.
Usually, the Cauchy Residue Theorem is applied to X(z) in order
to reduce the contour integral to an algebraic evaluation of residues.
The inverse z-transform is equivalent to the Laurent series expansion of X(z).
The command 68 implements the inverse z-transform
of one-dimensional and multidimensional expressions.
In two or more dimensions, the general inverse is quite difficult because it
involves multiple contour integrations, which <#1162#>Mathematica<#1162#> cannot perform.
The rule base, however, covers many of these cases.
<#415#>InvZTransform<#415#> calls a one-dimensional rule base for each dimension
to be transformed.
The rules are organized in a manner similar to the
one-dimensional forward z-transform rules (Table 9), the main
difference being that only two multidimensional hooks are encoded
(which invert transforms corresponding to upsampled sequences).
Application of the one-dimensional rule bases automatically invert
transforms corresponding to line impulses and multinomials.
<#3170#>Table<#3170#>:
<#3171#>Synopsis of rules for one-dimensional inverse z-transforms<#3171#>
| Type of rule |
Number |
Table |
| multidimensional hooks |
2 |
;SPMnbsp; |
| rational transform pairs |
26 |
5 |
| non-rational transform pairs |
9 |
6 |
| transform properties |
12 |
7 |
| transforms of operators |
4 |
8 |
| transform strategies |
9 |
10 |
|
Six of the inverse transform pairs concern <#424#>Impulse<#424#>, <#425#>Step<#425#>,
and sinusoidal forms.
The other pairs return the inverse transform of
rational <#426#>z-transform<#426#>s (with first-order and second-order denominators) and
non-rational <#427#>z-transform<#427#>s (containing logarithmic, factorial and square-root
forms).
The properties of the transform and the transforms of
operators are the same as in the forward case.
Transforms of exponentials in the time domain are inverse-transformed by the
exponential property rule, not by table lookup.
Some strategies for inverting <#428#>z-transform<#428#>s (Table 10) are similar to those
applied in taking forward <#429#>z-transform<#429#>s, but some new ones are also needed.
Two such strategies are partial fractions and power series expansion.
The goal of partial fractions is to break up rational functions into a
sum of terms involving only first-order and second-order denominators.
Then, each individual term can be easily transformed. Power series
expansion can be used to approximate the first N polynomial terms of
the inverse transform.
The repeated application of either of these
two strategies, however, would cause an infinite loop.
In order to avoid this, local state information ensures that any strategy
with this behavior is applied at most once per expression.
<#3180#>Table<#3180#>:
<#3181#>Strategies for inverse z-transforms. An asterisk
means that once the rule is applied to an expression,
it will no longer be applied to any part of that expression.<#3181#>
| #table430# |
As an example, consider the complex cepstrum of a sum of two impulses,
#math227#δ[n] + δ[n - 3].
(The <#1163#>complex cepstrum<#1163#> of a sequence is the
inverse z-transform of the logarithm of the z-transform of the sequence.)
Thus, we are interested in the inverse z-transform of
#math228#log(1 + az-3).
In the first pass through the rule base, none of the transform pairs
and none of the property rules apply.
However, the upsampling structure rule does apply, since
the greatest common divisor of all exponents of <#451#>z<#451#> in
<#452#>Log [ 1 + a z69-3 ]<#452#> is 3.
This rule fires and rewrites the inverse transform as
the upsampling by three of the inverse z-transform of
<#453#>Log [ 1 + a z70-1 ]<#453#>.
This can now be inverse-transformed by table lookup (Table 5), giving
71.
In addition to the 72 option, 73
accepts the option 74.
If set to a positive integer, this causes 75
to attempt a series expansion as a last resort, as shown
in Figure 1.
<#3191#>Figure<#3191#>:
<#3192#>An example of series expansion<#3192#>
<#1020#>
verbatim182# <#1020#>
|
<#1021#>
verbatim183# <#1021#>
|
|