Project Part II_16914285

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Project Part II Sumanth Sridhar, 16914285

TABLE OF CONTENTS

Contents Page No

BPSK Modulation 3

1

Assignment Cover Sheet

School of Engineering

Student Name

Student Number

Sumanth Sridhar

16914285

Unit Name and

Number

300513 Engineering Software Applications

Lecturer Dr Upul Gunawardana

Title of AssignmentSimulation Model of a Coded Baseband BPSK

Communication System in AWGN

Length 21 Pages

Due Date June 6 2011

Date Submitted June 6 2011

Campus Enrolment Kingswood

DECLARATION

I hold a copy of this assignment that I can produce if the original is lost or damaged. I hereby certify that no part of this

assignment/product has been copied from any other students work or from any other source except where due

acknowledgement is made in the assignment. No part of this assignment/product has been written/produced for me by

another person except where such collaboration has been authorised by the subject lecturer/tutor concerned.

Signature:

Note: An examiner or lecturer/tutor has the right not to mark this assignment if theabove declaration has not been signed)

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QPSK Modulation 4

Theoretical BER BPSK 6

Theoretical BER QPSK 7

Baseband System 8

Scatter Plot 9

Pulse Shaping 9

Eye Diagram 9

Description of the Communication System 11

Convolutional Encoder and Viterbi Decoder 12

Hard Decision & Soft Decision Decoding 13

Performance Gain in Hard & Soft Decision Decoding 15

Description of the Simulation Model 15

Simulation Results 21

SIMULATION MODEL OF A CODED BASEBAND BPSK

COMMUNICATION SYSTEM IN

AWGN

BPSK

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Binary Phase Shift Keying (BPSK) is the simplest form of Phase Shift

Keying. It uses two opposite phases (0 and 180). The Digital Signal is

broken up into individual bits (binary digits). The state of each bit is

determined according to the state of the preceding bit. If the phase of the

wave does not change, then the signal state stays the same (0 or 1). Thesignal state changes (from 0 to 1 or from 1 to 0) if the phase of the wave

changes by 180 degrees (phase reverses). Because there are two possible

wave phases, BPSK is sometimes called biphasic modulation.

In Binary Phase Shift Keying (BPSK) only one sinusoid is considered for

function modulation. Modulation is achieved by varying the phase of the

basis function depending on the message bits. A BPSK modulator can be

implemented by NRZ coding the message bits (1 represented by +ve

voltage and 0 represented by -ve voltage) and multiplying the output by a

reference oscillator running at carrier frequency . The following figureshows a BPSK transmitter and receiver.

Truth Table for BPSK

Implementation of BPSK:

The general form for BPSK is

3

Binary Input Output phase

Logic 0

Logic 1

180

0

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This yields two phases, 0 and . In the specific form, binary data is often

conveyed with the following signals:

For binary "0"

For binary "1"

Where fc is the frequency of the carrier-wave

Hence, the signal-space can be represented by the single basis function

Where 1 is represented by and 0 is represented by .

Pros & Cons:

BPSK modulation is the most robust of all the PSK modulation techniquessince it takes the highest level of noise or distortion to make thedemodulator reach an incorrect decision. But, it is only able to modulateat 1 bit/symbol. Hence, it is unsuitable for high data-rate applicationswhen bandwidth is limited. With an arbitrary phase-shift introduced by thecommunications channel, the demodulator is unable to identify

constellation points. As a result, the data is often differentially encodedbefore modulation.

QPSK

Quadrature Phase Shift Keying (QPSK) or Quadrature PSK is another formof angled-modulated, constant-amplitude digital modulation. QPSK usesfour points on the constellation diagram, equispaced around a circle. WithQPSK four Output phases are possible for a single carrier frequency.Because there are four different output phases, there must be four

different input conditions. Because the digital input to a QPSK modulatoris a binary (base 2) signal, to produce four different input conditions it

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takes more than a single bit. With 2 bits, there are four possibleconditions: 00,01,10,11.

