Feedback Delay Networks (FDN)

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**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 by the product of a diagonal matrix ${\bm \Gamma}$ times an orthogonal matrix $\mathbf{Q}$ , as shown in Fig.2.28 for . 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 in Eq. $\,$ (2.10), the FDN (Fig.2.28) reduces to a normal state-space model (§1.3.7),

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

. 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 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 , the input vector $\mathbf{u}(n)$ in Fig.2.28 can be defined as the scalar input 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

components of $\mathbf{y}(n)$ . An example single-input, single-output (SISO) FDN for

is shown in Fig.2.29.

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

Note that when , 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 such that the three delay lines are replaced by 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

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 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

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

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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