1. The Performance of Turbo codes for Wireless Communication Systems
Grace Oletu Predrag Rapajic
Department of Computer and Communication Systems Department of Computer and Communication Systems
University of Greenwich University of Greenwich
Chatham,United Kingdom Chatham,United Kingdom
Og14@gre.ac.uk p.rapajic@gre.ac.uk
Abstract—Turbo codes play an important role in complexity. The paper is arranged as follows Section II is the
making communications systems more efficient and Channel Model, decoding algorithm is described in section
reliable. This paper provides a description of two turbo III, section IV is on Log-MAP Algorithm.Principles of
codes algorithms. Soft-output Viterbi algorithm and Iterative Decoding in section V, and Section VI compares the
logarithmic-maximum a posteriori turbo decoding simulation results and performance for both algorithms for
algorithms are the two candidates for decoding turbo
codes. Soft-input soft-output (SISO) turbo decoder based different block length, before concluding in section VII.
on soft-output Viterbi algorithm (SOVA) and the II. CHANNEL MODEL
logarithmic versions of the MAP algorithm, namely,
Log-MAP decoding algorithm. The bit error rate (BER) The transmitted symbols +1/-1, corresponding to the
performances of these algorithms are compared. code bits of 1/0 pass through additive white Gaussian
Simulation results are provided for bit error rate channel. By scaling the random number of distribution N (0,
performance using constraint lengths of K=3, over 1) with the standard deviation ı, an AWGN noise of
AWGN channel, show improvements of 0.4 dB for log- distribution N (0, ı2) is obtained. This is added to the symbol
MAP over SOVA at BER 10-4. to emulate the noisy channel effect.
Keywords- Turbo codes, Iterative decoding III. DECODING TURBO CODE
I. INTRODUCTION Let the binary logical elements 1 and 0 be represented
electronically by voltages +1 and -1, respectively. The
The near Shannon limit error correction performance of
variable d is used to represent the transmitted data bit as
Turbo codes [1] and parallel concatenated convolutional
shown in figure (1), whether it appears as a voltage or as a
codes [2] have raised a lot of interest in the research
logical element. Sometimes one format is more convenient
community to find practical decoding algorithms for
than the other. Let the binary 0 (or the voltage value - 1) be
implementation of these codes. The demand of turbo codes
the null element under addition.
for wireless communication systems has been increasing
Signal transmission over an AWGN channel, A well-
since they were first introduced by Berrou et. al. in the early
known hard-decision rule, known as maximum likelihood
1990s [1]. Various systems such as 3GPP, HSDPA and
(ML), is to choose the data dk = +1 or dk = -1 associated with
WiMAX have already adopted turbo codes in their standards
the larger of the two intercept values. For each data bit at
due to their large coding gain. In [3], it has also been shown
time k, this is tantamount to deciding that dk = +1 if xk falls
that turbo codes can be applied to other wireless
on the right side of the decision line, otherwise deciding that
communication systems used for satellite and deep space
dk = -1.
applications.
The MAP decoding also known as BCJR [4] algorithm is d u
not a practical algorithm for implementation in real systems.
The MAP algorithm is computationally complex and a
sensitive to SNR mismatch and inaccurate estimation of the a
noise variance [5]. MAP algorithm is not practical to
implement in a chip. The logarithmic version of the MAP
algorithm [6-8] and the Soft Output Viterbi Algorithm
(SOVA) [9-10] are the practical decoding algorithms for
implementation in this system.