Therefore, with QPSK, the binary input data are combined into groups of 2bit called dibits. Each dibit code generates one of the four possible output

phases. Therefore, for each 2-bit dibit clocked into the modulator, a singleoutput change occurs. Hence, the rate of change at the output (baud rate)is one-half of the input bit rate.

QPSK can be used either to double the data rate compared with a BPSKsystem while maintaining the same bandwidth of the signal, or tomaintain the data-rate of BPSK but halving the bandwidth needed. Theadvantage of QPSK over BPSK is that QPSK transmits twice the data ratein a given bandwidth compared to BPSK, at the same BER. But, QPSKtransmitters and receivers are more complicated than the ones for BPSK.As with BPSK, there are phase ambiguity problems at the receiving end,and differentially encoded QPSK is often used in practice.

Implementation of QPSK

The implementation of QPSK is indicates the implementation of higher-order PSK.

This yields the four phases /4, 3/4, 5/4 and 7/4.

This results in a two-dimensional signal space with unit basis functions

And

The first basis function is used as the in-phase component of the signaland the second as the Quadrature component of the signal.

Hence, the signal constellation consists of the signal-space 4 points

The factors of 1/2 indicate that the total power is split equally between thetwo carriers.

By comparing the basis functions with that of BPSK, it can be inferred thatQPSK can be considered as two independent BPSK signals.

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A schematic diagram of the QPSK transmitter & Receiver:

Theoretical BER for

BPSK

For BPSK modulation the channel can be modelled as

where y is the received signal at the input of the BPSK receiver, x is themodulated signal transmitted through the channel , a is a channelamplitude scaling factor for the transmitted signal usually 1. 'n' is theAdditive Gaussian White Noise random variable with zero mean andvariance 2.

For AWGN the noise variance in terms of noise power spectral density (N0)is given by,

The symbol energy is given by

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where Es =Symbol energy per modulated bit (x), Rm = log2 (M), (for BPSKM=2, QPSK M=4. Rc is the code rate of the system if a coding scheme isused. In our case since no coding scheme is used Rc = 1. Eb is the Energy

per information bit.

Assuming Es=1 for BPSK (Symbol energy normalized to 1) ,then Eb/N0 canbe represented as (using above equations),

From the above equation the noise variance for the given Eb/N0 can becalculated as

Theoretical & simulated BER Curve-BPSK:

QPSK

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Although QPSK can be viewed as a quaternary modulation, it is easierconsidering it as two independently modulated Quadrature carriers.Hence, even (or odd bits) are used to modulate the in-phase componentof the carrier, while the odd (or even) bits are used to modulate theQuadrature phase component of the carrier. Therefore BPSK is used on

both carriers and they can be independently demodulated.

As a result, the probability of bit-error for QPSK is the same as for BPSK:

Theoretical & Simulated BER Curve-QPSK:

Baseband System

A baseband system is a lowpass communication channel that can transfer

frequencies that are very near zero. Examples are serial cables and local

area networks, as opposed to passband channels such as radio frequency

channels and passband filtered wires of the analogue telephone network.

A Baseband signal is represented as

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SIMULATION MODEL OF A CODED BASEBAND BPSK

COMMUNICATION SYSTEM IN

AWGN

BLOCK DIAGRAM OF THE COMMUNICATION SYSTEM

A BRIEFDESCRIPTIONOFTHEOVERALLCOMMUNICATIONSYSTEM

In the designed communication system, the data is sought from the user

and encoded using the convolutional encoder. Here, code rate k/n is set to

and constraint length k used is 5. The constraint length used representsthe number of bits present in the encoder memory. Addition of random

bits to the original input takes place here. The encoded data is modulated

in the BPSK modulator and transmitted through the communication

channel where additive white Gaussian noise (AWGN) is added to the

transmitted data. At the receiver end, first the data received is

demodulated with a BPSK demodulator and fed to the Viterbi decoder to

remove the random bits added during encoding. There are three decoding

algorithms used here. After decoding, the original data input is obtained.

The bit error rate for unquantised input, hard decision decoding and soft

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decision decoding is plotted against Eb/N0 to obtain the BER curve for

each decoding technique.

