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7/22/2019 Communications and Networks (311SE) Course Work - B.Shakeal 4750820
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FACULTY OFENGINEERING AND COMPUTING
BACHELOR OF ENGINEERING (HONS)
IN
ELECTRICAL AND ELECTRONICS ENGINEERING
A311SE COMMUNICATIONS AND NETWORK
Course Work
STUDENT NAME : M.BADURDEEN SHAKEAL
STUDENT ID : 4750820
SUPERVISOR : Dr. ROHAN MUNASINGHE
SUBMISSION DATE : 2012.10.01
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PROJECT/ASSIGNMENT SUBMISSION ACKNOWLEDGEMENT SLIPName of Student: _________________________________ Student No: _____________________________
Home Address: __________________________________________________________________________
Date of Submission: _______________________________ Name of Tutor: LE LE YI
Program/ Module: Bachelor / A311SE (Communications & Networks)
Individual Assignment: Design based CAD/ Implementation Assignment
Received By: _________________________ Date: ___________________________________
Individual Projects (30%)
Marks
Learning OutcomeWeightage 1
stmarker
2nd
marker/moderator
Finalmark
Q1. Appraise multi-level digital modulationtechniques
Principle of operations of multilevel digitalmodulations
Principles of M_QAMmodulation technique
Parameters for Performance measurement of M-QAM modulation
Design selection and calculations for M_QAM
MATLAB simulation
Analysis/ discussion on Results
45%
3%
4%3%
11%14%10%
Q2. Apply the principles of error correction coding.
Principles of FEC coding Performance measurement parameters for FEC
coding
Principles of Hammingcoding and decoding Design selection and Calculations for Hamming
coding and decoding
MATLAB Simulation
Analysis / discussion on Results
45%
3%
3%3%
11%
15%10%
Quality and structure of report10%
Total Marks 100%
M.Badurdeen Shakeal 4750820
2012.10.01
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1stmarkers comment
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
2nd
markers/ moderator comment
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
Plagiarism Check
Turnitin checkCollusion check
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Page - i
Declaration of Originality
I Badurdeen Shakeal declare that this thesis is my own work and contains no material which has
been accepted for a degree or diploma by the University or any other institution, except by way
of background information and duly acknowledged in the thesis, and to the best of the my
knowledge and belief no material previously published or written by another person except where
due acknowledgement is made in the text of the thesis, nor does the thesis contain any material
that infringes copyright.
List of reference is given at the end of the assignment report using Coventry University Harvard
style referencing. This project is submitted in partial fulfilment of the requirements of the award
of Bachelor of Engineering (Hons) in Electrical & Electronic Engineering.
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Page - ii
Acknowledgment
First and foremost, I would like to thank to our Lecture Dr. Rohan Munasingheof this course
work, Mr. Duminda Wijesinghefor the valuable guidance, advice and supervising. He inspired
us greatly to work in this course work. His willingness to motivate us contributed tremendouslyto our course work. I also would like to thank Mr. Sujeewa Perera the librarian cum academic
support executive of Auston Institute of Management Ceylon for helping us with library
resources. Besides, I would like to thank the management of Auston Institute for providing us
with a good environment and facilities to complete this course work. Also, I would like to take
this opportunity to thank to the Coventry University for offering this program, computing course
work. It gave us an opportunity to learn about the operation of multilevel digital modulation and
forward error correction. Finally, an honorable mention goes to our families and friends for their
understandings and supports on us in completing this project. Without helps of the particular that
mentioned above, we would face many difficulties while doing this.
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Contents
Abstract .................................................................................................................... 4
1.1 Motivation................................................................................................................................... 4
1.2 Problem statement and scope of project...................................................................................... 5
1.3 Methodologies............................................................................................................................. 5
1.4 Result and conclusions................................................................................................................ 5
Introduction ............................................................................................................. 6
2.1. Background.................................................................................................................................. 6
2.2. Objectives.................................................................................................................................... 6
2.3. Scope/limitation of project.......................................................................................................... 6
3. Literature Review ............................................................................................... 7
3.1 Principle of operations of multilevel digital modulations.................................................................. 7
3.2 Principle of operations of M-QAM modulation................................................................................ 8
3.3 Parameters for performance measurement of M-QAM modulation................................................ 11
3.4 Principles for Forward Error Correction (FEC) coding................................................................... 14
3.5 Principles for hamming coding and decoding................................................................................. 17
4 Design criteria and calculations ...................................................................... 22
4.1 Design selection, criteria for M-QAM modulation......................................................................... 22
4.2 Design calculations for M-QAM modulation.................................................................................. 23
4.3 Design selection / criteria for (n, k) hamming coding and decoding............................................... 25
4.4 Design calculations for (n, k) hamming coding and decoding........................................................ 26
5. MATLAB Programming .................................................................................. 30
5.3 MATLAB program for M-QAM digital modulation................................................................. 30
5.4 MATLAB program for (7, 4) hamming coding and decoding.................................................. 32
5.4.1. MATLAB code for (7.4) Hamming coding....................................................................... 32
5.4.2. MATLAB simulation code for hard decision decoding.................................................... 32
5.4.3. MATLAB simulation code comparison of Bit Error Rate (BER) for coded and uncoded
signals. 33
5.5 Detail simulation steps and procedures..................................................................................... 35
5.5.1. Detail simulation steps of 8-QAM modulation.................................................................. 35
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5.5.2. Detail simulation steps of Hamming coding...................................................................... 37
5.5.3. Detail simulation steps of hard decision decoding............................................................ 40
5.5.4. Detail simulation steps of Bit Error Rate (BER) Comparison........................................... 44
6. Critical Analysis /Discussion and comparison on results .............................. 45
6.1 Analysis, discussion and comparison on different simulation results of M_QAM modulation 46
6.2 Analysis, discussion and comparison on different simulation results of Hamming coding and
decoding................................................................................................................................................. 46
6.3 Theoretical interpretation and Verification................................................................................ 46
7. Conclusions ........................................................................................................ 47
References ..................................................................Error! Bookmark not defined.
