The packages making up the context 87
implement the Laplace transform and the (continuous-time) Fourier transform.
Given a continuous-time function f (t), its bilateral Laplace
transform F(s) is defined as
#math246#
F(s) = #tex2html_wrap_indisplay3266# f (t) e-st dt <#1#>and<#1#> f (t) = #tex2html_wrap_indisplay3267##tex2html_wrap_indisplay3268# F(s) est ds
[Oppenheim and Willsky 1983].
The region of convergence for the Laplace transform is a strip, of the
form #math247##tex2html_wrap_inline3270#s∈#tex2html_wrap_inline3271# : R- ;SPMlt; ℜe s ;SPMlt; R+}.
In the inverse transform, the contour of integration includes the
section of the s-plane whose real components are bounded by the
constant σ, i.e., either #math248#ℜe s ;SPMlt; σ or
#math249#ℜe s ;SPMgt; sigma.
The continuous-time Fourier transform is a function of a real variable
ω:
#math250#
F(ω) = #tex2html_wrap_indisplay3278# f (t) e-jωt dt <#1#>and<#1#> f (t) = #tex2html_wrap_indisplay3279##tex2html_wrap_indisplay3280# F(ω) ejωt dω
The corresponding commands are 88, 89,
90 and
91.
Like their digital counterparts, these commands are implemented using
rule bases.
The organization of the forward Laplace transform rule
base follows the same outline as the forward z-transform rule base (see Table
4).
There are more transform pairs and property rules than in the
z-transform rule base, but fewer structure (operator) rules.
The strategies are the same except that strategy 4 of Table 8 does not
apply in the continuous case.
Similarly, the <#564#>InvLaPlace<#564#> rule base has the same
structure as the <#565#>InvZTransform<#565#> rule base, but contains more transform
pairs and fewer operator rules.
<#566#>LaPlace<#566#> and <#567#>InvLaPlace<#567#> are more comprehensive than the
Laplace-transform commands provided with the 1.2 release of <#1166#>Mathematica<#1166#>,
which support only one-dimensional, one-sided Laplace transforms, and
do not keep track of the region of convergence.
Both versions of the Laplace transform, however, can be loaded into
<#1167#>Mathematica<#1167#> since the bilateral and unilateral transforms have different
naming conventions.
The continuous-time analogs of <#568#>DTFTransform<#568#>, <#569#>InvDTFTransform<#569#>,
and <#570#>ZSolve<#570#> are
<#571#>CTFTransform<#571#>, <#572#>InvCTFTransform<#572#>, and
<#573#>LSolve<#573#>.
The forward Fourier transform command <#574#>CTFTransform<#574#>
relies on <#575#>LaPlace<#575#>, and the inverse Fourier transform command
<#576#>InvCTFTransform<#576#> is built on top of <#577#>InvLaPlace<#577#>.
Because their definitions are very similiar, <#578#>CTFTransform<#578#> and
<#579#>InvCTFTransform<#579#> share the same rule base.
<#580#>LSolve<#580#> solves linear differential equations with constant
coefficients; it takes the equation and the unknown function as arguments
and the initial conditions and justification level as options.
For example, the differential equation
#math251#y''(t) + #tex2html_wrap_inline3284#y'(t) + #tex2html_wrap_inline3285#y(t) = exp(at)u-1(t)
with initial conditions #math252#y'(0) = #tex2html_wrap_inline3287# and y(0) = 4 is solved as
follows:
verbatim190#
The function 92 is the analog counterpart of 93.
As an example of its use, consider the function
#math253#
te-atcos#tex2html_wrap_indisplay3290#3πt#tex2html_wrap_indisplay3291#16#tex2html_wrap_indisplay3292#u-1(t).
To be able to plot the function, <#589#>ASPAnalyze<#589#> substitutes a
default value of 1 for all unknown parameters (<#590#>a<#590#> in this case).
The Laplace transform is a rational polynomial in s with two poles,
each with multiplicity 2.
In the pole-zero diagram, a single zero is represented as <#591#>O<#591#>
and a single pole as <#592#>X<#592#>; multiple poles or zeroes appear as
<#593#>*X*<#593#> or <#594#>*O*<#594#>.
verbatim191#
verbatim192#
Finally, here is an example with Dirac delta functions.
The time-domain plot represents them with arrowheads,
a common engineering convention.
The value of the global variable $DeltaFunctionScaling indicates how
Delta functions are to be plotted: <#599#>None<#599#> means that all Delta functions
will have identical heights equal to the height of the plot, and
<#600#>Scaled<#600#> means that the height of each Delta function will be equal
to its area (the default).
This function has a Laplace transform but no poles or zeroes,
so a pole-zero diagram is not displayed.
<#601#>ASPAnalyze<#601#> does plot its magnitude and phase frequency responses,
although we have omitted them here.
verbatim193#
<#604#>PoleZeroPlot::notrational:<#604#>
<#606#>Transform is not a rational polynomial.<#606#>
<#607#>PoleZeroPlot::noplot:<#607#>
<#609#>A pole-zero plot cannot be generated.<#609#>
verbatim194#
<#612#>ASPAnalyze::notinteresting:<#612#>
<#614#>Could not determine the important section of the frequency response.<#614#>
verbatim195#