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Downsampling $ \leftrightarrow$ Aliasing

The downsampling operation $ \hbox{\sc Downsample}_M$ selects every $ M^{th}$ sample of a signal:

$\displaystyle \zbox{\hbox{\sc Downsample}_{M,n}(x) \;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\;x(Mn)}

In the DFT case, $ \hbox{\sc Downsample}_M$ maps $ \mathbb{C}^N$ to $ \mathbb{C}^{\frac{N}{M}}$ , while for the DTFT, $ \hbox{\sc Downsample}_M$ maps $ \mathbb{C}^\infty$ to $ \mathbb{C}^\infty$ .

The Aliasing Theorem states that downsampling in time corresponds to aliasing in the frequency domain:

$\displaystyle \zbox{\hbox{\sc Downsample}_M(x) \leftrightarrow \frac{1}{M} \hbox{\sc Alias}_M(X)}

where the $ \hbox{\sc Alias}$ operator is defined for $ X\in\mathbb{C}^N$
(DFT case) as

$\displaystyle \zbox{\hbox{\sc Alias}_{M,l}(X) \;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\;\sum_{k=0}^{M-1} X\left(l+k\frac{N}{M}\right),}\quad
l = 0,1,\ldots,\frac{N}{M}-1

For $ X\in\mathbb{C}^\infty$ (DTFT case), the $ \hbox{\sc Alias}$ operator is

\hbox{\sc Alias}_{M,\omega}(X)
\sum_{k=0}^{M-1} X\left(e^{j(\frac{\omega}{M} + k\frac{2\pi}{M})}\right),
\quad -\pi\leq \omega < \pi\\
&\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \sum_{k=0}^{M-1} X\left(W_M^k z^\frac{1}{M}\right)

where $ W_M\mathrel{\stackrel{\mathrm{\Delta}}{=}}e^{j2\pi/M}$ is a common notation for the primitive $ M$ th root of unity, and $ z=e^{j\omega}$ as usual. This normalization corresponds to $ T=1$ after downsampling. Thus, $ T=1/M$ prior to downsampling.

The summation terms above for $ k\neq 0$ are called aliasing components.

The aliasing theorem points out that in order to downsample by factor $ M$ without aliasing, we must first lowpass-filter the spectrum to $ [-\pi f_s / M, \pi f_s / M]$ . This filtering essentially zeroes out the spectral regions which alias upon sampling.

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``Review of the Discrete Fourier Transform (DFT)'', by Julius O. Smith III, (From Lecture Overheads, Music 421).
Copyright © 2020-06-27 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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