MAP Decoding Algorithm for Extended Turbo
Product Codes over Flat Fading Channel
Changlong Xu, Ying-Chang Liang, and Wing Seng Leon
Institute for Infocomm Research
21 Heng Mui Keng Terrace, Singapore 119613.
Abstract— In this paper, an efficient MAP iterative decoding
algorithm is proposed for extended turbo product codes over flat
fading channel. The overflow avoidance of extrinsic information
and the stop criteria of iterative decoder are also discussed.
Simulation results are presented to verify the performance and
show the efficacy of the proposed decoding algorithm.
be x = (x0 , . . . , xi , . . . , xN ) with xi ∈ {−1, 1} by mapping
the bit 0 to symbol 1 and bit 1 to symbol −1.
After transmission over a flat fading channel, the received
signal y = (y0 , . . . , yi , . . . , yN ) may be expressed as
I. I NTRODUCTION
where α = diag (α0 , . . . , αi , . . . , αN ) is a (N +1)×(N +1)
diagonal matrix with αi , the unit-variance i.i.d. fading coefficient, and g = (g0 , . . . , gi , . . . , gN ) is the zero-mean AWGN
vector, with each of its elements having variance N0 .
If the receiver has perfect knowledge of the fading channel
or channel state information (CSI), then coherent detection
may be performed by eliminating the phase rotations caused by
the channel. Thus, the phase recovered signal may be written
as
r = yαH
(2)
Because of low complexity and high data rate, turbo product
codes (TPC) have already been adopted in many wireless standards, such as IEEE 802.16, satellite communication systems
and digital storage systems. Extended TPC can be obtained by
appending an additional column and one row of single parity
check bits to conventional TPC. The minimum Hamming
distance of extended TPC is much larger than that of original
TPC, thus, almost all the TPC used in practical systems
are extended TPC. In 1994, Pyndiah proposed an iterative
decoding algorithm based on the Chase algorithm for TPC
[1]. However, the reliability factors and weighting factors in
the algorithm are optimized by experiment. It is not convenient
to implement the algorithm since the values of two factors
vary with the sizes of TPC and the number of iterations.
Several iterative decoding algorithms are presented for twodimensional binary block and convolutional codes in [2]. But
the structure of the two dimensional binary block codes is not
same as that of TPC defined in IEEE 802.16, and the extended
block codes are not considered either. In this paper, we propose
a maximum a posteriori probability (MAP) decoding algorithm
for extended TPC over flat fading channel. We also investigate
the effects of the extrinsic information having excessive large
magnitudes or overflowing, and the stopping criteria for the
iterative decoder. The performance comparison between the
MAP and the Chase algorithms are presented by computer
simulations.
y = xα + g
The output of the MAP decoder is defined as the a posteriori
log-likelihood ratio for a transmitted “+1” and a transmitted
“-1” in the information sequence
L(x̂k )
=
log
x∈C,xk =+1
x∈C,xk =−1
=
log
P (x|r)
P (x|r)
N
x∈C,xk =+1 j=0
N
x∈C,xk =−1 j=0
=
log
P (xj |rj )
P (xj |rj )
P (xk = +1)
P (rk |xk = +1)
+ log
P (xk = −1)
P (rk |xk = −1)
N
P (xj |rj )
+ log
II. MAP D ECODING A LGORITHM FOR EXTENDED TPC
OVER FLAT FADING C HANNELS
x∈C,xk =+1 j=0,j=k
N
x∈C,xk =−1 j=0,j=k
The encoding and decoding process of TPC are the same
for all columns and all rows. For simplicity and brevity, we
only need to consider the encoding and decoding algorithm
for any one row. We assume that the source information is
an extended binary linear block codeword in C(N, K) and
expressed as c = (c0 , . . . , ci , . . . , cN ) with ci ∈ {0, 1}, and
cN is the extended single parity check bit. Thus, the transmit
signal with binary phase-shift keying (BPSK) modulation may
1­4244­0785­0/06/$20.00
(1)
P (xj |rj )
= L(xk ) + Lc rk + Le (xk )
(3)
where L(xk ) and Le (xk ) are a priori information and an
extrinsic information of bit xk , respectively. Lc = 4αk Es/N 0
is the reliability value of the channel.
To calculate the extrinsic information, we define the following log-likelihood ratio
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L(xi |ri ) = log
P (xi = +1|ri )
= L(xk ) + Lc rk .
P (xi = −1|ri )
(4)
Using the defined log-likelihood, the a priori probability
P (xi |ri ) can be expressed as
e±L(xj |rj )
1 + e±L(xj |rj )
−L(xj |rj )/2 e
eL(xj |rj )xj /2 . (5)
=
1 + e−L(xj |rj )
=
So the extrinsic information can be rewritten as
Q Q P
Q
P
Q
N
x∈C,xk =+1 j=0,j=k
N
x∈C,xk =−1 j=0,j=k
−L(xj |rj )/2
−L(xj |rj )
1+e
e
e
−L(xj |rj )/2
−L(xj |rj )
1+e
N
= log
e
x∈C,xk =+1 j=0,j=k
N
x∈C,xk =−1 j=0,j=k
L(xj |rj )xj /2
eL(xj |rj )xj /2
eL(xj |rj )xj /2
eL(xj |rj )xj /2
−1
.