BPSK & QPSK Modulation

The explanation for these topics is given in pages 5 to 9 of this report.

CONVOLUTIONAL ENCODERAND VITERBIDECODER

The idea of channel coding is to improve the capacity of a channel by

adding some carefully designed redundant information to the data being

transmitted through the channel. Convolutional codes operate on serial

data, one or a few bits at a time. Convolutional codes accept a continuous

stream of bits and map them into an output stream introducing

redundancies in the process. The efficiency or data rate of a convolutional

code is measured by the ratio of the number of bits in the input, k, and

the number of bits in the output, n. In a convolutional code, there is some

`memory' that remembers the stream of bits that flow by. This

information is used to encode the following bits. The number of the

preceding bits used in the encoding process is called the constraint length

m (that is similar to the memory in the system). Typically the values of k,

n, m are 1-2, 2- 3, and 4-7 in commonly employed convolutional codes.

Convolutional coding with Viterbi decoding has been the predominant FEC

technique used in space communications, particularly in geostationary

satellite communication networks, such as VSAT (very small aperture

terminal) networks. The most common variant used in VSAT networks is

rate 1/2 convolutional coding using a code with a constraint length K = 7.

With this code, BPSK or QPSK signals can be transmitted with at least 5 dB

less actual power. This is very useful in reducing transmitter and/or

antenna cost or permitting increased data rates given the same

transmitter power and antenna sizes. But there's a trade-off-the same

data rate with rate 1/2 convolutional coding takes twice the bandwidth of

the same signal without it, given that the modulation technique is thesame. That is because with rate convolutional encoding, two channel

symbols are transmitted per data bit. If the modulation technique stays

the same, the bandwidth expansion factor of a Convolutional code is n/k.

Convolutional codes are usually described using two parameters: the code

rate and the constraint length. The code rate, k/n, is expressed as a ratio

of the number of bits into the convolutional encoder (k) to the number of

channel symbols output by the convolutional encoder (n) in a given

encoder cycle. The constraint length parameter, K, denotes the "length" of

the convolutional encoder, i.e. how many k-bit stages are available to feedthe combinatorial logic that produces the output symbols. Closely related

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to K is the parameter m, which indicates how many encoder cycles an

input bit is retained and used for encoding after it first appears at the

input to the convolutional encoder. The m parameter can be thought of as

the memory length of the encoder. In this tutorial, and in the example

source code, I focus on rate 1/2 convolutional codes. Viterbi decoding wasdeveloped by Andrew J. Viterbi, a founder of Qualcomm Corporation. The

Viterbi decoding algorithm is also used in decoding trellis-coded

modulation, the technique used in telephone-line modems to squeeze

high ratios of bits-per-second to Hertz out of 3 kHz-bandwidth analogue

telephone lines. Viterbi decoding is one of two types of decoding

algorithms used with convolutional encoding-the other type is sequential

decoding. Viterbi decoding has the advantage that it has a fixed decoding

time. It is well suited to hardware decoder implementation. But its

computational requirements grow exponentially as a function of the

constraint length, so it is usually limited in practice to constraint lengths

of K = 9 or less.

HARDDECISIONDECODINGANDSOFTDECISIONDECODING

The output codeword is transmitted through the channel. For instance, "0"is transmitted as "0 Volt and "1" as "1 Volt". The channel attenuates thesignal that is being transmitted and the receiver sees a distortedwaveform (Red colour waveform"). The hard decision decoder makes adecision based on the threshold voltage. At each sampling instant in thereceiver (as shown in the figure above) the hard decision detector

determines the state of the bit to be "0" if the voltage level falls below thethreshold and "1" if the voltage level is above the threshold. Therefore,

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the output of the hard decision block is "001". Perhaps this "001" output isnot valid, which implies that the message bits cannot be recoveredproperly. The decoder compares the output of the hard decision blockwith the all possible code words and computes the minimum Hammingdistance for each case. The decoder's job is to choose a valid codeword

which has the minimum Hamming distance. When the hard decisiondecoding is employed the probability of recovering data is 1/3.