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List of Figures, Tables and Equations
Figure 1 Diagram of multilevel modulation .................................Error! Bookmark not defined.
Figure 2 Dagram of QAM Modulaion ............................................................................................ 8
Figure 3 Dagram of QAM Demodulaion ........................................................................................ 9
Figure 4 Bandwidth efficiency diagram ....................................................................................... 13
Figure 5- FEC System Diagram (A.B.Carlson, 2001) .................................................................. 16
Figure 6- Hamming Distance counting cube ................................................................................ 17
Figure 7 Block diagram for 8-ary QAM modulation .................... Error! Bookmark not defined.
Figure 8 Constellation Diagram for selected 8-ary QAM Modulation ......................................... 24
Figure 9 Programming code in MATLAB simulation software for 8-QAM Digital Modulation
for the output of the signal together with related input binary signal ........................................... 35Figure 10 Result shown in MATLAB simulation software for 8-QAM Digital Modulation for
the output of the signal together with related input binary signal ................................................ 35
Figure 11 Result of MATLAB simulation show the output of the signal together with related
input binary signal ........................................................................................................................ 36
Figure 12 Programming code in MATLAB simulation software for (7, 4) hamming coding ..... 37
Figure 13 Result shown in MATLAB simulation software for (7, 4) hamming coding .............. 37
Figure 14 Programming code in MATLAB simulation software for (7, 4) hamming codes hard
decision decoding ......................................................................................................................... 40
Figure 15 Result Shown in MATLAB simulation software for (7, 4) Hard Decision decoding. . 40
Figure 16 MATLAB coding in MATLAB simulation software for comparison of Bit Error Rate
(BER) for coded and uncoded signals .......................................................................................... 44
Figure 17 Result Show in MATLAB simulation software for comparison of Bit Error Rate
(BER) for coded and uncoded signals. ......................................................................................... 44
Figure 18 Result on MATLAB for Selected hamming coding..................................................... 45
List of Tables
Table 1 Bit Allocation Table for 8-ary QAM Modulation ........................................................... 24
Table 2 Codeword table ................................................................................................................ 27
Table 3 Decoding Table (Syndrome) ........................................................................................... 29
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Abstract
1.1MotivationFirst of all the actual motivation of accomplishing this course work is to get wide range of
knowledge withinside the regarding telecommunication transmission. Over time, the world
remains enhancing and advancing around us, there is lots heart and soul and personas wanting to
alter points about all of us all in the field of communication.
But alternatively tiny success is attained with this scientific stride, that's a similar in all human
beings they have a tendency to reside and assist a lesser amount of effort and time and much
more efficiently and effectively. This particular course work continues to be emotively dedicated
to communication strategies. Because we are inside a fast growing globe, where it really is
called an international village for people to stay in the actual reduces costs of that technology.
We have to have huge and also intricate communication national facilities is usually to be used
where a couple of primary modulation schemes has been used and tested throughout technique
environment with regard to appropriate operate. Which may end up being QAM (Quadrature
Amplitude Modulation) modulation technique.
Currently the communication have come up with numerous facilities and so there are several
processes to discover in the diverse kind of modulation techniques. Also on that point currently
certainly there exist wise the actual communication will have numerous noise full express and
not able to use properly where there will be more errors because of having difficulty and thus
that individuals should use computer simulation program such as MATLAB as it is an Advanced
programing language additionally numerical processing environment. Its used to change matrix
and also plotting regarding features and information.
The main aim with this venture is always to clearly realize and theoretically assess each and
every problem given under. The initial would be the comprehension of QAM digital modulation
techniques as well as the design of the Hamming code and its particular error recognition
techniques.
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1.2Problem statement and scope of projectIn this very initial Question, believe taking amplitudes as well as phase angles regarding
provided M value to plot the constellation diagram. Since the course work states utilize
MATLAB simulation software to simulate design the actual MQAM digital modulation and
possess showing regulated output signal as well as associated input binary sign.
Withinside the second question, to begin with must find the Block code and using that discover
parity check matrix (H), generator matrix (G) as well as codeword table. then find the syndrom
table. Issue informs to utilize difficult decision technique and to replicate the design utilize
Matlab software. With this locate overacting hamming code and also decoding.
1.3MethodologiesThe strategy utilized first of all is the calculating just about almost most exact numbers as well
as variables and then finally stepping into MATLAB software for implementation as well as
running purposes, where the accuracy of mathematical computed data could possibly be tested.
Similarly thing about this course work focuses the building of the Hamming code using basic
theory after this portion has been completed. Also finally MATLAB software has been used here
for implementing this interrogation.
1.4Result and conclusionsBenefits in this course work would that most mathematical components could be elucidated and
graphically represented using MATLAB computing software simulation. The main results of
this particular task accomplishment a comprehensive understanding is actually acquired
whenever completing this particular course work.
When utilizing this MATLAB simulation software program in 1st case we are able to check the
accuracy and reliability from the plotting diagram and the determined solutions we all do. Inside
second difficulty furthermore we use MATLAB in order to be able for you to help imitate the
record and to confirm the accuracy. In any case when using this particular software we could
discover the graphical answer and comprehension of the subject nicely as we want to operate in
useful every day in the life.
