The 81 packages provide a high-level
function, 82, which exercises most of the features
discussed above.
This function, like 83, requires two arguments: the expression
to analyze and the variable(s) involved.
Optionally, a third and fourth arguments specify the domain over which
to plot the discrete-time function.
Only one- and / signals can be plotted.
84 finds the signal's z-transform, if it exists,
and gives a stability criterion based on the transform's region of
convergence.
If the z-transform does not exist, the frequency
response is obtained using <#515#>DTFTransform<#515#>.
Using extensions to <#516#><#1165#>Mathematica<#1165#><#516#>'s graphics abilities, 85
plots the expression, its magnitude response, and its phase response.
For / and separable / expressions, it plots the pole-zero diagram,
and for non-separable / expressions, it plots the pole and zero root
locus in each dimension.
As an example, consider the / signal 86.
This expression does not have a z-transform (which causes the generation
of many warning messages), but it does have a Fourier
transform, so the magnitude and phase response can be computed and plotted.
The magnitude response is periodic of period 2π/3 (a period
of 2π/k is characteristic of signals upsampled by a factor of k),
and the phase response is zero, so it is omitted here, although
<#517#>DSPAnalyze<#517#> plots it.
<#3252#>
verbatim186#
<#523#>ZTransform::notvalid:<#523#>
<#525#>The forward z-transform could not be found.<#525#>
verbatim187# <#3252#> |
As a second example, consider the two-variable signal
#math242#
#tex2html_wrap_indisplay3254#an1bn2u[n1, n2],
with #math243#a = #tex2html_wrap_inline3256# and #math244#b = #tex2html_wrap_inline3258#.
The increasing nature of the discrete domain plot hints at instability,
but the z-transform cannot determine this.
Both pole root maps, however, do indicate instability because
some of the poles in the loci reside outside the unit circle.
Dudgeon and Mersereau [1984] have shown that a function of this form
is stable if and only if #math245#| a| + | b| ;SPMlt; 1.
Because the time function is not stable, and because the z-transform is used to
compute the Fourier transform, the Fourier transform and the
frequency response are physically meaningless.
For this reason we omit the magnitude and phase response graphs from
the output.
verbatim188#
verbatim189#