Properties of transfer function, linear systems

Hello,

Consider a linear transfer function with N-1 real and negative poles 
(p_n) and N real and negative zeros (z_n) factorized in the standard way:

	  (z-z_1)*(z-z_2)*...*(z-z_N)
Q(z) = K -----------------------------
	  (z-p_1)*(z-p_2)*...*(z-p_{N-1}),

where lim z-> infinity => Q(z)->infinity.

Are there then any theorems regarding such a transfer function? I am 
looking for properties of such transfer functions.

Thanks,

Peter

"Peter" <anders_noSPAM@kommtek.com> wrote in message 
news:eo31sr$f3u$2@orkan.itea.ntnu.no...
> Hello,
>
> Consider a linear transfer function with N-1 real and negative poles (p_n) 
> and N real and negative zeros (z_n) factorized in the standard way:
>
>   (z-z_1)*(z-z_2)*...*(z-z_N)
> Q(z) = K -----------------------------
>   (z-p_1)*(z-p_2)*...*(z-p_{N-1}),
>
> where lim z-> infinity => Q(z)->infinity.
>
> Are there then any theorems regarding such a transfer function? I am 
> looking for properties of such transfer functions.

Peter,

There are many.  You will probably get better answers if you focus your 
question a bit more.
- what is your situation, objective, ..... ?

Fred 

Peter wrote:
> Hello,
>
> Consider a linear transfer function with N-1 real and negative poles
> (p_n) and N real and negative zeros (z_n)

i don't know what you mean be "negative poles" or "negative zeros'
since they are all possible complex numbers.  if this were the s-plane
instead of the z-plane, i would expect "negative" applied to the real
part of the poles or zeros making the transfer function one of a stable
and minimum phase filter.  being that this is the z-plane i might
assume "negative poles and zeros" mean poles and zeros inside the unit
circle.  dunno what else it might mean.

> factorized in the standard way:
>
> 	  (z-z_1)*(z-z_2)*...*(z-z_N)
> Q(z) = K -----------------------------
> 	  (z-p_1)*(z-p_2)*...*(z-p_{N-1}),
>
> where lim z-> infinity => Q(z)->infinity.
>
> Are there then any theorems regarding such a transfer function? I am
> looking for properties of such transfer functions.

sounds like a homework problem.  one property of a system of transfer
function with more zeros than poles is that there is an anticipatory
advance (negative delay) of the output to the input due to a left over
z^(+1) term after partial fraction expansion.  that means the impulse
response will react before the impulse happens and the filter is
non-causal.

Fred Marshall wrote:
> "Peter" <anders_noSPAM@kommtek.com> wrote in message 
> news:eo31sr$f3u$2@orkan.itea.ntnu.no...
>> Hello,
>>
>> Consider a linear transfer function with N-1 real and negative poles (p_n) 
>> and N real and negative zeros (z_n) factorized in the standard way:
>>
>>   (z-z_1)*(z-z_2)*...*(z-z_N)
>> Q(z) = K -----------------------------
>>   (z-p_1)*(z-p_2)*...*(z-p_{N-1}),
>>
>> where lim z-> infinity => Q(z)->infinity.
>>
>> Are there then any theorems regarding such a transfer function? I am 
>> looking for properties of such transfer functions.
> 
> Peter,
> 
> There are many.  You will probably get better answers if you focus your 
> question a bit more.
> - what is your situation, objective, ..... ?
> 
> Fred 
> 
> 
Thank you for looking into my problem.

To specify the matters a bit more, my problem is not at all related to 
filtering, but the mathematical function I am looking can be written in 
the form above. So I am hoping to apply some theorems derived in linear 
systems theory to my problem.

Also, the poles and zeros are all real and negative, i.e., Q(z) is a 
ratio of two strictly increasing positive polynomials.

