National Conference on Recent Trends in Engineering & Technology
New Channel Coding Technique to Achieve The
Ultimate Shannon Limit
Mahesh R Patel
E-mail: [email protected]
Abstract — After Shannon’ s 1948 channel coding theorem, many
contributions have led to significant improvements in
performance versus complexity for practical applications,
particularly on the additive white Gaussian noise channel
(AWGN). This paper exhibits the new channel coding technique
and its simulation results, which can achieve near ultimate
Shannon limit error correction performance with moderate
decoding complexity.
Keywords-Channel Coding for Deep-Space Communications,
The Ultimate Shannon Limit, Capacity Approaching Code,
Forward Error Correction Code
I.
INTRODUCTION
The field of channel coding started with Claude Shannon’s
1948 landmark paper [1]. Since then there are so many channel
coding methods evolved to achieve the channel capacity. There
is wide range of channel coding techniques available with
different performance and complexity level, but none of them
is capable of achieving the ultimate Shannon limit. There is
still a gap of 1dB to the Ultimate Shannon Limit. The task here
is to provide better error correction ability at the cost of
minimal spectral efficiency and moderate complexity level.
The deep-space communications application is the field in
which the most powerful coding schemes for the power-limited
additive white Gaussian noise (AWGN) channel have been
deployed. Turbo codes represent a major paradigm shift in the
approach to coding systems for deep-space communications.
Researchers believe that the future deep-space coding system
will likely be based on the families of turbo codes. Progress in
development of more powerful code has been hampered
because researchers are trying to improve the performance by
altering the present coding techniques. The proposed new
channel coding technique, which works on hard decision, can
achieve a reliable bit error rate (BER) performance very close
to the ultimate Shannon limit (-1.59 dB Eb/No) for quadrature
phase shift keying (QPSK) modulation over AWGN channel.
This paper reveals the simulation results of this new coding
technique and compares its performance with the turbo codes.
II.
THE ULTIMATE SHANNON LIMIT
Performance of coding scheme over AWGN channel may
be characterized by two simple parameters: its signal-to-noise
ratio (SNR) and its spectral efficiency (). The SNR is the ratio
of average signal power to average noise power, a
dimensionless quantity. Spectral efficiency is the ratio of bit
transmission rate to the channel bandwidth. If coding scheme
transmits R bits per second over an AWGN channel of
bandwidth W Hertz then the spectral efficiency  = R/W bits
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per second per hertz (b/s/Hz). Shannon showed that for an
AWGN channel with given SNR and bandwidth W Hertz, the
rate of reliable transmission is upperbounded by
R < W log2 (1+ SNR)
Now closely related normalized SNR parameter that has been
traditionally used in the power-limited regime is Eb/No, which
may be defined as
Eb/No = SNR / ;   2r (r = coding rate)
For given spectral efficiency , Eb/No is lower bounded by
Eb / No > (2 –1) / 
Thus, the Shannon limit (lower bound) Eb/No as a function
of  is (2-1) / . This function decreases monotonically with 
and approaches ln 2 as   0, so the ultimate Shannon limit
(lower bound) on Eb / No is ln 2 (–1.59 dB). Means it is not
possible to carryout reliable communication below -1.59 dB
Eb/No.
III.
ENCODING METHOD OF NEW CODING TECHNIQUE
The encoding scheme for the new coding technique is quite
similar to serially concatenated convolutional codes. Block
diagram of the encoding scheme is as shown in Fig. 1.
Encoding process is performed in two different stages: inner
coding stage and outer coding stage. Inner coding stage is the
heart of the new coding technique. Coding rate for the Inner
code is 1/2 and coding rate for the outer code is variable.
Encoding logic of both the stages is somewhat similar to
convolutional codes [2].
I/P
Outer code
+ Interleaver
(Code rate:
Variable)
Inner code
(Code rate:
1/2)
O/P
(Code rate:
Variable)
Figure 1. Encoding scheme of new coding technique
IV. DECODING METHOD OF NEW CODING TECHNIQUE
The decoding technique is unique and entirely different
from all the present channel coding techniques. The New
coding technique works on hard-decision decoding so there is
no need of multiple iterations to decode the bits, which can
help to reduce the processing time. The demodulator provides
the hard decisions (logic 1 or 0) when regenerating the
B.V.M. Engineering College, V.V.Nagar,Gujarat,India
National Conference on Recent Trends in Engineering & Technology
transmitted sequence. These binary bits are then processed by
the inner decoding block and outer decoding block to retrieve
the original information bits. Decoding process is carried out in
two stages, inner decoding stage reduces the BER
considerably, which makes it easy for the outer decoding stage
to decode the bits correctly.
Outer decoding process is bit complicated and can closely
be associate with the logic of Viterbi algorithm (VA) [3]
decoding. Viterbi realized that if the channel errors are random
then the paths which are nonoptimal at any stage can never
become optimal in the future, he utilized this logic to correctly
decode the erroneous bits. This logic works well for low error
rate but as the uncoded errors increases the BER performance
of VA decoding degrades. Proposed new coding technique
solves this problem by reducing the error rate at the inner
decoding stage.
