Iterative Timing Recovery

1
Iterative Timing Recovery

• Aleksandar Kavcic
• Division of Engineering and Applied Sciences
• Harvard University
• based on a tutorial by
• Barry, Kavcic, McLaughlin, Nayak Zeng
• And on research by
• Motwani and Kavcic

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

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

4
Perfect timing
5
System Under Timing Uncertainty

• difference between transmitter and receiver
  clock
• basic assumption clock mismatch always present

6
A More Realistic Case
Sample instants kT ? kT?k
7
Properties of the timing error

• Brownian Motion Process (slow varying).
• Discrete samples form a Markov chain.

8
Timing recovery strategies
turbo equalization (inner loop)
a)
c)
timing recovery
symbol detection
timing recovery
symbol detection
decoding
decoding
?
?
free running oscillator
free running oscillator
iterative timing recovery (outer loop)
turbo timing/equalization
b)
d)
turbo equalization
timing recovery
symbol detection
joint soft timing recovery and symbol detection
decoding
decoding
?
?
free running oscillator
free running oscillator
9
Traditional Phase Locked Loop
10
Simplest iterative timing reovery
11
Simulation results
12
Convergence speed
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

14
Quantizing the Timing Offset
Uniformly quantize the interval ((k-1)T, kT to Q
levels.
15
Math Model for Timing Error
State Transition Diagram
State Transition Probability
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States for Timing Error
Semi-open segment ((k-1)T, kT Q 1-sample
states 1i i1, 2, , Q 1 deletion states 0 1
2-sample state 2
17
Example timing error realization
18
0
T
2T
3T
4T
5T
6T
7T
8T
9T
10T
0th interval
1st interval
2nd interval
3rd interval
4th interval
5th interval
8th interval
6th interval
7th interval
10th interval
9th interval
15
0
0
2
15
14
14
15
11
11
12
t
0-?0
3T- ?3
2T- ?2
5T- ?5
4T- ?4
T- ?1
6T- ?6
7T- ?7
8T- ?8
9T- ?9
0
11
12
13
14
15
2
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Single trellis section
0
0
11
11
12
12
13
13
14
14
15
15
2
2
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Source Model
Second order Markov chain
21
Full Trellis
Total number of states at each time interval
Trellis length n (block length). (note that
each branch may have different number of outputs).
22
Joint Trellis Example
a) pulse example
1
h(t)
0
-2T
0
-T
T
2T
3T
3T/5
-2T/5
8T/5
23
Soft-Output Detector
24
Definition of Some Functions
Definition
25
Calculation of the Soft-outputs
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Recursion of a(t,m,i)
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Recursion of ß(t,m,i)
28
known timing
conventional 10 iterations
10-1
after 2 iterations
after 4 iterations
after 10 iterations
10-2
bit error rate
10-3
10-4
2
3
4
5
6
29
Cycle-slip correction results
true timing error timing error estimate after 1
iteration timing error estimate after 2
iterations timing error estimate after 3
iterations
T
0
timing error
-T
-2T
1000
2000
3000
4000
5000
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

31
known timing
conventional 10 iterations
10-1
after 2 iterations
after 4 iterations
after 10 iterations
10-2
bit error rate
10-3
loss due to timing error Can we compute this loss?
10-4
2
3
4
5
6
32
Open Problems

• Information Theory for channels with
  synchronization error
• Capacity
• Capacity achieving distribution
• Capacity achieving codes

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 ?

34
Deletion channels

• Some results
• Ulmann 1968, upper bounds on the capacities of
  deletion channels
• DiggaviGrossglauser 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)
Prob/xt
Prob/yt
st-1
st
st-1
st
P00/0
Q00/0
0
0
0
0
P01/1
P10/0
Q01/1
Q10/0
1
1
1
1
P11/1
Q11/1
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Trellis for Y X
Prob/y1
Prob/y2
s0
s1
s2
11
02
(1-?)/1
(1-?)/0
0

(1-?)?/0
(1-?)?/0
Run a reduced-state BCJR algorithm on tis
trellis to upper-bound H(YX)
02
03
(1-?)/0

(1-?)?2/1
(1-?)?2/0
03
14
(1-?)/1

(1-?)?3/1
(1-?)?/1
14
15
(1-?)/1



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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)