Uppebounding the impulse response of an IIR filter?

Hi Everybody,

Suppose that you've been provided with a stable IIR filter (which
guarantees that its impulse response will become stable). the impulse
response can be generated by dividing the numerator by denominator and
continuing it (long division) as long as desired.

There's a point that, based on the IIR filter coefficients, can anybody
give a function like C*r^-n in which n is the time index, so that it
behaves like an upperbound for the absolute value of the impulse
response? Let me know if you have any ideas.

Vahid

<vraissi@gmail.com> wrote in message 
news:1110914268.819143.257260@l41g2000cwc.googlegroups.com...
> Hi Everybody,
>
> Suppose that you've been provided with a stable IIR filter (which
> guarantees that its impulse response will become stable). the impulse
> response can be generated by dividing the numerator by denominator and
> continuing it (long division) as long as desired.
>
> There's a point that, based on the IIR filter coefficients, can anybody
> give a function like C*r^-n in which n is the time index, so that it
> behaves like an upperbound for the absolute value of the impulse
> response? Let me know if you have any ideas.
>
> Vahid
>

Hello Vahid,

From your filter, find the transfer function. Then use partial fraction 
decomposition to break the transfer function into a sum of linear and 
quadratic terms, and finally inverse Z transform each of the terms to get 
the time domain functions.

IHTH,
Clay

<vraissi@gmail.com> wrote in message 
news:1110914268.819143.257260@l41g2000cwc.googlegroups.com...
> Hi Everybody,
>
> Suppose that you've been provided with a stable IIR filter (which
> guarantees that its impulse response will become stable). the impulse
> response can be generated by dividing the numerator by denominator and
> continuing it (long division) as long as desired.
>
> There's a point that, based on the IIR filter coefficients, can anybody
> give a function like C*r^-n in which n is the time index, so that it
> behaves like an upperbound for the absolute value of the impulse
> response? Let me know if you have any ideas.
>
> Vahid
>
Vahid,

It seems to me that you might do something like this:

First, recognize that the impulse response will be a superposition of terms 
that come from conjugate pole pairs.

Next, recognize that each pole pair results in a damped sinousoid in the 
impulse response.  The closer to the unit circle, the slower the decay.

Since you don't know how the sinusoids will add without generating the 
impulse response itself (which it appears you don't want to do) then perhaps 
you assume that all the sinusoids could add at their peaks at one time or 
another.  This suggests adding their exponentially decaying envelopes to get 
an upper bound.

How you do this with proper scaling is an exercise I'll leave to the 
student.

If this is too conservative an upper bound then that is also an exercise 
I'll leave to the graduate student.

If I knew a "canned" answer, I might have given it..... but I don't.

Fred