Downsampling and Aliasing

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

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

(3.32)

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

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

(3.33)

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

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

(3.34)

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 [287]

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

(3.35)

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

th root of unity. On the unit circle of the

plane, this becomes

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

(3.36)

The frequency scaling corresponds to having a sampling interval 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 [50,287]. Conceptually, a stretch-by- is followed by a lowpass filter cutting off at $\omega_c \isdeftext \pi/(L\;\max\;M)$ , followed by downsample-by- , i.e.,

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

(3.37)

In practice, there are more efficient algorithms for sampling-rate conversion [270,135,78] based on a more direct approach to bandlimited interpolation.

Proof of Aliasing Theorem

To show:

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

$\fbox{$x(nN) \;\longleftrightarrow\;\dfrac{1}{N} \displaystyle\sum_{m=0}^{N-1} X\left(e^{j2\pi m/N} z^{1/N}\right)$}$

From the DFT case [264], we know this is true when and are each complex sequences of length , in which case and are length . Thus,

$\displaystyle x(nN) \;\longleftrightarrow\; Y(\omega_k N) \eqsp \frac{1}{N} \sum_{m=0}^{N-1} X\left(\omega_k + \frac{2\pi}{N} m \right), \; k\in [0,N_s/N)$

(3.38)

where we have chosen to keep frequency samples $\omega_k$ in terms of the original frequency axis prior to downsampling, i.e., $\omega_k = 2\pi k/ N_s$ for both

and

. This choice allows us to easily take the limit as $N_s\to\infty$ by simply replacing $\omega_k$ by $\omega$ :

$\displaystyle x(nN) \;\longleftrightarrow\; Y(\omega N) \eqsp \frac{1}{N} \sum_{m=0}^{N-1} X\left(\omega + \frac{2\pi}{N} m \right), \; \omega\in[0,2\pi/N)$