impulse response / phase

porterboy76@yahoo.com wrote:
> 
> I agree with you of course. I was just making the point that the
> "frequency response" is known, technically, but doesn't tell you very
> much. Extra computation is needed to obtain the phase response. But
> this will generally give you the magnitude response as well.

I don't understand. If you don't consider the multiplication, addition,
etc. "computation" when evaluating the frequency response for any given
value of w then why does it become "computation" when evaluating the
phase response of a particular value of w?

-jim

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Rick Lyons wrote:
...
> >Assume a digital system with impulse response h(n). Take the
> >z-transform of that to compute the transfer function H(z). Then the
> >phase response is arg( H(e^(i w) ) ) and the magnitude response is |
> >H(e^(i w)) |. One doesn't need to compute the magnitude response to
> >compute the phase response (which was the question that I answered
with
> >"yes").
> >
> >Regards,
> >Andor
>
> Hi Andor,
>
>   Ha ha.  Words words words.
>
> The original poster asked if we compute the
> phase response *without* computation of the frequency
> response.
>
> You answered with "One doesn't need to compute the
> magnitude response to compute the phase response",
> which I believe is correct.  But it seems to me that
> to compute the phase response we must compute the
> frequency response.

Granted. I was thinking of the case where the frequency response H(e^i
w) was computed as real and imaginary part. If H(z) is some rational
function, this would be a viable approach.

>
> I could be wrong.

I think you are right. Perhaps the correct answer to the OP's question
would have been to ask _why_ he did not want to compute the whole
frequency response?

An interesting related question is how to go about this in practice.
Let's assume you have some digital LTI system, and you input an impulse
and measure the response. It might well be that the system is IIR.
Let's assume further that one truncates the impulse response after it
has decayed below a certain threshold below the main impulse (the main
impulse might have a large delay, think of a large order linear-phase
system in series to the IIR system). Now you are stuck with an impulse
response that might well be several thousand samples long -- this means
that you have a trigonometric function with several thousand sin- and
cos-terms for the frequency response. There might be some numerical
issue with exactly evaluting this.

It would seem more appropriate to fit an ARMA-model to the impulse
response, ie. find an IIR-filter with low order and fit the
coefficients such that the error of the impulse response is minimized
(in whatever criterion). The delay of the main impulse might be a good
indicator for the order of the MA part. There must be some literature
on this -- I remember there is a digital audio unit that does just
about that: it samples the impulse response of other audio units to
"clone" them. Their webpage is http://www.sintefex.com, they once wrote
a paper for the AES on how their algorithm worked (they also model
non-linear units with dynamic impulse response "swapping" -- quite
sophisticated).


Regards,
Andor