Repeat Operator

Like the $\hbox{\sc Stretch}_L()$ and $\hbox{\sc Interp}_L()$ operators, 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 $M\isdef LN$, and indexing of $x$ is modolu $N$ (periodic extension). Thus, the $\hbox{\sc Repeat}_L()$ operator simply repeats its input signal $L$ times.^7.7 An example of $\hbox{\sc Repeat}_2(x)$ is shown in Fig.7.8. 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 7.8: Illustration of $\hbox{\sc Repeat}_2(x)$.

A frequency-domain example is shown in Fig.7.9. Figure 7.9a shows the original spectrum $X$, Fig.7.9b shows the same spectrum plotted over the unit circle in the $z$ plane, and Fig.7.9c 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 7.9: 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

$$\hbox{\sc Stretch}_L \;\longleftrightarrow\;\hbox{\sc Repeat}_L,$$

where $\hbox{\sc Stretch}_L$ is defined in §7.2.6. That is, when you stretch a signal by the factor $L$, its spectrum is repeated $L$ times around the unit circle. The simple proof is given on page page .