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Re: [Tftb-help] Question about TF representations as images

From: Eric Chassande-Mottin
Subject: Re: [Tftb-help] Question about TF representations as images
Date: Thu, 11 Oct 2012 09:38:08 +0200 (CEST)
User-agent: Alpine 2.02 (DEB 1266 2009-07-14)

dear Jaidev,

let me recall the mathematical definition of the Wigner-Ville

  W(t,f)= int dtau x(t+tau) x^*(x-tau) exp(-2 pi i tau f)

which can be rewritten as

        = int dtau R_x(t,tau) exp(-2 pi i tau f)

i.e., the Fourier transform of the local autocorrelation function R_x.

tfrwv.m implements exactly this last equation.
it computes R_x and takes the Fourier transform over tau using FFT.

for icol=1:tcol,
                  % for every instance of time
ti= t(icol);
                     % ti is the time instance
      % What is taumax?

on a computer, we are working with sampled signals of finite duration. the above integral becomes a sum and the limits of sum have to be computed. tfrwv.m assumes that x(.) is zero before the first sample and after the last one. this leads to the value for taumax. tfrwv.m also allows to restrict the length N of the FFT which explains the last term in the min().

tau=-taumax:taumax; indices= rem(N+tau,N)+1;
% Why are we interested only in the indices in the 'indices' variable?
Windowing the input, maybe?

tfr(indices,icol) = x(ti+tau,1) .* conj(x(ti-tau,xcol));

as you can see, 'indices' contains the indices where the local autocorrelation components (the function R_x) will be stored in the matrix tfr. the lag axis tau runs from -taumax to +taumax. this has to be converted into a set of indices that matches the FFT convention. zero corresponds to index=1. positive indices runs from from 1 to N/2 and negative indices correspond to N/2+1:N.


octave:1> N=4
N =  4
octave:2> tau=[-1 0 1]
tau =

  -1   0   1

octave:3> rem(N+tau,N)+1
ans =

   4   1   2

          % Product in the neighbourhood of -tau:tau
                % Why this step?

the sample tau=N/2 leads to a slightly different calculation due the Hilbert symmetry of FFT.

if (ti<=xrow-tau)&(ti>=tau+1),
 tfr(tau+1,icol) = 0.5 * (x(ti+tau,1) * conj(x(ti-tau,xcol))  + ...
                          x(ti-tau,1) * conj(x(ti+tau,xcol))) ;

I'd be grateful if someone could walk me through this code, step by
step. I notice that this is script is common to most TFR scripts in
the toolbox, with minor modifications. I'm not getting what exactly it
does. I'm particularly interested in answering the questions, "Given a
signal f and a TFR method like STFT or WVD, how do you make the TFR
matrix?" If you can point me to a basic tutorial, that would be great

The TFTB includes a tutorial available at this URL:


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