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Chapters

Physical Audio Signal Processing
    Acoustic Modeling with Digital Delay
       Feedback Delay Networks (FDN)
          Single-Input, Single-Output (SISO) FDN

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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:

$\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.1.23.

Figure 1.23: Order 3 SISO Feedback Delay Network (FDN).
\begin{figure}\input fig/FDNSISO.pstex_t \end{figure}

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 [460, 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.1.23 can be handled in one of two ways: (1) the matrices $ \mathbf{A}, \mathbf{B}, \mathbf{C}$ can be augmented to order