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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}
\...
...}{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 [218].


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