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LINEAR BLOCKCODES

LINEAR BLOCKCODES

AIM

To detect and correct the error accruing in the transmitted data with the help

of linear block codes.

THEORY

A code is said to be linear if any two code word in the code, can be added in module 2 arithmetic to Produce a third codeword in the code. The (n-k) bits in the remaining portion are computed from the message bits in the accordance with the prescribed encoding rule that determines a mathematical structure of the code, then use frame to get the syndrome calculation.

The hamming weight of the code vector C is defined in the number of non zero elements in the code vector. The minimum distance of the linear block code is defiend in the smallest hamming distance between any pair of code vector in code.

Algorithm

Get the generator matrix value.
Calculate the order of given matrix.
calculate the code word.
calculate the minimum ha
Get the received code word.
Display the hamming code
Display the syndrome of the code word.
Calculate the error in the bit.
Display the correlated code word.

PROGRAM

#######################################

clc;

clear all;

close all;

%input generator matrix

g=input('enter the generator matrix:')

disp('G=')

disp('the order of linear block code for given generator mathrix is:');

[n,k]=size(transpose(g))

for i=1:2^k

for j=k:-1:1

if rem(i-1,2^(-j+k+1))>=2^(-j+k)

u(i,j)=1;

else

u(i,j)=0;

end

end

end

u;

disp('the possible code word are:');

c=rem(u*g,2);

disp('the minimum hamming distance dmin for given block code is:');

d_min=min(sum((c(2:2^k,:))'))

%code word

r=input('enter the recieved code word:');

p=[g(:,n-k+2:n)];

h=[transpose(p),eye(n-k)];

disp('hamming code');

ht=transpose(h)

disp('syndrome of a given code word is:');

s=rem(r*ht,2)

for i=1:1:size(ht)

if(ht(i,1:3)==5)

r(i)=1-r(i);

break;

end

end

disp('the error is in bit:');

i;

disp('the correct codeword is:');

r;

###########################################################################

OUTPUT

Enter the generator matrix:[1000101;0100111;0010110;0001011]

g =1000101

0100111

0010110

0001011

G = The order of linear block code for given generator mathrix is:

n = 7

k = 4

The possible code word are:

0000000

0001011

0010110

0011101

0000111

0101100

0110001

0111010

1000100

1001110

1010011

1011000

1100010

1101001

1110100

1111111

The minimum hamming distance dmin for given block code is:

d_min =3

Enter the received code word:[1000100]

r=1000100

Hamming code

H=

101

111

110

011

100

101

001

Syndrome of t given codeword is:

s=001

The error is in bit :

i=7

The corrected code word is : r=100010.

Result

Thus the error is detected and corrected n the transmitted code by using linear block codes.

ESTIMATION OF PERIODOGRAM

ESTIMATION OF PERIODOGRAM

AIM

To estimate the power spectral density of a given signal using periodogram

in MATLAB.

THEORY

The power spectral density (PSD) of a WSS process is the Fourier transform of the autocorrelation sequence. Periodogram is a non-parametric method to estimate PSD

() = (k)

For an autocorrelation ergodic process and an unlimited amount of data, the autocorrelation sequence may be detemined by using the time average

(k) = (n+k)x*(n)

If x(n) is only measured over a finite interval, say n=1,2,…N-1, then the autocorrelation sequence must be estimated using with a finite sum

(r) = () (n+k)x*(n)

In order to ensure that the value of x(n) that is fully outside the interval [0,N-1] are excluded and written as follows

(k) = () (n+k)x*(n) k=0,1,2….,N-1.

Taking the discrete Fourier transform of rx^(k) leads to an estimation of the power spectrum known as the periodogram.

() = (k)

The periodogram

() = ()() = ()

Where XN(ejw) is the discrete time Fourirer transform of the N-point data sequence XN(n)

() = (n) =

ALGORITHM

STEP 1: Compute the value of x.

STEP 2: Perform periodogram function for x signal.

STEP 3: Using pwelch function, smoothen the output of periodogram signal.

