2014 IEEE 22nd Signal Processing and Communications Applications Conference (SIU 2014)
1
New Early Termination Methos For Turbo Decoders
Orhan Gazi
Abstract— Turbo codes have large latency due to iterative
decoding steps and complex decoding algorithms. To decrease
the decoding delay it is essential to terminate the decoding
operation as soon as there is sufficient information for data
bits. Early termination can be achieved by defining stopping
criteria beyond which negligible achievement is observed. In this
article, we investigate some efficient stopping criteria based on
the variance of extrinsic information and population of the bits
below a predetermined threshold. We show that variance based
method is better than its previously suggested counterparts, and
threshold based method is the least complex one considering the
available methods.
I. INTRODUCTION
T
Urbo codes were introduced in 1993 by [1]. This was
a breakthrough in coding society. Turbo codes employ
complex soft decision decoding algorithms. For practical use
of turbo codes in communication systems, it is essential
to find high speed implementation techniques. High speed
implementation can be supported either by using parallel
processors or by reducing the complexity of the decoding
algorithms. The latter can be accomplished by reducing total
iteration number. This can be done by early termination of the
decoding operation without degrading the code performance.
Recently, there are a number of stopping criteria proposed.
Kullback in [2] introduced cross-entropy (CE) which is used
to compare a true distribution to a reference one. CE is used
as a stopping criteria in [3]-[4] where output distributions of
two component decoders are inspected by CE rule. Based
on CE criterion another stopping criteria called sign change
ration (SCR) is studied in [4]. Some less complex stopping
criteria such as mean estimate (ME) [5], sign difference ratio
(SDR) [6] and variance estimate (VE) [7] are also available
in the literature. ME method considers the absolute values of
LLRs. SDR criteria accounts the number of sign differences
of the extrinsic information between consecutive iterations at
a component decoder. In VE method variance of the LLR of
the extrinsic information is taken into consideration for early
termination.
The aim of the stopping criteria is to reduce the computation
amount, therefore the complexity of the stopping criteria
should be as low as possible otherwise there may not be a big
advantage of the stopping rule. Stopping rules are especially
useful from medium to high SNR regions. Since at high SNR
regions for a medium size interleaver only 1 or 2 iterations
are necessary. On the other hand at low SNR regions due to
poor performance too many iteration numbers are required.
The general algebraic stopping models designed for turbo
decoders usually are complex and requires extra hardware
Orhan Gazi (phone: +90 312 284 45 00-4015, e-mail:
o.gazi@cankaya.edu.tr), is with Electronics and Communication Engineering
Department, Cankaya University, Ankara, Turkiye.
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and additional calculations. Since, communication systems
are constructed with known encoder and decoder units, our
approach in this article is to propose stopping method for
communication systems with certain encoder decoder units.
We propose two new early termination technique. One is
based on the variance of the extrinsic information, and it
is shown that the proposed technique is better than the one
proposed in [7] which is also based on variance of the extrinsic
information.In our second approach, we define a threshold
zone for the extrinsic information, and monitor the population
in the zone between thresholds. Thresholds are determined by
simulations for certain turbo codes. When the population in
the threshold zone goes below a predefined number, stopping
decision is taken. We compare our approach with the other
ones by simulation results. Considering the complexity of
stopping criteria our second approach has the least complexity
and simple to implement.
The outline of the paper is as follows. Section II is devoted
to the review of MAP algorithm. In Section III, we first
introduce the variance and threshold-zone approaches and then
present the simulation results. Finally, conclusions are drawn
in Section IV.
II. T HE MAP A LGORITHM W ITH T URBO D ECODER
In this section we review the turbo decoding operation
with MAP algorithm [3] and point out expressions which are
used for variance computation and population based stopping
criteria. The turbo codes under consideration have component
codes RSC(1, 5/7)octal . BPSK modulation is used. The aposteriori probability for information bit uk = +1 given the
received signal y is
P (uk = +1|y) =
αk (s )γk (s , s)βk+1 (s)
(1)
s ,s
uk =1
where αk (s ), γk (s , s), and βk+1 (s) are the forward state,
transition, and backward state probabilities. The log values of
the a-posteriori, forward state, backward state, and transition
probabilities are defined as,
P (uk = +1|y) = log(P (uk = +1|y))
α
k (s ) = log(αk (s ))
βk+1 (s) = log(βk+1 (s))
γ
k (s , s) = log(γk (s , s)).
Using the above definitions, equation (1) can be written as,
P (uk = +1|y) = log
eαk (s )+γk (s ,s)+βk+1 (s)
(2)
s ,s
uk =1
which can further be expressed for a turbo code with rate 1/2
constituent codes RSCs(1, 5/7)octal as,
2014 IEEE 22nd Signal Processing and Communications Applications Conference (SIU 2014)
2
TABLE I
α (s )+ yp cp +β (s)
yu
k+1
σ2
e k
,
P (uk = +1|y) = 2 + L1e21 + log
σ
AVERAGE ITERATION NUMBER AND BER RESULTS FOR DIFFERENT
VARIANCE THRESHOLDS , GM MEANS GENIE M ETHOD , K IS THE
MULTIPLYING FACTOR USED IN
s ,s
uk =1
(3)
where yu and yp are the received signals for BPSK modulated
data and parity bits, σ 2 is the noise variance, cp is the BPSK
modulated parity symbol in the trellis diagram. L1e21 is the
extrinsic information for bit ’1’ supplied by the accompanying
component decoder. In a similar manner the log a-posteriori
probability for information bit ’0’ is expressed as
α (s )+ yp cp +β (s)
−yu
k+1
σ2
P(uk = 0|y) = 2 + L0e21 + log
e k
.