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Introduction
2.1.BackgroundBackground of this course work is whenever we come to telecom it is not easier compared to we
all thought. As it associated with of procedure going through the senders and also devices.
Therefore, there should be ways to handle and also correct whenever transmitting signals are
usually incorrect so that you can have got perfect as well as quick communication. Thus,
Forward error correction [FEC] may be the great way to perform that kind of scenario and also
FEC coding were created not merely to detect yet correct errors in the communication to avoid
requirement for retransmission.
FEC is actually techniques are often charge in wireless transmission, exactly in which
retransmission is strategies tend to be highly unproductive as well as error rate might be large.
Consequently this really is gaining popularity through other error correction techniques.
2.2.ObjectivesIn this course work really aim is actually examine the particular calculations and also results
which receive utilizing MATLAB encoding. As well as reading good results expression about
the techniques utilized in these systems and getting good results knowledge in hard choice
method in hamming programming and also decoding. Consequently simulation and comparison
of obtained value of coded and uncoded signals.
2.3.Scope/limitation of projectWhile In the first chapter M value is given by the lecturer. So through the use of M value k value
can be determine. Then through which have to prepare the particular binary bit assignment table.
From then on need to pull the constellation diagram regarding of QAM modulation. Then Utilize
the MATLAB software that can simulate the answer. Despite the fact that QAM effectiveness
will be high it has many downsides in fact. A very important factor is always that is much more
prone minimizing degree of noise is necessary to move the actual signal to various places. Not
only must that hen QAM having amplitude aspects one-linearly be maintained.
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3. Literature Review
3.1 Principle of operations of multilevel digital modulations.
In order to be able to help first view the functioning of multilevel digital modulation must realize
first why digital modulation is used in the field of telecommunication and also researches in entireworld.
Especially in telecommunication field the entire transmission to be achievable there has to be a
possible transmission media in which the transmission is actually generated (sender) and may
travel for the meant destination had been throughout communication it is known as the receiver.
In individuals transmitting media with inside conversation there is certainly merely a static range
of frequencies which can be found with regard to transmission purpose, as this can be obtained
we're able to not really perhaps transmit in all frequencies this may be devastating, if your option
would be not necessarily identified and the date that we sent wouldn't be suitable for that channel.
A prosperous plan was found following considerable research is in which to improve any
transmitting signal based on the information which is give feed to as input data, this particular
alteration of the signal is named as modulation. This information that is modulated might be
received by the receiver and also the information that's in the signal could be retrieved because
the original data that has been encoded this is known as demodulation. (Sklar, 2001)
Figure 1 Block Diagram of Multilevel Digital Modulation
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3.2 Principle of operations of M-QAM modulation.
Before viewing of M-QAM (Multiple Quadrature Amplitude Modulation) modulation will realize
first the understanding of functioning of the (QAM) Quadrature Amplitude Modulation and
demodulation.
QAM modulation and demodulation.
Quadrature Amplitude Modulation is really a combination of Phase-Shift-Key (PSK) and
Amplitude-Shift-Key (ASK) modulation techniques. Additionally QAM can be a form of digital
modulation much like Phase-Shift-Key other than digital information is within the amplitude and
also phase of the transmitting carrier. It is possible to deliver 2 different signals simultaneously
on the same carrier frequency, through the use of two copies of the carrier frequency, one altered
through 900 with regards to the additionalIn this modulation each carrier is ASK modulated.
(Sklar, 2001)
The above mentioned figure 2 shows the actual QAM modulation scheme generally. The
particular input can be a flow associated with binary numbers coming to a rate of R bps. This kind
of supply will be converted into two separate bit streams. Withinside the diagram, top of the
stream is actually ASK modulated over a carrier regarding regularity fc only multiplying the bit
stream by the carrier. Therefore, a binary "0" is represented by the absence of the carrier wave
2-bit serial to
parallel
converter
-/2
Binary Input
D(t) R bps
d1(t)
R/2 bps
d2(t)
R/2 bps
Carrier
Oscillator
Phase
Shift
cos(2fct)
sin(2fct)
QAM signalOut
S(t)
+
+
Figure 2 Dagram of QAM Modulaion
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and a binary 1 is represented by the presence of the actual carrier wave at a continuous
amplitude.
This exact identical carrier wave is shifted simply by simply 900 as well as used for ASK
associated with the lower binary stream. Both modulated signals are added together and
transmitted.
The transmitted signal can be expressed as follows:
() () () () ()---------Equation 1If a couple of- level ASK is used, then each of the two streams can be in one of two states and the
combined stream could be in one of4 = 2 2 states. That is fundamentally QPSK. If four level
ASK is used (i.e. four different amplitude levels), then the combined stream can be in one of 16
= 4 4 states. Systems using 64 and even 256 states have been implemented. The greater the
number of states, the higher the data rate that is feasible within a given bandwidth. The higher the
number of states, the larger the potential error rate due to noise and attenuation.
-/2
QAM
signal in
s(t) R bps
d1(t)
R/2 bps
d2(t)
R/2 bps
Carrier
Oscillator
Phase
Shift
cos(2fct)
sin(2fct)
Low Pass Filter
Low Pass Filter
y1(t)= d1(t)/2
y2(t)= d2(t)/2
Figure 3 Dagram of QAM Demodulaion
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The above figure 2 shows the QAM demodulation scheme in general terms. The input is a stream of QAM
signal arriving at a rate of R bps. This stream is converted into two separate bit streams. Both upper and
lower part signal go through two separate Low Pass Filters.