V.
SIMULATION RESULTS
During the decoding process, Inner coding stage is decoded
first. This decoding stage is the heart of the new coding
technique because during this decoding process, bits with
higher error probability are discarded and only accurate bits are
selected for the further processing. Thus, the decoded useful
bits, called “Accurate bits”, has very low error rate. These
“Accurate bits” are then processed by the outer decoding stage
to obtain the original information bits. Simulation results for
BER of “Accurate bits” over AWGN channel are listed in
Table I. The simulation is carried out on MATLAB for 1200
bits burst length using QPSK modulation technique.
TABLE I.
To reveal the effectiveness of the new coding technique,
please check the simulated results of very simple architecture
as shown in Fig. 2.
I/P
Outer code
Convolution type code (1/4) +
Repetition code (1/2)
+Interleaver
Inner code
(Code rate:
1/2)
O/P
(Code rate:
1/16)
Figure 2. Encoding architecture for code rate 1/16
It uses outer code of rate 1/8 (1/4 convolution type code +
1/2 repetition code) and rate 1/2 inner code. The overall coding
rate is thus 1/8*1/2 = 1/16. Fig. 3 shows the comparison
between the performance of new coding technique (code rate
1/16) and the performance of turbo codes with different code
rate (code rate: 1/2, 1/3, 1/4, 1/15) and different number of
iterations (m: 11, 12, 13, 18). In Fig. 3 the performance of
proposed coding technique (rate: 1/16) is superimposed on the
performance plot of turbo codes. D. Divsalar and F. Pollara
achieved this turbo code performance during their research “On
the Design of Turbo Codes” [4]. Size of the interleaver for the
simulated turbo codes was 16,384 bits and the size of the
interleaver for the simulation of new coding technique is just
1,200 bits.
SIMULATION RESULTS FOR “ACCURATE BITS”
Eb/No in dB
Uncoded BER
“Accurate bits” BER
–1.59
0.1196
0.0366
–1
0.1067
0.0286
– 0.5
0.0912
0.02
0
0.0781
0.0144
0.5
0.063
0.0102
1
0.0562
0.0072
2
0.0378
0.0031
3
0.0231
0.0011
0.0127
0.00033
4
provide some idea about the uncoded error rate at the receiver
end, which can be useful to trace the channel condition and to
track down the presence of jammer.
Burst Length: 1200 bits; Channel: AWGN; Error function: “awgn”
From “Table I” it is clear that the BER of “Accurate bits”/
after first decoding stage is always lower than the uncoded
BER and it holds true even at –1.59 dB Eb/No. This is a unique
feature of the new coding technique because for most of the
coding techniques as we move towards the ultimate Shannon
limit the coded BER becomes poorer than the uncoded BER,
which stops the further improvement in the performance. The
new coding technique capitalizes on the special property of
“Accurate bits” and these “Accurate bits” are then decoded by
the outer decoding stage, which uses VA kind of logic to
retrieve the original information bits. “Accurate bits” can also
13-14 May 2011
Figure 3. Performance comparison between Turbo codes and New coding
technique (code rate: 1/16)
It is clear from Fig. 3 that results of the new coding
technique, with code rate 1/16, are better than the turbo codes.
The same or even better performance can be achieved with the
code rate 1/12, but it may need a bit complicated decoding
logic. However, with the advancement in the signal-processing
field there should not be any issue with the level of decoding
complexity.
B.V.M. Engineering College, V.V.Nagar,Gujarat,India
National Conference on Recent Trends in Engineering & Technology
For one simple architecture, with code rate of 1/40 and
QPSK modulation technique, performance of 5.1X10-6 (BER)
is being achieved at –1.7 dB Eb/No. It practically proves the
violation of the ultimate Shannon limit because according to
the ultimate Shannon limit it is not possible to operate below
–1.59 dB Eb/No.
VI.
REFERENCES
[1]
C. E. Shannon, “A mathematical theory of communication,” Bell Syst.
Tech. J., vol. 27, pp. 379-423 and 623-656, 1948.
CONCLUSION
[2]
From the simulation results, it is clear that the new coding
technique of forward error correcting code can provide better
performance at moderate decoding complexity. It can be very
useful for the deep-space communications application and
military application. It can give almost zero BER if the facility
of the retransmission of the burst is available. With few
modifications, this coding technique may provide better
13-14 May 2011
performance for bandwidth limited regime as well. It also
raises the question on the concept of the ultimate Shannon
limit.
[3]
[4]
P. Elias, “Coding for noisy channels,” IRE Conv. Rec., pt. 4, pp.37-46,
Mar. 1955.
A. J. Viterbi, “Error bounds for convolutional codes and an
asymptotically optimum decoding algorithm,” IEEE Trans. Inform.
Theory, vol. IT-13, no. 4, pp. 260–269, Apr. 1967.
D. Divsalar and F. Pollara, “On the Design of Turbo Codes,”
Communications Systems and Research Section, TDA Progress Report
42-123, pp. 111–113, Nov. 1995.
B.V.M. Engineering College, V.V.Nagar,Gujarat,India