division of engineering and applied sciences dimacs-04 iterative timing recovery aleksandar kavčić...
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Division of Engineering and Applied Sciences
DIMACS-04
Iterative Timing Recovery
Aleksandar Kavčić
Division of Engineering and Applied Sciences
Harvard University
based on a tutorial by
Barry, Kavčić, McLaughlin, Nayak & Zeng
And on research by
Motwani and Kavčić
Division of Engineering and Applied Sciences
slide 2
Outline
• Motivation• Timing model• Conventional timing recovery• Simple iterative timing recovery• Joint timing and intersymbol interference trellis• Soft decision algorithm• Performance results• Conclusion• Future challenge: capacity of channels with
synchronization error
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slide 3
Motivation
• In most communications (decoding) scenarios, we assume perfect timing recovery
• This assumption breaks down, particularly at low signal-to-noise ratios (SNRs)
• But, turbo-like codes work exactly at these SNRs
• Need to take timing uncertainty into account
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slide 5
System Under Timing Uncertainty
t
ChannelXn YL S R
• difference between transmitter and receiver clock
• basic assumption: clock mismatch always present
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slide 6
A More Realistic Case
0-T T 2T 3T t
1
Sample instants: kT kT+k
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slide 7
Properties of the timing error
• Brownian Motion Process (slow varying).
• Discrete samples form a Markov chain.
t
t
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slide 8
Timing recovery strategies
timingrecovery
symboldetection
decoding
timingrecovery
symboldetection
decoding
free runningoscillator
free runningoscillator
turbo equalization
timingrecovery
symboldetection
decoding
joint soft timing recovery and symbol detection
decoding
free runningoscillator
free runningoscillator
turbo timing/equalization
turbo equalization(inner loop)
iterative timing recovery (outer loop)
a)
b)
c)
d)
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slide 13
Strategy to solve the problem
1. Set up math model for timing error (Markov).
2. Build separate stationary trellis to characterize the channel and source.
3. Form a full trellis.
4. Derive an algorithm to perform the Maximum a posteriori probability (MAP) estimation of the timing offset and the input bits
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slide 14
Quantizing the Timing Offset
Uniformly quantize the interval ((k-1)T, kT] to Q levels.
0-T T 2T 3T t
1
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slide 15
Math Model for Timing Error
State Transition Diagram:
State Transition Probability:
0 θ 2θ-2θ -θ
δ
δ
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slide 16
States for Timing Error
0-T T 2T 3T t
1
Semi-open segment : ((k-1)T, kT]:
Q 1-sample states: 1i i=1, 2, …, Q
1 deletion states: 0
1 2-sample state: 2
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slide 17
Example: timing error realization
k
k10 2 3
4 5 6 7 8 9 10 11 12 13 14 150
T/Q
-T/Q
-2T/Q
-3T/Q
-4T/Q
-5T/Q = -T
Q = 5
0 T 2T 3T 4T 5T 6T 7T 8T 9T 10T
0-0 3T- 32T- 2 5T- 54T- 4T- 1 6T- 6 7T- 7 8T- 8 9T- 9
t
0thinterval
1stinterval
2ndinterval
3rdinterval
4thinterval
5thinterval
8thinterval
6thinterval
7thinterval
10thinterval
9thinterval
15 0 0215 14 14 15 11 11 12
0
11
12
13
14
15
2
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slide 19
Single trellis section
0
11
12
13
14
15
2
0
11
12
13
14
15
2
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slide 20
Source Model
Second order Markov chain
-1, -1 -1, -1
-1, -1, 1
1, -1
1, 1
-1, -1 -1, -1
-1, -1, 1
1, -1
1, 1
-1, -1 -1, -1
-1, -1, 1
1, -1
1, 1
-1, -1 -1, -1
-1, -1, 1
1, -1
1, 1
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slide 21
Full Trellis
Full states set:
Total number of states at each time interval:
Trellis length = n (block length). (note that each branch may have different number of outputs).
(-1,-1)
(-1,1)
(1,-1)
(1,1)
(-1,-1)(-1,1)
(1,-1)
(1,1)
b) ISI trellis
(-1,-1,0)
(-1,-1,11)
(-1,1,11)
(1,-1,11)
(1,1,11)
(1,-1,2)
(1,1,2)
(-1,-1,12)
(-1,-1,0)
(-1,-1,11)
(-1,1,11)
(1,-1,11)
(1,1,11)
(1,-1,2)
(1,1,2)
(-1,1,12)
……
…
……
…
……
…c) joint ISI-timing trellis
-2T 0-T T 2T 3T
1
0
h(t)
3T/5-2T/5 8T/5
a) pulse example
Joint Trellis Example
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slide 24
Definition of Some Functions
Notation:
Definition:
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slide 28
2 3 4 5 6
10-4
10-3
10-2
10-1
known timing
after 10 iterations
after 2 iterations conventional 10 iterations
after 4 iterations
bit
err
or r
ate
SNR per bit (dB)
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slide 29
Cycle-slip correction results
1000 2000 3000 4000 5000
-2T
-T
0
T true timing errortiming error estimate after 1 iterationtiming error estimate after 2 iterationstiming error estimate after 3 iterations
time
timin
g e
rro
r
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slide 30
Conclusion
• Conventional timing recovery fails at low SNR because it ignores the error-correction code.
• Iterative timing recovery exploits the power of the code.
• Performance close to perfect timing recovery
• Only marginal increase in complexity compared to system that uses conventional turbo equalization/decoding
Division of Engineering and Applied Sciences
slide 31
2 3 4 5 6
10-4
10-3
10-2
10-1
known timing
after 10 iterations
after 2 iterations conventional 10 iterations
after 4 iterations
bit
err
or r
ate
SNR per bit (dB)
loss due to timing error
Can we compute this loss?
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slide 32
Open Problems
• Information Theory for channels with synchronization error:– Capacity– Capacity achieving distribution– Capacity achieving codes
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slide 33
Deletion channels
• Transmitted sequence x1, x2, x3, ….
– Xk { 0, 1 }
• Received sequence y1, y2, y3, ….– Sequence y is a subsequence of sequence x
• Symbol xk is deleted with probability
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slide 34
Deletion channels
• Some results:– Ulmann 1968, upper bounds on the capacities of
deletion channels– Diggavi&Grossglauser 2002, analytic lower bounds on
capacities of deletion channels– Mitzenmacher 2004, tighter analytic lower bounds
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slide 37
Received symbols per transmitted symbol
Let K(m) denote the number of received symbols per m transmitted symbols
K(m) is a random variable
Asymptotically, we have A received symbols per transmitted symbol
For the deletion channel,
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slide 38
Capacity per transmitted symbol
upper boundcompute
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slide 39
Markov sources
0
1
0
1
P00/0
P11/1
P01/1 P10/0
st-1 st
Prob/xt
0
1
0
1
Q00/0
Q11/1
Q01/1 Q10/0
st-1 st
Prob/yt
If X is a first-order Markov source (transition matrix P), then Y is also a first-order Markov source (transition matrix Q)
Division of Engineering and Applied Sciences
slide 40
Trellis for Y | X
011
02
(1-)/1
s0 s1
Prob/y1
s2
(1-)/0
(1-)2/0
(1-)3/1
03
14
02
03
14
15
(1-)/0
(1-)/0
(1-)2/1
(1-)/0
(1-)/1
(1-)/1
(1-)/1
Prob/y2
… ……
…
…
…
Run a reduced-stateBCJR algorithm on tis trellisto upper-bound H(Y|X)
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