Quadrature amplitude modulation is a modulation scheme which has digital and analogue
The mathematical description of the new modulated signal
() () ()
-----------Equation 2The level parameter for in-phase component and quadrature component are independent of each other for
all i. M-ary QAM is a hybrid form of M-ary modulation.
M-ary amplitude-shift keying (M-ary ASK)
If bi=0 for all i, the modulated signal si(t) of above equation is reduces to
() () ( )}--------Equation 3
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3.3 Parameters for performance measurement of M-QAM modulationAverage signal to noise ratio (SNR)
Signal-to-noise ratio is understood to be the power ratio between signal and the background noise.
The particular ideas regarding signal-noise ratio and energetic range tend to be closely connected.
Powerful variety measures the particular proportion between the strongest un-distorted
transmission channel on a route and the bare lowest discernable signal. Which usually for most
reasons will be the noise level. SNR actions the actual percentage in among an arbitrary signal
and noise.
SNR is normally taken up indicate the average signal-to-noise ratio, as it is feasible for
instantaneous signal-noise ratio proportions will be significantly different. The idea may berecognized as normalizing the particular noise level to a single (0 dB) and calculating how far the
transmission. (R.S., 2004)
------Equation 4Exactly in which P is typical power. Both transmission noise power must be assessed at the very
similar and equal points in a system, and also inside the exact identical system bandwidth. If the
transmission as well as the noises are usually assessed throughout the exact identical impedance,
then the SNR can be obtained by establishing the actual square of the amplitude voltage. (Tocci,
2007)
All the received power Pris in the modulating signal
Relationship of Eb/N0and SNR as follow.
-------------Equation 5
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Bit error probability
For a rectangular constellation, a Gaussian channel, and matched filter reception, the probability
of bit error P is expressed by.
Where L represents the number of amplitude levels in one dimension. We assume that a sequence
of log L (k) bits are assigned to an L-ary symbol using a Gray code.
Noise power spectral density
Eb/N0 is the ratio of signal energy per bit to noise power density per Hertz and it is standard
quality to measure for digital communication system performance.
Power (W) = Energy/time ( J/s)
Eb= STb--------------Equation 7
------------Equation 8(A.B.Carlson, 2001)
Signal power
The average signal power = A2/2 ----------------Equation 9
------------Equation 6
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Bandwidth requirements
Operating points for coherent quadrature amplitude modulation (QAM) is plotted in Figure
Of the modulations shown, QAM is clearly the most bandwidth efficient.
Figure 4 Bandwidth efficiency diagram (Sklar, 2001)
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3.4 Principles for Forward Error Correction (FEC) coding.
FEC
Two main categories of FEC codes are block code and convolutional code
Throughout telecommunication, information theory, and code principle, forward error correction
(FEC) of channel is a technique used for controlling errors in data transmission over unreliable or
noisy communication channels. The particular central idea is the sender scribes their massage in
a redundant way approach by utilizing a good error-correcting code (ECC). The actual United
States math wizard Rich Overacting developed this field in the nineteen forties as well as invented
the very first error-correcting code for Hamming (7, 4) signal in 1950. (Anon., 2009)
The redundancy permits the particular receiver in order to be able for you to help identify arestricted quantity of error that may take place around what it's all about, and sometimes to fix
these errors without retransmission. FEC gives the receiver the ability to appropriate errors
without resorting to the change route so that you can ask for retransmission of knowledge,
however at the expense of a fixed, increased forwards channel bandwidth. FEC is therefore
applied in circumstances where retransmissions are costly or impossible, for example one-way
communication links when broadcasting in order to be able for you to help several shower
receiver throughout multicast. FEC details are usually added mass storage devices to allow
recuperation associated with corrupted information, and is widely used in modems.
FEC digesting in the receiver could be placed on a digital bit stream or perhaps the particular
demodulation of your digitally modulated carrier. For the last option, FEC is an integral part of
the preliminary analog-to-digital conversion withinside the receiver. The Viterbi decoder tools
the soft-decision algorithm to demodulate digital information through an analog signal damaged
through noises. Several FEC coders also can produce bit-error rate (BER) signal transmission
which may be utilized since comments so that you can refine the particular analog getting
electronics. (Anon., 2009)
The utmost fragments of errors or of missing bits that may be corrected is determined by the
design of the particular FEC code, thus different forward error correcting codes are suitable for
diverse ailments.
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Two main principle type of FEC codes are
Block code Convolutional code
Block code
Error-correcting codes are employed to dependably transmit digital information over variable
connections and communication channel subject to channel noise. Each time transmitter needs to
transfer a possibly very long data stream using a block code, the particular transmitter breaks or
cracks the actual stream upwards in to piece of a few fixes size. Every such piece is called messageand also the procedure given by the actual block code encodes each massage individually right in
to a codeword, also referred to as the block poor obstruct rules. The particular transmitter next
transfers just about almost most blocks for the recipient, who is able to subsequently use some
decoding mechanism to recover the original communications from the probably damaged
received blocks. The particular performance and also success of the total transmitting depends
upon the parameters of the channel and the block code. (A.B.Carlson, 2001)
Convolution Code
Convolutional codes are used thoroughly in various applications to have trustworthy data
transmission, such as digital electronic video, radio, microwave mobile communication, and also
satellite communication. These types of codes in many cases are implemented in concatenation
using a hard-decision code, particularly Reed Solomon. Just before turbo codes, this kind of
improvements had been probably one of essentially by far the best efficient, returning best
towards the Claude Shannon limit. (Sklar, 2001)
Add redundant bits to a massage so that error can be detected and corrected
Rate of code is a measure of redundancy = ratio of massage bits to total bits
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Maybe effected on fixed length data blocks (block codes) or on a continuous data stream
(convolutional codes)
Figure 5- FEC System Diagram (A.B.Carlson, 2001)
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3.5 Principles for hamming coding and decoding.
