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Embedded SystemsFPGAElectronics

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.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].

Previous: FDN and State Space Descriptions
Next: FDN Stability

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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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