Downsampling (Decimation) Operator

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Downsampling (Decimation) Operator

**Figure:** Downsampling by .
$\begin{figure}\input fig/downsample.pstex_t \end{figure}$

Figure 11.3 shows the symbol for downsampling by the factor . The downsampler selects every th sample and discards the rest:

$\begin{eqnarray*} y(n) &=& \hbox{\sc Downsample}_{N,n}(x)\\ &\isdef & x(Nn), \quad n\in{\bf Z} \end{eqnarray*}$

In the frequency domain, we have

$\begin{eqnarray*} Y(z) &=& \hbox{\sc Alias}_{N,z}(X)\\ &\isdef & \frac{1}{N} \... ...(z^\frac{1}{N}e^{-jm\frac{2\pi}{N}} \right), \quad z\in{\bf C}. \end{eqnarray*}$

Thus, the frequency axis is expanded by factor , wrapping times around the unit circle, adding to itself times. For , two partial spectra are summed, as indicated in Fig.11.4.

**Figure:** Illustration of **$\hbox {\sc Alias}_2$** in the frequency domain.
$\includegraphics[scale=0.8]{eps/dnsampspec}$

Using the common twiddle factor notation

$\displaystyle W_N \isdef e^{-j2\pi/N},$

the aliasing expression can be written as

$\displaystyle Y(z) = \frac{1}{N} \sum_{m=0}^{N-1} X(W_N^m z^{1/N}).$

Subsections

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written by 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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Spectral Audio Signal Processing
    Multirate Polyphase Filter Banks
       Upsampling and Downsampling
          Downsampling (Decimation) Operator