Hamming distance
The number of symbol positions in which two code words vary is denoted as a Hamming distance,
d. e.g. 1001 and 1000 are separated by Hamming distance 1, while 1001 and 0110 are separated
by distance 4. Codes with larger minimum Hamming distance dmin are more powerful. Binary
words of the length n carrying only k information bits contain redundancy which, if properly
structured can provide. Redundancy for error detection and correction in an (n, k) code. The ratio
R is the code rate is k/n. (Tocci, 2007)
Hamming Distancecount
3-bit binary words arranged in 3-dimensional space on a cube. Minimum Hamming distance
between words is 1. Words 000 and 111 have distance 3 and represent EEC. Code rate is 1/3,
single errors can be corrected.
Example:
Then hamming distance is = 3
Figure 6- Hamming Distance counting cube
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The minimal Hamming distance between any two correct codeword is 3, and received words can
be correctly decoded if they are at distance at most one from the codeword that was transmitted
by the sender.
Hamming Distance = Minimum Weight
Parity Checks
We can add a parity bit to a word to ensure that it has a defined parity 1110 has odd parity (odd
number of 1s). A single error will then disturb the parity and thus be detected, this is the basis
for error correction coding. By introducing parity bits applied across subsets of the information
bit positions it is possible to both detect and correct errors the basis of error correction codes
(ECCs). The parity bits are sometimes mixed in with the information bits rather than attached at
the end of the code word.
Block Codes
A block code is a code that operates upon fixed-length blocks of information bits. In (n,k) block
codes each sequence of k information bits is mapped into a sequence of n(n >k). In order to
achieve high error correction and detection capability the code minimum distance and code rate
is should be large. Systematic block code is the block code which has the following structure
where: (A.B.Carlson, 2001)
First k elements are the same as the message bits, c=n-k bits are the check bits
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3.6 Performance measurement parameters for FEC coding.
Code rate
The Shannons theorem defined the code rate as the ratio of the data bits k to total bits n. Its
the measure of how much additional bandwidth is required to carry data at the same data rate as
without code
SNR
The actual SNR benefit indicates the number of energy required to obtain a specific BER price. Typically
a bigger SNR value results in lower BER. Programming obtain will be defined because the quantity of
enhancement as well as level of superior throughout SNR, whenever a particular programming structure is
employed. The normal way of discovering code acquire is to plot the actual BER versus SNR for equally
numbered and united nations touch pad program, (Tocci, 2007)
SNR calculation Equation
----------Equation 10
------------Equation 11
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Minimum Distance
The actual minimum distance provides all of us all a security program against noise and also attenuations.
This particular mechanism the minimal range in among code phrases can be utilized for problem diagnosis
and/or error a static correction. The actual fail-safe system regarding problem diagnosis, and or a static
correction capacity for a FEC are determined by just about the minimum distance of the FEC. This might
be worked out in two methods can be that we can help to eliminate the number of error corrections and or
raise the number of mistake detections or even the other way circular. This can be the hamming distance.
The hamming distance will be acquired by the bare lowest length between of two codewords. (R.S., 2004)
Minimum distance calculation equation
( ) ( ) ( ) { }
Coding gain
The coding gain defined as reduction in Eb/No(dB) to achieve a specific BER of and error
correcting coded system compared to encoded system using the same modulation
---------Equation 13
---------------Equation 14
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Error Detecting Capability
The maximum number of errors t that can be detected satisfies.
--------------Equation 15Error Correction Capability
The maximum number of guaranteed correctable errors per code word satisfies
Closure property of a code
The closure property describes that the sum of any two codewords in the space must yield a valid
codeword in the space.
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4 Design criteria and calculations
4.1 Design selection, criteria for M-QAM modulation
This is an 8-ary QAM
Block Diagram for 8-QAM transmitter
Bit Splitter
Q I C
2- to-4 Level
Converter
2- to-4 Level
Converter
Product
Modulator
Product
Modulator
Reference
Oscillator
Linear
Summer
Band pass
Filter
+900
Input Data
fb
fb/3
fb/3
fb/3
A Channel
B Channel PAM
PAM
8-QAM
Output
Sin wct
Figure 7 Block Diagram for given 8-ary QAM modulation
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4.2 Design calculations for M-QAM modulation
Calculations
Value of M=8, thus this is 8-ary QAM
Calculation of k using given value M
( )------------Equation 16
( )
In line with this QAM there are 8 signals to be allocate. Thus here 2 two different amplitudes and 4
different phases has been assigned.
Amplitude and phase allocation
1. Amplitude A1 = 1.5V,2. Amplitude A2 = 3.0V
Phase allocation
1. Phase 1 = 00,2. Phase 1 = 900,3. Phase 1 = 1800,4. Phase 1 = 2700
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Bit assignment table
Bit Assignment Output
Q I C Amplitude (v) Phase
0 0 0 1.5 0
0 0 1 3 0
0 1 0 1.5 90
0 1 1 3 90
1 0 0 1.5 180
1 0 1 3 180
1 1 0 1.5 270
1 1 1 3 270
Table 1 Bit Allocation Table for 8-ary QAM Modulation
Constellation Diagram this 8-QAM modulation
Symbol rate will be 1/3 of bitrate.