Now, what I am after is some facts about the variation / oscillations of 
this function. Ideally, I wanted it to have no local maximums for z>0, 
but unfortunately this is not true. I am thinking something along the 
lines of that since all the poles and zeros are for z<0, the 
oscillations should more or less be "used" up. Also, I am only 
interested in the properties of z>0. Also, only real z's are of interest.

Looking it as a transfer function, I then would be interested in results 
on the magnitude, since everything is positive, i.e. |Q| = Q.


Thanks,

Peter.

robert bristow-johnson wrote:
> i don't know what you mean be "negative poles" or "negative zeros'
> since they are all possible complex numbers.  if this were the s-plane
> instead of the z-plane, i would expect "negative" applied to the real
> part of the poles or zeros making the transfer function one of a stable
> and minimum phase filter.  being that this is the z-plane i might
> assume "negative poles and zeros" mean poles and zeros inside the unit
> circle.  dunno what else it might mean.

It doesn't matter what letter you use for the variable. It's simply a
rational function with negative poles and zeros. As Peter doesn't say
what he wants to prove, there is no way to help him.

Regards,
Andor

Andor wrote:
>  As Peter doesn't say what he wants to prove, there is no way to help him.
> 

Andor, thank you for looking at this. To see more about what I want to 
prove, check my follow-up post "above".

Peter

Peter schrieb:

> Andor wrote:
> >  As Peter doesn't say what he wants to prove, there is no way to help him.
> >
>
> Andor, thank you for looking at this. To see more about what I want to
> prove, check my follow-up post "above".

I can't make anything out of your follow-up either. You want "facts"
about the "variation / oscillation" without saying what kind of facts,
or what you mean by "variation". You are interested in the "properties"
for z>0. Which properties? I can't read your mind, sorry.

Why don't you just come out and say what you really want?

Regards,
Andor

Andor wrote:
> robert bristow-johnson wrote:
> > i don't know what you mean be "negative poles" or "negative zeros'
> > since they are all possible complex numbers.  if this were the s-plane
> > instead of the z-plane, i would expect "negative" applied to the real
> > part of the poles or zeros making the transfer function one of a stable
> > and minimum phase filter.  being that this is the z-plane i might
> > assume "negative poles and zeros" mean poles and zeros inside the unit
> > circle.  dunno what else it might mean.
>
> It doesn't matter what letter you use for the variable.

but it *does* matter if it's the s-plane (as in Laplace Transform of
the impulse response) or the z-plane (as in Z Transform of the unit
pulse response).

> It's simply a
> rational function with negative poles and zeros.

negative *what*?  negative real part?  he didn't say "real part".
poles and zeros are, in general, complex numbers and you can't
meaningful say that a complex number is negative without a little more
qualification of what you mean.

r b-j

r b j wrote:
> Andor wrote:

> > It's simply a
> > rational function with negative poles and zeros.negative *what*?  negative real part?  
> he didn't say "real part".
> poles and zeros are, in general, complex numbers and you can't
> meaningful say that a complex number is negative without a little more
> qualification of what you mean.

he said real and negative poles and zeros - poles and zeros are on the
real line - their imaginary part is zero - the rational function can be
factored into first order sections using real numbers only - if a
complex number is negative it is also real - I think I drank too much
beer at dinner - ...

Andor wrote:
> r b j wrote:
> > Andor wrote:
>
> > > It's simply a
> > > rational function with negative poles and zeros.negative *what*?  negative real part?
> > he didn't say "real part".
> > poles and zeros are, in general, complex numbers and you can't
> > meaningful say that a complex number is negative without a little more
> > qualification of what you mean.
>
> he said real and negative poles and zeros

oh!  you're right!  i missed that word.  so "negative" makes sense.
then, if this is on the z-plane, each of those poles, assuming they're
stable, has a resonant frequency of Nyquist and the output samples will
be alternating signs with a collection of different exponential
envelopes.  and since there is one more zero than pole, the output will
also be non-causal.  the system will react to an impulse one sample
before the impulse hits.

r b-j