Coefficients for A-weighting filter

On Thu, 21 Dec 2006 17:44:00 -0500, Randy Yates <yates@ieee.org> wrote:

>What I'm wondering is if it can be proven that the algorithm in
>general will produce a stable IIR transfer function, i.e., how can you
>guarantee the poles it generates are always inside the unit circle?

As far as I know, you can't.  However, in my experience, with only some
pathological exceptions, stable prototype systems have always yielded
stable models and unstable prototype systems have always yielded
unstable models.  The exceptions were cases where pole(s) were extremely
close to the jw-axis (like real part > -0.0001) or to the unit circle
(like radius > .9999), where the model pole might slip to an unstable
position.  That's why I was so surprised when the A-weighting filter
produced an unstable model -- it's a fairly benign transfer function.

The problem in this case had to do with the way that I implemented the
"artificial delay" mentioned in the magazine article.  While the
explanation is probably too complex for a brief comp.dsp post, I can
summarize:

I use a "cosine" input instead of a "sine" input because of
observability problems with "sine" near half the sampling frequency.
(This is also mentioned in the article.)  If the frequency of a sine
wave is "Fs/2", and you sample it at "Fs", you only capture the
zero-crossings.  So, assuming that the output of the system at Fs/2 is
nonzero, the algorithm interprets the situation as
"zero-input/nonzero-output".  It deals with this by placing one or more
poles at an angle of PI on the Z-plane, and those poles are often
outside the unit circle.

In my computer program I made the mistake of adding the "artificial
delay" not only to the output sequence, but to the input sequence as
well.  Well, if you delay a cosine wave you phase-shift it.  Trig
identities will tell you that a phase-shifted cosine is equal to a
cosine plus a sine.  That little bit of sine wave on the input led to
the unstable pole(s) in the transfer function.  I solved the problem by
eliminating the phase shift from the input sequence, allowing it only in
the output sequence.

The net effect of this is that if you inject "delta" samples of
"artificial delay" into the frequency response, the first "delta"
coefficients of the numerator of the transfer function will be
approximately zero.  (Approximately, not exactly, because of the least
squares nature of the model fit.)

Now a teaser, for future study:  It can be shown that a cosine input
formulation models the real part of the frequency response, and that a
sine input formulation models the imaginary part (and is almost always
unstable because of the observability problems mentioned above).  When I
was in graduate school I was trying to find a way to eliminate that sine
formulation instability.  But I had to move on with my life before I
solved the problem.

Greg