International Journal of Engineering Trends and Technology (IJETT) – Volume 11 Number 7 - May 2014
Comparison of Hamming & Reed-Solomon Block
Codes in AWGN Channel Using Different
Modulation Techniques
Hardeer Kaur#1, Seema Luhach*2
1#
Department of Electronics and Communication,
Manav Rachna international University, Faridabad (India)
2*
Asst. Professor, Department of Electronics and Communication,
Manav Rachna international University, Faridabad (India)
Abstract— This paper is for comparing the BER of Hamming &
Reed Solomon Block Codes when used in combination with
different modulation techniques namely BPSK, QPSK, 8PSK,
16PSK in a AWGN channel. To perform the comparative study
we had taken help of computer simulation software. Simulink
and Ber tool in the communication tool box of Matlab is used to
model the block codes and plot comparison graphs.
In first part of the paper a basic introduction is given for all the
block codes and the modulation techniques used in the
comparative study. Secondly, a comparison is drawn between
various modulation techniques to determine which modulation
technique has the lowest BER when used along with AWGN
channel. Then different block codes are modelled in Simulink
and the BER is plotted for different modulation techniques to
judge the effect of introducing the block codes on the BER
performance of the communication system. In last part the
comparison is drawn between various block codes with the best
suitable modulation technique, to determine the best possible
combination of the block code and modulation technique.
Keywords— BPSK, QPSK, 8PSK, 16PSK, AWGN, Reed Solomon
Code , Hamming code .
I. INTRODUCTION TO COMMUNICATION SYSTEM
The most basic definition of the communication system
can be a channel or a pathway using which a signal is
transmitted from one location to another.
The most basic communication system has three components
as shown in fig.1.
Source: Source of the signal which is to be transmitted.
Channel: Communication medium or a system through which
the signal is passed.
Receiver: End or the destination location of the signal.
Source
Channel
Receiver
Fig. 1 Basic Communication system
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In communication channel there is always some noise present
and when the signal is passed through it the input signal is
mixed with the present noise. Hence, the output signal
received at the receiver is scrambled.
To improve the efficiency of the communication system
different modulation and coding techniques are used, as
shown in the fig.2.
Source
Encoder
Modulator
Channel
Receiver
Decoder
Demodulator
Fig. 2 Communication system with encoder-decoder & modulatordemodulator
With the use of different encoder-decoders & modulatordemodulators near perfect signal can be transmitted from the
source to the receiver.
II. MODULATION TECHNIQUES
Modulation is the process by which some characteristics of
a carrier are varied in accordance with a modulating wave. It
can be better understood by using a sinusoidal signal.
We can see that this sinusoidal signal has 3 parameters that
can be altered, to affect the shape of the graph. The first term,
A, is called the magnitude, or amplitude of the signal. The
next term,  is known as the frequency, and the last term,
 is known as the phase angle. All 3 parameters can be
altered to transmit data. There are 3 basic types of modulation:
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International Journal of Engineering Trends and Technology (IJETT) – Volume 11 Number 7 - May 2014
1. Amplitude modulation: a type of modulation where the
amplitude of the carrier signal is modulated (changed) in
proportion to the message signal while the frequency and
phase are kept constant.
QPSK: Quadrature Phase Shift Keying (QPSK) uses four
points on the constellation diagram, equally spaced around a
circle. With four phases, QPSK can encode two bits per
symbol, shown in the fig.5
2. Frequency modulation: a type of modulation where the
frequency of the carrier signal is modulated (changed) in
proportion to the message signal while the amplitude and
phase are kept constant.
3. PHASE MODULATION: a type of modulation where the
phase of the carrier signal is varied accordance to the low
frequency of the message signal is known as phase
modulation.
Fig. 5 Constellation diagram example for QPSK
8PSK & 16PSK: These are called higher order phase shift
keying modulation techniques. These are implemented using 8
or 16 point on the constellation diagram as shown in the fig. 6
Fig. 3 Three basic modulation techniques
In this paper we will look in details of Phase modulation or
rather phase shift modulation technique.
Phase modulation works by modulating the phase of the signal,
i.e. changing the rate at which the point moves around the
circle. This changes the phase of the signal from what it would
have been if no modulation was applied.
