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Re: Inverse Z-Transform and/or Long Division
*To*: mathgroup at smc.vnet.net
*Subject*: [mg2189] Re: Inverse Z-Transform and/or Long Division
*From*: Paul Abbott <paul at earwax.pd.uwa.edu.au>
*Date*: Fri, 13 Oct 1995 02:24:19 -0400
*Organization*: Dept of Physics, University of WA
degennar at bmsrs.usc.edu (Raymond Anthony Ralph DeGennaro II) wrote:
>I have a rather nasty Z-Transform of a transfer function that looks >like:
> a0 + a1*z^-1 + a2*z^-2
> H(z) = ----------------------
> b0 + b1*z^-1 + b2*z^-2
>
>Is there an easy way to bring this back into the time domain in either >of these packages?
>
Try MathSource (http://www.wri.com/mathsource/) and search for keywords. For example, MathSource contains the following items relevant to `Z-Transform':
0200-080: Laplace Transforms
0205-513: The Window Method for FIR Digital Filter Design
0206-738: Approximate Inversion of Laplace Transform
0205-502: The Kaiser Window
0202-981: Evaluating Powers of (-1) (Technical Note)
0206-468: List To Array Package
0204-499: Harmonic Function Theory and Mathematica
0200-068: Fourier Transforms
0205-276: An Efficient Implementation of the Patterson-Holdsworth
Auditory Filter Bank
0207-144: The Discrete Periodic Wavelet Transform in 1D
0203-724: Basic Sets of Polynomial Solutions for the Iterated Laplace and Wave Equations
0205-041: ExtendGraphics Packages by Tom Wickham-Jones
0205-401: Geometry in Motion
0207-324: Mathematica Graphics: Techniques and Applications by Tom
Wickham-Jones, Electronic Supplement
0201-979: Integration over Polytopes
0205-052: Notebooks for Partial Differential Equations with Mathematica
0202-611: COSY-PAK: A Symbolic Control Systems Analysis Package V0.9
0202-240: Signal Processing Packages and Notebooks Version 2.9.5
>Alternately, if I could just do a symbolic long division, I would get
>terms in the form:
> a0 + a1*z^-1 + a2*z^-2 + a3*z^-3 + ...
>which are super-simple to bring into the time-domain. But I can't seem >to get Matlab to do this (I haven't tried Mathematica, yet).
In Mathematica, after defining
H[z_] = (a0 + a1*z^-1 + a2*z^-2)/
(b0 + b1*z^-1 + b2*z^-2);
a2 a1
a0 + -- + --
2 z
z
------------
b2 b1
b0 + -- + --
2 z
z
you can use Taylor series (about Infinity) to get the expansion you want:
H[z] + O[z,Infinity]^3 // Simplify
2 2
a0 a1 b0 - a0 b1 a2 b0 - a1 b0 b1 + a0 b1 - a0 b0 b2
-- + ------------- + ------------------------------------- +
b0 2 3 2
b0 z b0 z
1 3
O[-]
z
Cheers,
Paul
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