STEP 4: Plot the graph for input and output signal

PROGRAM

##########################################################

clc;

clear all;

close all;

fs=1000;

t=0.1:1/fs:0.3;

x=cos(2*pi*t*200)+0.1*randn(size(t));

figure(1);

plot(x);

title('input signal');

xlabel('time');

ylabel('amplitude');

figure(2);

periodogram(x,[],'one sided',512,fs);

figure(3);

pwelch(x,30,10,[],fs,'one sided');

#############################################################

RESULT

Thus the MATLAB program to estimate the power spectral density of given signal using periodogram is executed and output is plotted.

SIMULATION OF BLOCK CODES HAMMING AND CYCLIC CODES

SIMULATION OF BLOCK CODES

AIM

To simulate linear block coding techniques for hamming and cyclic code using MATLAB.

THEORY

HAMMING CODES

Consider a family of (n,k) linear block codes that have the following parameters.

Block length,n=2^m-1

No.of.message bits k=2^m-m-1

No. of parity bits, n-k=m, where m>=3.

These are so called Hamming codes. To illustrate the relations between the minimum distance dmin and the structure of the parity check matrix H. Consider the codeword 0110100,In the matrix multiplication is done, and the non-zero elements of this codeword “shift” out the second, third and fifth column of the matrix if yielding. An important property of hamming codes is that they satisfy the condition. t=1.This means that hamming codes are single error correcting binary perfect codes.

CYCLIC CODES

Cyclic codes form a sub class of linear block codes.A binary code is said to be cyclic code,if it exhibits two fundamental properties.

LINEARITY PROPERTY

The sum of two codeword is also a codeword.

CYCLIC PROPERTY

Any cyclic shift of codeword is also called a codeword.

ALGORITHM

HAMMING CODE

STEP 1: Start the program

STEP 2: Assign the number of parity bits m=4

STEP 3: Calculate the block length n from m=2^m-1

STEP 4: Assign the number of message bits k such that n-k=m so k=11

STEP 5: The hamming code is (5,11)

STEP 6: Obtain the input signal message randomly the input message is in binary format

STEP 7: The parity bits are calculated for input message taken.

STEP 8: The parity bits are appended along the message bit to form the codeword.

STEP 9: The codeword formed is transmitted through AWGN channel

STEP10: The received signal is then decoded to retrieve the message

STEP 11: The BER is calculated for the retrieved message

STEP 12: For the various values of the SNR and its corresponding BER,the graph is

Plotted.

CYCLIC CODE

STEP 1: Start the program.

STEP 2: Assign the block length n=7.

STEP 3: Assign the message bits,k=4.

STEP 4: The cyclic code is (7,4)

STEP 5:Generate the polynomial.

STEP 6: Obtain the input message randomly.

STEP 7: The input message is in binary format.

STEP 8: The codeword is formed by appending the parity bits with the message bits.

STEP 9: The parity bits are calculated from the generation polynomial.

STEP10: The codeword formed is transmitted through AWGN channel.

STEP11: The received signal is then decoded with the help of generator polynomial

Knowledge to retrieve the message.

PROGRAM

HAMMING CODE

####################################

clc;

clear all;

close all;

m=4;

n=2^m-1;

k=11;

berf=[];

for i=1:10

b=0;

for j=1:50

msg=randint(500,k,[0,1]);

code=encode(msg,n,k,'hamming/binary');

t=0:0.1:10;

snr=0;

y=awgn(code,i);

y(find(y>0))=1;

y(find(y<0))=0;

msgop=decode(y,n,k,'hamming/binary');

[number,b1]=biterr(msgop,msg);

b=b+b1;

end

berf(i)=b/50;

end

semilogy(1:10,berf);

title('performance analysis in awgn for hamming codes');

xlabel('snr(db)');

ylabel('BER');

