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Feedback Delay Networks (FDN)

Figure 2.28: Order 3 MIMO Feedback Delay Network (FDN).
\includegraphics[width=\twidth]{eps/FDNMIMO}

The FDN can be seen as a vector feedback comb filter,3.10obtained by replacing the delay line with a diagonal delay matrix (defined in Eq.$ \,$(2.10) below), and replacing the feedback gain $ g$ by the product of a diagonal matrix $ {\bm \Gamma}$ times an orthogonal matrix $ \mathbf{Q}$, as shown in Fig.2.28 for $ N=3$. The time-update for this FDN can be written as

$\displaystyle \left[\begin{array}{c} x_1(n) \\ [2pt] x_2(n) \\ [2pt] x_3(n)\end...
...gin{array}{c} u_1(n) \\ [2pt] u_2(n) \\ [2pt] u_3(n)\end{array}\right] \protect$ (3.6)

with the outputs given by

$\displaystyle \left[\begin{array}{c} y_1(n) \\ [2pt] y_2(n) \\ [2pt] y_3(n)\end...
...array}{c} x_1(n-M_1) \\ [2pt] x_2(n-M_2) \\ [2pt] x_3(n-M_3)\end{array}\right],$ (3.7)

or, in frequency-domain vector notation,
$\displaystyle \mathbf{X}(z)$ $\displaystyle =$ $\displaystyle {\bm \Gamma}\mathbf{Q}\mathbf{D}(z)\mathbf{X}(z) + \mathbf{U}(z)$ (3.8)
$\displaystyle \mathbf{Y}(z)$ $\displaystyle =$ $\displaystyle \mathbf{D}(z) \mathbf{X}(z)$ (3.9)

where

$\displaystyle \mathbf{D}(z) \isdef \left[\begin{array}{ccc} z^{-M_1} & 0 & 0\\ [2pt] 0 & z^{-M_2} & 0\\ [2pt] 0 & 0 & z^{-M_3} \end{array}\right]. \protect$ (3.10)

FDN and State Space Descriptions

When $ M_1=M_2=M_3=1$ in Eq.$ \,$(2.10), the FDN (Fig.2.28) reduces to a normal state-space model1.3.7),

$\displaystyle \mathbf{x}(n+1)$ $\displaystyle =$ $\displaystyle \mathbf{A}\, \mathbf{x}(n) + \mathbf{u}_+(n)$  
$\displaystyle \mathbf{y}_+(n)$ $\displaystyle =$ $\displaystyle \mathbf{x}(n)
\protect$ (3.11)

The matrix $ \mathbf{A}={\bm \Gamma}\mathbf{Q}$ is the state transition matrix. The vector $ \mathbf{x}(n) = [x_1(n), x_2(n), x_3(n)]^T$ holds the state variables that determine the state of the system at time $ n$. The order of a state-space system is equal to the number of state variables, i.e., the dimensionality of $ \mathbf{x}(n)$. The input and output signals have been trivially redefined as

\begin{eqnarray*}
\mathbf{u}_+(n) &\isdef & \mathbf{u}(n+1)\\
\mathbf{y}_+(n) &\isdef & \mathbf{y}(n+1)
\end{eqnarray*}

to follow normal convention for state-space form.

Thus, an FDN can be viewed as a generalized state-space model for a class of $ N$th-order linear systems--``generalized'' in the sense that unit delays are replaced by arbitrary delays. This correspondence is valuable for analysis because tools for state-space analysis are well known and included in many software libraries such as with matlab.


Single-Input, Single-Output (SISO) FDN

When there is only one input signal $ u(n)$, the input vector $ \mathbf{u}(n)$ in Fig.2.28 can be defined as the scalar input $ u(n)$ times a vector of gains:

$\displaystyle \mathbf{u}(n) = \mathbf{B}u(n)
$

where $ \mathbf{B}$ is an $ N\times 1$ matrix. Similarly, a single output can be created by taking an arbitrary linear combination of the $ N$ components of $ \mathbf{y}(n)$. An example single-input, single-output (SISO) FDN for $ N=3$ is shown in Fig.2.29.

Figure 2.29: Order 3 SISO Feedback Delay Network (FDN).
\includegraphics[width=\twidth]{eps/FDNSISO}

Note that when $ M_1=M_2=M_3=1$, this system is capable of realizing any transfer function of the form

$\displaystyle H(z) = \frac{\beta_1z^{-1}+\beta_2z^{-2}+\beta_3z^{-3}}{1+a_1z^{-1}+a_2z^{-2}+a_3z^{-3}}.
$

By elementary state-space analysis [449, Appendix E], the transfer function can be written in terms of the FDN system parameters as

$\displaystyle H(z) = \mathbf{C}^T(z\mathbf{I}- \mathbf{A})^{-1}\mathbf{B}
$

where $ \mathbf{I}$ denotes the $ 3\times 3$ identity matrix. This is easy to show by taking the z transform of the impulse response of the system.

The more general case shown in Fig.2.29 can be handled in one of two ways: (1) the matrices $ \mathbf{A}, \mathbf{B}, \mathbf{C}$ can be augmented to order $ N=M_1+M_2+M_3$ such that the three delay lines are replaced by $ N$ unit-sample delays, or (2) ordinary state-space analysis may be generalized to non-unit delays, yielding

$\displaystyle H(z) = \mathbf{C}^T \mathbf{D}(z)\left[\mathbf{I}- \mathbf{A}\mathbf{D}(z)\right]^{-1}\mathbf{B}
$

where $ \mathbf{C}^T$ denotes the matrix transpose of $ \mathbf{C}$, and

$\displaystyle \mathbf{D}(z) \isdef \left[\begin{array}{ccc} z^{-M_1} & 0 & 0\\ [2pt] 0 & z^{-M_2} & 0\\ [2pt] 0 & 0 & z^{-M_3} \end{array}\right]. \protect$

In FDN reverberation applications, $ \mathbf{A}={\bm \Gamma}\mathbf{Q}$, where $ \mathbf{Q}$ is an orthogonal matrix, for reasons addressed below, and $ {\bm \Gamma}$ is a diagonal matrix of lowpass filters, each having gain bounded by 1. In certain applications, the subset of orthogonal matrices known as circulant matrices have advantages [385].


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.


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Comb Filters