The difference between hard and soft decision decoder is as follows

In Hard decision decoding, the received codeword is compared withthe all possible code words and the codeword which givesthe Minimum Hamming Distance is selected

In Soft decision decoding, the received codeword is compared withthe all possible code words and the codeword which givesthe Minimum Euclidean Distance is selected.

Thus the soft decision decoding improves the decision makingprocess by supplying additional reliability information ( calculatedEuclidean distance or calculated log-likelihood ratio)

Soft decision decoders use all of the information ( voltage levels inthis case) in the process of decision making whereas the harddecision decoders does not fully utilize the information available inthe received signal

The usage of Soft decision decoding scheme will improve theperformance of the receiver by approx 2 dB when compared to hard

decision scheme.

For the same encoder and channel combination the effect of replacing thehard decision block with a soft decision block.

Voltage levels of the received signal at each sampling instant are shown14

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in the figure. The soft decision block calculates the Euclidean distancebetween the received signal and the all possible code words and selects acodeword as the output. Even though the encoder cannot correct errors,the soft decision scheme helped in recovering the data in this case. Thisfact delineates the improvement that will be seen when this soft decision

scheme is used in combination with forward error correcting (FEC)schemes like convolution codes, LDPC etc. Soft decision decoders use allof the information ( voltage levels in this case) in the process of decisionmaking whereas the hard decision decoders does not fully utilize theinformation available in the received signal (evident from calculatingHamming distance just by comparing the signal level with the thresholdwhereby neglecting the actual voltage levels). Soft decision decodingViterbi decoders utilize Soft Output Viterbi Algorithm (SOVA) whichconsiders the probabilities of the input symbols producing a soft outputindicating the reliability of the decision.

PERFORMANCEGAINCOMPARISONWITHSOFTDECISIONANDHARDDECISIONDECODING

In the simulated communication system it can be inferred from the plotthat the usage of Soft decision decoding will improve the performance ofthe receiver by approx 2 dB when compared to hard decision decoding.

SIMULATIONMODEL

The convolutional encoder used in the communication system with code

rate k/n =1/2

Convolutional Encoder:

To generate random bits x = (sign(rand(1,k))+1)/2;Here encoding takes place y = convenc(x,trellis);

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BPSK Modulation before transmitting the signal z = -2*y+1;

AWGN:

To add additive white Gaussian noise to the channel :

z=z+sqrt(Var)*randn(1,n);

Viterbi Decoder for unquantised input:

xhat = vitdec(z,trellis,tblength,'cont','unquant');[errors,ratio]=biterr(xhat(tblength+1:end),x(1:end-tblength));

TotalNumError = TotalNumError + errors;TotalN = TotalN + k;

Viterbi Decoder hard decision:

xhat = vitdec(z,trellis,tblength,'cont','hard');[errors,ratio]=biterr(xhat(tblength+1:end),x(1:end-tblength));

TotalNumError = TotalNumError + errors;TotalN = TotalN + k;

Viterbi decoder-soft decision:

clear

N = 10^6 ; % number of bits or symbolsEb_N0_dB = [0:1:10]; % multiple Eb/N0 valuesEc_N0_dB = Eb_N0_dB - 10*log10(2);refHard = [0 0 ; 0 1; 1 0 ; 1 1 ];refSoft = -1*[-1 -1; -1 1 ;1 -1; 1 1 ];ipLUT = [ 0 0 0 0;...

0 0 0 0;...1 1 0 0;...

