Comparative study of turbo codes in AWGN

channel using MAP and SOVA decoding

K. M. S. Soyjaudah and M. T. Russool
Department of Electrical and Electronic Engineering, University of Mauritius, Re duit, Mauritius
E-mail: s.soyjaudah@uom.ac.mu
Abstract A final-year undergraduate project in error-control coding is discussed. The project consists of the study,
software implementation and computer simulation of turbo codes using two different decoding approaches, the
Maximum a Posteriori (MAP) and Soft-Output Viterbi Algorithm (SOVA). It combines theory with computer-aided design
and provides hands-on experience in the field of channel coding.
Keywords MAP and SOVA decoding; turbo coding
A software-based project in error-control coding combines theory and design
with computer simulation. It helps engineering undergraduate students to grasp
the principles of communication systems, the need to add controlled redun-
dancy via a channel encoder and to exploit this redundancy at the receiver
side, in order to recover the transmitted information through a channel decoder.
Further, it helps students to understand Shannons theorem and the concept
of the Shannon limit. This paper presents a project which consists of a study
of turbo codes as an error-control code and the software implementation of
two dierent decoders, namely the Maximum a Posteriori (MAP)1 and Soft-
Output Viterbi Algorithm (SOVA) decoders. A comparison of their perform-
ances is made against that of the classical Viterbi decoder.
Turbo codes were introduced in 1993 by Berrou et al.2 and are perhaps the
most exciting and potentially important development in coding theory in recent
years. They achieve near-Shannon-limit error correction performance with
relatively simple component codes and large interleavers. They can be con-
structed by concatenating at least two component codes in a parallel fashion,
separated by an interleaver. One feature of turbo codes is that the constituent
codes need not be complex. Simple (2,1,4) convolutional codes can achieve
very good results. In order for a concatenated scheme such as a turbo code to
work properly, the decoding algorithm must eect an exchange of soft infor-
mation between component decoders. The concept behind turbo decoding is
to pass soft information from the output of one decoder to the input of the
succeeding one, and to iterate this process several times to produce better
decisions. Turbo codes are still in the process of standardisation but future
applications will include mobile communication systems, deep space communi-
cations, telemetry3 and multimedia. Hence, the understanding of turbo encoding
and decoding and their performance is of utmost importance to engineering
students.
All software implementation of the turbo codecs has been done in C since
it is a exible and structured programming language. Furthermore, undergrad-
International Journal of Electrical Engineering Education 38/2
104 K. M. S. Soyjaudah and M. T. Russool
uate students have already acquired a systematic approach to C programming
earlier in their course.
The paper is partitioned as follows. The next section describes the constituent
encoder, the turbo encoder and the interleaver. The MAP and SOVA decoding
algorithms are then outlined, followed by a discussion of the educational
benets derived from such a project. Typical results obtained are then presented,
followed by a short conclusion.
Turbo code encoder
A turbo encoder is the parallel concatenation of recursive systematic convol-
utional (RSC) codes, separated by an interleaver, as shown in Fig. 1. The data
ow d
k
goes into the rst elementary RSC encoder, and after interleaving, it
feeds a second elementary RSC encoder. The input stream is also systematically
transmitted as X
k
, and the redundancies produced by encoders 1 and 2 are
transmitted as Y
1k
and Y
2k
. For turbo codes, the main reason of using RSC
encoders as constituent encoders instead of the traditional non-recursive non-
systematic convolutional codes, is to use their recursive nature and not the fact
that they are systematic.5
The global rate of the turbo encoder in Fig. 1 is one-third. To achieve a
higher rate, the parity outputs can be punctured. In order to obtain a rate half
encoder, the parity bits at the encoder outputs are selected alternately, i.e., one
bit from encoder 1 and then one bit from encoder 2, as illustrated in Fig. 2.
The interleaver is an important design parameter in a turbo code. It takes a
particular stream at its input and produces a dierent sequence as output. Its
main purpose at the encoder side is to increase the free distance of the turbo
code,5 hence improving its error-correction performance.6
There are dierent types of interleavers, such as the block, pseudo-random,
simile, and odd-even interleavers. They dier in the way they shue the input
symbols. As an example, the block interleaver is explained below. A sequence
of L bits is written into an NM matrix row by row starting from the rst
Encoder 1
Encoder 2
Interleaver
information
bits. d
k
X
k
. svstematic
outputd
k
Y
1k
Y
2k
paritv
outputs
Fig. 1 A fundamental turbo encoder.
International Journal of Electrical Engineering Education 38/2
105 Turbo codes in AWGN channel
Encoder 1
Encoder 2
Interleaver
d
k

