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Investigation of the effect of LDPC coding on the sparseness of a

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					      Investigation of the effect of LDPC coding on the
  sparseness of a data source in AWGN channel conditions
                                                    J. Schoeman† and L.P. Linde‡
                 Department of Electrical, Electronic and Computer Engineering, University of Pretoria, Pretoria, 0002, South Africa
                                               Email: † jschoeman@up.ac.za, ‡ llinde@postino.up.ac.za


   Abstract— This paper implements a measure for sparseness in a
simple antipodal modulation system employing LDPC coding and a                                              p(ai )
message passing algorithm (MPA) based decoder in AWGN channel
                                                                                                ηs =       n−1            = p(ai )      (1)
conditions. The authors proceed to show the effect of the LDPC en-                                         k=0   p(ak )
coder on the statistics and CDF of the source, as well as the PDF of        with ai the symbol that is least likely to occur. This does not
the received data. Comparative results for various levels of sparseness
and average signal to noise ratio are presented, as well as simulation
                                                                            account for the source alphabet, bh . Lets define the prob-
results showing that more detailed source analysis should be made be-       ability of alphabet letter bh in symbol ai as P rob[bhi ] =
                                                                                                                                     a
fore the assumption of equiprobable input symbols can be justified.          p(bhi ). The measure for sparseness on a letter level, rather
                                                                               a
Also presented is simulation results of bit error performance for the       than a symbol level, can now be given (when weighted with
coded system, yielding a Pe = 9.8 × 10−5 at Eb /N0 = 5 dB for the
(10, 5, 0.5) MPA decoder with 10 iterations, as well as a additional 0.1    p(ai ) and bh is least likely to occur) as
dB gain when the number of iterations are increased to 100.
                                                                                                           n−1          h
                                                                                                           k=0 p(ak )p(bak )
                                                                                             ηα =      m−1    n−1            j
                                                                                                                                        (2)
                        I. I NTRODUCTION                                                               j=0    k=0 p(ak )p(bak )

                                                                                      m−1      n−1
Recent years have seen an incredible flourish in the fields of                with      j=0      k=0   p(ak )p(bj k ) = 1.
                                                                                                              a
lossy data compression techniques for multimedia types of
data (like images, video, audio and voice) that is specifically              This letter level (or bit level, if m = 2 for a binary source)
matched for such a source. However, given a scenario where                  measure is more practical, considering detectors will gener-
the average code length of the compressed data stream is                    ally receive and process data on a chip, sample or bit level,
not matched to the entropy of the source, it is regularly                   and not initially on a symbol level. This is true especially in
found that the data stream transmitted to the channel appear                circumstances where data is fragmented into frames in sys-
sparse. This leads to non-optimal detection of the received                 tems employing time division multiplexing. It is also im-
signal, given that the regular assumption of equiprobable                   portant to note that an infinite blocklength is not a practical
inputs to the channel is not strictly true. It has been shown               consideration, but if the blocklength chosen, l, is sufficiently
recently that the decision region of the detected sparse sig-               large, ηα can be accurately approximated within the frames.
nal can be adjusted to yield improved results, given that a
measure exists to quantify the sparseness of a data stream                              III. T HE LDPC       CODER AND DECODER
[1]. The derivation of this measure is revisited in Section II.             A linear LDPC error correcting code is best described by an
                                                                            (N, K) binary generator matrix, GT . Assuming that the a-
Once the sparseness of a data stream can be reliably quan-                  priori probability of s is uniformly distributed and indepen-
tified, we proceed to present a low density parity check                     dent of the probability of the noise vector, n, it is convenient
(LDPC) channel coding setup in Section III. These codes                     to define the (N − K, N ) parity check matrix, H, such that
have undergone a rebirth since their discovery in the 1960’s                HGT = 0. Encoding a K-bit input vector, s, using either
by Gallager [2] and they have been extensively researched                   GT or H, yields an N -bit vector given by x = GT · s. All
for the last decade as an alternative to the very powerful                  arithmetic operations are restricted to GF(2). System per-
turbo codes [3][4].                                                         formance is increased by allowing N → ∞.
Finally, the simulation setup and comparative results are                   Optimal decoding is performed when maximizing the pos-
presented and discussed in Section IV, concluding remarks                   terior probability using Bayes’ theorem as
are presented in Section V and some acknowledgements are
made in Section VI.                                                                                              P (r|s, G) P (s)
                                                                                              P (s|r, G) =                              (3)
                                                                                                                     P (r|G)
              II. A   MEASURE FOR SPARSENESS
                                                                            The received vector, r, can be described as r = x + n,
Consider a source with n symbols ai = {a0 , a1 , ..., an−1 },               and n a zero mean Gaussian process with variance, σn . A
with ai constructed from any of m possible alphabet letters                 probability of receiving a 1, p1 is given by the distribution
                                                                                                            l
                                                                                         1                  0        1
bh = {b0 , b1 , ..., bm−1 }. Each symbol, ai , has a probability            f (n) =     −2r/σn2 , yielding pl = 1 − pl .
                                                                                       1+e
of occurrence given as P rob[ai ] = p(ai ). A measure of
sparseness for a blocklength l → ∞ can then be defined                       The decoding of Eq. (3) can be accurately approximated
as the ratio of p(ai ) and p(a0 ), ..., p(an−1 ), with p(ai ) <             by implementing a sum-product algorithm, also referred
p(aj ) for j [0, n − 1], but j = i. This yields a measure for               to as the message-passing algorithm decoder. The imple-
sparseness for symbols, given simply by                                     mented 3 step decoding algorithm is described in detail in
[4], with the main emphasis of this paper only on initializ-                                                                            ηα              ηα                        ηα cd
      0     1
ing {qml , qml } to {p0 , p1 }, after which a horizontal step is
                      l    l
                                                                                                                                        0.1          0.09900                     0.17000
performed to determine                                                                                                                  0.2          0.19937                     0.29032
                                                                  x
                                                                                                                                        0.3          0.29974                     0.37573
   0
  rml =                P (zm |xl = 0, {xl : l ∈ χ})              qml
                                                                   l
                                                                                                                                        0.4          0.40115                     0.44466
          {xl :l ∈χ}                                      l ∈χ
                                                                                                                                        0.5          0.49931                     0.49949
   1                                                              x
  rml =                P (zm |xl = 1, {xl : l ∈ χ})              qml
                                                                   l
                                                                                                                                         Probability Density Function of Sparse Data Source (LDPC Coding)
          {xl :l ∈χ}                                      l ∈χ
                                                                                                                         0.5
                                                        (4)
                                                                                                                        0.45                                                                        E /N =0 dB
with χ = L (m) \l. Finally, a vertical step is performed by                                                                                                                                             b
                                                                                                                                                                                                    η =0.1
                                                                                                                                                                                                        α
                                                                                                                                                                                                            0



