Punctured Self-Concatenated Trellis Codes with Iterative Decoding

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(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 8, No. 6, September 2010

Punctured Self-Concatenated Trellis Codes
with Iterative Decoding
Labib Francis Gergis
Misr Academy for Engineering and Technology
Mansoura City, Egypt
drlabeeb@yahoo.com

Abstract-A special concatenated code structure second time after a scrambling of the
called self-concatenated trellis code (SCTC) is information bits.
presented. This scheme based on only one Concatenated trellis codes are classified as
recursive convolutional code(RSC), followed by serially concatenated convolutional codes
a mapping modulator. The union bounds of (SCCC), these codes were analyzed in [4].
SCTC are derived for communications over Using the same ingredients, another type of
Additive White Gaussian Noise (AWGN) and concatenated codes named parallel
Rayleigh fading channels. Asymptotic results concatenated convolutional codes (PCCC), was
for large interleavers are extended to M-ary described in [5]. A third choice is defined as a
bandwidth efficient modulation schemes by hybrid concatenation of convolutional codes
puncturing process. The combination of self- (HCCC) was described in [4] and [6]. Self-
concatenated codes with powerful bandwidth- concatenated convolutional codes proposed in
efficient component codes leads to a [7], [8], and [9] constitute another attractive
straightforward encoder structure, and allows iterative detection aided code-family for their
iterative decoding. The scheme has been low complexity, since they invoke only a single
investigated for 4-PSK, 8-PSK, 16-PSK, and encoder and a single decoder.
16-QAM modulation schemes with varying Puncturing is the process of deleting some
overall bandwidth efficiencies. The choice parity bits from the codeword according to a
based on the rate of RSC and puncturer puncturer code rate. The redundant bits in
encoder component. coding decrease the bandwidth efficiency.
Puncturing increases code rate without
key words ;Self-Concatenated codes, trellis-coded increasing complexity and decreases free
modulation, uniform interleaved coding, distances of code. The advantage of punctured
convolutional coding, iterative decoding codes for binary transmission is that the
encoders and decoders for the entire class of
codes constructed easily by modifying the
1. INTRODUCTION single encoder and decoder for the rate 1/2
binary convolutional code from which the high
Trellis coded modulation (TCM) [1] was rate punctured code was derived [10].
originally proposed for transmission over The construction of self-concatenated trellis
AWGN and fading channels due to its codes (SCTC) is described in section 2. Section
attractive bandwidth efficiency. 3, derives analytical upper bounds to the bit-
Concatenated trellis-coded modulation is an error probability of SCTC using the concept of
alternative to TCM. Different approaches to uniform interleavers. Factors that affect the
concatenated trellis-coded modulations were performance of SCTC are described in section
presented in [2], and [3]. The main principle in 4. Finally results for some examples depicted in
the concatenated coding schemes is to use two section 4, have been stated in section 5.
codes in series (or parallel) joined through one
or more interleavers. This means that the
information sequence is encoded twice, the

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2. SCTC MODEL where Eb/No is the bit energy to noise density
ratio, Acw,h for block code C represents the
number of codewords of the block code with
The basic concept of self-concatenated
output weight h associated with an input
scheme is shown in Figure. 1, the input bit
sequence of weight w, and N is the size of the
sequence {b1} of the self-concatenated encoder
interleaver. The Acw,h is the input-output weight
is interleaved to yield the bit sequence {b2}.
coefficient (IOWC). The function Q (√2R h
After the parallel-to-serial (P/S) conversion, the
Eb/No) represents the pairwise error
information sequence is defined as b(1) = {b1,1
probability which is a monotonic decreasing
b2,1 b1,2 b2,2 …. }. The resultant bit sequences
function of the signal to noise ratio and the
are input to a recursive systematic
output weight h.
convolutional (RSC) encoder. At the output of
For a fading channels, assuming coherent
the encoder the interleaved bit sequence is
detection, and perfect Channel State
punctured. The encoder output is composed of
Information (CSI), the conditional pairwise
the combined systematic bit sequence and
error probability is given by
parity bit sequence.
h