#######################################################################

CYCLIC CODES

clc;

clear all;

close all;

n=7;

k=4;

genpoly=cyclpoly(n,k,'max');

berf=[];

for i=1:10

b=0;

for j=1:50

msg=randint(500,k,[0,1]);

code=encode(msg,n,k,'cyclic/binary',genpoly);

t=0:0.1:10;

snr=0;

y=awgn(code,i);

y(find(y>0))=1;

y(find(y<0))=1;

msgop=decode(y,n,k,'cyclic/binary',genpoly);

[number,b1]=biterr(msgop,msg);

b=b+b1;

end

berf(i)=b/50;

end

semilogy(1:10,berf);

title('performance analysis in awgn for cyclic codes');

xlabel ('snr (db)');

ylabel ('BER');

#########################################################

RESULT

Thus the linear block coding technique for hamming code and cyclic code has been simulated using MATLAB.

SIMULATION OF QUADRATURE MIRROR FILTER

SIMULATION OF QUADRATURE MIRROR FILTER

AIM:

To simulate the frequency response of quadrature mirror for a two channel filter band.

THEORY:

The QMF filter is used in the sub-band coding. This filter can be used for reducing aliasing. This is a multirate digital filter structure that employes 2 decimeter in signal synthesis section. The low pass and high pass filters in the analysis section have impulse response filters (n) and (n) respectively.

Similarly the low pass filter and high pass filters contained in the synthesis section have impulse response filters (n) and (n) respectively. To reduce aliasing the synthesis section have impulse response (n) and (n) respectively,

(ω)= (ω)

(ω)=- (ω-π)

Since (ω) and (ω)is a mirror image filters

H0(ω)=H(ω)

H1(ω)=H(ω- π)

G0(ω)=2H(ω)

This is due to the above design, aliasing effects cancels.

ALGORITHM:

1. Generate the low pass filter

2. Generate the high pass filter

3. Compute the gain response of two filters

4. Plot the gain response of two filters.

QUADRATURE MIRROR FILTER:

X(ω)

FILTER CHARACTERISTICS FOR SUB-BAND CODING

Gain

H0 (ω) H1 (ω)

PROGRAM

####################################################

clc;

clear all;

%generation of complimentary lpf

b1=fir1(50,0.5);

%generation of complimentary hpf

l=length(b1);

for k=1:l

b2(k)=((-1)^k)*b1(k)

end

%computation of gain response of two filters

[H1Z,W]=freqZ(b1,1,256);

H1=abs(H1Z);

g1=20*log10(H1);

[H2Z,W]=freqZ(b2,1,256);

H2=abs(H2Z);

g2=20*log10(H2);

%PLOT OF GAIN RESPONSE OF TWO FILTERS

plot((W*180)/pi,g1,'-',(W*180)/pi,g2,'-');

grid on

xlabel('normalized freq');

ylabel('gain');

#############################################################

RESULT:

Thus the frequency response of quadrature mirror filter for a two channel filter bands was simulated.

FREQUENCY HOPPING SPREAD SPECTRUM

FREQUENCY- HOP SPREAD SPECTRUM

AIM

To simulate the frequency-hop spread spectrum modulation techniques using MATLAB program and to calculate the bit error rate.

THEORY

Spread Spectrum is a means of transmission in which the data of interest occupies a bandwidth in excess of the minimum bandwidth is necessary to send the data.

The primary advantage of a spread-spectrum communication system is its ability to reject interference whether it is the unintentional interference of another user simultaneously attempting to transmit through the channel, or the intentional interference of a hostile transmitter attempting to jam transmission. Spread Spectrum modulation was originally developed for military applications where resistance to jamming is of major concern. Another application is in multi access communication in which a number of independent users are required to share a common channel without an external synchronizing mechanism. Principles of spread spectrum modulation emphasis on direct sequence and frequency hopping techniques. In a direct sequence spread spectrum technique two stages of modulations are used. First, the incoming data sequence is used to modulate a wide band code. This code transforms the narrow band data sequence into a noise-like wide band signal. The resulting wide band signal undergoes a second modulation using a phase shift keying technique.

