GABFIRDUAL - Compute FIR dual window

Usage

gd=gabfirdual(Ldual,g,a,M);
gd=gabfirdual(Ldual,g,a,M, varagin);

Input parameters

`Ldual`	Length of dual window
`g`	Window function
`a`	Time shift
`M`	Number of Channels
`alpha1`	Weight of \(l^1\)-norm in the time domain
`alpha2`	Weight of \(l^1\)-norm in the freq. domain

Output parameters

`gd`	Dual window

Description

gabfirdual(Ldual,g,a,M) computes an FIR window gd which is an approximate dual window of the Gabor system defined by g, a and M. The FIR dual window will be supported on Ldual samples.

This function solve a convex optimization problem that can be written as:

\begin{equation*} \begin{split} \text{gd} = & \text{arg} \min_x \| \alpha x \|_1 + \| \beta \mathcal{F}x\|_1 \\\\ & + \| \omega (x - g_l) \|_2^2 \\\\ & \delta \| x \|_{S0}+ \mu \| \nabla x \|_2^2 +\gamma \| \nabla \mathcal{F} x \|_2^2 \\\\ & \text{such that } x \text{ is a dual window of }g \end{split} \end{equation*}

Note: This function require the unlocbox. You can download it at http://unlocbox.sourceforge.net

The function uses an iterative algorithm to compute the approximate FIR dual. The algorithm can be controlled by the following flags:

`'alpha',alpha`	Weight in time. If it is a scalar, it represent the weights of the entire L1 function in time. If it is a vector, it is the associated weight assotiated to each component of the L1 norm (length: Ldual). Default value is \(\alpha=0\). Warning: this value should not be too big in order to avoid the the L1 norm proximal operator kill the signal. No L1-time constraint: \(\alpha=0\)
`'beta',beta`	Weight in frequency. If it is a scalar, it represent the weights of the entire L1 function in frequency. If it is a vector, it is the associated weight assotiated to each component of the L1 norm in frequency. (length: Ldual). Default value is \(\beta=0\). Warning: this value should not be too big in order to avoid the the L1 norm proximal operator kill the signal. No L1-frequency constraint: \(\beta=0\)
`'omega',omega`	Weight in time of the L2-norm. If it is a scalar, it represent the weights of the entire L2 function in time. If it is a vector, it is the associated weight assotiated to each component of the L2 norm (length: Ldual). Default value is \(\omega=0\). No L2-time constraint: \(\omega=0\)
`'glike',g_l`	\(g_l\) is a windows in time. The algorithm try to shape the dual window like \(g_l\). Normalization of \(g_l\) is done automatically. To use option omega should be different from 0. By default \(g_d=0\).
`'mu',mu`	Weight of the smooth constraint Default value is 1. No smooth constraint: \(\mu=0\)
`'gamma',gamma`	Weight of the smooth constraint in frequency. Default value is 1. No smooth constraint: \(\gamma=0\)
`'delta',delta`	Weight of the S0-norm. Default value is 0. No S0-norm: \(\delta=0\)
`'dual'`	Look for a dual windows (default)
`'painless'`	Construct a starting guess using a painless-case approximation. This is the default
`'zero'`	Choose a starting guess of zero.
`'rand'`	Choose a random starting phase.
`'tol',t`	Stop if relative residual error is less than the specified tolerance.
`'maxit',n`	Do at most n iterations. default 200
`'print'`	Display the progress.
`'debug'`	Display all the progresses.
`'quiet'`	Don't print anything, this is the default.
`'fast'`	Fast algorithm, this is the default.
`'slow'`	Safer algorithm, you can try this if the fast algorithm is not working. Before using this, try to iterate more.
`'printstep',p`	If 'print' is specified, then print every p'th iteration. Default value is p=10;
`'hardconstraint'`	Force the projection at the end (default)
`'softconstaint'`	Do not force the projection at the end