NSGAITERF - Iterative nonstationary Gabor filterbank analysis

Usage

[c,Ls,res,Nit]=nsgaiterf(f,g,shift,M,varargin)
[c,Ls,res,Nit]=nsgaiterf(f,g,shift,M)
[c,Ls,res]=nsgaiterf(...)
[c,Ls]=nsgaiterf(...)
c=nsgaiterf(...)

Input parameters

f Input signal
g Cell array of filters
shift Vector of shifts between the center frequencies
M Number of time steps
varargin Optional input pairs (see table below)

Output parameters

c Filterbank coefficients
Ls Input signal length
res Vector of relative residuals
Nit Number of iterations

Description

Given a function f and nonstationary Gabor filterbank frame specified by g, shift and M, this routine approximates the frame coefficients associated to the canonical dual frame.

The approximated coefficients are obtained by first applying the inverse frame operator to f iteratively using the conjugate gradients method, followed by computing the analysis coefficients of \(S^{-1}f\) with respect to g, shift and M with nsgtf, followed. The following equivalence is used:

\begin{equation*} c\{n\}(m) = \langle f, \mathbf{S}^{-1} g_{n,m} \rangle = \langle \mathbf{S}^{-1} f, g_{n,m} \rangle \end{equation*}

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:

nsgaiterf(f,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.