Stretch Operator

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Stretch Operator

Unlike all previous operators, the $\hbox{\sc Stretch}_L()$ operator maps a length signal to a length $M\isdeftext LN$ signal, where and are integers. We use ``'' instead of ``'' as the time index to underscore this fact.

**Figure 7.6:** Illustration of $\hbox{\sc Stretch}_3(x)$ .
$\includegraphics[width=4in]{eps/stretch}$

A stretch by factor is defined by

$\displaystyle \hbox{\sc Stretch}_{L,m}(x) \isdef \left\{\begin{array}{ll} x(m/... ...mbox{ an integer} \\ [5pt] 0, & m/L\mbox{ non-integer.} \\ \end{array}\right.$

Thus, to stretch a signal by the factor

, insert

zeros between each pair of samples. An example of a stretch by factor three is shown in Fig.7.6. The example is

$\displaystyle \hbox{\sc Stretch}_3([4,1,2]) = [4,0,0,1,0,0,2,0,0].$

The stretch operator is used to describe and analyze upsampling, that is, increasing the sampling rate by an integer factor. A stretch by followed by lowpass filtering to the frequency band $\omega\in(-\pi/K,\pi/K)$ implements ideal bandlimited interpolation (introduced in Appendix D).

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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.

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Mathematics of the DFT
    Fourier Theorems for the DFT
       Signal Operators
          Stretch Operator