0 0 1 1 ];for yy = 1:length(Eb_N0_dB)

% Transmitterip = rand(1,N)>0.5; % generating 0,1 with equal

probability

cip1 = mod(conv(rand(1,N),[1 1 1 ]),2);cip2 = mod(conv(rand(1,N),[1 0 1 ]),2);

cip = [cip1;cip2];cip = cip(:).';

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s = 2*cip-1; % BPSK modulation 0 -> -1; 1 -> 0

n = 1/sqrt(2)*[randn(size(cip)) + j*randn(size(cip))]; % white gaussiannoise, 0dB variance

% Noise additiony = s + 10^(-Ec_N0_dB(yy)/20)*n; % additive white gaussian

noise

% receivercipHard = real(y)>0; % hard decisioncipSoft = real(y); % soft decision

% Viterbi decodingpmHard = zeros(4,1); % hard path metricsvHard_v = zeros(4,length(y)/2); % hard survivor pathpmSoft = zeros(4,1); % soft path metricsvSoft_v = zeros(4,length(y)/2); % soft survivor path

for ii = 1:length(y)/2

rHard = cipHard(2*ii-1:2*ii); % taking 2 hard bitsrSoft = cipSoft(2*ii-1:2*ii); % taking 2 soft bits

% computing the Hamming distance and euclidean distancerHardv = kron(ones(4,1),rHard);

rSoftv = kron(ones(4,1),rSoft);hammingDist = sum(xor(rHardv,refHard),2);euclideanDist = sum(rSoftv.*refSoft,2);

if(ii == 1) || (ii == 2)

% branch metric and path metric for state 0bm1Hard = pmHard(1,1) + hammingDist(1);pmHard_n(1,1) = bm1Hard;svHard(1,1) = 1;

bm1Soft = pmSoft(1,1) + euclideanDist(1);pmSoft_n(1,1) = bm1Soft;svSoft(1,1) = 1;

% branch metric and path metric for state 1bm1Hard = pmHard(3,1) + hammingDist(3);pmHard_n(2,1) = bm1Hard;svHard(2,1) = 3;bm1Soft = pmSoft(3,1) + euclideanDist(3);pmSoft_n(2,1) = bm1Soft;

svSoft(2,1) = 3;

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% branch metric and path metric for state 2bm1Hard = pmHard(1,1) + hammingDist(4);pmHard_n(3,1) = bm1Hard;svHard(3,1) = 1;

bm1Soft = pmSoft(1,1) + euclideanDist(4);pmSoft_n(3,1) = bm1Soft;svSoft(3,1) = 1;

% branch metric and path metric for state 3bm1Hard = pmHard(3,1) + hammingDist(2);pmHard_n(4,1) = bm1Hard;svHard(4,1) = 3;bm1Soft = pmSoft(3,1) + euclideanDist(2);pmSoft_n(4,1) = bm1Soft;svSoft(4,1) = 3;

else% % branch metric and path metric for state 0

bm1Hard = pmHard(1,1) + hammingDist(1);bm2Hard = pmHard(2,1) + hammingDist(4);[pmHard_n(1,1) idx] = min([bm1Hard,bm2Hard]);svHard(1,1) = idx;bm1Soft = pmSoft(1,1) + euclideanDist(1);bm2Soft = pmSoft(2,1) + euclideanDist(4);[pmSoft_n(1,1) idx] = min([bm1Soft,bm2Soft]);

svSoft(1,1) = idx;

% branch metric and path metric for state 1bm1Hard = pmHard(3,1) + hammingDist(3);bm2Hard = pmHard(4,1) + hammingDist(2);[pmHard_n(2,1) idx] = min([bm1Hard,bm2Hard]);svHard(2,1) = idx+2;bm1Soft = pmSoft(3,1) + euclideanDist(3);bm2Soft = pmSoft(4,1) + euclideanDist(2);[pmSoft_n(2,1) idx] = min([bm1Soft,bm2Soft]);svSoft(2,1) = idx+2;

% branch metric and path metric for state 2bm1Hard = pmHard(1,1) + hammingDist(4);bm2Hard = pmHard(2,1) + hammingDist(1);[pmHard_n(3,1) idx] = min([bm1Hard,bm2Hard]);svHard(3,1) = idx;bm1Soft = pmSoft(1,1) + euclideanDist(4);bm2Soft = pmSoft(2,1) + euclideanDist(1);[pmSoft_n(3,1) idx] = min([bm1Soft,bm2Soft]);svSoft(3,1) = idx;

