Downsampling and Aliasing

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

$\displaystyle \zbox {\hbox{\sc Downsample}_{M,n}(x) \isdef x(Mn)}$

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

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

where the $\hbox{\sc Alias}$ operator is defined as

$\displaystyle \zbox {\hbox{\sc Alias}_{M,\omega}(X) \isdef \sum_{k=0}^{M-1} X\left(\omega+k\frac{2\pi}{M}\right)}$

for $\omega\in[-\pi,\pi)$ . The summation terms for $k\neq 0$ are called aliasing components.

In z transform notation, the $\hbox{\sc Alias}$ operator can be expressed as [266]

$\displaystyle \hbox{\sc Alias}_{M,z}(X) = \sum_{k=0}^{M-1} X\left(W_M^k z^\frac{1}{M}\right)$

where $W_M\isdef e^{j2\pi/M}$ is a common notation for the primitive

th root of unity. On the unit circle of the

plane, this becomes

$\displaystyle \hbox{\sc Alias}_{M,\omega}(X) = \sum_{k=0}^{M-1} X\left(e^{j\le... ...frac{\omega}{M} + k\frac{2\pi}{M}\right)}\right), \quad -\pi\leq \omega < \pi.$

The frequency scaling corresponds to having a sampling inverval of

after downsampling, which corresponds to the interval

prior to downsampling.

The aliasing theorem makes it clear that, in order to downsample by factor without aliasing, we must first lowpass-filter the spectrum to $(-\pi / M, \pi / M)$ . This filtering (when ideal) zeroes out the spectral regions which alias upon downsampling.

Note that any rational sampling-rate conversion factor $\rho = L/M$ may be implemented as an upsampling by the factor followed by downsampling by the factor [50,266]. Conceptually, a stretch-by- is followed by a lowpass filter cutting off at $\omega_c \isdef \pi/(L\;\max\;M)$ , followed by downsample-by-, i.e.,

$\displaystyle x^\prime = \hbox{\sc Downsample}_M\{\hbox{\sc Lowpass}_{\omega_c}[\hbox{\sc Stretch}_L(x)]\}$

In practice, there are more efficient algorithms for sampling-rate conversion [254] based on a more direct approach to bandlimited interpolation.

Subsections

Proof of Aliasing Theorem

Previous: Stretch/Repeat (Scaling) Theorem
Next: Proof of Aliasing Theorem

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About the Author: 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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See Also

Spectral Audio Signal Processing
    Fourier Transforms for Continuous/Discrete Time/Frequency
       Fourier Theorems for the DTFT
          Downsampling and Aliasing