% 1-D Hilber transform and amplitude demodulation

% writen by Lu, 7/28/2015

clc; clear all; close all;

% setup

X=linspace(0,1,1024);

ac=255/2; dc=ac;

Y=255/2+(255/2)*sin(2*pi*5*X);

Yac=Y-dc;

% Hilbert transfrom (-90 degree phase shifter), x=xr+i*xi

% G(k)=F(k)*H(k), H(k)=-i*sgn(k); k>0, sgn=1 and k<0, sgn=-1

% If xr=b*cos(phai), then xi=b*sin(phai);

% If xr=b*sin(phai), then xi=-b*cos(phai);

% Transfrom steps: x=ac+dc*b*cos(phai)

% 1) fft on x, F=fft(x)

% 2) frequency processing： F(1)=0, remove dc component;

% 2) F(p+1:end)=F(p+1:end)*(-1), p=ceil(length(F)/2),

% 2) that is, mutiply the negative frequencies by -1

% 3) apply ifft, the obtained signal is i*xi

% Those can easily proofed by inspection of dirac functions forms of

% Fourier transfrom of cosine and sine functions

P=ceil(length(Y)/2); % note how fftshift/ifftshift works

Y_fft(1)=0;

Y_fft(P+1:end)=Y_fft(P+1:end)*(-1);

Yht=ifft(Y_fft);

Yreal=real(Yht);

Yimag=imag(Yht);

% Amplitude demodulation

% Amp=abs(Yac+Yht);

Amp=sqrt(Yac.^2+Yimag.^2);

figure; plot(X, Yreal,'r-',X,Yimag,'g-','linewidth',2);

hold on;

plot(X,Yac,'b-','linewidth',2);

plot(X,Amp,'k-','linewidth',2);

xlabel('spatial distance'); ylabel('intensity');

title('Hilbert transform based demodulation');

%Comments:

% The 1-D transfrom can be extended to 2-D space by line by line processing

% But note boundary effect esists in the hilbert-transform based demodulation

% Further cutoff should be considered to mitigate the boundary effects

% compare with spiral-phase-operator based transfrom (vortex transform)