracking Topic
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• PLL (FLL) and DLL are explained in sections
on carrier and code tracking
• Each section will also cover a set of error
detectors applicable in a given tracking loop
• Tracking loop filter is explained in a separate
section as the same theory is used for all
types of tracking loops
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The Carrier Tracking Loop
The Phase Locked Loop (PLL)
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Carrier Tracking Loop
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• The goal of the Carrier Tracking Loop is to produce a
perfectly aligned carrier replica. The most common
way is to use a PLL:
In-phase arm
D(n) cos(ωif n)
cos(ωif n + φ )
sin(ωif n + φ )
Quadraturephase arm
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Carrier Tracking Loop
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• The demodulation in the In-phase (I) branch:
D(n) cos(ωif n) cos(ωif n + φ ) = 12 D(n) cos(φ ) + 12 D(n) cos(2ωif n + φ )
• The demodulation in the Quadrature-phase (Q)
branch:
D(n) cos(ωif n) sin(ωif n + φ ) = 12 D(n) sin(φ ) + 12 D(n) sin( 2ωif n + φ )
• The I signal:
• The Q signal:
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I = 12 D(n) cos(φ )
Q = 12 D(n) sin(φ )
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Carrier Tracking Loop
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• Costas loop:
I = 12 D(n) cos(φ )
cos(ωif n + φ )
D(n) cos(ωif n)
sin(ωif n + φ )
Q = 12 D(n) sin(φ )
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Carrier Tracking Loop
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• To find the phase error:
Q 12 D(n) cos(φ )
= 1
= tan(φ )
I
2 D ( n) sin(φ )
φ = tan
1
−1 2
1
2
D(n) sin(φ )
D(n) cos(φ )
φ
-2π
2π
Phase
delay
Q
φ
I
Q
φ = tan
I
−1
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Carrier Tracking Loop
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• The Costas loop is independent on the phase shifts
caused by the data bits
• Phasor diagram:
An output example:
Discrete-Time Scatter Plot
Q
4000
φ
φ
I
Q prompt
2000
0
-2000
-4000
-8000 -6000 -4000 -2000
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0
2000 4000 6000 8000
I prompt
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Carrier Tracking Loop
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• Different kinds of phase lock loop discriminators:
– Arctan
D = tan −1
Q
I
• Much time consuming (not a big problem today)
• The output is the real phase error
– Sign product
D = Q • sign(I )
• Fast method
• The discriminator output is proportional to sin(φ)
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Carrier Tracking Loop
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• Different kinds of phase lock loop discriminators:
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Carrier Tracking Loop
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• Costas loop:
I = 12 D(n) cos(φ )
D(n) cos(ωif n)
cos(ωif n + φ )
sin(ωif n + φ )
φ = tan −1
I
Q
Q = 12 D(n) sin(φ )
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Carrier Tracking Loop
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• The signal energy in I and Q when PLL has locked on
the signal:
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The Carrier Tracking Loop
The Frequency Locked Loop (FLL)
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Frequency Lock Loop
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• The discrimators are a bit different than in
PLL. They measure change in carrier phase
over an interval of time.
• Less noise sensitive than PLL – it can track at
lower SNR
• The tracking loop has more noise than PLL
• Can be used for the re-acquisition or pull-in
states due to bigger frequency lock range
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Loop Filters
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Why The Filter Is Needed
Anyway?
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• The error measurement is there, so just
correct the generator frequency and job is
done, right?
• The answer is NO:
– There is an error measurement noise (even at good
SNR)
– There is a stady state error caused by Doppler
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The Typical Tracking Loop
• Phase error detector has
gain Kd. The transfer
function is showed in
figure a)
• The VCO has a center
frequency ω0 and gain
K0. The transfer function
is showed in figure b)
• The filter coefficients
depend on Kd and K0
Phase detector
Θ1(s)
+
Σ
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Θe(s)
–
Θ2(s)
Filter
Kd
F(s)
VCO
K0/s
ud
ω
ω0
Θe
u
a)
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b)
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An Example Of Filter Output
vs. Input
• There is an initial
frequency between
tracked and generated
signals (27Hz here)
• Figures show:
Raw PLL discriminator
Amplitude
0.2
0.1
0
-0.1
-0.2
0.05
0.1
0.15
Time (s)
0.2
0.25
0.3
0.25
0.3
Filtered PLL discriminator
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20
Amplitude
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0
– The filter “accumulates”
offset over time and keeps
it
– The result of damping –
different convergence
times
-10
-20
0.05
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0.1
0.15
Time (s)
0.2
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A Simple Digital Loop Filter
• The C1 and C2 depend
on loop noise bandwidth
BL, VCO and PD gains
and loop damping factor
ζ.
• Damping factor controls
how fast the filter
reaches its settle point
• Noise bandwidth
controls the amount of
allowed noise in the
filter
Filter
input
c1
C1 =
Σ
Σ
Filter
output
z-1
8ζω nT
1
K 0 K D 4 + 4ζω nT + (ωnT ) 2
4(ζω nT ) 2
1
C2 =
K 0 K D 4 + 4ζω nT + (ωnT ) 2
ωn =
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c2
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8ζ BL
4ζ 2 + 1
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Damping Factor
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Different loop responses
depending on the damping factor
• Determines how much
the loop filter ”resists”
to the control signal:
(first 20ms are due to loop filter initialization)
80
minator output°][
60
2008
40
20
ζ = 0.3
ζ = 0.5
ζ = 0.7
ζ = 0.9
ζ = 1.1
– On one hand – how fast
the loop will ”fix” the
tracking error
– On other hand – how
much the loop will
overshoot 0 error point
• A compromise value is
used or few values are
used for different
receiver modes
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Noise Bandwidth
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• Narrow noise bandwidth decreases noise in
the tracking loop, AND – response speed
• At 100ms the loop noise bandwidth is
switched from about 100Hz to 15Hz
Frequency offset (Hz)
Phase offset [degrees]
-2520
45
0
-45
-90
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Carrier frequency offset
PLL discriminator
90
-2540
-2560
-2580
-2600
0
100
200
300
Time (ms)
400
500
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100
200
300
Time (ms)
400
500
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Noise Bandwidth
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PLL discriminator
90
45
0
-45
-90
Frequency offset (Hz)
• Loop noise
bandwidth also
determines
maximum
Doppler offset
and rates
tolerated by the
loop
• Figures show a
case of too big
initial frequency
error in
acquisition
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0
50
100
150
200
250
Time (ms)
300
350
400
450
500
350
400
450
500
Carrier frequency offset
-98
-99
-100
-101
0
50
100
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150
200
300
250
Time (ms)
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Loop Order
Filter
input
c2
c1
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Σ
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Σ
z-1
Filter
output
• The filter is a first order
filter
• The tracking loop
(excluding filter) is a
first order system,
therefore the tracking
loop is second order
• Higher order filters
approximate the error
dynamics better (e.g. to
be used for ships etc.)
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An Example Of A PLL
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• ω=20/0.7845; T=0.02;
–
–
–
k1= 2.4 * ω
k2= (1.1 * ω ^2)/20
k3= ω ^3/20
• ω =20/0.53; T=0.02;
– k1= 1.414 * ω
– k2= ω ^2/20
– k3= 0
This slide
contents is only
available to the
listeners of our
courses
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2-nd Order Loop Responce
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3-rd Order Loop Responce
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Questions and Exercises
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