The performance of certain binary block codes with soft-decision decoding is evaluated by simulation. A construction is proposed to introduce memory on block codes, and simulation results are shown for a trellis code derived from the Nordstrom-Robinson code

Pollara, F.

English

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TDA Progress Report 42-90
ORIGINAL PAGE IS
OF POOR QUALITY
N87-28786
Aprii-Jun e
Soft-Decision Decoding of Some Block Codes
F. Pollara
Communications Systems Research Section
The performance of certain binary block codes with soft-decision decoding is evalu-
ated by simulation. A construction is proposed to introduce memory on block codes, and
simulation results are shown for a trellis code derived from the Nordstrom-Robinson
code.
I. Introduction
Recent technological advances make soft-decision decoders,
for moderate length block codes, a practical alternative to con-
volutionally coded systems. In particular, in order to improve
the performance of current concatenated coding systems used
in deep space communication, it is interesting to find the per-
formance of some block codes, which might be used as inner
codes with soft-decision decoders. Soft-decision decoding is
mandatory for inner codes to avoid a penalty of approxi-
mately 2 dB incurred by hard quantization of channel outputs.
A well-known upper bound on the word error probability
Pw of an (n, k) linear binary block code with soft-decision
decoder is given by [1]
M-1
1
P _ 2 2 erfc(x/RWmEb/No) (1)
m=l
where M = 2k, R = k/n, Ez,/N O is the bit signal-to-noise ratio,
and w m is the weight of the ruth code word. This bound is
tight for large E b/No, and the coding gain is proportional to
the product Rdmin, where dmi n is the minimum distance of
the block code.
Unfortunately, Eq. (1), which is based on the union b_
yields a very loose performance estimate at low E b/N
therefore resort to simulation techniques to find the trm
decision, maximum likelihood performance of some sel
binary block codes.
II. Soft-Decision Decoding
We assume a channel with no output quantization.
fore, each binary waveform is obtained by the op
demodulator (a matched filter followed by a sampler), i
code word is represented by a sequence of n random vari
Let E denote the energy of one of these n binary
forms. Then each binary decision variable can be writte
to an irrelevant constant, as
= _ n/E-+ n i, ifithbit = 1
r.
.-_/E-+ ni, if ith bit = 0
and i = 0, 1 ..... n - 1, where the variables ni are saml
additive white Gaussian noise (AWGN) with zero mea
variance N O/2.
76
https://ntrs.nasa.gov/search.jsp?R=19870019353 2020-03-20T10:15:31+00:00Z
From knowledge of the M = 2 k code words, and upon
reception of the sequence ro, rl,..., rn_l, the decoder com-
putes M squared Euclidean distances
n-I
d: =
/=0
i=0,1 ..... M-1
where xij = -+1corresponds to the j th bit of the i th code word.
The decoded code word is the one corresponding to the mini-
mum distance.
III. Simulation Results for Various
Block Codes
A few rate 1/2 binary block codes are considered:
(a) The cxtcnded Golay (24, 12) code with dmin = 8.
(b) The first-order Reed-Muller (8, 4) code with dmi n = 4.
_1 _-, ,r 1 , lr-,_1_: ...... E'er "r_'_(c) lne nonlinear (16,_ l_om_tlom-txuum_u, tl_-lxy
code [2] with dmi n = 6.
The word error probability Pw of these codes, obtained by
simulation, is shown in Fig. 1, in the range of interest for use
as inner codes. The results are also compared to the perfor-
