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Mathematics of the DFT
    Fourier Theorems for the DFT
       Signal Operators
          Convolution
             Commutativity of Convolution

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Commutativity of Convolution

Convolution (cyclic or acyclic) is commutative, i.e.,

$\displaystyle \zbox {x\circledast y = y\circledast x .} $



Proof:

\begin{eqnarray*} (x\circledast y)_n &\isdef & \sum_{m=0}^{N-1}x(m) y(n-m) = \s... ...=& \sum_{l=0}^{N-1}y(l) x(n-l) \\ &\isdef & (y \circledast x)_n \end{eqnarray*}

In the first step we made the change of summation variable $ l\isdeftext n-m$, and in the second step, we made use of the fact that any sum over all $ N$ terms is equivalent to a sum from 0 to $ N-1$.

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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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