kasami_example.m

%This demonstrates the small set of Kasami seqences.  Kasami sequences are
%sequences with good periodic auto-correlation and cross-correlation
%properties.

%For more details, see Pingzhui Fan and Michael Darnell, "Sequence Design
% for communications applications," Research  Studies Press, Tauton, 1996.

m=10;%order of sequence (must be even)

%Kasami sequences are generated from a maximal length sequence, a, of length
%2^m-1 where m is even.  A second sequence, b, is generated by sampling a
%by a factor of 1+2^(m/2), which is a maximal length sequence of period
%2^(m/2)-1. Kasami sequences are the set of sequences of
%xor(a,circshift(b)).  2^(m/2) sequences are calculated.

%The function kasami() performs this calculation, with each row of K being
%a sequence.

K=kasami(m);

%Now we check the autocorrelation
[N_seq N]=size(K);

acorr=zeros(N_seq,N);

for i=1:N_seq
    acorr(i,:)=xcorr_fft(K(i,:),K(i,:));
end

%The autocorrelation should have a peak of N at lag=0.  All other values
%should be equal to one of the possible values of R_max:
R_max=-1+[0 -2^(m/2) 2^(m/2)];

figure(1)
plot(0:(N-1),acorr','.')
hold on
h=plot([0 N-1],[R_max;R_max],'k');
hold off
xlabel('Lag')
ylabel('Auto Correlation')
legend(h(1),'R_{max}')

%check cross-correlation for one sequence

k_check=2;%index of sequence to check

xcorr=zeros(N_seq,N);

for i=1:N_seq
    xcorr(i,:)=xcorr_fft(K(i,:),K(k_check,:));
end
xcorr(k_check,:)=[];%clear autocorr

%Again, all values should be one of R_max
figure(2)
plot(0:(N-1),xcorr','.')
hold on
h=plot([0 N-1],[R_max;R_max],'k');
hold off
xlabel('Lag')
ylabel('Cross Correlation')
legend(h(1),'R_{max}')

%Now, a practical example.  Can we pick out the delay of one sequence, when
%the other sequences are added to it (as noise).

k_target=2;%index of target sequence

delays=round(N*rand(1,N_seq));%delay to apply to each sequence

signal=zeros(1,N);

for i=1:N_seq
    signal=signal+circshift(K(i,:),[0 delays(i)]);
end

sigma_noise=1;%add some Gaussian noise (this can be larger with larger m)

signal=signal+sigma_noise*randn(size(signal));

%the peak of the cross-correlation between the noisy signal and the target
%sequence should reveal the delay

xc_delay=xcorr_fft(signal,K(k_target,:));

figure(3)

plot(0:(N-1),xc_delay','.')
hold on

[xc_max idx_max]=max(xc_delay);
h2=plot(idx_max-1,xc_max,'ro');
h1=plot(delays(k_target)*[1 1],xc_max*[-1 1],'k');
hold off
xlabel('Lag')
ylabel('Cross Correlation')
legend([h1(1) h2],'Target Lag','Measured Lag','Location','Best')

%notice that although the noise (interfering signals plus Gaussian) is much
%larger than the signal, the larger number of samples allows us to pick out
%the transmitted signal.


%
%Copyright (c) 2009, Travis Wiens
%All rights reserved.
%
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