X
k
Y
k
Puncturer
Fig. 2 Punctured rate 1/2 turbo encoder.
row of the matrix. Block interleaving then consists of reading the matrix
elements column by column starting from the rst one. The resulting sequence
is written to an array of length L as shown in Fig. 3.
Turbo decoding
A turbo code is far too complex to decode with a single decoder. Instead, each
convolutional code in the turbo code is decoded separately with soft infor-
mation being passed from one decoder to the next. The decoding scheme of a
turbo code is shown in Fig. 4. The above decoder consists of two serially
interconnected soft-in soft-out (SISO) decoders, which can be SOVA or MAP
decoders. x
k
and y
k
are the channel outputs with x
k
corresponding to the
systematic encoder output, and y
k
is a multiplexed stream of the two punctured
encoder outputs. Hence, a demultiplexer is necessary at the receiver. z
k
is called
the a priori information and is equal to zero for the rst iteration.
As described in Ref. 1, the decoder soft output, L (d
k
), also called log-likeli-
hood ratio (LLR), can be separated into three components:
L (d
k
)=L
sys
+L
apr
+L
ext
(1)
d
k
denotes the actual hard decision of the decoder at step k. L
ext
is called the
extrinsic information. It is a function of the redundant information introduced
by the encoder and has the same sign as d
k
.7 (L
sys
+L
apr
) constitutes the
0 1 2 3
4 5 6 7
8 9 10 11
12 13 14 15
Input Sequence 0,1,2,3,.,12,13,14,15
Output Sequence 0,4,8,12,.,3, 7, 11,15
Fig. 3 Block interleaving.
International Journal of Electrical Engineering Education 38/2
106 K. M. S. Soyjaudah and M. T. Russool

LLR1
LLR2
v
2k

v
1k

v
k

x
k

z
k

d
k

Fig. 4 Iterative turbo decoder.
intrinsic information. L
sys
is the LLR for the channel output and is the received
systematic input to the decoder, scaled by the channel reliability. L
apr
is equal
to the extrinsic information produced by the previous decoder.
The intrinsic information is subtracted from the soft decoder output (LLR1,
LLR2). The resulting output is the extrinsic information, which is passed on
to the next decoder. This process is repeated until a desired performance is
attained after a number of iterations.
It is the decoding method that gives turbo codes their name, since the
feedback action of the decoder is reminiscent of a turbo-charged engine.
Principle of the SOVA
The SOVA is a modied Viterbi algorithm which produces soft outputs associ-
ated with the decoded bit sequence. These modications include a modied
metric computation and a reliability value update along the maximum likeli-
hood (ML) path.
Let m denote a trellis branch into a node and M(m)
k
denote the accumulated
metric at time k for branch m. Also, let u(m)
k
be the systematic encoder output
(in bipolar voltage form) for path m and x(m)
k
be the corresponding parity
output (Fig. 5).
If y
k,1
and y
k,2
are the channel outputs corresponding to the systematic and
parity outputs of the RSC encoder, then the metric used for the SOVA algorithm
becomes4
M(m)
k
=M(m)
k1
+u(m)
k
L
c
y
k,1
+x(m)
k
L
c
y
k,2
(2)
RSC
ENCODER
dk
) ( m
k
u
) ( m
k
x
Fig. 5 An RSC encoder.
International Journal of Electrical Engineering Education 38/2
107 Turbo codes in AWGN channel
Time k-1 k k

