IDGT - Inverse discrete Gabor transform

Usage

f=idgt(c,g,a);
f=idgt(c,g,a,Ls);
f=idgt(c,g,a,Ls,lt);

Input parameters

`c`	Array of coefficients.
`g`	Window function.
`a`	Length of time shift.
`Ls`	Length of signal.
`lt`	Lattice type (for non-separable lattices)

Output parameters

f

Signal.

Description

idgt(c,g,a) computes the inverste discrete Gabor transform of the input coefficients c with respect to the window g and time shift a. The number of channels is deduced from the size of the coefficients c.

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

[f,g]=idgt(...) additionally outputs the window used in the transform. This is useful if the window was generated from a description in a string or cell array.

For perfect reconstruction, the window used must be a dual window of the one used to generate the coefficients.

The window g may be a vector of numerical values, a text string or a cell array. See the help of gabwin for more details.

If g is a row vector, then the output will also be a row vector. If c is 3-dimensional, then idgt will return a matrix consisting of one column vector for each of the TF-planes in c.

Assume that f=idgt(c,g,a,L) for an array c of size \(M \times N\). Then the following holds for \(k=0,\ldots,L-1\):

\begin{equation*} f(l+1) = \sum_{n=0}^{N-1}\sum_{m=0}^{M-1}c(m+1,n+1)e^{2\pi iml/M}g(l-an+1) \end{equation*}

Non-separable lattices:

idgt(c,g,a,'lt',lt) computes the Gabor expansion of the input coefficients c with respect to the window g, time shift a and lattice type lt. Please see the help of matrix2latticetype for a precise description of the parameter lt.

Assume that f=dgt(c,g,a,L,lt) for an array c of size \(M\times N\). Then the following holds for \(k=0,\ldots,L-1\):

\begin{equation*} f(l+1) = \sum_{n=0}^{N-1}\sum_{m=0}^{M-1}c(m+1,n+1)e^{2\pi iml/M}g(l-an+1) \end{equation*}

Additional parameters:

idgt takes the following flags at the end of the line of input arguments:

`'freqinv'`	Compute an IDGT using a frequency-invariant phase. This is the default convention described above.
`'timeinv'`	Compute an IDGT using a time-invariant phase. This convention is typically used in FIR-filter algorithms.

Examples:

The following example demostrates the basic pricinples for getting perfect reconstruction (short version):

f=greasy;            % test signal
a=32;                % time shift
M=64;                % frequency shift
gs={'blackman',128}; % synthesis window
ga={'dual',gs};      % analysis window

[c,Ls]=dgt(f,ga,a,M);    % analysis

% ... do interesting stuff to c at this point ...

r=idgt(c,gs,a,Ls); % synthesis

norm(f-r)          % test

The following example does the same as the previous one, with an explicit construction of the analysis and synthesis windows:

f=greasy;     % test signal
a=32;         % time shift
M=64;         % frequency shift
Ls=length(f); % signal length

% Length of transform to do
L=dgtlength(Ls,a,M);

% Analysis and synthesis window
gs=firwin('blackman',128);
ga=gabdual(gs,a,M,L);

c=dgt(f,ga,a,M); % analysis

% ... do interesting stuff to c at this point ...

r=idgt(c,gs,a,Ls);  % synthesis

norm(f-r)  % test