In a frequency-hop spread spectrum technique, the spectrum of a data modulator carrier is widened by changing the carrier frequency in a pseudo random modulator manner. For their operation both of this techniques rely on the ability of a noise like spreading code called a pseudo random or pseudo noise sequence.

PN SEQUENCE

A Pseudo Noise (PN) sequence is defined as a coded sequence of 0’s & 1’s with certain autocorrelation properties. The maximum length sequence, a type of cyclic code is commonly used as a periodic PN sequence.

In case of spread spectrum a Periodic PN sequence is used with a period of N = 2^m-1, where m is the length of the shift register. Such sequences have long periods and require simple instrumentation in the form of a linear feedback shift register. PN sequence may also be a periodic. Such sequence is known as Barker sequence.

FREQUENCY HOP SPREAD SPECTRUM

In a frequency-hop spread spectrum technique, the spectrum of a data modulated carrier is widened by changing the carrier frequency in a pseudo random manner. In this the carrier hops randomly from one frequency to another. There are two frequency hopping.

Slow-frequency hopping

In this the symbol rate R_s is an integer multiple of the hop rate R_h. ie. Several

Symbols are transmitted on each frequency hop.

R_s = nR_h

Fast-frequency hopping

In this the hop rate R_h is an integer multiple of the symbol rate R_s.ie. Carrier frequency will change or hop several times during the transmission of one symbol.

R_h = nR_s

ALGORITHM

Generate the signal which is to be spread.
Generate the PN sequence.
Multiply the PN sequence with message signal.
Display the pseudo sequence and frequency-hop spread sequence.

FREQUENCY HOPPING SPREAD SPECTRUM

clc;

clear all;

close all;

% Generation of bit pattern

s=round(rand(1,25));%generating 20 bits

signal=[];

carrier=[];

t=[0:2*pi/119:2*pi];

for k=1:25

if s(1,k)==0

sig=-ones(1,120);%120 minus ones for bit0

else

sig=ones(1,120);

end

c=cos(t);

carrier = [carrier c];

signal = [signal sig];

end

subplot(4,1,1);

plot(signal);

axis([-100 3100 -1.5 1.5]);

title('|bf|it original bit sequence');

% BPSK modulation of the signal

bpsk_sig=signal.*carrier; %modulating the signal

subplot(4,1,2);

plot(bpsk_sig);

axis([-100 3100 -1.5 1.5]);

title('|bf|itbpsk modulated signal');

% Preparation of 6 new carrier frequencies

t1=[0:2*pi/9:2*pi];

t2=[0:2*pi/19:2*pi];

t3=[0:2*pi/29:2*pi];

t4=[0:2*pi/39:2*pi];

t5=[0:2*pi/59:2*pi];

t6=[0:2*pi/119:2*pi];

c1=cos(t1);

c1=[c1 c1 c1 c1 c1 c1 c1 c1 c1 c1 c1 c1];

c2=cos(t2);

c2=[c2 c2 c2 c2 c2 c2];

c3=cos(t3);

c3=[c3 c3 c3 c3];

c4=cos(t4);

c4=[c4 c4 c4];

c5=cos(t5);

c5=[c5 c5];

c6=cos(t6);

% Random frequency hops to form a spread_signal

spread_signal=[];

for n=1:25

c=randint(1,1,[1,6]);

switch(c);

case(1)

spread_signal = [spread_signal c1];

case(2)

spread_signal = [spread_signal c2];

case(3)

spread_signal = [spread_signal c3];

case(4)

spread_signal = [spread_signal c4];

case(5)

spread_signal = [spread_signal c5];

case(6)

spread_signal = [spread_signal c6];

end

subplot(4,1,3);

plot(spread_signal);

axis([-100 3100 -1.5 1.5]);

title('|bf}if spread signal with 6 frequencies');

% Spreading BPSK signal into wideband with total of 12 frequencies

freq_hopped_sig = bpsk_sig.*spread_signal;

subplot(4,1,4);

plot(freq_hopped_sig);

axis([-100 3100 -1.5 1.5]);

title('|bf|if frequency hopped spread spectrum signal');

% Expressing the FFT's

figure,subplot(2,1,1);

plot([1:3000],freq_hopped_sig);

axis([-100 3100 -1.5 1.5]);

title('|bf|if frequency hopped spread spectrum signal or in fft');

subplot(2,1,2);

plot(abs(fft(freq_hopped_sig)));

################################################################################

RESULT

Thus the MATLAB programs for frequency hopping spread spectrum modulation techniques were executed and the waveforms were obtained.

