Stretch Operator

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

Figure 7.6: Illustration of $\hbox{\sc Stretch}_3(x)$ .

A stretch by factor $L$ is defined by

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

Thus, to stretch a signal by the factor $L$ , insert $L-1$ zeros between each pair of samples. An example of a stretch by factor three is shown in Fig.7.6. The example is

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

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