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
DIMACS-04
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Outline
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•
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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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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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Perfect timing
S
Xn
Channel
Yn
R
Xn*
t
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System Under Timing Uncertainty
t
S
Xn
Channel
YL
R
• difference between transmitter and receiver clock
• basic assumption: clock mismatch always present
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A More Realistic Case
1
-T
0
T
2T
3T
Sample instants: kT  kT+k
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t
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Properties of the timing error
t
t
• Brownian Motion Process (slow varying).
• Discrete samples form a Markov chain.
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Timing recovery strategies
a)
turbo equalization
(inner loop)
c)
timing
recovery
symbol
detection
timing
recovery
decoding


free running
oscillator
free running
oscillator
turbo equalization
b)
timing
recovery
symbol
detection
d)
symbol
detection
iterative timing
recovery (outer loop)
turbo timing/equalization
joint soft timing recovery
and symbol detection
decoding


free running
oscillator
free running
oscillator
decoding
decoding
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Traditional Phase Locked Loop
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Simplest iterative timing reovery
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Simulation results
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Convergence speed
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Strategy to solve the problem
1.
2.
3.
4.
Set up math model for timing error
(Markov).
Build separate stationary trellis to
characterize the channel and source.
Form a full trellis.
Derive an algorithm to perform the
Maximum a posteriori probability (MAP)
estimation of the timing offset and the
input bits
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Quantizing the Timing Offset
1
-T
0
T
2T
3T
t
Uniformly quantize the interval ((k-1)T, kT] to Q levels.
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Math Model for Timing Error
State Transition Diagram:
δ
-2θ
-θ
θ
0
2θ
δ
State Transition Probability:
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States for Timing Error
1
-T
0
T
2T
3T
t
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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Example: timing error realization
k
Q=5
T/Q
0
-T/Q
4
0
1
2
5
6
3
7
8
9 10 11 12 13 14 15
k
-2T/Q
-3T/Q
-4T/Q
-5T/Q = -T
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0
T
2T
3T
4T
5T
6T
7T
8T
9T
10T
0th
1st
2nd
3rd
4th
5th
6th
7th
8th
9th
10th
interval interval interval interval interval interval interval interval interval interval interval
15
15
14
14
15
0
2
0
11
11
12
t
0-0
0
11
12
13
14
15
2
T- 1 2T- 2
3T- 3
4T- 4
5T- 5 6T- 6
7T- 7
8T- 8
9T- 9
Single trellis section
0
11
12
13
14
15
2
0
11
12
13
14
15
2
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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
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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).
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Joint Trellis Example
a) pulse example
c) joint ISI-timing trellis
1
h(t)
-T
0
-2T/5
T
3T/5
2T
8T/5
b) ISI trellis
3T
(-1,-1,11)
(-1,1,11)
(1,-1,11)
(1,1,11)
(-1,-1,11)
(-1,1,11)
(1,-1,11)
(1,1,11)
(-1,1,12)
…
…
(1,-1,2)
(1,1,2)
…
(-1,-1,12)
…
(-1,-1)
(-1,1)
(1,-1)
(1,1)
…
…
(-1,-1)
(-1,1)
(1,-1)
(1,1)
(-1,-1,0)
…
…
0
-2T
…
(-1,-1,0)
(1,-1,2)
(1,1,2)
Soft-Output Detector
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Definition of Some Functions
Notation:
Definition:
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Calculation of the Soft-outputs
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Recursion of α(t,m,i)
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Recursion of β(t,m,i)
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known timing
conventional 10 iterations
after 2 iterations
after 4 iterations
after 10 iterations
bit error rate
10-1
10-2
10-3
10-4
2
3
SNR per bit
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4
5
(dB)
6
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Cycle-slip correction results
timing error
T
true timing error
timing error estimate after 1 iteration
timing error estimate after 2 iterations
timing error estimate after 3 iterations
0
-T
-2T
1000
2000
3000
4000
5000
time
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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
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known timing
conventional 10 iterations
after 2 iterations
after 4 iterations
after 10 iterations
bit error rate
10-1
10-2
10-3
loss due to timing error
10-4
Can we compute this loss?
2
3
SNR per bit
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4
5
(dB)
6
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Open Problems
• Information Theory for channels with
synchronization error:
– Capacity
– Capacity achieving distribution
– Capacity achieving codes
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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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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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Numerical capacity computation methods
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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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Capacity per transmitted symbol
compute
upper bound
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Markov sources
If X is a first-order Markov source (transition matrix P),
then Y is also a first-order Markov source (transition matrix Q)
st-1
0
Prob/xt
P00/0
P01/1
1
P11/1
st
0
st-1
0
P10/0
Prob/yt
Q00/0
Q01/1
1
1
st
0
Q10/0
Q11/1
1
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Trellis for
Prob/y1
Prob/y2
s0
0
Y|X
s1
11
(1-)/1
s2
02
(1-)/0
…
(1-)/0
(1-)/0
02
(1-)/0
(1-)2/1
(1-)2/0
03
03
(1-)/1
14
…
(1-)3/1 (1-)/1
14
(1-)/1
…
Run a reduced-state
BCJR algorithm on tis trellis
to upper-bound H(Y|X)
15
…
…
…
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Future research
• Upper bounds for insertion/deletion channels?
• Channels with non-integer timing error?
• Codes?
(long run-lengths are favored in deletion channels)
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