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Downsampling Operator

Downsampling by $ L$ (also called decimation by $ L$) is defined for $ x\in{\bf C}^N$ as taking every $ L$th sample, starting with sample zero:

\hbox{\sc Downsample}_{L,m}(x) &\isdef & x(mL),\\
m &=& 0,1,2,\ldots,M-1\\

The $ \hbox{\sc Downsample}_L()$ operator maps a length $ N=LM$ signal down to a length $ M$ signal. It is the inverse of the $ \hbox{\sc Stretch}_L()$ operator (but not vice versa), i.e.,

\hbox{\sc Downsample}_L(\hbox{\sc Stretch}_L(x)) &=& x \\
...L(\hbox{\sc Downsample}_L(x)) &\neq& x\quad \mbox{(in general).}

The stretch and downsampling operations do not commute because they are linear time-varying operators. They can be modeled using time-varying switches controlled by the sample index $ n$.

An example of $ \hbox{\sc Downsample}_2(x)$ is shown in Fig.7.10. The example is

$\displaystyle \hbox{\sc Downsample}_2([0,1,2,3,4,5,6,7,8,9]) = [0,2,4,6,8].

Figure 7.10: Illustration of $ \hbox{\sc Downsample}_2(x)$. The white-filled circles indicate the retained samples while the black-filled circles indicate the discarded samples.

Note that the term ``downsampling'' may also refer to the more elaborate process of sampling-rate conversion to a lower sampling rate, in which a signal's sampling rate is lowered by means of bandlimited interpolation (discussed in Appendix D). To distinguish these cases, we can call this bandlimited downsampling, because a lowpass-filter is needed, in general, prior to downsampling so that aliasing is avoided. This topic is address in Appendix D. Early sampling-rate converters were in fact implemented using the $ \hbox{\sc Stretch}_L$ operation, followed by an appropriate lowpass filter, followed by $ Select_M$, in order to implement a sampling-rate conversion by the factor $ L/M$.

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[How to cite this work] [Order a printed hardcopy]

``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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