Repeat Operator

Like the $\hbox{\sc Stretch}_L()$ operator, the $\hbox{\sc Repeat}_L()$ operator maps a length $N$ signal to a length $M\isdeftext LN$ signal:

Definition: The repeat $L$ times operator is defined for any $x\in{\bf C}^N$ by

$$\hbox{\sc Repeat}_{L,m}(x) \isdef x(m), \qquad m=0,1,2,\ldots,M-1,$$

where $M\isdef LN$. Thus, the $\hbox{\sc Repeat}_L()$ operator simply repeats its input signal $L$ times.^7.5 An example of $\hbox{\sc Repeat}_2(x)$ is shown in Fig. 7.6. The example is

$$\hbox{\sc Repeat}_2([0,2,1,4,3,1]) = [0,2,1,4,3,1,0,2,1,4,3,1]$$

Figure: Illustration of $\hbox{\sc Repeat}_2(x)$.

A frequency-domain example is shown in Fig. 7.7. Figure 7.7a shows the original spectrum $X$, Fig. 7.7b shows the same spectrum plotted over the unit circle in the $z$ plane, and Fig. 7.7c shows $\hbox{\sc Repeat}_3(X)$. The $z=1$ point (dc) is on the right-rear face of the enclosing box. Note that when viewed as centered about $k=0$, $X$ is a somewhat ``triangularly shaped'' spectrum. We see three copies of this shape in $\hbox{\sc Repeat}_3(x)$.

Figure: Illustration of $\hbox{\sc Repeat}_3(X)$. a) Conventional plot of $X$. b) Plot of $X$ over the unit circle in the $z$ plane. c) $\hbox{\sc Repeat}_3(X)$.

The repeat operator is used to state the Fourier theorem

$$\hbox{\sc Stretch}_L \leftrightarrow \hbox{\sc Repeat}_L.$$

That is, when you stretch a signal by the factor $L$, its spectrum is repeated $L$ times around the unit circle.