rate R = ½ and generators G = [7 5]
v
Figure 1 Recursive systematic convolutional Encoder with memory two,
This paper describes Turbo decoding Algorithms, SOVA
has the least computational complexity and the worse bit A similar decision rule, known as maximum a posteriori
error rate (BER) performance, while the Log-MAP algorithm (MAP), which can be shown to be a minimum probability of
[6] has the best BER performance but high computational error rule, takes into account the a priori probabilities of the
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978-1-61284-840-2/11/$26.00 ©2011 IEEE

2. data. The general expression for the MAP rule in terms of L (d) = Lc(x) + L (d) + Le (d) (9)
APPs is as follows:
Equation (9) shows that the output LLR of a systematic
‫ͳܪ‬ decoder can be represented as having three LLR Elements: a
P (d) = +1| x) ൐ P (d = -1|x) (1) channel measurement, a priori knowledge of the data, and an
൏ extrinsic LLR stemming solely from the decoder. To yield
H2 the final L (d), each of the individual LLRs can be added as
shown in Equation (9), because the three terms are
Equation (1) states that you should choose the hypothesis statistically independent. This soft decoder output L(d) is a
H1, (d = +1), if the APP P (d = +1|x), is greater than the APP real number that provides a hard decision as well as the
P (d = -1|x). Otherwise, you should choose hypothesis H2, reliability of that decision. The sign of L(d) denotes the hard
(d= -1). Using the Bayes’ theorem, the APPs in Equation (1) decision; that is, for positive values L(d) of decide that d =
can be replaced by their equivalent expressions, yielding the +1, and for negative values decide that d = -1. The
following: magnitude of denotes the reliability of that decision. Often,
‫ͳܪ‬ the value L(d) due to the decoding has the same sign as Lc(x)
P (x|d) = +1) P (d = +1) ൐ P (x|d = -1) P (d = -1) (2) + L(d), and therefore acts to improve the reliability of L(d).
൏
H2
IV. LOG-MAP ALGORITHM
Equation (2) is generally expressed in terms of a ratio, This algorithm, called the log-MAP algorithm [11-15],
yielding the so-called likelihood ratio test, as follows: gives the same error performance as the MAP algorithm but
is easier to implement. The Log-MAP algorithm computes
‫ͳܪ‬ ‫ͳܪ‬ the MAP parameters by utilizing a correction function to
௉ሺ௫ȁௗୀ ାଵሻ ௉ሺௗୀ ିଵሻ ௉ሺ௫ȁௗୀ ାଵሻ ௉ሺௗୀ ାଵሻ (3)
௉ሺ௫ȁௗୀ ିଵሻ
൐ ௉ሺௗୀ ାଵሻ Or ௉ሺ௫ȁௗୀ ିଵሻ ௉ሺௗୀ ିଵሻ ൐ 1 compute the logarithm of sum of numbers. More precisely
൏
H2
൏ for A1 = A + B, then
H2 ሚ ሚ ෨ ሚ ෨
‫ = 1ܣ‬ln (A + B) = max (‫ + )ܤ,ܣ‬fc (|‫)|ܤ – ܣ‬ (11)
By taking the logarithm of the likelihood ratio, we obtain
a useful metric called the log-likelihood ratio (LLR). It is a ሚ ෨ ሚ ෨
where fc(|‫ )| ܤ — ܣ‬is the correction function. fc (|‫)| ܤ— ܣ‬
real number representing a soft decision output of a detector, can be computed using either a look-up table [8] or simply a
designated as follows: threshold detector [12] that performs similar to look-up table.
௉ሺௗୀ ାଵȁ௫ሻ ௉ሺ௫ȁௗୀ ାଵሻ ௉ሺௗୀ ାଵሻ The simple equation for threshold detector is
L (d|x) = log [௉ሺௗୀ ିଵȁ௫ሻ] =log [ ௉ሺ௫ȁௗୀ ିଵሻ ௉ሺௗୀ ିଵሻ] (4)
ሚ ෨ ͲǤ͵͹ͷ ሚ ෨
fc (|‫ = )| ܤ—ܣ‬ቄ if |‫ 2 | ܤ—ܣ‬otherwise
௉ሺ௫ȁௗୀ ାଵሻ ௉ሺௗୀ ାଵሻ
Ͳ
L (d|x) = log [௉ሺ௫ȁௗୀ ିଵሻ] + log [ ] (5) can be extended recursively. If A2 =A+ B + C, then A2
௉ሺௗୀ ିଵሻ
ሚ ሚ ሚ ሚ
ln(A1 + C) = max(‫ + ) ܥ ,ܣ‬fc(|‫)| ܥ – ܣ‬ (12)
L (d|x) = L (x|d) + L(d) (6) This recursive operation is specially needed for
computation of the soft output decoded bits. At each step, the
To simplify the notation, Equation (6) is rewritten as logarithm of addition of two values by maximization
follows: operation is accommodated by additional correction value
መ
L'(݀) = Lc (x) + L (d) which is provided by a look-up table or a threshold detector
(7) in the Log-MAP algorithm. The Log-MAP parameters are
where the notation Lc(x) emphasizes that this LLR term very close approximations of the MAP parameters and
is the result of a channel measurement made at the receiver. therefore, the Log-MAP BER performance is close to that of
The equations above were developed with only a data the MAP algorithm.