b2 Q( 2R Eb/ No ∑ ρi2 ) (3)
π
i=1
P RSC Puncturer
/ Encoder
The fading samples ρ are independent
S R1=1/2 R2=1/2 c(1) identically distributed (i.i.d.) random variables
b1 b1 b(1) with Rayleigh density of the form
- ρ2
Fig 1. The Self-Concatenated
f(ρ)= 2 ρ e (4)
Code Encoder
The structure of a SCTC, as shown in
The overall code rate, R, can be derived Figure .1, is composed of q-1 interleavers each
based on [9] as: of size N bits, and a single systematic recursive
trellis code C with rate (bq/bq+1),where only
R = R1 / 2 R2 = (1/2) / 2 (1/2) = 1/2 the b + 1 outputs of the encoder are mapped to
(1) 2b+1 modulation levels.
It can be observed that different codes can The average input-output weight
be designed by changing R2. coefficients Acw,h for SCTC with q-1
interleavers can be obtained by averaging
equation (2) over all possible interleavers. A
3. PERFORMANCE OF uniform interleaver is defined as a probabilistic
SELF-CONCATENATED device that maps a given input word of weight
TRELLIS CODES w into all its distinct N permutations with
w
equal probability 1/ N .
Consider a linear block code C with code w
rate R, and minimum distance hm. An upper Thus, the expression for IOWC of SCTC is
bound on the conditional bit-error probability derived as [7]
of the block code C over AWGN channels,
assuming coherent detection, maximum Acw,w,….,w,h
likelihood decoding, can be obtained in the c
A w,h = (5)
form [4] N
N/R N
w
Pb(e/ρ) ≤ ∑ ∑ (w/N) Acw,h ·
h= dmin w=1 where Acw,w,….,w,h is the number of code words of
the trellis encoder of weight h, which is
Q R h (Eb / No) (2) determined in [5], and

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N Nw
≈ (6)
1.e-4
w w!
1.e-5
Substituting equation (6) in equation (2) yields
[3] 1.e-6

Pb (e/ρ ) = Bm N-q+1 Q ( √ 2 R hm Eb/ No ) 1.e-7

BER
(7)
1.e-8
where the constant Bm is independent of N, and
is derived in [8], and hm is the minimum 1.e-9
Euclidean distance of the SCTC scheme . N = 10
N = 50
1.e-10
N = 100
4. SCTC: PERFORMANCE N = 200
1.e-10 N = 300
FACTORS
0 1 2 3 4 5
Eb / No dB
It is shown from equation (7), that there
are many factors that affect the performance of
Fig. 3. Upper Bounds to the Bit Error
SCTC. The most influential parameter is the
Probability for SCTC with QPSK
interleaver size N. The bit error probabilities
using different Interleaver Lengths
for self-concatenated trellis code with overall
rate R=1/2, is shown in Fig. 2, with various
interleaver lengths N= 10, 50, 100, 200, and 300
Applying the upper bound of equation (7),
are plotted versus the signal-to-noise ratio we obtain the results reported in Fig. 3.It is
Eb/No. The systematic and parity bits, bo and b1, also clear from equation (7) that, the minimum
are mapped to 4-ary Phase Shift Keying Euclidean distance of the SCTC code (hm) is an
(QPSK) modulation. The figure shows the another main parameter affecting the
beneficial gain that can be achieved through performance of SCTC. Different values of hm
increasing N. could be obtained by a variety of modulation
schemes. Puncturing is used in order to
increase the achievable bandwidth efficiency.
Different codes could be designed by changing
the rates R1 and R2. The output of the encoder
is then mapped to the Gray-code mapping
function. The various coding schemes
considered in this paper are characterized in
Table 1, that defines both R1,R2, the overall
code rate R, and the associated mapped
modulation scheme to R. The BER versus
Eb/No performance curves of the various
QPSK, 8-PSK, 16-PSK, and 16-QAM are
shown in Fig.4.

Fig. 2. Self-Concatenated Trellis Encoder with
rate R = 1/2

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R1 R2 R Modulation
Scheme

1/2 1/2 1/2 QPSK

1/3 1/4 2/3 8-PSK

16-PSK
1/3 2/3 1/4 16-QAM

Table 1.
Various Modulation Schemes Obtained
from Varying R1 and R2. Fig 5. Self-Concatenated Trellis Decoder

