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[Tftb-help] Algorithm for discrete ZAM distribution

 From: Sean A Fulop Subject: [Tftb-help] Algorithm for discrete ZAM distribution Date: Thu, 13 Aug 2009 18:10:31 -0700 (PDT)

Greetings,

I am writing a book for speech scientists about methods for spectrum analysis.  I have found the ZAM distribution to be a useful method in many cases, but there is a real scarcity of literature documenting the method in discrete form.

There is a chapter in the signal processing compilation by Boashash (2003) that describes a general algorithm for discrete implementation of Cohen-class distributions.  This involves a discrete convolution of the kernel function with the instantaneous autocorrelation matrix.   There is a brief discussion of the theoretical discrete ZAM kernel, which mentions that this is a difficult kernel to perform a discrete convolution with.  There is mention of a need to "approximate" the theoretical discrete kernel for this distribution.

The algorithm employed in the tftb code, by contrast, does not appear to compute this discrete convolution directly.  It seems instead to implement some sort of discrete form of the continuous ZAM distribution that appears in the documentation (also found in Cohen 1995).  This equation "compiles" the time-lag form of the kernel into the equation more directly, implementing the cone-shape as special limits on the inner time summation.  It appears that the tftb algorithm uses this method somehow, but I can find no literature presenting this algorithm or discussing how it differs from the "approximate" discrete convolution advocated by Boashash 2003.

Accordingly I am a little confused about what the algorithm is in fact doing and how its parameters work.  I see that the "frequency smoothing window" refers to the lag window; I am not sure what is the "time smoothing window."   Is this something that has to be added to the discrete implementation, which is not found in the continuous equation for the ZAM distribution?  The ZAM is an example of joint smoothing rather than separable smoothing, is it not?

Any helpful comments or explanation of the tftb algorithm for ZAM will be appreciated.

-Sean

Sean A. Fulop, PhD
Associate Prof. of Linguistics
California State University Fresno
5245 N. Backer Ave. PB92
Fresno, CA 93740-8001