Repeat Operator

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Repeat Operator

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Repeat Operator

Like the $\ operator, the $\ operator maps a length$N$ signal to a length $M\ signal:



Definition: The repeat $L$ times operator is defined by

\

where $M\. Thus, the $\ operator simply repeats its input signal$L$ times.8.4An example of $\ is shown in Fig. 8.6. The example is

\

Figure:Illustration of $\.
\

A frequency-domain example is shown in Fig. 8.7. Figure 8.7a shows the original spectrum $X$, Fig. 8.7b shows the same spectrum plotted over the unit circle in the $z$ plane, and Fig. 8.7c shows $\. 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. The repeating block can be considered to extend from the point at $z=1$ to the point far to the left, or it can be considered the triangularly shaped “baseband” spectrum centered about $z=1$.

Figure:Illustration of $\. a) Conventional plot of $X$. b) Plot of $X$ over the unit circle in the $z$ plane. c) $\.
\

The repeat operator is used to state the Fourier theorem

\

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

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