Y-axis
X-axis000
010
100
110
001
011
101
111
Figure 8 Constellation Diagram for selected 8-ary QAM Modulation
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4.3 Design selection / criteria for (n, k) hamming coding and decoding
Value for mprovided is 8
( )-----------Equation 17So value of n
Value of k Here number of check bit
By the determination from above equations the hamming code ( ) ()So here there are 4 data bits has been added with 3 parity bits to build a 7 bit codeword. So there
are 16 combination can be used for this 4 bit parity matrix. Those are 000, 001, 010, 011, 100,
101, 110, and 111.
Allocated Parity Equations
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4.4 Design calculations for (n, k) hamming coding and decoding
Construction of matrix algebra Parity Check Matrix
----------------Equation 18
Generator Matrix
-------------Equation 19
Codeword Calculation
For data sequence 0000
------Equation 20
For data sequence 0001
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Similarly codeword table is generated by possible codeword
Codeword Table
No. M1 M2 M3 M4 P1 P2 P3 Code weigh
0 0 0 0 0 0 0 0 0
1 0 0 0 1 0 1 1 3
2 0 0 1 0 1 0 1 3
3 0 0 1 1 1 1 0 4
4 0 1 0 0 1 1 0 3
5 0 1 0 1 1 0 1 4
6 0 1 1 0 0 1 1 4
7 0 1 1 1 0 0 0 3
8 1 0 0 0 1 1 1 4
9 1 0 0 1 1 0 1 4
10 1 0 1 0 0 1 0 3
11 1 0 1 1 0 0 1 4
12 1 1 0 0 0 0 1 3
13 1 1 0 1 0 1 0 4
14 1 1 1 0 1 0 0 4
15 1 1 1 1 1 1 1 7
Table 2 Codeword table
Minimum distance = Minimum code weight =
----------Equation 21
Error detecting capability
---------Equation 22
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Error correcting capability
--------Equation 23 Syndrome
Syndrome for e =0000000
Syndrome []
-----------Equation 24
Syndrome for e =1000000
[
]
Syndrome for e =0100000
[
]
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Similarly decoding table is generated by possible syndromes
Decoding Table
Error Pattern Syndrome
0000000 000
1000000 111
0100000 110
0010000 101
0001000 011
0000100 100
0000010 010
0000001 001
Table 3 Decoding Table (Syndrome)
Illustration of decoding
Massage is 1111, transmitted codeword is 1111111 and received codeword is 1011111
So e = 1011111 and s = 110
Correction is 1111111 and decoded massage is 1111.
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5. MATLAB Programming
5.3 MATLAB program for M-QAM digital modulationMATLAB programfor simulate the designed 8-ary QAM digital modulation in the simulationresult, output modulated signal together with related input binary signal
Input data Stream is 001011010000100101111110
format long;clcclear allM=8;data='001011010000100101111110';A1=1.5; A2=3;
f=2;t=linspace(0,1,900);time=[];Digital_signal=[];QAM_signal=[];
N=length(data)/log2(M);phase=[0 0 pi/2 pi/2 pi pi (3*pi)/2 (3*pi)/2];phase_data=[];fork=1:3:length(data)
phase_data=[phase_data bin2dec(data(k:k+2))+1];
endforkk=1:1:N
QAM_signal=[QAM_signal (phase_data(kk)==1)*A1*sin(2*pi*f*t + phase(1))+...
(phase_data(kk)==2)*A2*sin(2*pi*f*t + phase(2))+...(phase_data(kk)==3)*A1*sin(2*pi*f*t + phase(3))+...(phase_data(kk)==4)*A2*sin(2*pi*f*t + phase(4))+...(phase_data(kk)==5)*A1*sin(2*pi*f*t + phase(5))+...(phase_data(kk)==6)*A2*sin(2*pi*f*t + phase(6))+...(phase_data(kk)==7)*A1*sin(2*pi*f*t + phase(7))+...(phase_data(kk)==8)*A2*sin(2*pi*f*t + phase(8))];
time=[time t];t=t+1;
endt1=linspace(0,1/3,300);forjj=1:1:length(data)
Digital_signal = [Digital_signal (str2num(data(jj))==0)*zeros(1,length(t1))+...(str2num(data(jj))==1)*ones(1,length(t1))];
t1=t1+(1/3);endsubplot(2,1,2);
plot(time,QAM_signal,'.');title('8QAM Signal'); xlabel('time'); ylabel('Amplitude');axis([0 time(end) -8 8]);text(0.3,5.2,'\bf001'); %001text(1.3,5.2,'\bf011'); %011text(2.3,5.2,'\bf010'); %010text(3.3,5.2,'\bf000'); %000text(4.3,5.2,'\bf100'); %100
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text(5.3,5.2,'\bf101'); %101text(6.3,5.2,'\bf111'); %111text(7.3,5.2,'\bf110'); %110grid on;subplot(2,1,1);
plot(time,Digital_signal,'LineWidth',3);
title('Digital Input Signal'); xlabel('time'); ylabel('Amplitude');axis([0 time(end) 0 2]);grid on;text(0.3,1.3,'\bf001');%001text(1.3,1.3,'\bf011');%011text(2.3,1.3,'\bf010');%010text(3.3,1.3,'\bf000');%000text(4.3,1.3,'\bf100');%100text(5.3,1.3,'\bf101');%101text(6.3,1.3,'\bf111');%111text(7.3,1.3,'\bf110');%110
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5.4 MATLAB program for (7, 4) hamming coding and decoding5.4.1. MATLAB code for (7.4) Hamming coding%Simulation for encoding and decoding of a [7,4] Hamming code. The decoder
clearn = 7%# of codeword bits per blockk = 4%# of message bits per blockA = [ 1 1 1;1 1 0;1 0 1;0 1 1 ]; %Parity submatrix-Need binary(decimal combination of 7,6,5,3)
G = [ eye(k) A ]%Generator matrixH = [ A' eye(n-k) ]%Parity-check matrix
% ENCODER%msg = [ 1 1 1 1 ] %Message block vector-change to any 4 bit sequencecode = mod(msg*G,2)%Encode message
code(2)= ~code(2);
recd = code %Received codeword with error
5.4.2.MATLAB simulation code for hard decision decoding% DECODER%
syndrome = mod(recd * H',2)%Find position of the error in codeword (index)find = 0;forii = 1:n
if~finderrvect = zeros(1,n);errvect(ii) = 1;search = mod(errvect * H',2);ifsearch == syndrome
find = 1;index = ii;
endend
enddisp(['Position of error in codeword=',num2str(index)]);correctedcode = recd;correctedcode(index) = mod(recd(index)+1,2)%Corrected codeword%Strip off parity bitsmsg_decoded=correctedcode;msg_decoded=msg_decoded(1:4)
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5.4.3.MATLAB simulation code comparison of Bit Error Rate (BER) for codedand uncoded signals.
