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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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