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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

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

                                                                                      ISSN 1947-5500
                                                 (IJCSIS) International Journal of Computer Science and Information Security,
                                                 Vol. 8, No. 6, September 2010

              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.

         b2                                                               Q(      2R Eb/ No ∑ ρi2 )                             (3)
              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
                                                                   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
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

                                                                                            ISSN 1947-5500
                                          (IJCSIS) International Journal of Computer Science and Information Security,
                                          Vol. 8, No. 6, September 2010

       N               Nw
                ≈                           (6)
       w               w!
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

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
                                                                                  N = 100
     4. SCTC: PERFORMANCE                                                         N = 200
                                                                     1.e-10       N = 300
                                                                              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

                                                                                               ISSN 1947-5500
                                                                     (IJCSIS) International Journal of Computer Science and Information Security,
                                                                     Vol. 8, No. 6, September 2010

R1              R2                 R               Modulation

1/2             1/2                1/2              QPSK

1/3             1/4                2/3              8-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
                                                                    N = 100
                                                                                        aposteriori probability (MAP) algorithm [9]. It
                                                                                        first calculates the extrinsic logliklihood Ratio
                                                                                        (LLR) of the information bits, namely Le(b1)
                                                                                        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

                                                                                        proceeds, until a fixed number of iterations is
       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,
                      8 PSK
                                                                                        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
        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
                                                                                        punctured codes together with higher order
   A functional diagram of the iterative
decoding algorithm for SCTC is presented in
Fig. 5.

                                                                                                                ISSN 1947-5500
                                                        (IJCSIS) International Journal of Computer Science and Information Security,
                                                        Vol. 8, No. 6, September 2010

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

        1.e-11                                                                 on Turbo Codes and Related Topics,
                                                                               Brest, France 1997
                                                                           [5] S. Benedetto, and G. Montorsi,"
        1.e-13                                                                 Unveiling Turbo Codes: Some Results
                     No Iteration
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                     2 Iteration                                               Schemes,"IEEE Transactions on
        1.e-15       3 Iteration                                               Information Theory, Vol. 42, No. 2,
                     4 Iteration                                               March 1996
                                                                           [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
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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,
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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
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     67, Jan. 1982                                                          [10] R. Deshmukh, and S. Ladhake, "
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                                                                                                   ISSN 1947-5500
                                        (IJCSIS) International Journal of Computer Science and Information Security,
                                        Vol. 8, No. 6, September 2010


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.

                                                                                   ISSN 1947-5500

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