Pole/Zero/Impulse response

Greetings,

Are there any set rules for figuring out the effect of the zeros and
poles location in the zplane on the impulse response. for example I
think that the larger the pole then the decay in impulse response
should be faster.
One could find the effect of zeros and poles location on the frequency
response by simply moving on the unit circle and then the frequency
response goes to infinity when getting close to a pole and to zero
when you get close to a zero. I am looking for a similar way to find
the effect of pole zero location on the impulse response.

thanks
mehdi

mehdi wrote:
> Greetings,
>
> Are there any set rules for figuring out the effect of the zeros and
> poles location in the zplane on the impulse response. for example I
> think that the larger the pole then the decay in impulse response
> should be faster.
> One could find the effect of zeros and poles location on the frequency
> response by simply moving on the unit circle and then the frequency
> response goes to infinity when getting close to a pole and to zero
> when you get close to a zero. I am looking for a similar way to find
> the effect of pole zero location on the impulse response.

You won't find many people here willing to answer your university questions
for you...

Do some research, buy a book or two, find the DSPGuide and read it -
something like that.

Ben
-- 
I'm not just a number.  To many, I'm known as a string...

Hello Mehdi,

Yes, there is a rule. Go look up a neat theorem by Oliver Heaviside. It will
let you directly write down the impulse response given the poles and zeroes
of the transfer function.

Clay


"mehdi" <mehdi01@penguin.poly.edu> wrote in message
news:faa6ff00.0308071237.6ebe04a3@posting.google.com...
> Greetings,
>
> Are there any set rules for figuring out the effect of the zeros and
> poles location in the zplane on the impulse response. for example I
> think that the larger the pole then the decay in impulse response
> should be faster.
> One could find the effect of zeros and poles location on the frequency
> response by simply moving on the unit circle and then the frequency
> response goes to infinity when getting close to a pole and to zero
> when you get close to a zero. I am looking for a similar way to find
> the effect of pole zero location on the impulse response.
>
> thanks
> mehdi

mehdi wrote:
> 
> Greetings,
> 
> Are there any set rules for figuring out the effect of the zeros and
> poles location in the zplane on the impulse response. for example I
> think that the larger the pole then the decay in impulse response
> should be faster.

A pole is a pole. Waht do you mean by larger?

> One could find the effect of zeros and poles location on the frequency
> response by simply moving on the unit circle and then the frequency
> response goes to infinity when getting close to a pole and to zero
> when you get close to a zero. I am looking for a similar way to find
> the effect of pole zero location on the impulse response.

The impulse and frequency (with phase) responses are linked. If you
figure one out, you found the other. Are you asking about how to do
that?
> 
> thanks
> mehdi

Jerry
-- 
Engineering is the art of making what you want from things you can get.
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mehdi01@penguin.poly.edu (mehdi) wrote in message news:<faa6ff00.0308071237.6ebe04a3@posting.google.com>...
> Greetings,
> 
> Are there any set rules for figuring out the effect of the zeros and
> poles location in the zplane on the impulse response. for example I
> think that the larger the pole then the decay in impulse response
> should be faster.
> One could find the effect of zeros and poles location on the frequency
> response by simply moving on the unit circle and then the frequency
> response goes to infinity when getting close to a pole and to zero
> when you get close to a zero. I am looking for a similar way to find
> the effect of pole zero location on the impulse response.
> 
> thanks
> mehdi

It seems that non of the people who replied understood the question,
of course given the zeros and poles one could calculate and obtain the
impulse response. But that is not the question. (as Ben noted this is
in every dsp book).

What I am asking is if one knows the effect on the impulse response
for example as the pole location moves from the origin to -1
(similarly for zeros). That is without calculating the impulse
response every time!!!

mehdi

"Clay S. Turner" <physicsNOOOOSPPPPAMMMM@bellsouth.net> wrote in message news:<ldCYa.1251$YQ4.61@fe04.atl2.webusenet.com>...
...
> Yes, there is a rule. Go look up a neat theorem by Oliver Heaviside. It will
> let you directly write down the impulse response given the poles and zeroes
> of the transfer function.

... to within a constant factor, i think.

but what neat theorem is this, Clay?  the Heaviside partial fraction
expansion theorem?

poles go directly to "alpha" parameters in the exponential impulse
response, but the zeros map to the impulse response more obscuredly,
don't they?

just curious.

r b-j

I think its a very thought provoking question.
Assuming that you are referring to impulse response in time domain,My
reasoning would be the following:

THe magnitude of the pole or zero does not affect the decay time of
impulse response. What matters here is the argument(angle) of the
zeros i.e. omega=2*pi*F/Fs
The smaller the value of omega for the zeros, the faster the frequency
response decays. Now, this leads to the opposite effect in the time
domain.
Because,contraction in frequency domain is expansion in time domain
and vice versa. Thus, as omega value is small, impulse response decay
in the time domain would be slower and vice versa.

I welcome more discussion on this.



