Single-Input, Single-Output (SISO) FDN

When there is only one input signal $u(n)$, the input vector $\mathbf{u}(n)$ in Fig.1.22 can be defined as the scalar input $u(n)$ times a vector of gains:

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

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.1.23.

Figure 1.23: Order 3 SISO Feedback Delay Network (FDN).

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

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

By elementary state-space analysis [460, Appendix E], the transfer function can be written in terms of the FDN system parameters as

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

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.1.23 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

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

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

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

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 [391].