Forums

Obtaining inverse z-transform of given expression

Started by Mackan 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?

Any comments would be appreciated.
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 and get the answer symbolically (MathCad uses the Maple symbolic engine, 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