The decoder is a self-concatenated scheme
using a soft-input soft-output (SISO) maximum
1.e-6
N = 100
aposteriori probability (MAP) algorithm [9]. It
first calculates the extrinsic logliklihood Ratio
(LLR) of the information bits, namely Le(b1)
1.e-7
and Le(b2). Then they are appropriately
interleaved to yield the a priori LLRs of the
information bits, namely La(b1) and La(b2), as
1.e-8 shown, in Fig. 5.Self-concatenated decoding
BER

proceeds, until a fixed number of iterations is
reached.
1.e-9 The performance of SCTC with QPSK
16QAM modulation schemes considered are shown in
16 PSK Fig .6. The SCTC has an overall rate R = 1/2,
1.e-10
8 PSK
QPSK
the interleaver length N of this code = 100 bits.
The performance after various numbers of
0 1 2 3 4 5 iteration is shown. It is clear that performance
Eb / No dB improves as the number of decoder iterations
increases.
Fig. 4. Upper Bounds to the Bit Error
Probability for SCTC versus
Different Modulation Schemes
5. CONCLUSIONS
In this paper, a channel coding scheme
The choice of decoding algorithm and
(SCTC) that is bandwidth efficient and allows
number of decoder iterations also influences
iterative decoding of codes built around
performance.
punctured codes together with higher order
A functional diagram of the iterative
signaling.
decoding algorithm for SCTC is presented in
Fig. 5.

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[3] A. Amat, G. Montorsi, and S.
Benedetto, " New High-rate
Convolutional Codes for Concatenated
1.e-6
QPSK Scheme Schemes", Proceeding of IEEE
N = 100
1.e-7 International Conference on
Communication, ICC 2002,Vol. 3, pp.
1.e-8
1661-1666, April 2002.
1.e-9 [4] D. Divsalar, and F. Pollara, " Serial and
Hybrid Concatenated Codes with
1.e-10
Applications,"International Symposium
BER

1.e-11 on Turbo Codes and Related Topics,
Brest, France 1997
1.e-12
[5] S. Benedetto, and G. Montorsi,"
1.e-13 Unveiling Turbo Codes: Some Results
No Iteration
On Parallel Concatenated Coding
1.e-14
2 Iteration Schemes,"IEEE Transactions on
1.e-15 3 Iteration Information Theory, Vol. 42, No. 2,
4 Iteration March 1996
1.e-16
[6] A. Amat, and E. Rosnes," Good
0 1 2 3 4 5
Concatenated Code Ensembles for the
Eb / No dB
Binary Erasure Channel", IEEE
Fig. 6. Upper Bounds to the Bit Error Journal on Selected Areas in
Probability for SCTC versus Communications, Vol. 27, No. 6, pp.
Different Decoding Iterations 928-943, August 2009.
[7] S. Ng, M. Butt, and L. Hanzo, " On the
The SCTC schemes consists of binary RSC Union Bounds of Self-Concatenated
codes and different puncturing rates. The Convolutional Codes", IEEE Signal
puncturer is used to increase the achievable Processing Letters, Vol. 16, No. 16, No.
bandwidth efficiency. A search for good rates 9, September 2009.
was performed, taking into account the [8] S. Benedetto, D. Divsalar, G. Montorsi,
puncturing at the transmitter. It is also and F. Pollara, " Self-Concatenated
demonstrated the significant in the Codes with Self-Iterative Decoding for
performance and the decrease of the bit error Power and Bandwidth Efficiency"
rate and probability of errors to SCTC within International Symposium on
increasing: the interleaver size N, and the Information Theory, ISIT 1998,
number of decoder iterations Cambridge, MA, USA, August 1998.
[9] M. Butt, R. Riaz, S. Ng, and
L. Hanzo, Distributed Self-
REFERENCES Concatenated Codes for Low-
Complexity Power-Efficient
[1] G. Ungerboeck, "Channel coding Cooperative Communication", IEEE
with multilevel phase signaling," VTC 2009, Anchorage, Alasks, USA,
IEEE Trans. Inf. Th., Vol. 25, pp. 55- 2009.
67, Jan. 1982 [10] R. Deshmukh, and S. Ladhake, "
[2] F. Brannstrom, A. Amat, and L. Analysis of Various Puncturing
Rasmussen, " A General Structure for Patterns and Code Rates: Turbo
Rate-Compatible Concatenated Code", International Journal of
Codes," IEEE Trans on Communication Electronic Engineering Research
Letters, Vol. 11, pp. 437-439, May 2007. Volume 1, Number 2, pp.79-88, India,
2009.

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AUTHOR PROFILE

Labib F.Gergis received the Bsc, Msc, and
Ph.D from faculty of engineering, Mansoura
University, Egypt, in 1980, 1990, and 2000,
respectively. He is presently in Misr Academy
for Engineering and Technology, Egypt. His
areas of interest include digital
communications, Coding, and Multiple Access.

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