mance of the NASA (7, 1/2) convolutional code, assuming a
t word length of 8 bits. Care must be exercised in comparing the
results, due to different word lengths.
Figure 2 shows the performance of the first-order Reed-
Muller (16, 5) code and the (7, 1/3) convolutional code with
8-bit words. Figure 3 shows a similar comparison between the
first-order Reed-Muller (32,6) code and the (14, 1/4) and
(14, 1/55 convolutional codes with 10-bit words.
The bit error probabilities of two of the codes considered in
Fig. 1, the Golay (24, 12) and the NASA (7, 1/2) codes, are
compared in Fig. 4.
In Fig. 1 it appears that the NASA (7, 1/2) code is still the
best, except at very low E b/No, and that longer block codes,
longer than those considered here, are necessary to compare
favorably. Nevertheless, the simulation of the Nordstrom-
Robinson code gave rise to a new approach.
Instead of pursuing the path of longer codes we devised
some simple methods to introduce limited memory on a block
code. Given that the Nordstrom-Robinson code is composed
of a first-order Reed-Muller (16, 5) code and 7 of its cosets
[3], we were able to construct a 4-state trellis code by pro-
perly assigning these 8 cosets to the branches of the state
Ulitgl?lIIl iil rig. 5. llllb conshuctioil, uc_cllocu dllt.l
generalized in [4], yields in this case a 4-state (16, 7) code
with free distance df = 8. Notice that even though the product
(Rdy = 3.55 for this code is lower than that for the Golay code
(Rdf = 4) and for the NASA (7, 1/25 code (Rdf = 5), the code
performs better at low Eb/No, and has lower complexity than
the Golay code. Figures 6 and 7 show simulation results for
the word error probabilities Pw and bit error probabilities Pb
respectively, for the three codes just mentioned.
References
[1 ] J.G. Proakis, Digital Communications. New York: McGraw-Hill, 1983.
[2] A.W. Nordstrom and J. P. Robinson, "An optimum nonlinear code," Inform. and
Control, vol. 11, pp. 613-616, 1968.
[3] F.J. MacWilliams and N. J. Sloane, The Theory of Error-Correcting Codes. New
York" North-Holland, 1977.
[4] F. Pollara, R. J. McEliece, and K. Abdel-Ghaffar, "Construction for finite-state
codes," TDA Progress Report 42-90, vol. April-June 1987 (this issue), Jet Propul-
sion Laboratory, Pasadena, Calif., August 15, 1987.
77
10-1
8
10-2
[ I I I
-R (16, 8)
AY (24, 12)
,,,,sA(;2;2)
I I I I
0 0.5 1.0 1.5 2.0 2.5
Eb/No, dB
Fig. 1. Soft-decoding performance of three rate 112 block codes
10-1
8
6
10-2
I I I I
ULLER (16, 5)
I I I \ I
0.5 1.0 1.5 2.0
Eb/N O, dB
2.5
Fig. 2. Soft-decoding performance of first-order Reed-Muller
(16, 5) code
6 I I I I
4
2
R (32, 6)
10-1
8
61
\ (14, 1/4)
4 _O-bit WO R D
\\
(14, 1/5) \
2 lO-bit wORD__
10-2 I I _ I_ I
-1.0 -0,5 0 0.5 1.0
Eb/No,dB
Fig. 3. Soft-decoding performance of first-order Reed-Muller
(32, 6) code
78
i I I
I I
10-2
8
6
4
2
10-3
0
NASA (7, 1/2)_
I I I I
0.5 1.0 1.5 2.0
Eb/N0, dB
2.5
/,-,,
0
o
J
Fig. 5. Diagram of a 4-state trellis
°1
4
I I I I
10-1
8
\
4-STATE (16, 7) _,
\
\
\
GOLAY (24, _2)
NASA (7,
\
\
\
\
\
\
Fig. 4. Bit error probability of Golay code
Eb/N0, dB
Fig, 6. The Pw of new code compared to Golay and (7, 1/2) codes
79
10-1
8
6:
10-2
8
6
4-STATE (16,
3OLAY (24, 12)
NASA (7,
\
\
\
lO-3 I I I I
0 0.5 1.0 1.5 2.0
Eb/N0, dB
Fig. 7. The Pb of new code compared to Golay and (7, 1/2) codes
8O


Soft-decision decoding of some block codes

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