1
) 1 (
1 k
M
) 2 (
1 k
M
) 1 ( ) 1 (
k k
x u
) 2 ( ) 2 (
k k
x u
) (m
k
M
) (m
k
u
) (m
k
x
: systematic encoder
output for path m
: parity bit
Fig. 6 T rellis section illustrating SOVA metric computation.
Referring to the trellis section in Fig. 6, there are two branches merging into
the central node (i.e. the node at time k). Therefore, m=1, 2. In the rst step,
the metric associated with each trellis node is computed by performing a
forward sweep through the trellis. In addition, the metric dierence (which
represents the reliability value) between the paths merging in any node is
calculated and stored. Then, the ML path is found. Finally, the reliability
update procedure is performed as follows:
Referring to Fig. 7, where the ML path is shown in full lines, the non-
surviving paths obtained by making an initial opposite decision at each stage
along the ML path, are considered one at a time, until they merge with the
ML path. Only one such non-surviving path is shown ( broken lines) in Fig. 6
for the sake of illustration. If the bit on the survivor path at any step is the
same as the associated bit on the competing path, the reliability value at that
bit position remains unchanged. However, if they dier, for example, at times
k0 k1 k2 k3 k4 k5
1
1
-1
1
1 -1
-1
-1
1
Fig. 7 Example of reliability update in SOVA.
International Journal of Electrical Engineering Education 38/2
108 K. M. S. Soyjaudah and M. T. Russool
1 and 4, the reliability values at the corresponding steps needs to be updated.
The soft output is then the nal updated reliablity values of each decoded bit.
Principle of the MAP algorithm
The MAP algorithm proceeds by estimating the log-likelihood ratios (LLR) of
the decoded bits, based on the received symbols. The MAP decoder produces
estimates, d
k
, of the information bits based on the comparison of the LLR,
L (d
k
), to a threshold. For equiprobable binary input values over an AWGN
channel,
d
k
=
G
1 if L (d
k
)0
0 if L (d
k
)<0
(3)
Dening ai
k
(m) and bi
k
(m) as forward and backward state metrics respectively,
where m is a state on the trellis diagram, the LLR is given by8
L (d
k
)=log

m
a1
k
(m)b1
k
(m)

m
a0
k
(m)b0
k
(m)
, where 0m2n1 (4)
ai
k
(m) represents a state likelihood at time k for input bit i, based on a forward
sweep through the trellis, and bi
k
(m) also represents a state likelihood but it is
based on a backward sweep through the trellis.
The expression for ai
k
(m) is
ai
k
(m)=d
i
(R
k
, m)
1
j=0
aj
k1
[Sj
b
(m)] (5)
where Sj
b
(m) is the state going backwards in time which leads to state m with
input j. d
i
(R
k
, m) is the branch metric and depends on the channel. It is given
by
d
i
(R
k
, m) =exp
G
2
s2
[x
k
i+y
k
Y i
k
(m)]
H
(6)
where Y i
k
(m) is the encoder output with input bit i and encoder state m.
Also,
bi
k
(m) =
1
j=0
bj
k+1
[Si
f
(m)]d
j
[R
k+1
, Si
f
(m)] (7)
where Si
f
(m) is the state going forward in time starting from state m with input
bit i.
Using (5) and (7), the state metrics ai
k
(m) and bi
k
(m) can be computed
recursively. On a trellis, d
i
(R
k
, m) corresponds to the branch of the transitions
from time k to k+1, with an initial encoder state m and an information bit i
as shown in Fig. 8.
The forward state metric ai
k
(m) represents the state metric for the transition
from the state m to the next state, at time k and with a transition bit i, as
illustrated in Fig. 9.
International Journal of Electrical Engineering Education 38/2
109 Turbo codes in AWGN channel
00 00 00
01 01 01
10 10 10
11 11 11
) 00 , (
2 0
R
k 2 k 3 k 4
) 01 , (
2 1
R
) 10 , (
2 0
R
) 00 , (
3 1
R
) 11 , (
3 1
R
Fig. 8 T rellis representation of d
i
(R
k
, m).
00 00 00
01 01 01
10 10 10
11 11 11
k 2 k 3 k 4
) 10 (
1
3

