Repeat or Scale

Similarly, the $\hbox{\sc Repeat}_L$ operator, defined on the unit circle, frequency-scales its input spectrum by the factor $L$ :

$$\omega \leftarrow L\omega$$

The original spectrum is repeated $L$ times as $\omega$ traverses the unit circle. This is illustrated in the following diagram for $L=3$ :

Using these definitions, we have the Stretch Theorem:

$$\zbox{\hbox{\sc Stretch}_L(x) \leftrightarrow \hbox{\sc Repeat}_L(X)}$$

Application: Upsampling by any integer factor $L$ : Passing the stretched signal through an ideal lowpass filter cutting off at $\omega \geq \pi/L$ yields ideal bandlimited interpolation of the original signal by the factor $L$ .