linear phase iir filters

allnor@tele.ntnu.no (Rune Allnor) wrote in message news:<f56893ae.0309120934.46b22238@posting.google.com>...
> Jerry Avins <jya@ieee.org> wrote in message news:<3F61CC5E.BF7E91DD@ieee.org>.>..
>
> > 
> > What is an irrational transfer function? Surely, it can't be h(x) =
> > x&#4294967295;sqrt(2)! Why would I care if a transfer function were rational or not?
> 
> A transfer function that is rational (i.e. is a polynomial in frequency 
> domain) separates easily into a feed-forward part and a feed-back loop.
> Check out the link between difference/differential equations and 
> representations in the discrete/continuous Fourier domains.

Practically implementable discrete-time systems have rational transfer
functions, i.e., H(z) is a polynomial in z (or equivalently in 1/z). 
This means that they can be implented in the time-domain by using
simple delays.

Any zeros of rational transfer functions are isolated.  If the TF is
zero over a band of frequencies, it cannot be rational, the ideal
brickwall LPF being a case in point (which is a linear phase
non-causal IIR filter).

allnor@tele.ntnu.no (Rune Allnor) wrote in message news:<f56893ae.0309120934.46b22238@posting.google.com>...
> A transfer function that is rational (i.e. is a polynomial in frequency 
> domain) 

A rational function is, I believe, one that is expressed as a *fraction*
of polynomials in frequency domain. The denominator in that fraction is 
related to feedback, the numerator is related to feed-forward.

Rune

Rune Allnor wrote:
> 
> Jerry Avins <jya@ieee.org> wrote in message news:<3F61CC5E.BF7E91DD@ieee.org>...
> > Vanamali wrote:
> > >
> > > allnor@tele.ntnu.no (Rune Allnor) wrote in message
> > >
> > > > That's a succint formulation of my immediate reaction... I always thought
> > > > that "[causal] IIR filter" and "linear phase" were contradicions in terms?
> > >
> > > I browsed through comp.dsp after a long gap and came across this topic
> > > and the above quoted post.  In case others have not already pointed
> > > this out, as the Clements paper shows,  you can have causal IIR filter
> > > with precise linear phase.  The catch is that the system transfer
> > > function is not rational.  If one is restricted to the class of
> > > rational transfer functions, then causal, stable IIR system with
> > > precise linear phase is not possible.
> >
> > What is an irrational transfer function? Surely, it can't be h(x) =
> > x&#4294967295;sqrt(2)! Why would I care if a transfer function were rational or not?
> 
> A transfer function that is rational (i.e. is a polynomial in frequency
> domain) separates easily into a feed-forward part and a feed-back loop.
> Check out the link between difference/differential equations and
> representations in the discrete/continuous Fourier domains.
> 
> Rune

I know what is meant. I wasn't aware of the terminology.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
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