Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books

Sponsor

Industry's highest performing at the lowest power DSPs now as low as $5.00*
Start development today!
*volume pricing for 10ku

Chapters

See Also

Embedded SystemsFPGAElectronics

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

Chapter Contents:

Search Spectral Audio Signal Processing

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

Downsampling (Decimation) Operator

figure[htbp] \includegraphics{eps/downsample}

Figure 10.3 shows the symbol for downsampling by the factor $ N$. The downsampler selects every $ N$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 $ 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.10.4.

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
Previous: Upsampling (Stretch) Operator
Next: Example: Downsampling by 2

Order a Hardcopy of Spectral Audio Signal Processing

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.


Comments

No comments yet for this page


Add a Comment
You need to login before you can post a comment (best way to prevent spam). ( Not a member? )