% Script for computing BER with Hamming (7,4) code and maximal% likelihood hard decision decoding
clearN = 10^6 ;% number of bits
Eb_N0_dB = [0:1:10]; % multiple Eb/N0 valuesEc_N0_dB = Eb_N0_dB - 10*log10(7/4);
h = [ 1 1 1 ; 1 1 0; 1 0 1; 0 1 1];ht = [h ;eye(3)];g = [eye(4) h];synRef = [ 5 7 6 3 ];
bitIdx = [ 7 7 4 7 1 3 2].';
foryy = 1:length(Eb_N0_dB)
% Transmitterip = rand(1,N)>0.5; % generating 0,1 with equal probability
% Hamming coding (7,4)ipM = reshape(ip,4,N/4).';ipC = mod(ipM*g,2);cip = reshape(ipC.',1,N/4*7);
% Modulations = 2*cip-1; % BPSK modulation 0 -> -1; 1 -> 0
% Channel - AWGNn = 1/sqrt(2)*[randn(size(cip)) + j*randn(size(cip))]; % white gaussian noise, 0dB variance
% Noise additiony = s + 10^(-Ec_N0_dB(yy)/20)*n; % additive white gaussian noise
% Receiver
cipHard = real(y)>0; % hard decision
% Hamming decodercipHardM = reshape(cipHard,7,N/4).';syndrome = mod(cipHardM*ht,2); % find the syndromesyndromeDec = sum(syndrome.*kron(ones(N/4,1),[4 2 1]),2); % converting the three bit syndrom to decimalsyndromeDec(find(syndromeDec==0)) = 1; % to prevent simulation crash, assigning no error bits to parity
bitCorrIdx = bitIdx(syndromeDec); % find the bits to correctbitCorrIdx = bitCorrIdx + [0:N/4-1].'*7; % finding the index in the arraycipHard(bitCorrIdx) = ~cipHard(bitCorrIdx); % correcting bitsidx = kron(ones(1,N/4),[1:4]) + kron([0:N/4-1]*7,ones(1,4)); % index of data bitsipHat = cipHard(idx); % selecting data bits
% counting the errorsnErr(yy) = size(find([ip- ipHat]),2);
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end
theoryBer = 0.5*erfc(sqrt(10.^(Eb_N0_dB/10))); % theoretical ber uncoded AWGNsimBer = nErr/N;
close allfiguresemilogy(Eb_N0_dB,theoryBer,'bd-','LineWidth',2);hold onsemilogy(Eb_N0_dB,simBer,'ms-','LineWidth',2);axis([0 10 10^-5 0.5])grid onlegend('theory - uncoded', 'simulation - Hamming 7,4 (hard)');xlabel('Eb/No, dB');ylabel('Bit Error Rate');title('BER for BPSK in AWGN with Hamming (7,4) code');
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Figure 10Result shown in MATLAB simulation software for 8-QAM Digital
odulation for the output of the signal together with related input binary signal
5.5Detail simulation steps and procedures5.5.1.Detail simulation steps of 8-QAM modulation.Result of M-QAM digital modulation obtained by MATLAB simulations
Figure 9 Programming code in MATLAB simulation software for 8-QAM Digital Modulation
or the output of the signal together with related input binary signal
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MATLAB simulation result. This figure 11 shows the output amplitude and phase combinations
related to the input Signal 001011010000100101111110
Figure 11 Result of MATLAB simulation show the output of the signal together with
related input binary signal
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5.5.2.Detail simulation steps of Hamming codingMATLAB simulation of hamming coding
Figure 12 Programming code in MATLAB simulation software for (7, 4) hamming coding
Figure 13 Result shown in MATLAB simulation software for (7, 4) hamming coding
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Result Obtain from MATLAB Simulation Software.
Hamming Code
n =
7
k =
4
G =
1 0 0 0 1 1 1
0 1 0 0 1 1 0
0 0 1 0 1 0 1
0 0 0 1 0 1 1
H =
1 1 1 0 1 0 0
1 1 0 1 0 1 0
1 0 1 1 0 0 1
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msg =
1 1 1 1
code =
1 1 1 1 1 1 1
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5.5.3.Detail simulation steps of hard decision decodingMATLAB simulation of hard decision decoding
Figure 14Programming code in MATLAB simulation software for (7, 4) hamming codes hard
decision decoding
Figure 15 Result Shown in MATLAB simulation software for (7, 4) Hard Decision decoding.
