The objective of this section is to use time-frequency analysis methods to reconstruct contaminated or masked auditory spectra. In many acoustic applications, a spectral estimate of the source is found to be masked in a particular frequency band because of filtering in the environment, or because of jamming by competing sound sources. In these cases, it can be shown that certain techniques involving early auditory spectral representations, combined with time frequency analysis methods, may be able to recover the lost spectral information. One such approach can be formulated as follows, and is based on the assumption that the cochlear analysis of sound signal f can be thought of as the wavelet transform [.Yang Wang Shamma 1992.]
where t is time, 
is convolution, g is the impulse response for the
cochlear filter G [.Allen 1985.], 
is a scale channel for some
a>1, and 
. Including the effects of the
nonlinear transduction and membrane leakage in the haircell, the cochlear
output at each channel s is given by:
where 
is an instantaneous sigmoidal nonlinearity applied to 
followed by a lowpass filter H with impulse response h, and 
designates composition. The sound signal can be fully reconstructed from
the cochlear outputs in the case of a lateral inhibitory network
[.Morishita Yajima 1972, Shamma 1985 spee2.], by means of a Gabor irregular
sampling theory developed in [.Benedetto Heller 1990, Benedetto 1992.].
This irregular sampling solution for sound reconstruction was given in
[.Benedetto Teolis 1993.] cf., the solution in terms of alternating
projections given in [.Yang Wang Shamma 1992.].
To specify this formulation for the case of omitted spectral data, and
because of the form of 
, we shall study operators L defined by
, where the filter H corresponding to h is no
longer necessarily low pass. Our goal is to reconstruct f
from cochlear-output-data Lf in the case when some spectral information is
omitted from this data.
The first step in this project is to develop nonlinear wavelet packet
algorithms for operators 
and 
where 
are essentially quadrature mirror
filters. When these algorithms are formulated we shall design filters and
dilation operators so that the frequency domain is decomposed in prescribed
ways for an appropriate level of the nonlinear wavelet packet pyramid.
This means that our frequency decompositions must include intervals
associated with poor signal reconstruction, as might occur in a masking
problem.
Technically, the aforementioned algorithms and filter design must provide a
method of signal reconstruction. We have accomplished this in special cases
[.Benedetto Saliani 1994.]. A full solution will involve the inversion of
nonlinear ``cochlear output'' operators compatible with linear wavelet packet
theory. Formally, the computation of f from available data 
and
at this first level of our nonlinear wavelet packet pyramid is
where 
and 
designate the Fourier transform and inverse
Fourier transform, respectively. The validity of this inversion formula
depends on natural constraints imposed on the nonlinearity 
. A
priori, there are problems of computational complexity especially when the
omitted frequency information for the error estimation problem is at a low
level of the pyramid. However, the linear wavelet packet reconstructions
are 
-order algorithms, and there are reasons to believe some of
this efficiency can be transmitted to the nonlinear case.