detector in mind. Next, the introduction of a decoder will V. PRINCIPLES OF ITERATIVE DECODING
typically yield decision-making benefits. For a systematic
code, it can be shown that the LLR (soft output) L (d) out of In a typical communications receiver, a demodulator is
the decoder is equal to Equation (8): often designed to produce soft decisions, which are then
መ መ
L (݀) = L'(݀) + Le (݀)መ transferred to a decoder. The improvement in error
(8) performance of systems utilizing such soft decisions is
typically approximated as 2 dB, as compared to hard
Where L(d) is the LLR of a data bit out of the
decisions in AWGN. Such a decoder could be called a soft
demodulator (input to the decoder), and Le(d) is called the
input/ hard output decoder, because the final decoding
extrinsic LLR, represents extra knowledge gleaned from the
process out of the decoder must terminate in bits (hard
decoding process. The output sequence of a systematic
decisions). With turbo codes, where two or more component
decoder is made up of values representing data bits and
codes are used, and decoding involves feeding outputs from
parity bits. From Equations (7) and (8), the output LLR L (d)
one decoder to the inputs of other decoders in an iterative
of the decoder is now written as follows:

3. fashion, a hard-output decoder would not be suitable. That is both algorithms. For figure 5, LOG MAP shows better
because a hard decision into a decoder degrades system performance than SOVA for constraint length of three and
performance (compared to soft decisions). for block length of 1024 and 4096 respectively.
Hence, what is needed for the decoding of turbo codes is
a soft input/ soft output decoder. For the first decoding CONCLUSIONS
iteration of such a soft input/soft output decoder , we Our Simulation results shows that the Log-MAP
generally assume the binary data to be equally likely, performs better in terms of block length compared to SOVA,
yielding an initial a priori LLR value of L(d)=0. The channel and thus it is more suitable for wireless communication.
LLR value, Lc(x), is measured by forming the logarithm of
the ratio of the values for a particular observation which
appears as the second term in Equation (5). The output L(d) REFERENCES
of the decoder in Figure 3 is made up of the LLR from the [1] C. Berrou, A. Glavieux, and P. Thitimajshima, "Near Shannon Limit
detector, L’(d) , and the extrinsic LLR output, Le(d) , Error-Correcting Coding and Decoding: Turbo Codes,“Proceeding of
representing knowledge gleaned from the decoding process. IEEE ICC 93, pp. 1064-1070.
As illustrated in Figure 2, for iterative decoding, the extrinsic [2] S. Benedetto, G. Montorsi, “Design of Parallel Concatenation
likelihood is fed back to the decoder input, to serve as a Convolutional Codes: IEEE Trans. on communication, vol. 44,
No.5, May 1996.
refinement of the a priori probability of the data for the next
[3] C. Berrou, “The Ten-Year-Old Turbo Codes are Entering into
iteration. Service,” IEEE Commun. Mag. vol. 41, no. 8, pp.110-116, Aug 2003.
[4] L. Bahi, J. Cocke, F. Jelinek, and J. Raviv, "Optimum decoding of
linear codes for minimizing symbol error rate," IEEE Trans.on Inf.
Theory, vol. IT-20, pp. 284-287, Mar. 1974.
[5] T.A. Summers and S.G. Wilson, "SNR Mismatch and Online
Estimation in Turbo Decoding, "IEEE Trans. On Comm. vol.46, no.
4, pp. 421-424, April 1998.
[6] P. Robertson, P. Hoeher, and E. Villebrun, "Optimal and Sub-Optimal
Maximum A Posteriori Algorithms Suitable for Turbo Decoding,
“European Trans. on Telecomm. vol. 8, no. 2, pp. 119-126, March-
April 1997.
[7] P. Robertson, E. Villebrun, and P. Hoeher, "A Comparison of
Optimal and Sub-optimal MAP Decoding Algorithms Operating in
the Log Domain,” International Conference on Communications, pp.
