Stretch Operator

We define the stretch operator in the time domain by

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 (see §2.4.4 for the continuous-time case).

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.