DIRECT SEQUENCE SPREAD SPECTRUM

AIM

To simulate the direct sequence spread spectrum modulation techniques using MATLAB program and to calculate the bit error rate

THEORY

Spread Spectrum is a means of transmission in which the data of interest occupies a bandwidth in excess of the minimum bandwidth is necessary to send the data.

Spread Spectrum modulation was originally developed for military applications where resistance to jamming is of major concern. Another application is in multi access communication in which a number of independent users are required to share a common channel without an external synchronizing mechanism.

Principles of spread spectrum modulation emphasis on direct sequence and frequency hopping techniques. In a direct sequence spread spectrum technique two stages of modulations are used. First, the incoming data sequence is used to modulate a wide band code. This code transforms the narrow band data sequence into a noise-like wide band signal. The resulting wide band signal undergoes a second modulation using a phase shift keying technique.

PN SEQUENCE

DIRECT SEQUENCE SPREAD SPECTRUM

The data sequence b (t) is a narrow band sequence whereas the PN sequence c(t) is a wideband sequence. The product signal m (t) i.e.b (t)*c (t) will have a spectrum that is nearly same as that of the PN sequence.

In this the PN sequence performs the role of spreading the code by multiplying the information bearing signal b (t) by the spreading code c (t). Each information bit is chopped up into a number of small time increments. This small time increment is called as chip.

ALGORITHM

Generate the signal which is to be spread.
Generate the PN sequence.
Multiply the PN sequence with message signal.
Display the pseudo sequence and frequency-hop spread sequence.

OUTPUTS:

DIRECT SEQUENCE SPREAD SPECTRUM

ORIGINAL SEQUENCE

PSEUDO RANDOM BIT SEQUENCE

DSSS SIGNAL

FFT OF DSSS SIGNAL

RESULT

Thus the MATLAB program for direct sequence spread spectrum technique was executed and the waveforms were obtained.

PROGRAM

clc;

clear all;

s=round(rand(1,20));

Pattern=[];

for i=1:20

if(s(1,i)==1)

SS=zeros(1,6);

else

SS=ones(1,6);

end

Pattern=[Pattern SS];

end

figure,plot(Pattern);

axis([1,120,-1,2]);

title('Original Signal');

Pr=round(rand(1,120));

figure,plot(Pr);

axis([1,120,-1,2]);

title('Pseudo Random Number');

Hopped_sig=xor(Pattern,Pr);

% Modulating Hopped Signal

DSSS_sig=[];

t=1:120;

fc=0.1;

f1=cos(2.*t.*pi.*fc);

f2=cos(2.*t.*pi.*fc+pi);

for j=1:120

if(Hopped_sig(j)==true)

DSSS_sig=[DSSS_sig f1];

else

DSSS_sig=[DSSS_sig f2];

end

figure,plot(DSSS_sig);

axis([1,1200,-1,2]);

title('DSSS Signal');

figure,plot(abs(fft(DSSS_sig)));

% axis([1,120,-1,2]);

title('FFT');

Services | Embedded Support | Reviews | Virtual Technologys

SERVFORU

LINEAR BLOCKCODES

ESTIMATION OF PERIODOGRAM

SIMULATION OF BLOCK CODES HAMMING AND CYCLIC CODES

SIMULATION OF QUADRATURE MIRROR FILTER

FREQUENCY HOPPING SPREAD SPECTRUM

DIRECT SEQUENCE SPREAD SPECTRUM

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