If N − K > K, the computing complexity in Equation (6) can
be reduced by using the dual code C of the extended code C
[3]. In this way, the Equation (3) can be expressed as
L(x̂k ) = L(xk ) + Lc rk +
1+
log
1−
N
i=2
j=0,j=k
i=2
−2
10
−3
10
(7)
2N +1−K
2N +1−K
10
(6)
BER
Le (xk ) = log
P
P
We use Monte Carlo simulations are used to evaluate the
effectiveness of the proposed MAP decoding scheme for
extended TPC. The square extended TPCs are constructed by
using extended Hamming codes (8, 4, 4), (16, 11, 4),(32, 26, 4)
and (64, 57, 4) as the component codes. Perfect CSI is assumed and the number of iterations is set to 4. We also included the simulation results of the Chase decoding algorithm
for extended TPC in [4] for comparison purposes. The number
of the test patterns is 16 and the value of the scaling factor α
is fixed as 0.5 for all the iterations. The modulation used is
QPSK.
(−xik )
tanh
N
j=0,j=k
L(xj |rj )
2
tanh
(1−xij )
2
L(xj |rj )
2
−4
10
(1−x )
ij
2
−2
(8,4) CHS
(8,4) MAP
(16,11) CHS
(16,11) MAP
(32,26) CHS
(32,26) MAP
(64,57) CHS
(64,57) MAP
−1
0
1
2
3
4
5
SNR(dB)
where x represents the codewords of the dual code C , and
the index i = 1 denotes the all “+1” codeword.
It is easy to realize the iterative decoder for extended TPC
after implementing the soft-input and soft-output decoder of
the component codes. However, the overflow problem of extrinsic information and the correct stop criteria may affect the
decoding performance significantly. We suggest the following
solutions to these problems:
1) Overflow Avoidance
The magnitude of extrinsic information may become
excessively large and overflow with the increased number of iterations, especially for high signal to noise
ratio (SNR). It will consequently degrade the decoding
performance. To circumvent the overflow, we set z =
0.9999z, when z = tanh(L(xj |rj )/2) > 0.9999.
2) Stopping Criteria
The assumption that the received signal and the extrinsic
information are statistical independent may not be valid
with increasing number of iterations. This will cause
performance degradation, if the decoding procedure is
not terminated at an appropriate time. In this paper,
we use the syndrome values of the of the component
codes as an indication to terminate the decoding process.
That is, if the values of the syndrome are all zeros, the
decoder will, otherwise, the iterative decoding procedure
will continue.
Fig. 1. The TPC performance comparison between the Chase and MAP
decoding algorithm with QPSK under AWGN channel.
−1
10
−2
10
BER
P (xj = ±1|rj )
III. S IMULATION R ESULTS
−3
10
−4
10
−5
10
0
(8,4) CHS
(8,4) MAP
(16,11) CHS
(16,11) MAP
(32,26) CHS
(32,26) MAP
(64,57) CHS
(64,57) MAP
2
4
6
SNR(dB)
8
10
12
Fig. 2. The TPC performance comparison between the Chase and MAP
decoding algorithm with QPSK under flat fading channel.
The performances of the Chase and proposed MAP decoding algorithms for extended TPC transmitted over AWGN
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channel is shown in Figure 1. Signal-to-noise ratio (SNR)
is defined as the ratio of symbol energy to noise variance.
The results show that the TPC performance is degraded with
increased code rate for both the proposed MAP and Chase
decoding algorithms. For square TPC constructed by extended
Hamming codes (8, 4, 4) and (16, 11, 4), the performances
of Chase algorithm are better than that of MAP algorithm.
However, the situation is reversed for square TPC constructed
by extended Hamming codes (32, 26, 4) and (64, 57, 4). The
reason is that the Chase decoding algorithm with 16 test patterns is more efficient for smaller sizes of extended Hamming
codes. Figure 2 displays performances of the proposed MAP
and Chase algorithm to decode extended TPC transmitted over
a flat fading channel. It is also observed that the performance
of MAP algorithm is superior to the Chase algorithm for
extended TPC with high code rates.
R EFERENCES
[1] R. Pyndiah, A. Glavieux, A. Picart, and S. Jacq, “Near optimum decoding
of product codes,” in Proc.of GLOBECOM, vol. 1, San Francisco, CA,
Nov. 1994, pp. 339–343.
[2] J. Hagenauer, E. offer, and L. Papke, “iterative decoding of binary block
and convolutional codes,” IEEE Trans. Inform. Theory, Vol. 42, pp. 429445, March 1996.
[3] G. Battail, M. C. Decouvelaere, and P. Goldlewski, “Replication decoding,” IEEE Trans. Inform. Theory, vol. IT-25, pp.332-345, May 1979.
[4] Changlong Xu, Wing Seng Leon and Ying-Chang Liang, “Chase decoding
algorithm for shortened turbo product codes in flat fading channels,” in
Radio and Wireless Symposium, 2006 IEEE, San Diego, CA, Juan. 2006,
pp. 15 - 18
IV. C ONCLUSIONS
In this paper, we propose an efficient MAP iterative decoding algorithm for extended turbo product codes over flat
fading channel. The performance comparison between the
proposed MAP algorithm and Chase algorithm is investigated
by simulation results. They show that the performance of MAP
algorithm is significantly better than that of Chase algorithm
for extended TPC with high code rates.
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