              0     1                                                                                                    0.4                                                                        ηα cd =0.17
determining {qml , qml } for each value of l as                                                                                                                                                     rd =0.39641




                                                                                    Probability Density Distribution
                                                                                                                        0.35

                 0
                qml    =   αml p0
                                l
                                                    0
                                                   rm l                                                                  0.3

                                      m ∈M (l)\m
                                                                       (5)                                              0.25
                 1
                qml = αml p1
                           l
                                                    1
                                                   rm l                                                                  0.2
                                      m ∈M (l)\m
                                                                                                                        0.15


                              0       1
with αml scaled such that qml + qml = 1. At this stage the                                                               0.1

                           0 1
posterior probabilities, {ql , ql }, can be approximated with                                                           0.05


                                                                                                                              0
                                                                                                                              −4   −3          −2          −1             0             1       2                   3      4
                        0
                       ql = α l p 0
                                  l
                                                 0
                                                rml                                                                                                             Input [normalized Eb]

                                      m∈M (l)
                                                                       (6)
                        1
                       ql = α l p 1
                                  l
                                                 1
                                                rml                           Fig. 1. PDF for an LDPC coded system with ηα = 0.1, Eb /N0 = 0dB
                                      m∈M (l)


The algorithm now repeats from the horizontal step and will                                                                              Probability Density Function of Sparse Data Source (LDPC Coding)



continue to do so until either a valid codeword is determined                                                           0.35
                                                                                                                                                                                                    Eb/N0 =0 dB
or the maximum number of iterations are exceeded.                                                                                                                                                   ηα =0.3
                                                                                                                         0.3                                                                        η           =0.37573
                                                                                                                                                                                                        α cd
                                                                                                                                                                                                    rd =0.12692
                                                                                    Probability Density Distribution




                                                                                                                        0.25
                        IV. S IMULATIONS
                                                                                                                         0.2

A. Simulation Platform
                                                                                                                        0.15


A single simulation platform was implemented. This test                                                                  0.1

platform was configured for BPSK modulation with rectan-
gular pulse shaping and a symbol rate of 1000 symbols/s.                                                                0.05