1009-1013, June 1995.
[8] S. Benedetto, G. Montorsi, D. Divsalr, and F. Pollara "Soft- Output
Decoding Algorithm in Iterative Decoding of Turbo Codes," TDA
Progress Report 42-124, pp. 63-87, February 15, 1996.
[9] J. Hagenauer, and P. Hoeher, "A Viterbi Algorithm with Soft-
Decision Outputs and Its applications, "Proc. Of GLOBECOM, pp.
1680-1686, November 1989.
[10] J. Hagenauer, Source-Controlled Channel Decoding, "IEEE
Transaction on Communications, vol. 43, No. 9, pp.2449-2457,
VI. SIMULATION RESULTS September 1995.
The simulation curves presented shows the influence of [11] J.Hagenauer E.Offer, and L.Papke,”Iterative Decoding of Binary
iteration number, Block length, code rate and code generator. Block and Convolutional Codes,” IEEE Trans.Inform. Theory,
42:429-45, March 1996.
Rate ½ codes are obtained from their rate 1/3 counterparts by
[12] W.J. Gross and P.G. Gulak, "Simplified MAP algorithm suitables for
alternately puncturing the parity bits of the constituent implementation of turbo decoders" Electronic Letters, vol. 34, no.
encoders. For rate R = ½ encoder with Constraint Length 3 16, August 6, 1998.
and generators G1 = 7, G2 = 5. The BER has been computed [13] J. Hagenauer, L. Papke, “Decoding Turbo Codes With the Soft
after each decoding as a function of signal to noise ratio Output Viterbi Algorithm (SOVA),” in Proc. Int. Symp.On
Eb/No. Information Theory, p164, Norway, June 1994.
[14] J. Hagenauer, P. Robertson, L. Papke, “Iterative Decoding of
In figures (3-6) BER for SOVA and LOG MAP as a Systematic Convolutional Codes With the MAP and SOVA
Algorithms,” ITG Conf., Frankfurt,Germany,pp 1-9, Oct. 1994
function of Eb/No curves are shown for constituent codes of
[15] J. Hagenauer, E. Offer, L. Papke, “Iterative Decoding of Bloc and
constraint length three and code rate ½. Eight decoding Convolutional Codes,” IEEE Trans. Infor. Theory, Vol. IT. 42, No.2,
iterations were performed for Block length of 1024 and 4096. pp 429-445, March 1996.
From these figure it can be observed that a large block length
corresponds to a lower BER. Also the improvement achieved
when the block length is increased from 1024 to 4096 for

4. 0 0
10 10
-1
10
1 10
-1
1
10
-2
3 2 10
-2
2
4 3
BER
BER
-3
10
-3 iter 1
-4 iter 1
10
iter 3
4
10
iter 3 iter 6
-4
10 iter 8
-5 iter 6
10
iter 8
-5
-6 10
10 0 0.5 1 1.5 2
0 0.5 1 1.5 2
EbNo (dB)
EbNo (dB) Figure 4, BER of K = 1024 Turbo code with log MAP decoding in
Figure 3, BER of K = 4096 Turbo code with SOVA decoding in AWGN AWGN channel with various number of iteration
channel with various number of iterations. (1) Iteration 1, (2) Iteration 3, (3) Iteration 6, (4) Iteration 8
VII. Iteration 1, (2) Iteration 3, (3) Iteration 6, (4) Iteration 8
0
10
0
10
log map 10
-1
1
-1
10
sova
-2
2
-2 10
10
BER
3
BER
10
-3
2 1 10
-3
4
-4
10 -4 iter 1
10
iter 3
iter 6
-5
10 iter 8
-5
10
0 0.5 1 1.5
-6
10 EbNo (dB)
0 0.5 1 1.5 2
EbNo (dB)
Figure 5, BER of K = 4096 Turbo code of log MAP and SOVA after 8
decoder iteration in AWGN channel.
(1) SOVA 1, (2) LOG-MAP Figure 6, BER of K = 4096 Turbo code with log MAP decoding in
AWGN channel with various number of iteration.
(1) Iteration 1, (2) Iteration 3, (3) Iteration 6, (4) Iteration 8