σ
s ,s
uk =1
(4)
Eb /N0
1dB
1.2dB
V arth
0.880
0.988
0.990
0.992
0.992
0.994
0.997
-
log10 (BER)
−2.79
−4.64
−4.73
−4.98
−4.92(GM)
−4.23, K=10
−4.43, K=14
−4.57, K=16
−4.74, K=18
−5.30
−5.40
−5.51
−5.62(GM)
−4.83, K=10
−4.98, K=12
−5.06, K=14
−5.34, K=16
[7]
Avg. Itr. Num.
2.47
3.86
3.94
4.04
2.99(GM)
3.90
4.38
4.53
4.71
3.43
3.50
4.07
2.5(GM)
3.69
3.97
4.18
4.35
where yu , and yp are the received signals for BPSK modulated
data and parity bits, cp is the BPSK modulated parity symbol
in trellis diagram, L0e21 is the logarithmic extrinsic information
supplied by the second decoder to the first one. Using the aposteriori probabilities, log likelihood ratio is defined as
constituent codes in turbo decoder. Frame length is chosen as
1024. S-random (S = 15) interleaver is used. We compared
the simulation results with the one obtained using the GENIE
method [8] where it is assumed that the receiver knows the
transmitted frames, and compares them to the decoded frames,
P (uk = 1|y)
LLR = log
if exact matching occurs iteration is stopped, otherwise next
P (uk = 0|y)
iteration is performed. In [8] it is shown for Eb /N0 = 1dB
= P(uk = 1|y) − P (uk = 0|y)
that average iteration number with any stopping criteria is at
α (s )+ yp cp +β (s)
2yu
1
0
k
k+1
2
σ
= 2 + Le21 − Le21 + log
e
− least one more than the one obtained with GENIE method. We
σ
now explain both approaches in the following sub-sections.
s ,s
uk =1
yp cp
log
eαk (s )+ σ2 +βk+1 (s)
A. Variance Approach
s ,s,u =0
k
(5)
where L1e21 and L0e21 are the logarithmic extrinsic bit probabilities for bits ’1’ and ’0’, the actual extrinsic bit probabilities
1
0
e1
are Pe1 = eLe21 , Pe0 = eLe21 . This leads to Le21 = log( P
Pe0 ).
Let’s denote the extrinsic information supplied by the second
decoder to the first one L1e21 −L0e21 by Le21 and from equation
(5) the extrinsic information supplied by the first decoder to
the second one is:
Le12 = log
eαk (s )+
yp cp
σ2
k+1 (s)
+β
s ,s
uk =1
log
eαk (s )+
yp cp
σ2
k+1 (s)
+β
−
.
(6)
s ,s
uk =0
III. E ARLY T ERMINATION F OR T URBO D ECODERS
In this section we will consider two different approaches
for the early termination of the turbo decoders. In our first
Pi
i
i
approach variance of |Pe1
−Pe0
| and |log( Pe1
i )| are considered
e0
separately. The second approach considers the populations of
Pi
i
i
− Pe0
| and |log( Pe1
|Pe1
i )| below a predetermined threshold.
e0
We tested our early termination methods by turbo codes.
For our simulations, we employed RSCs (1, 5/7)octal for
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In [7] using the variance of extrinsic information the termination condition was suggested as V ar(Lie21 ) > K ×
V ar(L1e21 ), i.e., iteration is stopped when the variance of the
LLR of the extrinsic information at ith iteration is K times
greater than that of the first iteration. In our approach, we
i
i
compute the variance for |Pe1
− Pe0
| and check whether it is
greater than a predetermined threshold or not. The stopping
i
i
decision is taken when the variance(|Pe1
− Pe0
|) is above the
i
i
predetermined threshold. In Fig. 1 variance(|Pe1
− Pe0
|) v.s.
iteration number ’i’ is plotted. It is clear from Fig. 1 that for
different Eb /N0 values variance saturates for different iteration
numbers beyond which no change in variance is observed. It
is clear from Fig. 1 that we can determine a threshold for
variance of (|Pe1 − Pe0 |) and use it as a stopping criteria.
In Table 1 the performance of the variance-threshold based
approach along with average iteration numbers are depicted.
It is obvious from tabulated data that the proposed technique
achieves almost the same performance as that of the one
by GENIE method with one more iteration. In addition, the
proposed technique is better than the one in [7] in terms of
BER and average iteration number.
B. Population Approach
We have two sub categories for population approach.
