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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Artificial Reverberation
       FDN Reverberation
          Choice of Lossless Feedback Matrix
             Hadamard Matrix

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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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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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