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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Acoustic Modeling with Digital Delay
       Feedback Delay Networks (FDN)
          FDN Stability

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FDN Stability

Stability of the FDN is assured when some norm [451] of the state vector $ \mathbf{x}(n)$ decreases over time when the input signal is zero [220, ``Lyapunov stability theory'']. That is, a sufficient condition for FDN stability is

$\displaystyle \left\Vert\,\mathbf{x}(n+1)\,\right\Vert < \left\Vert\,\mathbf{x}(n)\,\right\Vert, \protect$ (3.12)

for all $ n\geq0$, where $ \left\Vert\,\mathbf{x}(n)\,\right\Vert$ denotes the norm of $ \mathbf{x}(n)$, and

$\displaystyle \mathbf{x}(n+1) = \mathbf{A}\left[\begin{array}{c} x_1(n-M_1) \\ [2pt] x_2(n-M_2) \\ [2pt] x_3(n-M_3)\end{array}\right]. $

Using the augmented state-space analysis mentioned above, the inequality of Eq.$ \,$(2.12) holds under the $ L2$ norm [451] whenever the feedback matrix $ \mathbf{A}$ in Eq.$ \,$(2.6) satisfies [473]

$\displaystyle \left\Vert\,\mathbf{A}\mathbf{x}\,\right\Vert _2 < \left\Vert\,\mathbf{x}\,\right\Vert _2 \protect$ (3.13)

for all $ \mathbf{x}$, where $ \left\Vert\,\cdot\,\right\Vert _2$ denotes the $ L2$ norm, defined by

$\displaystyle \left\Vert\,\mathbf{x}\,\right\Vert _2 \isdef \sqrt{x_1^2+x_2^2+\dots+x_N^2}. $

In other words, stability is guaranteed when the feedback matrix decreases the $ L2$ norm of its input vector.

The matrix norm corresponding to any vector norm $ \vert\vert\,\cdot\,\vert\vert $ may be defined for the matrix $ \mathbf{A}$ as

$\displaystyle \left\Vert\,\mathbf{A}\,\right\Vert \isdef \max_{\mathbf{x}\neq \... ...\Vert\,\mathbf{A}\mathbf{x}\,\right\Vert}{\left\Vert\,\mathbf{x}\,\right\Vert} $

where $ \left\Vert\,\mathbf{x}\,\right\Vert$ denotes the norm of the vector $ \mathbf{x}$. In other words, the matrix norm ``induced'' by a vector norm is given by the maximum of $ \vert\vert\,\mathbf{A}\mathbf{x}\,\vert\vert $ over all unit-length vectors $ \mathbf{x}$ in the space. When the vector norm is the $ L2$ norm, the induced matrix norm is often called the spectral norm. Thus, Eq.$ \,$(2.13) can be restated as

$\displaystyle \left\Vert\,\mathbf{A}\,\right\Vert _2 < 1 \protect$ (3.14)

where $ \left\Vert\,\mathbf{A}\,\right\Vert _2$ denotes the spectral norm of $ \mathbf{A}$.

It can be shown [167] that the spectral norm of a matrix $ \mathbf{A}$ is given by the largest singular value of $ \mathbf{A}$ (`` $ \left\Vert\,\mathbf{A}\,\right\Vert _2=\sigma_1(\mathbf{A})$''), and that this is equal to the square-root of the largest eigenvalue of $ \mathbf{A}\mathbf{A}^T$, where $ \mathbf{A}^T$ denotes the matrix transpose of the real matrix $ \mathbf{A}$.3.11

Since every orthogonal matrix $ \mathbf{Q}$ has spectral norm 1,3.12 a wide variety of stable feedback matrices can be parametrized as

$\displaystyle \mathbf{A}= {\bm \Gamma}\mathbf{Q} $

where $ \mathbf{Q}$ is any orthogonal matrix, and $ {\bm \Gamma}$ is a diagonal matrix having entries less than 1 in magnitude:

$\displaystyle {\bm \Gamma}= \left[ \begin{array}{cccc} g_1 & 0 & \dots & 0\\ 0... ...\\ 0 & 0 & \dots & g_N \end{array}\right], \quad \left\vert g_i\right\vert<1. $

An alternative stability proof may be based on showing that an FDN is a special case of a passive digital waveguide network (derived in §C.15). This analysis reveals that the FDN is lossless if and only if the feedback matrix $ \mathbf{A}$ has unit-modulus eigenvalues and linearly independent eigenvectors.

Previous: Single-Input, Single-Output (SISO) FDN
Next: Allpass Filters

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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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