# Obtaining inverse z-transform of given expression

Started by June 19, 2004
Hi group!

I need to determine the impulseresponse for a system with the
transferfunction
$$H(z)=\frac{1-0.5z^{-1}}{1-z^{-1}+0.5z^{-2}}$$
Knowing that the impulserespone is the inverse z-transform of H I
thought this would be no problem. However after performing
partialfraction decomposition im unable to locate any known
z-transforms (that I can look up in a table), so im kind of stuck.
Perhaps inverse transfrom by partialfraction decomposition isn't the
method of choice to solve this problem?

regards
mackan

Mackan wrote:
> Hi group!
>
> I need to determine the impulseresponse for a system with the
> transferfunction
> $$> H(z)=\frac{1-0.5z^{-1}}{1-z^{-1}+0.5z^{-2}} >$$
> Knowing that the impulserespone is the inverse z-transform of H I
> thought this would be no problem. However after performing
> partialfraction decomposition im unable to locate any known
> z-transforms (that I can look up in a table), so im kind of stuck.
> Perhaps inverse transfrom by partialfraction decomposition isn't the
> method of choice to solve this problem?
>
> Any comments would be appreciated.
> regards
> mackan

First multiply your numerator and denominator by z^2 to get a ratio of
polynomials in z instead of 1/z.  Then you'll find that the roots of
your denominator polynomial are at z = 0.5 +/- j0.5.  You can either
expand the result into the two complex forms and use the Euler identity
(which will magically combine back into a form with all real values) or
you can use the identities

z e^(-at) sin bT
--------------------------------  --->  e^(-akT) sin(bkT)
z^2 - 2ze^(-at)cos bT + e^(-2aT)

and

z^2 - z e^(-at) cos bT
--------------------------------  --->  e^(-akT) cos(bkT).
z^2 - 2ze^(-at)cos bT + e^(-2aT)

Personally I'd use the complex form, or I'd just stick it into MathCad
so anything that uses Maple would do for you).

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Tim Wescott <tim@wescottnospamdesign.com> wrote in message news:<10d8rnbr5a834c9@corp.supernews.com>...
> Mackan wrote:
> > Hi group!
> >
> > I need to determine the impulseresponse for a system with the
> > transferfunction
> > $$> > H(z)=\frac{1-0.5z^{-1}}{1-z^{-1}+0.5z^{-2}} > >$$
> > Knowing that the impulserespone is the inverse z-transform of H I
> > thought this would be no problem. However after performing
> > partialfraction decomposition im unable to locate any known
> > z-transforms (that I can look up in a table), so im kind of stuck.
> > Perhaps inverse transfrom by partialfraction decomposition isn't the
> > method of choice to solve this problem?
> >
> > Any comments would be appreciated.
> > regards
> > mackan
>
> First multiply your numerator and denominator by z^2 to get a ratio of
> polynomials in z instead of 1/z.  Then you'll find that the roots of
> your denominator polynomial are at z = 0.5 +/- j0.5.  You can either
> expand the result into the two complex forms and use the Euler identity
> (which will magically combine back into a form with all real values) or
> you can use the identities
>
>         z e^(-at) sin bT
> --------------------------------  --->  e^(-akT) sin(bkT)
> z^2 - 2ze^(-at)cos bT + e^(-2aT)
>
> and
>
>      z^2 - z e^(-at) cos bT
> --------------------------------  --->  e^(-akT) cos(bkT).
> z^2 - 2ze^(-at)cos bT + e^(-2aT)
>
> Personally I'd use the complex form, or I'd just stick it into MathCad
> and get the answer symbolically (MathCad uses the Maple symbolic engine,
> so anything that uses Maple would do for you).

Thank you for your informative response, ive solved the problem now.
regards
mackan