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Example:
For $ N=3$ we have, in general,

$\displaystyle \left<\underline{u},\underline{v}\right> = u_0 \overline{v_0} + u_1 \overline{v_1} + u_2 \overline{v_2}.
$

Let

\begin{eqnarray*}
\underline{u}&=& [0,j,1] \\
\underline{v}&=& [1,j,j].
\end{eqnarray*}

Then

$\displaystyle \left<\underline{u},\underline{v}\right> = 0\cdot 1 + j \cdot (-j) + 1 \cdot (-j) = 0 + 1 + (-j) = 1-j.
$

See §I.3.1 regarding computation of inner products in the matlab programming language.


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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