Most General Lossless Feedback Matrices

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Physical Audio Signal Processing

Choice of Lossless Feedback Matrix

Householder Reflections

Triangular Feedback Matrices

A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

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Choice of Lossless Feedback Matrix

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Most General Lossless Feedback Matrices

As shown in §C.15.3, an FDN feedback matrix $\mathbf{A}_N$ is lossless if and only if its eigenvalues have modulus 1 and its eigenvectors are linearly independent.

A unitary matrix is any (complex) matrix that is inverted by its own (conjugate) transpose:

$\displaystyle Q^{-1} = Q^H,$

where

denotes the Hermitian conjugate (i.e., the complex-conjugate transpose) of

. When

is real (as opposed to complex), we may simply call it an orthogonal matrix, and we write $Q^{-1} = Q^T$ , where

denotes matrix transposition.

All unitary (and orthogonal) matrices have unit-modulus eigenvalues and linearly independent eigenvectors. As a result, when used as a feedback matrix in an FDN, the resulting FDN will be lossless (until the delay-line damping filters are inserted, as discussed in §3.7.4 below).

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Most General Lossless Feedback Matrices

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University