) 00 (
0
3

Fig. 9 Trellis representation of ai

k
(m).
For example, a0
3
(00), which corresponds to the a value for path on
the trellis diagram, equals
{a0
2
[S0
b
(00)]+a1
2
[S1
b
(00)]}d
0
(R
3
, 00)

International Journal of Electrical Engineering Education 38/2

110 K. M. S. Soyjaudah and M. T. Russool
a1
3
(10), which corresponds to the value a for path on the trellis
diagram, equals
{a0
2
[S0
b
(10)]+a1
2
[S1
b
(10)]}d
1
(R
3
, 10)
{@{@{@{@{ {{{{{{{
The underline types in the equations above represent the corresponding trellis
path in Fig. 9.
The expression for bi
k
(m) is similar to that for ai
k
(m). However, as compared
to ai
k
(m), the bs are computed backward in time. They represent the value of
b for the transition from state m at time k and with a transition bit i. Again, a
trellis is used (Fig. 10) to concretely represent the backward state metric
bi
k
(m).
For example, b1
2
(01), which corresponds to the b value for path on
the trellis diagram, equals
{b0
3
[S1
f
(01)]d
0
[R
3
, S1
f
(01)]+b1
3
[S1
f
(01)]d
1
[R
3
, S1
f
(01)]}
{@{@{@{@{ {@{@{@{@{@{
b0
2
(11), which corresponds to the b value for path on the trellis
diagram, equals
{b0
3
[S0
f
(11)]d
0
[R
3
, S0
f
(11)]+b1
3
[S0
f
(11)]d
1
[R
3
, S0
f
(11)]}
{@{@{@{@{ {@{@{@{@{@{@{
All the values for ai
k
(m) and bi
k
(m) can be computed from (5) and (7), respect-
ively, on the condition that they are initialised correctly. The MAP algorithm
is as follows:
00 00 00
01 01 01
10 10 10
11 11 11
k 2 k 3 k 4
i

1
i

0
Fig. 10 Trellis representation of bi
k
(m).
International Journal of Electrical Engineering Education 38/2
111 Turbo codes in AWGN channel
Step 1
Starting at time k=0, the branch metrics d
i
(R
k
, m) are calculated and stored
for all received symbols (x
k
, y
k
) using (6).
Step 2
Since the encoder starts in state 00, ai
0
(00) is initialised to d
i
(R
0
, 00) for i=0, 1
and ai
0
(m) =0 for all m00. ai
k
(m) is computed for each k from 1 to N1
using (5).
Step 3
Since the trellis is terminated in the zero state, bi
N1
[Si
b
(00)] is initialised to 1
for i=0, 1 and bi
N1
(m)=0 for all other m00. bi
k
(m) is computed and stored
for each k from N2 to 0 using (7).
Step 4
Finally, using ( 4), L (d
k
) is computed for k=0 to N1 and the estimated bit
d
k
is obtained using (3).
Design and simulation
The aim of this project is to enable the student to have hands-on experience of:
(a) generation of a pseudorandom sequence,
( b) turbo encoding of bits using the parallel concatenated RSC encoders,
(c) generation and addition of AWGN to the encoded bits,
(d) iterative decoding of the corrupted information using SOVA and MAP
algorithms,
(e) computation of the BER at dierent levels of signal-to-noise ratio per
bit, and
(f ) evaluation of the eect of using dierent frame sizes.
In working through the project, the student understands the eect of channel
impairments on the data transmitted and how the errors are reduced through
the use of error-control codes. In addition, the student compares the perform-
ance of these codes using the Viterbi, SOVA and MAP decoding approaches.
Moreover, the eect of increasing the frame size is made clear.
The project is divided into three parts:
(a) the generation of AWGN and evaluation of its eect on the performance
during data transmission using hard decision Viterbi decoding,
( b) study of the eect of employing a convolutional code during data trans-
mission in a noisy channel, and
(c) study of the performance improvement produced by employing turbo
codes during data transmission in a noisy channel.
The three schemes to be implemented are illustrated in Fig. 11. Students are
expected to write programs in C to implement a pseudorandom sequence
International Journal of Electrical Engineering Education 38/2
112 K. M. S. Soyjaudah and M. T. Russool
Pseudo-
Random
Bit Sequence
Hard Decision
Decoder
Sink
Additive
White
Gaussian
Noise
Pseudo-
Random
Bit Sequence
ConvoIutionaI
Encoder
Sink
Additive
White
Gaussian
Noise