Jerry Avins <jya@ieee.org> wrote in message news:<3F33CF56.B5193468@ieee.org>...
> mehdi wrote:
> > 
> > Greetings,
> > 
> > Are there any set rules for figuring out the effect of the zeros and
> > poles location in the zplane on the impulse response. for example I
> > think that the larger the pole then the decay in impulse response
> > should be faster.
> 
> A pole is a pole. Waht do you mean by larger?
> 
> > One could find the effect of zeros and poles location on the frequency
> > response by simply moving on the unit circle and then the frequency
> > response goes to infinity when getting close to a pole and to zero
> > when you get close to a zero. I am looking for a similar way to find
> > the effect of pole zero location on the impulse response.
> 
> The impulse and frequency (with phase) responses are linked. If you
> figure one out, you found the other. Are you asking about how to do
> that?
> > 
> > thanks
> > mehdi
> 
> Jerry

mehdi01@penguin.poly.edu (mehdi) wrote in message news:<faa6ff00.0308081854.25e2b48a@posting.google.com>...
> 
> It seems that non of the people who replied understood the question,
> of course given the zeros and poles one could calculate and obtain the
> impulse response. But that is not the question. (as Ben noted this is
> in every dsp book).
> 
> What I am asking is if one knows the effect on the impulse response
> for example as the pole location moves from the origin to -1
> (similarly for zeros). That is without calculating the impulse
> response every time!!!
> 
> mehdi

But I think they did. You can write the generic impulse response
formula for a set of poles and zeros and calculate their effect as you
move them around. Why would you have to calculate the impulse response
each and every time?

For instance, suppose you have a single pole, -1 < a < 1 on the real
line. Then the impulse response is h(n) = a^n for n >= 0. If you
define the time constant as the time it takes the impulse response to
decline to 1/10 of its value at n = 0 the time constant is simply T =
log(0.1)/log(|a|). So, you can see that as "a" becomes smaller T
become bigger.

Hello Robert,

Yes, I'm referring to the Heaviside expansion theorem. While it is true that
it doesn't hand Medhi's answer over on a silver platter, but it does allow
one to go and dig into the problem some. Certainly the poles give the
exponential decay rates. So the immediate effect on moving the poles can be
ascertained. Likewise the zeroes (yes a little more obscure) effect a
combination of the exponentials.

The whole problem of relating a polynomial to its roots is quite nonlinear
and  very small changes in a coefficient in one representation can make for
huge changes in the other. So while my suggestion is not a panacea for the
Medhi's question, I think he will have to be happy with partial answers for
anything short of doing a complete transform.

Another limitation of this approach is the requirement of non-repeated real
roots.

Clay




"robert bristow-johnson" <rbj@surfglobal.net> wrote in message
news:4cbb922e.0308082033.707ac98b@posting.google.com...
> "Clay S. Turner" <physicsNOOOOSPPPPAMMMM@bellsouth.net> wrote in message
news:<ldCYa.1251$YQ4.61@fe04.atl2.webusenet.com>...
> ...
> > Yes, there is a rule. Go look up a neat theorem by Oliver Heaviside. It
will
> > let you directly write down the impulse response given the poles and
zeroes
> > of the transfer function.
>
> ... to within a constant factor, i think.
>
> but what neat theorem is this, Clay?  the Heaviside partial fraction
> expansion theorem?
>
> poles go directly to "alpha" parameters in the exponential impulse
> response, but the zeros map to the impulse response more obscuredly,
> don't they?
>
> just curious.
>
> r b-j

On Sat, 9 Aug 2003 09:34:48 -0400, "Clay S. Turner"
<physicsNOOOOSPPPPAMMMM@bellsouth.net> wrote:

>Hello Robert,
>
>Yes, I'm referring to the Heaviside expansion theorem. While it is true that
>it doesn't hand Medhi's answer over on a silver platter, but it does allow
>one to go and dig into the problem some. Certainly the poles give the
>exponential decay rates. So the immediate effect on moving the poles can be
>ascertained. Likewise the zeroes (yes a little more obscure) effect a
>combination of the exponentials.
>
>The whole problem of relating a polynomial to its roots is quite nonlinear
>and  very small changes in a coefficient in one representation can make for
>huge changes in the other. So while my suggestion is not a panacea for the
>Medhi's question, I think he will have to be happy with partial answers for
>anything short of doing a complete transform.
>
>Another limitation of this approach is the requirement of non-repeated real
>roots.
>
>Clay
>

   (snipped)


Hi,
  While you guys are discussing Heaviside.  I read somewhere 
(I can't recall where, at this time) that Oliver was the guy 
who developed the so-called Maxwell's equations for electromagnetic 
fields.  What I read was that Maxwell described 
electromagnetic fields with 20 equations, and Heaviside 
condensed the 20 equations down to four equations.
(Maxwell never saw what we now call "Maxwell's equations".)
Maybe the equations should be called the "Maxwell-Heaviside 
equations".

Ol' Oly Heaviside was one *serious* sharp character.
[I haven't even mentioned the Laplace transform, which as far 
as I know, was Heaviside's idea.]

[-Rick-]