Hadamard Matrix

A second-order Hadamard matrix may be defined by

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

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

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

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.lib^4.12contains 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].