Viterbi
Decoder
Pseudo-
Random
Bit Sequence
Turbo
Encoder
Sink
Iterative Turbo
Decoder
(SOVA/MAP)
Additive
White
Gaussian
Noise

(a) Uncoded Hard Decision Decoding

(b) Coded Viterbi Decoding
(c) Iterative Decoding of Turbo Codes
using SOVA or MAP aIgorithms
Fig. 11 Block diagrams of systems to be impemented.
generator, a Gaussian noise generator, a hard decision decoder, a convolutional
code encoder and a Viterbi decoder, a turbo encoder and SOVA and MAP
decoders.
Typical results and evaluation
The variation of the bit error rate with the signal-to-noise ratio per bit has
been measured for the two decoding methods, using dierent frame sizes. Using
bipolar (antipodal ) signalling scheme, 100 000 channel code symbols are
International Journal of Electrical Engineering Education 38/2
113 Turbo codes in AWGN channel
transmitted over a discrete noisy channel in each test. For all cases, perfect bit
and frame synchronisation is assumed. In each test, the average energy per
information bit is xed, and the variance of the AWGN is adjusted for a range
of average bit errors.
Figure 12 shows the result of a simulation carried out to compare the
performance of the Viterbi algorithm, the SOVA and the MAP algorithm using
a 4-state convolutional encoder and a frame size of 64 bits. From Fig. 12, it is
seen that the MAP algorithm gives the best performance in terms of BER
performance, followed by the SOVA and the Viterbi algorithm. At a BER of
104, the MAP algorithm gives a coding gain of 0.4 dB over the Viterbi
algorithm while the performance of the SOVA is similar to that of the Viterbi
algorithm since the only dierence between these two algorithms lies in the
reliability update in the SOVA.
Iterative decoding was performed on a rate half 16-state encoder using a
frame size of 192 bits for both the SOVA and MAP algorithms. The results
are shown in Figs. 13 and 14. It can be seen that as the number of iteration
increases, the error performance for both SOVA and MAP decoders is
improved. This is due to the sharing of the extrinsic information generated by
one decoder with the next one. Hence students understand that this process of
sharing of information enables the decoder to make a more accurate decision.
It can also be observed that the MAP algorithm performs better than the
SOVA decoder.
Figures 15 and 16 show the performance of the SOVA and MAP algorithms
for a frame size of 2048 bits, respectively. The simulations were carried out on
a 16-state punctured rate half turbo code. It can be noted that as the frame
size increases, the error performance of both SOVA and MAP decoders
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
0 1 2 3 4 5 6
Eb/No (dB)
P
r
o
b
a
b
i
I
i
t
y

o
f

B
i
t

E
r
r
o
r
Viterbi
SOVA
MAP
Uncoded
Fig. 12 Comparison of the performance of V iterbi, SOVA and MAP algorithms.
International Journal of Electrical Engineering Education 38/2
114 K. M. S. Soyjaudah and M. T. Russool
Fig. 13 Iterative turbo decoding using the SOVA algorithm.
Fig. 14 Iterative turbo decoding using the MAP algorithm.
improves. After 5 iterations, the coding gain of MAP decoding over SOVA
decoding is about 2 dB when a frame size of 2048 bits is used
Conclusions
A design and simulation project that allows students to acquire hands-on
experience on the development and analysis of a turbo codec in AWGN
channels has been presented in this paper. Being entirely software-based, such
a project provides a exible means for exploring a wide range of schemes
around the designated system. Thus, the eect of varying the frame size on the
error-correcting capability of the code can be readily investigated by modifying
International Journal of Electrical Engineering Education 38/2
115 Turbo codes in AWGN channel
Fig. 15 SOVA decoding using f rame size of 2048 bits.
Fig. 16 MAP decoding using f rame size of 2048 bits.
the software routines. It is also shown that the MAP algorithm gives a better
error performance than the SOVA decoder under similar conditions.
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