NSGSITERF - Iterative nonstationary Gabor filterbank synthesis

Usage

[fr,res,Nit]=nsgsiterf(c,g,shift,M,Ls,varargin)
[fr,res,Nit]=nsgsiterf(c,g,shift,M,varargin)
[fr,res,Nit]=nsgsiterf(c,g,shift,M,Ls)
[fr,res,Nit]=nsgsiterf(c,g,shift,M)
[fr,res]=nsgsiterf(...)
fr=nsgsiterf(...)

Input parameters

`c`	Nonstationary Gabor coefficients
`g`	Cell array of filters
`shift`	Vector of shifts between the center frequencies
`M`	Number of time steps
`Ls`	Original signal length
`varargin`	Optional input pairs (see table below)

Output parameters

`fr`	Synthesized output signal
`res`	Vector of relative residuals
`Nit`	Number of iterations

Description

Given coefficients c and nonstationary Gabor frame specified by g, shift and M, this routine approximates the synthesis formula associated to the canonical dual frame.

The synthesized signal fr is obtained by first synthesizing with respect to g, shift and M using nsigtf followed by iteratively applying the inverse frame operator to the result using the conjugate gradients method. The following equivalence is used:

\begin{equation*} fr = \sum_{n=0}^{N-1}\sum_{m=0}^{M(n)-1}c\{n\}(m)S^{-1}g_{n,m} = S^{-1}\left(\sum_{n=0}^{N-1}\sum_{m=0}^{M(n)-1}c\{n\}(m)g_{n,m}\right), \end{equation*}

where N=numel(shift). The conjugate gradients algorithm uses the frame operator, or rather its efficient realization by applying nsgtf and nsigtf consecutively.

Convergence speed of the conjugate gradients algorithm depends on the condition number of the frame operator, which can be improved by preconditioning. Currently, only a diagonal preconditioner using the inverse of the frame operator diagonal is implemented.

Note: The algorithm only converges if g, shift and M form a frame.

Optional input arguments arguments can be supplied like this:

nsgsiter(c,g,shift,M,'tol',tol)

The arguments must be character strings followed by an argument:

`'tol',tol`	Error tolerance
`'Mit',Mit`	Maximum number of iterations
`'prec',prec`	Preconditioning switch

References:

K. Gröchenig. Acceleration of the frame algorithm. IEEE Trans. SSP, 41/12:3331-3340, 1993.

T. Necciari, P. Balazs, N. Holighaus, and P. Søndergaard. The ERBlet transform: An auditory-based time-frequency representation with perfect reconstruction. In to appear in Proceedings the 38th International Conference on Acoustics, Speech and Signal Processing (ICASSP 2013), 2013.