#math229#X(ejω) of a continuous
variable ω, called frequency:
#math230#
X(ejω) = #tex2html_wrap_indisplay3203#x[n]e-jωn.
The inverse transformation is given by
#math231#
x[n] = #tex2html_wrap_indisplay3205##tex2html_wrap_indisplay3206#X(ejω)ejωndω
[Oppenheim and Schafer 1989].
Thus, the Fourier transform is a periodic function of period 2π.
The equation for the forward DTFT is the same as for the forward z-transform,
with #math232#z = exp(jω), so it is tempting to think that the DTFT
can always be obtained from the z-transform.
But this is not so for two reasons: first, some functions,
such as #math233#sinc#tex2html_wrap_inline3212#t, have a DTFT but not a z-transform.
Conversely, if the function does have a z-transform, the
substitution #math234#z = exp(jω) is only valid when the unit circle
is in the region of convergence.
The context 78 provides the command <#494#>DTFTransform<#494#>, which first checks a special DTFT rule base, then
applies the z-transform rule base (the substitution #math235#z = exp(jω)
rewrites the z-transform as a DTFT).
The specific DTFT rule base contains transform pairs of
forms that do not have <#495#>z-transform<#495#>s, and rules describing downsampling and
upsampling.
(A downsampled signal does not have a z-transform, and the
upsample operator behaves differently in the frequency domain.)
The only redundancy between the two rule bases is the recasting of three
property rules (homogeneity, additivity, and multiplication by n)
and one strategy rule (expand all terms).
The inverse discrete-time Fourier transform command, <#496#>InvDTFTransform<#496#>, is implemented in a similar manner.
Since both DTFT rule bases rely on the z-transform rule bases,
the DTFT works in several dimensions, for one- and multi-sided signals.
The inverse DTFT properly handles many non-separable transforms.