DemoFourierProp.m

% Show that the Fourier transform of the autocorrelation
% of a sequence is the magnitude squared of the Fourier transform of the
% sequence
%% Autocorrelation section

seq = [-1 -1]; % Original sequences

autoCorr = xcorr(seq)

% Tarek likes to trim the autocorrelation (good for US processing?)
% a = autoCorr(length(aseqPad)/2:length(seqPad)/2+length(seqPad)-1);
magFTACF = abs(fft(autoCorr))

%%
% Spectral power density
% Pad with zeroes (so circular convolution doesn't introduce extra terms)
% (do this earlier?)
seqPad = [zeros(1,length(seq)-1) seq ]; 
spd = abs(fft(seqPad)).^2 %fft(seqPad).*conj(fft(seqPad)) % Spectral power density
iftToGetACF = ifft(spd)

% So why do we have this implied shift in time?
% Also, why does padding zeroes in time work?

%% Demo finding symmetric code
% g = [0 0 -1 0], f = [0 0 0 -1]
% This is complementary pair, and g is symmetric
f = [0 0 0 -1];
abs(fft(xcorr(f)))

g = [0 0 -1 0]
abs(fft(xcorr(f)))

sqrt(2 - abs(fft(xcorr(f))))

% Probably need to padd zeroes somewhere
% But this does tell us the magnitudes we need for g, which is nice

%% Demo how cross correlation properties show up in Fourier domain

% Set up complementary orthogonal sequences
A = [1 1 3 4   % A1
    1 -1 5 6];  % A2

B = [-1 1 7 8   % B1
    -1 -1 9 1]; % B2

corrFirst = fft(conv(A(1,:),fliplr(B(1,:))));
orig = xcorr(A(1,:),B(1,:))

corrResult = ifft(corrFirst)

FFTFirst = fft([A(1,:) zeros(1,size(A,2))]) .* conj(fft([ B(1,:) zeros(1,size(B,2))]));
FTResult = ifft(FFTFirst)