NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University
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Like the operator, the operator maps a length signal to a length signal:
Definition: The repeat times operator is defined by
where . Thus, the operator simply repeats its input signal times.8.4An example of is shown in Fig. 8.6. The example is
A frequency-domain example is shown in Fig. 8.7. Figure 8.7a shows the original spectrum , Fig. 8.7b shows the same spectrum plotted over the unit circle in the plane, and Fig. 8.7c shows . The point (dc) is on the right-rear face of the enclosing box. Note that when viewed as centered about , is a somewhat ''triangularly shaped'' spectrum. The repeating block can be considered to extend from the point at to the point far to the left, or it can be considered the triangularly shaped ''baseband'' spectrum centered about .
Figure:Illustration of . a) Conventional plot of . b) Plot of over the unit circle in the plane. c) .
The repeat operator is used to state the Fourier theorem
That is, when you stretch a signal by the factor , its spectrum is repeated times around the unit circle.