Stretch/Repeat (Scaling) Theorem
Using these definitions, we can compactly state the stretch theorem:
![]() |
(3.31) |
Proof:
![\begin{eqnarray*}
\hbox{\sc DTFT}_\omega[\hbox{\sc Stretch}_L(x)]
&\isdef & \sum_{n=-\infty}^{\infty}\hbox{\sc Stretch}_{L,n}(x)e^{-j\omega n}\\
&=& \sum_{m=-\infty}^{\infty}x(m)e^{-j\omega m L}\qquad \hbox{($m\isdef n/L$)}\\
&\isdef & X(\omega L)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img199.png)
As
traverses the interval
,
traverses the unit circle
times, thus implementing the repeat
operation on the unit circle. Note also that when
, we have
, so that dc always maps to dc. At half the sampling
rate
, on the other hand, after the mapping, we may have
either
(
odd), or
(
even), where
.
The stretch theorem makes it clear how to do
ideal sampling-rate conversion for integer upsampling ratios
:
We first stretch the signal by the factor
(introducing
zeros
between each pair of samples), followed by an ideal lowpass
filter cutting off at
. That is, the filter has a gain of 1
for
, and a gain of 0 for
. Such a system (if it were realizable) implements ideal bandlimited interpolation of the original signal by the factor
.
The stretch theorem is analogous to the scaling theorem for continuous Fourier transforms (introduced in §2.4.1 below).
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Downsampling and Aliasing
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Repeat (Scaling) Operator