Most commonly used Phase shift modulation techniques are
elaborated below.
BPSK: Binary Phase Shift Key (BPSK) is the simplest form
of phase shift keying (PSK). It uses two phases which are
separated by 180° and so can also be termed 2-PSK.
Fig. 4 Constellation diagram example for BPSK
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Fig. 6 Constellation diagram example for 8PSK
III. BLOCK CODES
In coding theory, block codes comprise the large and
important family of error-correcting codes that encode data in
blocks. There is a vast number of examples for block codes,
many of which have a wide range of practical applications.
Block Codes are conceptually useful because they allow
coding theorists, mathematicians, and computer scientists to
study the limitations of all block codes in a unified way. Such
limitations often take the form of bounds that relate different
parameters of the block code to each other, such as its rate and
its ability to detect and correct errors.
Examples of block codes are Reed–Solomon codes, Hamming
codes, Hadamard codes, Expander codes, Golay codes, and
Reed–Muller codes. These examples also belong to the class
of linear codes, and hence they are called linear block codes.
More particularly, these codes are known as algebraic block
codes, or cyclic block codes, because they can be generated
using boolean polynomials.Coding is procedure for mapping s
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International Journal of Engineering Trends and Technology (IJETT) – Volume 11 Number 7 - May 2014
given set of massages [m1 m2 m3... mn] in such a way that
the transformation is one-to-one; i.e for each message, there is
only one encoded message. This is called “source coding”.
HAMMING CODE: Hamming codes are a family of linear
error-correcting codes that generalize the Hamming (7,4) code
which was invented by Richard Hamming in 1950. Hamming
codes can detect up to two-bit errors or correct one-bit errors
without detection of uncorrected errors. Hamming codes
are perfect codes as they can achieve the highest
possible rate for codes with their block length and minimum
distance 3.
Fig. 7 Simulation setup for simulation 16PSK with Reed-Solomon code
IV. AWGN CHANNEL
The transmitted waveform gets corrupted by noise ‘n’,
For each integer r ≥ 2, there is a code with block length n = typically referred to as Additive White Gaussian Noise
(AWGN). In communications, the AWGN channel model is
2r - 1 and message length k = 2r – r – 1.
one in which the only impairment is the linear addition of
wideband or white noise with a constant spectral density
(expressed as watts per hertz of bandwidth) and a Gaussian
Hence, the rate of Hamming codes is
distribution of amplitude. The model does not account for the
phenomena of fading, frequency selectivity, interference,
nonlinearity or dispersion. However, it produces simple,
R = k / n = 1 – r / (2r-1),
tractable mathematical models which are useful for gaining
which is highest possible for codes with minimum insight into the underlying behaviour of a system before these
distance 3 (i.e. the minimal number of bit changes needed to other phenomena are considered. AWGN is commonly used
go from any code word to any other code word is 3) and block to simulate background noise of the channel under study, in
length 2r-1. The parity-check matrix of a Hamming code is addition to multipath, terrain blocking, interference, ground
constructed by listing all columns of length r that are non-zero, clutter and self-interference that modern radio systems
which means that the dual code of the Hamming code is encounter in terrestrial operation.
the punctured Hadamard code. The parity-check matrix has Additive white Gaussian noise (AWGN) is a basic noise
the property that any two columns are pair wise linearly model used in Information theory to mimic the effect of many
independent.
random processes that occur in nature.
Due to the limited redundancy that Hamming codes add to the
data, they can only detect and correct errors when the error
rate is low.
V. RESULTS AND DISCUSSION
In this paper three basic sets of simulations were performed.
1. BER performance of all the modulation techniques
namely BPSK, QPSK, 8PSK, 16PSK without using
any block codes.
Fig. 7 Simulation setup for simulation 16PSK with Hamming code
REED SOLOMON: Reed-Solomon codes are non-binary
cyclic error-correcting codes invented by Irving S. Reed and
Gustave Solomon. They described a systematic way of
building codes that could detect and correct multiple random
symbol errors. By adding t check symbols to the data, an RS
code can detect any combination of up to t erroneous symbols,
or correct up to t/2 symbols. As an erasure code, it can correct
up to t known erasures, or it can detect and correct
combinations of errors and erasures. Furthermore, RS codes
are suitable as multiple-burst bit-error correcting codes, since
a sequence of b + 1 consecutive bit errors can affect at most
two symbols of size b. The choice of t is up to the designer of
the code, and may be selected within wide limits.
