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Downsampling (Decimation) Operator

Figure: Downsampling by $ N$ .

Figure 11.3 shows the symbol for downsampling by the factor $ N$ . The downsampler selects every $ N$ th sample and discards the rest:

y(n) &=& \hbox{\sc Downsample}_{N,n}(x)\\
&\isdef & x(Nn), \quad n\in\mathbb{Z}

In the frequency domain, we have

Y(z) &=& \hbox{\sc Alias}_{N,z}(X)\\
&\isdef &
\frac{1}{N} \sum_{m=0}^{N-1} X\left(z^\frac{1}{N}e^{-jm\frac{2\pi}{N}} \right),
\quad z\in\mathbb{C}.

Thus, the frequency axis is expanded by the factor $ N$ , wrapping $ N$ times around the unit circle, adding to itself $ N$ times. For $ N=2$ , two partial spectra are summed, as indicated in Fig.11.4.

Figure: Illustration of $ \hbox {\sc Alias}_2$ in the frequency domain.

Using the common twiddle factor notation

$\displaystyle W_N \isdefs e^{-j2\pi/N},$ (12.1)

the aliasing expression can be written as

$\displaystyle Y(z) \eqsp \frac{1}{N} \sum_{m=0}^{N-1} X(W_N^m z^{1/N}).

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University