% branch metric and path metric for state 3

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bm1Hard = pmHard(3,1) + hammingDist(2);bm2Hard = pmHard(4,1) + hammingDist(3);[pmHard_n(4,1) idx] = min([bm1Hard,bm2Hard]);svHard(4,1) = idx+2;bm1Soft = pmSoft(3,1) + euclideanDist(2);

bm2Soft = pmSoft(4,1) + euclideanDist(3);[pmSoft_n(4,1) idx] = min([bm1Soft,bm2Soft]);svSoft(4,1) = idx+2;

end

pmHard = pmHard_n;svHard_v(:,ii) = svHard;pmSoft = pmSoft_n;svSoft_v(:,ii) = svSoft;

end

% trace back unitcurrHardState = 1;currSoftState = 1;ipHatHard_v = zeros(1,length(y)/2);ipHatSoft_v = zeros(1,length(y)/2);

for jj = length(y)/2:-1:1prevHardState = svHard_v(currHardState,jj);ipHatHard_v(jj) = ipLUT(currHardState,prevHardState);

currHardState = prevHardState;prevSoftState = svSoft_v(currSoftState,jj);ipHatSoft_v(jj) = ipLUT(currSoftState,prevSoftState);currSoftState = prevSoftState;

end

% counting the errorsnErrHardViterbi(yy) = size(find([ip- ipHatHard_v(1:N)]),2);nErrSoftViterbi(yy) = size(find([ip- ipHatSoft_v(1:N)]),2);

endsimBer_HardViterbi = nErrHardViterbi/N; % simulated ber - hard decisionViterbi decoding BERsimBer_SoftViterbi = nErrSoftViterbi/N; % simulated ber - soft decisionViterbi decoding BERtheoryBer = 0.5*erfc(sqrt(10. (Eb_N0_dB/10))); % theoretical beruncoded AWGN

close allfigure

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semilogy(Eb_N0_dB,theoryBer,'bd-','LineWidth',2);hold on%semilogy(Eb_N0_dB,simBer_HardViterbi,'mp-','LineWidth',2);semilogy(Eb_N0_dB,simBer_SoftViterbi,'cd-','LineWidth',2);axis([0 10 10^-5 0.5])

grid onlegend('theory - uncoded', 'simulation - hard Viterbi', 'simulation - softViterbi');xlabel('Eb/No, dB');ylabel('Bit Error Rate');title('BER for BCC with Viterbi decoding for BPSK in AWGN');

To calculate BER:

P1 = TotalNumError/TotalN;P2 = [P2 P1];

SNR = [SNR SNRdB];fprintf('%5.2f\t%e\t%6.0f\t%6.0f\n', SNRdB, P1, TotalNumError, TotalN);

To plot the ber curve:

semilogy (SNR,P2,'-bo');title('Eb/N0 vs BER Curve')xlabel('Eb/N0');ylabel('Bit Error Rate');

hold onaxis tightgrid on

To define the trellis:

trellis = poly2trellis(5, [32 25]);here, 5 represents the value of k and [32,25] is the generator polynomial.

To define the number of simulations:

nsim = 1000;

To generate SNR:

SNRinit = 0;SNRinc = 1;SNRfinal = 8;

Here, the initial value of SNR will be 0 and it will increase by 1 up to 8

To define the length of the block

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n = 8000;

Here, k represents the number of message bits and code rate is given byk/n whose value is .

k = 4000;rate = k/n;tblength = 16;

To get the data:seed = input('Please Enter Input:');rand('state',seed);randn('state',seed);

SNR Calculation (Eb/N0)

for SNRdB = SNRinit:SNRinc:SNRfinalTotalNumError = 0;TotalN = 0;SNRdBs = SNRdB + 10*log(rate)/log(10);No = (10.^(-SNRdBs/10));Var = No/2;sigma = sqrt(Var);for Nsimulation = 1:nsim

fprintf('*');

if mod(Nsimulation,50) == 0fprintf('\n');end

SIMULATIONRESULTS

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0 1 2 3 4 5 6 7 8

10-6

10-5

10-4

10-3

10-2

10-1

Eb/N0 vs BER Curve

Eb/N0

BitErrorRate

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