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{\bf 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{\bf 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}).

Example: Downsampling by 2

For $ N=2$ , downsampling by 2 can be expressed as $ y(n) = x(2n)$ , so that (since $ W_2\isdef e^{-j2\pi/2}=-1$ )

Y(z) &=& \frac{1}{2}\left[X\left(W^0_2 z^{1/2}\right) + X\left(W^1_2 z^{1/2}\right)\right] \\ [5pt]
&=& \frac{1}{2}\left[X\left(e^{-j2\pi 0/2} z^{1/2}\right) + X\left(e^{-j2\pi 1/2}z^{1/2}\right)\right] \\ [5pt]
&=& \frac{1}{2}\left[X\left(z^{1/2}\right) + X\left(-z^{1/2}\right)\right] \\ [5pt]
&=& \frac{1}{2}\left[\hbox{\sc Stretch}_2(X) + \hbox{\sc Stretch}_2\left(\hbox{\sc Shift}_\pi(X)\right)\right].

Example: Upsampling by 2

For $ N=2$ , upsampling (stretching) by 2 can be expressed as
$ y=[x_0,0,x_1,0,\ldots]$ , so that

$\displaystyle Y(z) \eqsp X(z^2) \eqsp \hbox{\sc Repeat}_2(X),$ (12.2)

as discussed more fully in §2.3.11.

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Filtering and Downsampling
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Upsampling (Stretch) Operator