### Stretch Operator

We define the*stretch operator*in the

*time domain*by

(3.29) |

In other terms, we stretch a sampled signal by the factor by inserting

*zeros*in between each pair of samples of the signal. 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 is called upsampling the signal by the factor . (See §11.1.1 for the graphical symbol ( ) 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