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Embedded SystemsFPGAElectronics

Mathematics of the DFT
    Fourier Theorems for the DFT
       Signal Operators
          Repeat Operator

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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

$\displaystyle \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 modulo $ N$ (periodic extension). Thus, the $ \hbox{\sc Repeat}_L()$ operator simply repeats its input signal $ L$ times.7.10 An example of $ \hbox{\sc Repeat}_2(x)$ is shown in Fig.7.8. The example is

$\displaystyle \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)$.
\includegraphics[width=\textwidth]{eps/repeat}

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)$.
\includegraphics[width=4in]{eps/repeat3d}

The repeat operator is used to state the Fourier theorem

$\displaystyle \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$ (inserting zeros between the original samples), its spectrum is repeated $ L$ times around the unit circle. The simple proof is given on page [*].

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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