Downsampling and Aliasing

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

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

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

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

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

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

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 [264]

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

where $W_M\isdef e^{j2\pi/M}$ is a common notation for the primitive $M$th root of unity. On the unit circle of the $z$ plane, this becomes

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

The frequency scaling corresponds to having a sampling inverval of $T=1$ after downsampling, which corresponds to the interval $T=1/M$ prior to downsampling.

The aliasing theorem makes it clear that, in order to downsample by factor $M$ 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 sampling.

Note that any rational sampling-rate conversion factor $\rho = L/M$ may be implemented as an upsampling by the factor $L$ followed by downsampling by the factor $M$ [46,264]. Conceptually, a stretch-by-$L$ is followed by a lowpass filter cutting off at $\omega_c \isdef \pi/(L\;\max\;M)$, followed by downsample-by-$L$, i.e.,

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

In practice, there are more efficient algorithms for sampling-rate conversion [247] based on a more direct approach to bandlimited interpolation.