The platform does not support DS/SSMA CDMA capabili-                                                                          0
                                                                                                                              −4   −3          −2          −1             0             1       2                   3      4
ties. A sparse data source is used with adjustable levels of                                                                                                    Input [normalized E ]
                                                                                                                                                                                   b


sparseness. No source coding is performed to maximize the
average code length and entropy ratio. Channel coding is                      Fig. 2. PDF for an LDPC coded system with ηα = 0.3, Eb /N0 = 0dB
performed by employing an LDPC encoder and an optimal
message passing algorithm decoder, as described in Section                                                                              Probability Density Function of Sparse Data Source (LDPC Coding)

III. Traditionally, N would be allowed to be very big. How-                                                             0.9



ever, the simulation platform was restricted to a very small                                                            0.8                                                                         E /N =5 dB
                                                                                                                                                                                                        b   0
                                                                                                                                                                                                    η =0.1
(10, 5, R = 0.5) code, with parity check matrix given by                                                                0.7                                                                         η
                                                                                                                                                                                                     α

                                                                                                                                                                                                     α cd
                                                                                                                                                                                                               =0.16989

                                                                                                                                                                                                  rd =0.12542
                                                                                     Probability Density Distribution




                1 1 0 1 1 1 0 1 1 0                                                                                     0.6


              1 1 1 0 1 0 1 1 0 1                                                                                     0.5
                                                   
       H =  0 0 1 0 0 1 1 1 1 0  (7)
                                                                                                                      0.4

              1 0 0 1 1 1 1 0 0 1 
                                                                                                                        0.3

                0 1 1 1 0 0 0 0 1 1
                                                                                                                        0.2



The main motivation behind this restriction was to present                                                              0.1


results close to the worst case scenario. This is so that it can                                                         0
                                                                                                                         −4        −3         −2          −1             0              1      2                    3      4

be applied directly to 3G/4G communication systems as an                                                                                                        Input [normalized E ]
                                                                                                                                                                                   b


upper bound on bit error rate performance.
                                                                              Fig. 3. PDF for an LDPC coded system with ηα = 0.1, Eb /N0 = 5dB

B. Simulation Results                                                        From these figures and the table, a number of important ob-
                                                                             servations can be made: (1) It is clear that the sparseness
Fig. 1 through Fig. 5 show the simulated results obtained                    of the source, ηα , is not the same as the sparseness of the
from the test platform, while the table below presents some                  LDPC output, ηα cd . (2) The approximate measured sparse-
numerical data returned by the test platform.                                ness values, ηα , closely resemble the theoretical values, ηα .
                                                                                              Probability Density Function of Sparse Data Source (LDPC Coding)
                                                                                                                                                                                                            V. C ONCLUSION
                                                                              0.6                                                                        E /N =5 dB
                                                                                                                                                             b   0
                                                                                                                                                                                   This paper investigated the effect on sparseness as defined
                                                                                                                                                         ηα =0.3
                                                                                                                                                         η       =0.3757
                                                                                                                                                                                   in Section II introduced by a linear LDPC encoder. It has
                                                                                                                                                          α cd


