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The following is a function to generate a Walsh Hadamard Matrix of given codeword size. The codeword size has to be a power of 2.
function [H]=generateHadamardMatrix(codeSize)
%[H]=generateHadamardMatrix(codeSize);
% Function to generate Walsh-Hadamard Matrix where "codeSize" is the code
% length of walsh code. The first matrix gives us two codes; 00, 01. The second
% matrix gives: 0000, 0101, 0011, 0110 and so on
% Author: Mathuranathan for https://www.gaussianwaves.com
% License: Creative Commons: Attribution-NonCommercial-ShareAlike 3.0
% Unported
%codeSize=64; %For testing only
N=2;
H=[0 0 ; 0 1];
if bitand(codeSize,codeSize-1)==0
while(N~=codeSize)
N=N*2;
H=repmat(H,[2,2]);
[m,n]=size(H);
%Invert the matrix located at the bottom right hand corner
for i=m/2+1:m,
for j=n/2+1:n,
H(i,j)=~H(i,j);
end
end
end
else
disp('INVALID CODE SIZE:The code size must be a power of 2');
end
Example:
To Generate Walsh Codes used in IS-95 (which utilizes 64 Walsh codes of size 64 bits each, use : [H]=generateHadamardMatrix(64). This will generate 64 Walsh Codes of length 64-bits (for each code).
Test Program:
Click Here to download
Also given below is a program to test the cross-correlation and auto-correlation of Walsh code. A set of 8-Walsh codes are used for this purpose.
% Matlab Program to test Walsh Hadamard Codes and to test their orthogonality
% Plots cross-correlation and auto correlation of Walsh Hadamard Codes
% Author: Mathuranathan Viswanathan for https://www.gaussianwaves.com
% License: Creative Commons: Attribution-NonCommercial-ShareAlike 3.0
% Unported
codeSize=8;
[H]=generateHadamardMatrix(codeSize);
%-----------------------------------------------------------
%Cross-Correlation of Walsh Code 1 with rest of Walsh Codes
h = zeros(1, codeSize-1); %For dynamic Legends
s = cell(1, codeSize-1); %For dynamic Legends
for rows=2:codeSize
[crossCorrelation,lags]=crossCorr(H(1,:),H(rows,:));
h(rows-1)=plot(lags,crossCorrelation);
s{rows-1} = sprintf('Walsh Code Sequence #-%d', rows);
hold all;
end
%Dynamic Legends
% Select the plots to include in the legend
index = 1:codeSize-1;
% Create legend for the selected plots
legend(h(index),s{index});
title('Cross Correlation of Walsh Code 1 with the rest of the Walsh Codes');
ylabel('Cross Correlation');
xlabel('Lags');
%-----------------------------------------------------------
%AutoCorrelation of Walsh Code - 1
autoCorr2(H(2,:),8,2,1);
Simulation Results
From the plots below, it can be ascertained that the Walsh codes has excellent cross-correlation property and poor autocorrelation property. Excellent cross-correlation property (zero cross-correlation) implies orthogonality, which makes it suitable for CDMA applications.
See also:
[1] Codes used in CDMA
[2] Generation of Gold Codes and their cross-correlation
[3] Preferred Pairs m-sequences generation for Gold Codes
[4] Maximum Length Sequences ( m-sequences)
[5] Introduction to Spread Spectrum
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