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Downsampling Operator

Downsampling by $ L$ (also called decimation by $ L$ ) is defined for $ x\in\mathbb{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 \\
\hbox{\sc Stretch}_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$ .

Figure 7.10: Illustration of $ \hbox{\sc Downsample}_2(x)$ .

The following example of $ \hbox{\sc Downsample}_2(x)$ is illustrated in Fig.7.10:

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

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 resampling using bandlimited interpolation. 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 $ \hbox{\sc Downsample}_M$ , in order to implement a sampling-rate conversion by the factor $ L/M$ .

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``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 © 2022-09-05 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University