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Result Obtain from MATLAB Simulation Software for hard decision decoding.
n =
7
k =
4
G =
1 0 0 0 1 1 1
0 1 0 0 1 1 0
0 0 1 0 1 0 1
0 0 0 1 0 1 1
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H =
1 1 1 0 1 0 0
1 1 0 1 0 1 0
1 0 1 1 0 0 1
msg =
1 1 1 1
code =
1 1 1 1 1 1 1
recd =
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1 0 1 1 1 1 1
syndrome =
1 1 0
Position of error in codeword=2
correctedcode =
1 1 1 1 1 1 1
msg_decoded =
1 1 1 1
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5.5.4.Detail simulation steps of Bit Error Rate (BER) Comparison.MATLAB Simulations for comparison of Bit Error Rate (BER) for coded and uncoded signals.
Figure 16 MATLAB coding in MATLAB simulation software for comparison of Bit Error Rate
(BER) for coded and uncoded signals
Figure 17 Result Show in MATLAB simulation software for comparison of Bit Error Rate
(BER) for coded and uncoded signals.
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Graph for comparison of Bit Error Rate (BER) for coded and uncoded signals.
6. Critical Analysis /Discussion and comparison on results
In above (figure 18) the blue color curve shows the uncoded bit error rate and purple color curve
show the bit error rate for selected Hamming code.
Figure 18 Result on MATLAB for Selected hamming coding
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6.1Analysis, discussion and comparison on different simulation results ofM_QAM modulation
As we know that QAM is a combination of ASK and PSK. Here we do with 8-QAM
modulation. For this 8-ary we can allocate different combinations like
Can be taken+ Single amplitude and 8 different phases+ Two amplitude and 4 different phases+ Four amplitude and 2 different phases+ Eight amplitude and single phase
According to the allocation of amplitude and phase constellation diagram is varying.
6.2Analysis, discussion and comparison on different simulation results ofHamming coding and decoding
According to the allocation of m value the hamming code is coming deferent. Here we allocate 3
for m value. We can see here that according to the allocation of parity equation the hamming
code simulation is coming different and results are obtain also different.
As well as consistent with parity equation the parity check matrix and generator matrix is
changing. And also the result of MATLAB simulation is changing.
Also we can see here when we changing the parity equation the coded BER (Bit Error Rate) alsoaltering habitually.
6.3Theoretical interpretation and VerificationIn both situation (QAM and Hamming code) the MATLAB simulation shows equality of actual
calculation result. So theoretically both section can be design cad based implementation using
MATLAB simulation software.
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7. Conclusions
We will go over the particular computer simulation results. The actual hamming code is a seo'ed
bitwise FEC for that Gaussian channel. Numerous transmission schemes include broke errors
exactly in which they will happen really withinside surrounding bits. The actual channel inbuilt
dynamics at times is employed to correct short burst problems. The channel interleaves attempts
to set aside burst open errors arbitrarily within a obstruct of data where they may be remedied by
way of a appropriate haphazard FEC at the shower radios finish, for instance Hamming code
rule. In this training we view such a situation. The key portion is that the calculation of BER
exactly in which this exhibits people the mistake rate from particular Eb/No. wherever this
demonstrates a great uncoded program provides little error rate at commence where Eb/no tend
to be small. However, if the data movements away a sizable power must keep up with the
problem rate do i think the particular numbered program.
More than I'm able to point out which this QAM modulation is very important since it could be
followed in to numerous engineering with the present. As an example, when we all work with a
cable system the actual slandered of QAM modulation, its extensive for signal digesting. Here
we have examined the actual performance regarding 8-QAM modulation. Here we can easily
transmit more data rate actually. To ensure that we could view the QAM passage getting much
far greater overall efficiency than PSK and have transitions because when using dull program
rule methods this can possess a lot more performances. In this syllabus we understood how the
QAM is a combination of PSK and ASK modulations. Anyhow the QAM could be exact
identical in order to be able for you to help PSK in the event the electronic digital camera
information wasnt possessing 2 service providers.
The actual acting signal can be referred to in array form with creator matrix as well as parity
check matrix. We need to calculate syndromes with regard to problem examining. In fact, when
a legitimate codeword will be grow from the examine array the particular symptoms comes zero.
Each time a codeword multiply along together using examine enclosure and also arriving non-
zero affliction this means there is a error. All solitary touch mistakes could be remedied and all
two-bit mistakes may be recognized.
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The main advantage of forward error correction coding can it be needs a feed back. Yet its bad
for multicast. In any case that's appropriate only for a few programs. This kind of FEC has a
proven way tranny and avoid multicast problems. Anyway its computationally pricey as well as
over-transmission
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References
A.B.Carlson, 2001. Communication System, an Introduction to signals and nise in electrical
communication.. 4th ed. s.l.:Mc Graw-Hill.
Anon., 2009. Radio-Electronics.Com. [Online]
Available at: http://www.radio-electronics.com/info/rf-technology-design/pm-phase-
modulation/8qam-16qam-32qam-64qam-128qam-256qam.php
[Accessed 25 09 2012].
R.S., D., 2004. In: Digital Transmission Systems. s.l.:s.n.
Sklar, B., 2001. Digital Communications, Fundamentals of Applications. 2nd ed. s.l.:Pearson Education.
Tocci, 2007. In: 1. Edition, ed. Digital System. s.l.:Prentice Hall of India, p. 80.