                                           Probability Density Distribution
                                                                              0.5                                                                        r =0.040148
                                                                                                                                                         d                         been shown via simulation that the coded sparseness does
                                                                              0.4
                                                                                                                                                                                   in fact differ from the sparseness of the source and that the
                                                                                                                                                                                   coded sparseness tends to be less sparse than the sparseness
                                                                              0.3                                                                                                  of the original data source. It has also been shown that a
                                                                                                                                                                                   (10, 5, R = 0.5) LDPC code can be implemented, but that
                                                                              0.2
                                                                                                                                                                                   the BER performance can only be described as conserva-
                                                                              0.1
                                                                                                                                                                                   tive, with Pe = 9.8 × 10−5 at Eb /N0 = 5 dB. This is still
                                                                                                                                                                                   an incredible result, given that practical blocklenghts of 10
                                                                               0
                                                                               −4        −3         −2          −1            0              1       2               3     4
                                                                                                                                                                                   bits are uncommon. The implemented MPA algorithm has
                                                                                                                     Input [normalized E ]
                                                                                                                                        b                                          been shown to be very efficient when implemented with 10
                                                                                                                                                                                   iterations or more. There is, however, room for optimiza-
 Fig. 4. PDF for an LDPC coded system with ηα = 0.3, Eb /N0 = 5dB
                                                                                                                                                                                   tion and simplification, which will be considered in a later
                                                                                                                                                                                   research effot.
(3) It is clear from the measured coded sparseness results                                                                                                                                           VI. ACKNOWLEDGEMENTS
for Eq. (7) that the coded sparseness is monotonic over
0 < ηα < 0.5. (4) It is clear that the general assumption                                                                                                                          The authors wish to thank Intel for its donation towards the
of selecting the decision region as rd = 0 will not yield op-                                                                                                                      development of the Ipercube distributed computing system
timal results. Optimal rd values are given in the figures. (5)                                                                                                                      at the University of Pretoria, as well as P. Greeff for his
Various bit error performances from [2] and [4] show that                                                                                                                          invaluable knowledge and assistance with the Ipercube.
a large blocksize provides for very powerful channel cod-                                                                                                                                                     R EFERENCES
ing. It is clear that the much smaller (10, 5, R = 0.5) code
                                                                                                                                                                                   [1] J. Schoeman and L. Linde, “Performance investigation of a sparse
is outperformed by it’s (1008, 504, R = 0.5) counterparts,                                                                                                                             data compression technique with awgn channel effects.” Submitted for
but that it still provides for adequate error protection at                                                                                                                            IEEE Africon 2004, 2004.
Pe = 9.8×10−5 at Eb /N0 = 5 dB. (6) The (10, 5, R = 0.5)                                                                                                                           [2] R. Gallager, Low-density Parity-Check Codes. PhD thesis, Cambridge,
                                                                                                                                                                                       1963.
code is suitable for error correction in applications where                                                                                                                        [3] D. J. C. MacKay, “Information theory, inference and learning algo-
a small blocksize is required, and will yield improved re-                                                                                                                             rithms.” Textbook in preparation, 1997.
sults (although still not as good as the (1008, 504, R = 0.5)                                                                                                                      [4] D. J. C. MacKay, “Good error correcting codes based on very sparse
                                                                                                                                                                                       matrices.” Submitted to IEEE transactions on Information Theory.
codes) if the codelength is slightly increased to support mul-                                                                                                                         Available from http://wol.ra.phy.cam.ac.uk/, 1997.
timedia and IP applications. (7) It is clear that by increasing
the number of iterations of the MPA decoder, the bit error                                                                                                                                         Johan Schoeman holds a B.Eng (2001)
performance is increased. (8) The 0.1 dB gain obtained by                                                                                                                                          and B.Eng Hons. (2002) in Electronic
increasing the number of iterations from 10 to 100 does not                                                                                                                                        Engineering from the University of Preto-
seem justified.                                                                                                                                                                                     ria. At present, he is studying towards an
                                                                                                                                                                                                   MEng degree in Electronic Engineering
                                    BER Performance for a LDPC coded system with small block size using MPA decoding
                                      0                                                                                                                                                            and is a full time lecturer in the Depart-
                                    10
                                                                                                                                                                                                   ment of Electrical, Electronic and Com-
                                                                                                                                                                                                   puter Engineering (E,E&C Eng), Faculty
                                      −1
                                                                                                                                                                                                   of Engineering, University of Pretoria.
                                    10
                                                                                                                                                                                                   His research interests are in SWR devel-
                                                                                                                                                                                   opment, source and statistical channel coding techniques
  Simulated Bit Error Probability




                                                                                                                                                                                   and bandwidth efficient modulation techniques applicable
                                    10
                                      −2
                                                                                                                                                                                   for rural WCDMA 3G/4G systems.

                                                                                                                                                                                                     Louis P. Linde holds a B.Eng Hons (1973)
                                    10
                                      −3
                                                                                                                                                                                                     degree in Electrotechnical Engineering from
                                                                                                                                                                                                     the University of Stellenbosch and M.Eng
                                                                                                                                                                                                     (1980) and D.Eng (1983) degrees in Elec-
                                    10
                                      −4
                                                                                    Uncoded (Theory)
                                                                                    Coded (Theory), R = 1/2
                                                                                                                                                                                                     tronic Engineering from the University
                                                                                    Gallager (1008, 504, 1/2)
                                                                                    McKay MPA (1008, 504, 1/2)
                                                                                                                                                                                                     of Pretoria. He is presently the Group
                                                                                    MPA (10, 5, 1/2), 1 Iteration(s)                                                                                 Head of Signal Processing and Telecom-
                                                                                    MPA (10, 5, 1/2), 10 Iteration(s)
                                      −5
                                                                                    MPA (10, 5, 1/2), 100 Iteration(s)                                                                               munications in the Department of Electri-
                                    10
                                                                               −4             −2               0          2                      4                   6         8                     cal, Electronic and Computer Engineering
                                                                                                               Average Eb/N0 [dB]
                                                                                                                                                                                   (E,E&C Eng), Faculty of Engineering, University of Preto-
                                                                                                                                                                                   ria, as well as Director of both the Centre for Radio and
Fig. 5. Comparitive BER plot for the LDPC coded system with various
iterations                                                                                                                                                                         Digital Communication (CRDC) and the DigiMod Group
                                                                                                                                                                                   in RE at UP.

				
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