# The Large Time-Frequency Analysis Toolbox

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# ISGRAM - Spectrogram inversion

## Usage

f=isgram(c,g,a);
f=isgram(c,g,a,Ls);
[f,relres,iter]=isgram(...);


## Input parameters

 c Array of coefficients. g Window function. a Length of time shift. Ls length of signal.

## Output parameters

 f Signal. relres Vector of residuals. iter Number of iterations done.

## Description

isgram(s,g,a) attempts to invert a spectrogram computed by

s = abs(dgt(f,g,a,M)).^2;


using an iterative method.

isgram(c,g,a,Ls) does as above but cuts or extends f to length Ls.

If the phase of the spectrogram is known, it is much better to use idgt.

[f,relres,iter]=isgram(...) additionally return the residuals in a vector relres and the number of iteration steps iter.

Generally, if the spectrogram has not been modified, the iterative algorithm will converge slowly to the correct result. If the spectrogram has been modified, the algorithm is not guaranteed to converge at all.

isgram takes the following parameters at the end of the line of input arguments:

 'lt',lt Specify the lattice type. See the help on matrix2latticetype. 'zero' Choose a starting phase of zero. This is the default 'rand' Choose a random starting phase. 'int' Construct a starting phase by integration. Only works for Gaussian windows. 'griflim' Use the Griffin-Lim iterative method, this is the default. 'bfgs' Use the limited-memory Broyden Fletcher Goldfarb Shanno (BFGS) method. 'tol',t Stop if relative residual error is less than the specified tolerance. 'maxit',n Do at most n iterations. 'print' Display the progress. 'quiet' Don't print anything, this is the default. 'printstep',p If 'print' is specified, then print every p'th iteration. Default value is p=10;

The BFGS method makes use of the minFunc software. To use the BFGS method, please install the minFunc software from: http://www.cs.ubc.ca/~schmidtm/Software/minFunc.html.

## Examples:

To reconstruct the phase of 'greasy', use the following:

% Setup the problem and the coefficients
f=greasy;
g='gauss';
a=20; M=200;
c=dgt(f,g,a,M);
s=abs(c).^2;
theta=angle(c);

% Reconstruct and get spectrogram and angle
r=isgram(s,g,a);
c_r=dgt(r,g,a,M);
s_r=abs(c_r).^2;
theta_r=angle(c_r);

% Compute the angular difference
d1=abs(theta-theta_r);
d2=2*pi-d1;
anglediff=min(d1,d2);

% Plot the difference in spectrogram and phase
figure(1);
plotdgt(s./s_r,a,16000,'clim',[-10,10]);
colormap([bone;flipud(bone)])
title('Relative difference in spectrogram');

figure(2);
plotdgt(anglediff,a,16000,'lin');
colormap(bone);
title('Difference in angle');


## References:

R. Decorsière and P. L. Søndergaard. Modulation filtering using an optimization approach to spectrogram reconstruction. In Proceedings of the Forum Acousticum, 2011.

D. Griffin and J. Lim. Signal estimation from modified short-time Fourier transform. IEEE Trans. Acoust. Speech Signal Process., 32(2):236--243, 1984.

D. Liu and J. Nocedal. On the limited memory BFGS method for large scale optimization. Mathematical programming, 45(1):503--528, 1989.