Re: The natural spectrogram, (Julius Smith )

Subject: Re: The natural spectrogram,
From:    Julius Smith  <jos(at)CCRMA.STANFORD.EDU>
Date:    Thu, 29 Jan 2004 11:09:58 -0800

At 01:48 AM 1/28/2004, Eckard Blumschein wrote:
>...So far I can neither imagine the
>STFT itself to be natural nor a spectrogram based on it. Wouldn't this
>require to naturally choose size of the window?

Yes -- and as a function of frequency.  We normally call it a
"multiresolution" STFT.

>Wouldn't one have to decide further arbitrary parameters like the degree
>of overlap?

This is just a sampling-rate issue.  If computational cost is no object,
one can simply choose maximum overlap (i.e., a "sliding FFT" instead of a
"hopping FFT").  On the other hand, FFT filter banks can usually be
downsampled quite a lot and still give equivalent end results.  In this
context, your window is your anti-aliasing filter for
downsampling.  Reference: Jont B. Allen, "Short Term Spectral Analysis,
Synthesis, and Modification by Discrete Fourier Transform", IEEE ASSP-25(3).

>  Doesn't any usual spectrogram incompletely represent the information?

STFTs are normally invertible, in my experience, even in the presence of
aliasing due to downsampling (it gets canceled in the reconstruction).  The
classic spectrogram discards phase, so it is not exactly invertible.  Of
course, it is well known that phase can be reconstructed from STFT
magnitude to a large extent for typical signals and analysis conditions.

>Isn't the usual spectrogram subject to the notorious trade-off beween
>spectral and temporal resolution?

Well sure, but we can let the human ear tell us where to be on that trade-off.

>Was there any physiological justification for STFT which could include the
>rectification?
>Is there close similarity to measurement of BM motion and neural pattern?

I don't understand the first question.  My understanding of rectification
that this is the nature of how the hair cells respond to basilar membrane
vibration.  Firing increases when the membrane pushes one way, but not the
other.  The STFT implements a filter bank, and the output of that filter
bank can be rectified accordingly (applied to real time-domain signals at
the STFT filter-bank output, of course).

>I am sceptical in all of these and further details.

I suppose you're posting to the right list!

Julius