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Fig. 8 BER vs Eb/No plot for all modulation techniques without block codes.
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International Journal of Engineering Trends and Technology (IJETT) – Volume 11 Number 7 - May 2014
2.
BER performance of Reed-Solomon and Hamming
block codes with different modulation techniques.
Fig. 9 BER vs Eb/No plot for all modulation techniques with
Hamming codes.
Fig. 11 BER vs Eb/No plot for BPSK when used without and with block codes.
For comparison purpose the BER values of all the simulations
are measured at Eb/No of 6db and are tabulated below.
TABLE I
SUMMARY OF RESULTS AT 6 DB.
Modulation
Technique
Eb/No
[db]
BER
BPSK
6
0.0023
QPSK
6
0.0023
8Psk
6
0.0204
4
16PSK
6
0.0681
5
BPSK
6
0.0015
QPSK
6
0.0015
8Psk
6
0.0146
16PSK
6
0.0516
BPSK
6
0.0032
QPSK
6
0.0066
8Psk
6
0.0290
16PSK
6
0.1900
Sl no
Block Code
1
2
3
6
7
without
code
Hamming
code
8
9
10
Fig. 10 BER vs Eb/No plot for all modulation techniques with
Reed Solomon codes.
3.
BER performance of best performing modulation
technique with Reed-Solomon & Hamming block
codes to find the best suitable combination in
modulation and block codes.
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11
Reed
Solomon
code
12
VI. CONCLUSION
In this paper we used Simulink to calculate the BER of
Hamming and Reed Solomon block codes with Different
modulation techniques comprising of BPSK, QPSK, 8PSK,
16PSK along with AWGN channel. We were able to perform
the study and conclude the following outcomes.
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International Journal of Engineering Trends and Technology (IJETT) – Volume 11 Number 7 - May 2014
Fig. 8 & fig. 9 shows the BER v/s Eb/No plot for different
modulation techniques with hamming and without using any
block codes. It was noticed that the BER curve of BPSK and
QPSK are overlapping each other i.e. both have same BER.
This can be explained as the matter of fact that QPSK can be
viewed as a quaternary modulation, but it is easier to see it as
two independently modulated quadrature carriers, similar to
BPSK. With this interpretation, the even (or odd) bits are used
to modulate the in-phase component of the carrier, while the
odd (or even) bits are used to modulate the quadrature-phase
component. Or in other words BPSK is used on both carriers
independently. Hence the BER rates are same for both the
simulations.
[9]
[10]
Xinmiao Zhang. “An Efficient Interpolation-Based Chase BCH
Decoder.” IEEE, Vol 60, April 2013, 212:216.
Rajani Katiyar et al. “BER performance of BPSK & QPSK over
Rayleigh channel & AWGN channel.” Int. J. Electronics & Electrical
Engineering & Telecoms., Vol 3, April 2014.
Fig. 11 it can be concluded that BPSK is the best performing
modulation technique with the lowest BER for all the
simulations. Without code the BER of BPSK is 0.0023, with
Hamming code is 0.0015 and with Reed-Solomon is 0.0032
when measured at Eb/No = 6db.
From the result Table I, it is concluded that BPSK has the
lowest BER of 0.0015 when used with Hamming code. Hence,
it can be said that BPSK modulation technique when used
with Hamming Codes should be preferred.
VII.
FURTHER STUDY & FUTURE WORK.
The work in this paper is the comparison of Eb/N0 v/s BER
for different modulation techniques comprising of BPSK,
QPSK, 8PSK & 16PSK over AWGN channel when used
without any block codes and when used with Hamming &
Reed-Solomon codes.
The future work is intended towards performing same
experiment with different block codes such as LDPC, Goley
etc. Further the study can be extended to study the use of
different types of noises by changing the channel types.
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