Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books



Chapters

Spectral Audio Signal Processing
    Overlap-Add (OLA) STFT Processing
       Convolution of Short Signals
          Acyclic FFT Convolution

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?

  

Acyclic FFT Convolution

If we add enough trailing zeros to the signals being convolved, we can obtain acyclic convolution embedded within a cyclic convolution. How many zeros do we need to add? Suppose the signal $ x(n)$ consists of $ N_x$ contiguous nonzero samples at times 0 to $ N_x-1$, preceded and followed by zeros, and suppose $ h(n)$ is nonzero only over a block of $ N_h$ samples starting at time 0. Then the acyclic convolution of $ x$ with $ h$ reduces to

$\displaystyle (x\ast h)(n) \isdef \sum_{m=-\infty}^\infty x(m)h(n-m) = \sum_{m=0}^n x(m)h(n-m) $

which is zero for $ n<0$ and $ n>(N_x+N_h-1)-1$. Thus,
$\textstyle \parbox{0.8\textwidth}{\emph{the acyclic convolution of $N_x$\ samples with $N_h$\ samples produces at most $N_x+N_h-1$\ nonzero samples.}}$
The number $ N_x+N_h-1$ is easily checked for signals of length 1 since $ \delta\ast \delta = \delta$, where $ \delta$ is 1 at time zero and 0 at all other times. Similarly,

$\displaystyle [\delta+\hbox{\sc Shift}_1(\delta)] \ast [\delta+\hbox{\sc Shift... ...\delta)] = \delta + 2\hbox{\sc Shift}_1(\delta) + \hbox{\sc Shift}_2(\delta) $

and so on.

When $ N_x$ or $ N_h$ is infinity, the convolution result can be as small as 1. For example, consider $ x=[1,r,r^2,r^3,\ldots]$, with $ \left\vert r\right\vert<1$, and $ h=[1,-r,0,0,\ldots]$. Then $ x\ast h = [1, 0, 0, \ldots]$. This is an example of what is called deconvolution. In the frequency domain, deconvolution always involves a pole-zero cancellation. Therefore, it is only possible when $ N_x$ or $ N_h$ is infinite. In practice, deconvolution can sometimes be accomplished approximately, particularly within narrow frequency bands [104].

We thus conclude that, to embed acyclic convolution within a cyclic convolution (as provided by the FFT), we need to zero-pad both operands out to length $ N$, where $ N$ is at least the sum of the operand lengths (minus one).


Subsections

Order a Hardcopy of Spectral Audio Signal Processing

Previous: Cyclic FFT Convolution
Next: Acyclic Convolution in Matlab or Octave
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.


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? )