2014 IEEE 22nd Signal Processing and Communications Applications Conference (SIU 2014)
|P −P | population in threshold−zone
Variance(|P −P |) w.r.t Iteration Number
e1
3
e1
e0
e0
400
1
Eb/N0=1.2dB
Eb/N0=1dB
Eb/N0=0.8dB
Eb/N0=0.6dB
E /N =0.4dB
0.9
b
E /N =1.2dB
b 0
Eb/N0=1dB
Eb/N0=0.8dB
Eb/N0=0.6dB
350
0
300
|Pe1−Pe0| population
Variance
0.8
0.7
0.6
250
200
150
100
0.5
50
0.4
0
0
2
4
6
8
Iteration Number
10
12
14
16
Fig. 1. Variance of |Pe1 − Pe0 | vs iteration number. Interleaver Size=1024.
TABLE II
AVERAGE ITERATION NUMBER AND BER
POPULATION THRESHOLDS
Eb /N0
0.8dB
1dB
1.2dB
Np
36
20
6
10
4
6
-
2
4
6
8
Iteration Number
10
12
14
16
Fig. 2.
|Pe1 − Pe0 | population in threshold-zone vs iteration number.
Interleaver Size=1024. Pth = 0.5
TABLE III
RESULTS FOR DIFFERENT
AVERAGE ITERATION NUMBER AND BER RESULTS FOR DIFFERENT
POPULATION THRESHOLDS Ñth , AND P̃th = 3
Np , Pth = 0.75, |Pe1 − Pe0 | < Pth .
log10 (BER)
−3.64
−4.03
−4.00(GM)
−4.90
−4.81
−4.92(GM)
−5.62
−5.26
−5.62(GM)
0
Avg. Itr. Num.
4.04
4.33
3.71(GM)
4.06
3.82
2.99(GM)
3.63
3.43
2.50(GM)
Eb /N0
1dB
1.2dB
0.8dB
1) First Population Approach: In our first population approach, for each iteration i we check the inequality
i
i
− Pe0k
| < Pth
1 if |Pe1k
w(k) =
0 otherwise
where k denotes the information bit, and takes values k =
1 . . . 1024. With the w(k) definition,
the termination condition
can be stated as: terminate if
k w(k) < Np , otherwise
go on with next iteration, Pth and Np
are probability and
population thresholds. Figure 2 shows
k w(k) population
in threshold-zone for different Eb /N0 values and Pth = 0.5.
It is seen from Fig. 2 that initially there is a sharp drop in
population which saturates after certain iteration numbers. The
performance of the proposed technique is tabulated in table
II where it is that the proposed technique achieves GENIE
method’s performance with only one more iteration.
2) Second Population Approach: In our next population
approach, we check the inequality
Pi
1 if |log( P ie1k) | < P̃th ,
w̃(k) =
e0k
0 otherwise.
In a similar manner
the termination condition can be stated
as: terminate if k w̃(k) < Ñp , otherwise continue, Ñp is the
predetermined population threshold.
The simulation results are depicted in Table III where it
is seen than with one more iteration the proposed schemes
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p
N
15
17
8
10
22
32
100
-
log10 (BER)
−4.70
−4.60
−4.92(GM)
−5.40
−5.30
−5.62(GM)
−3.95
−3.83
−3.26
−4.00(GM)
Avg. Itr. Num.
3.92
3.87
2.99(GM)
3.57
3.48
2.50(GM)
4.63
4.46
3.76
3.71(GM)
approximately achieve the same performance as the one with
GENIE method. Since, in practical applications usually logMAP based turbo decoder is used, the second population
approach is the most suitable one for practical applications
considering the complexity and easy of implementation. In
[9] all the available stopping criteria are inspected, and shown
that all require some additional computation. Hence, there is
always an increase in total computation amount when stopping
algorithms are employed. However, in our second population
approach when log-MAP algorithm is employed the additional
computation amount is almost negligible. The only overhead
is to check the inequality and increase bit counter and when
it passes threshold level the stopping decision is taken. As a
result, it can be concluded that instead of using generalized
stopping methods it is logical to find some threshold levels for
known decoders units which are employed in communication
units.
IV. C ONCLUSION
In this article, we inspected two new early stopping criteria
for turbo codes which are based on variance and threshold. We
see from simulation results that, the previously proposed variance stop criteria in [7] is less accurate compared to threshold
based variance stopping criteria. For practical use, the second
2014 IEEE 22nd Signal Processing and Communications Applications Conference (SIU 2014)
population approach proposed is the best from implementation
perspective due to its much lower complexity compared to
other stopping criteria. From simulation results, it is concluded
that population based approach gives similar performance as
variance method, and its complexity is much smaller compared
to other early stopping criteria. For communication systems
where latency is a critical issue population threshold based
approach is more suitable due to its much lower complexity
compared to other stopping methods.
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[4] R. Y. Shao, S. Lin and M. P. C. Fossorier, ”Two simple stopping criteria
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[8] Y. S. Kim, S. W. Ra, ”A Simple Efficient Stopping Crterion for Turbo
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