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Stretch Operator

We define the stretch operator in the time domain by

$\displaystyle \hbox{\sc Stretch}_{L,n}(x) \isdefs \left\{\begin{array}{ll} x\left(\frac{n}{L}\right), & n = 0\;(\hbox{\sc mod}\ L) \\ [5pt] 0, & \hbox{otherwise} \\ \end{array} \right..$ (3.29)

In other terms, we stretch a sampled signal by the factor $ L$ by inserting $ L-1$ zeros in between each pair of samples of the signal.

Figure 2.1: Illustration of the stretch operator.
\includegraphics[width=4in]{eps/stretch2}

In the literature on multirate filter banks (see Chapter 11), the stretch operator is typically called instead the upsampling operator. That is, stretching a signal by the factor of $ K$ is called upsampling the signal by the factor $ K$ . (See §11.1.1 for the graphical symbol ( $ \uparrow K$ ) and associated discussion.) The term ``stretch'' is preferred in this book because ``upsampling'' is easily confused with ``increasing the sampling rate''; resampling a signal to a higher sampling rate is conceptually implemented by a stretch operation followed by an ideal lowpass filter which moves the inserted zeros to their properly interpolated values.

Note that we could also call the stretch operator the scaling operator, to unify the terminology in the discrete-time case with that of the continuous-time case (§2.4.1 below).

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Repeat (Scaling) Operator
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Power Theorem for the DTFT