Stretch/Repeat (Scaling) Theorem

Using these definitions, we can compactly state the stretch theorem:

$$\zbox {\hbox{\sc Stretch}_L(x) \;\longleftrightarrow\;\hbox{\sc Repeat}_L(X)}$$ (3.31)

Proof:

$$\begin{eqnarray*} \hbox{\sc DTFT}_\omega[\hbox{\sc Stretch}_L(x)] &\isdef & \sum_{n=-\infty}^{\infty}\hbox{\sc Stretch}_{L,n}(x)e^{-j\omega n}\\ &=& \sum_{m=-\infty}^{\infty}x(m)e^{-j\omega m L}\qquad \hbox{($m\isdef n/L$)}\\ &\isdef & X(\omega L) \end{eqnarray*}$$

As $\omega$ traverses the interval $[-\pi,\pi)$ , $X(\omega L)$ traverses the unit circle $L$ times, thus implementing the repeat operation on the unit circle. Note also that when $\omega = 0$ , we have $\omega L = 0$ , so that dc always maps to dc. At half the sampling rate $\omega=\pm\pi$ , on the other hand, after the mapping, we may have either $Y(\pi)=X(-\pi)$ ($L$ odd), or $X(0)$ ($L$ even), where $Y(\omega) \isdeftext X(\omega L)$ .

The stretch theorem makes it clear how to do ideal sampling-rate conversion for integer upsampling ratios $L$ : We first stretch the signal by the factor $L$ (introducing $L-1$ zeros between each pair of samples), followed by an ideal lowpass filter cutting off at $\pi/L$ . That is, the filter has a gain of 1 for $\left\vert\omega\right\vert <\pi/L$ , and a gain of 0 for $\pi/L < \left\vert\omega\right\vert \leq \pi$ . Such a system (if it were realizable) implements ideal bandlimited interpolation of the original signal by the factor $L$ .

The stretch theorem is analogous to the scaling theorem for continuous Fourier transforms (introduced in §2.4.1 below).