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To show:
or
From the DFT case [264], we know this is true when
and
are each complex sequences of length
, in which case
and
are length
. Thus,
![$\displaystyle x(nN) \;\longleftrightarrow\; Y(\omega_k N) \eqsp \frac{1}{N} \sum_{m=0}^{N-1} X\left(\omega_k + \frac{2\pi}{N} m \right), \; k\in [0,N_s/N)$](img229.png) |
(3.38) |
where we have chosen to keep frequency samples
in terms of
the original frequency axis prior to downsampling, i.e.,
for both
and
. This choice allows us to easily take
the limit as
by simply replacing
by
:
![$\displaystyle x(nN) \;\longleftrightarrow\; Y(\omega N) \eqsp \frac{1}{N} \sum_{m=0}^{N-1} X\left(\omega + \frac{2\pi}{N} m \right), \; \omega\in[0,2\pi/N)$](img232.png) |
(3.39) |
Replacing
by
and converting to
-transform
notation
instead of Fourier transform notation
,
with
, yields the final result.
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