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Hadamard Matrix

A second-order Hadamard matrix may be defined by

$\displaystyle \mathbf{H}_2 \isdef
\frac{1}{\sqrt{2}}
\left[\begin{array}{rr}
1 & 1\\
1 & -1
\end{array}\right],
$

with higher order Hadamard matrices defined by recursive embedding, e.g.,

$\displaystyle \mathbf{H}_4 \isdef
\frac{1}{\sqrt{2}}
\left[\begin{array}{rr}
\mathbf{H}_2 & \mathbf{H}_2\\
\mathbf{H}_2 & -\mathbf{H}_2
\end{array}\right].
=
\frac{1}{2}
\left[\begin{array}{rrrr}
1& 1& 1& 1\\
1&-1& 1&-1\\
1& 1&-1&-1\\
1&-1&-1& 1
\end{array}\right].
$

When $ n$ is a power of $ 4$ , the Hadamard matrix $ \mathbf{H}_n$ of that order requires no multiplies in fixed-point arithmetic. An $ n\times
n$ Hadamard matrix has the maximum possible determinant of any $ n\times
n$ complex matrix containing elements which are bounded by $ 1$ in magnitude. This can be seen as an optimal mixing and scattering property of the matrix.

As of version 0.9.30, Faust's math.lib4.12contains a function called hadamard(n) for generating an $ n\times
n$ Hadamard matrix, where $ n$ must be a power of $ 2$ . A Hadamard feedback matrix is used in the programming example reverb_designer.dsp (a configurable FDN reverberator) distributed with Faust.

A Hadamard feedback matrix is said to be used in the IRCAM Spatialisateur [219].


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA