Sign in

Search Online Books

Free Online Books

Sponsor

Industry's highest performing at the lowest power DSPs now as low as $5.00*
Start development today!
*volume pricing for 10ku

Chapters

Physical Audio Signal Processing
    Acoustic Modeling with Digital Delay
       Allpass Filters
          Gerzon Nested MIMO Allpass

Chapter Contents:

Search Physical Audio Signal Processing

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

Gerzon Nested MIMO Allpass

An interesting generalization of the single-input, single-output Schroeder allpass filter (defined in §2.8.1) was proposed by Gerzon [157] for use in artificial reverberation systems.

The starting point can be the first-order allpass of Fig.2.31a on page , or the allpass made from two comb-filters depicted in Fig.2.30 on page .^3.15In either case,

all signal paths are converted from scalars to vectors of dimension ,
the delay element (or delay line) is replaced by an arbitrary $N\times N$ unitary matrix frequency response.^3.16

Let $\underline{x}(n)$ denote the $N\times 1$ input vector with components $x_i(n), i=1,\dots,N$ , and let $\underline{X}(z)=[X_1(z),\dots,X_N(z)]$ denote the corresponding vector of z transforms. Denote the $N\times 1$ output vector by $\underline{y}(n)$ . The resulting vector difference equation becomes, in the frequency domain (cf. Eq. $\,$ (2.15))

$\displaystyle \underline{Y}(z) = \overline{g} \underline{X}(z) + \mathbf{U}(z)\underline{X}(z) - g \mathbf{U}(z)\underline{Y}(z)$

which leads to the matrix transfer function

$\displaystyle \mathbf{H}(z) = [\mathbf{I}+ g \mathbf{U}(z)]^{-1}[\overline{g}\mathbf{I}+ \mathbf{U}(z)]$

where $\mathbf{I}$ denotes the $N\times N$ identity matrix, and $\mathbf{U}(z)$ denotes any paraunitary matrix transfer function [500], [449, Appendix C].

Note that to avoid implementing $\mathbf{U}(z)$ twice, $\mathbf{H}(z)$ should be realized in vector direct-form II, viz.,

$\begin{eqnarray*} \underline{v}_d(n) &=& \mathbf{U}(d)\underline{v}(n) = {\cal Z... ...line{y}(n) &=& \underline{v}(n) + \overline{g}\underline{v}_d(n) \end{eqnarray*}$

where denotes the unit-delay operator ( $d^k x(n)\isdef x(n-k)$ ).

To avoid a delay-free loop, the paraunitary matrix must include at least one pure delay in every row, i.e., $\mathbf{U}(z) = z^{-1} \mathbf{U}^\prime(z)$ where $\mathbf{U}^\prime(z)$ is paraunitary and causal.

In [157], Gerzon suggested using $\mathbf{U}(z)$ of the form

$\displaystyle \mathbf{U}(z) = \mathbf{D}(z) \mathbf{Q}$

where $\mathbf{Q}$ is a simple $N\times N$ orthogonal matrix, and

$\displaystyle \mathbf{D}(z) = \left[ \begin{array}{ccccc} z^{-m_1} & 0 & 0 & \d... ...ts & \ddots& \vdots\\ 0 & 0 & 0 & \dots & z^{-m_N} \end{array} \right] \protect$

(3.17)

is a diagonal matrix of pure delays, with the lengths

chosen to be mutually prime (as suggested by Schroeder [417] for a series combination of Schroeder allpass sections). This structure is very close to the that of typical feedback delay networks (FDN), but unlike FDNs, which are ``vector feedback comb filters,'' the vectorized Schroeder allpass is a true multi-input, multi-output (MIMO) allpass filter.

Gerzon further suggested replacing the feedback and feedforward gains $\pm g$ by digital filters $\pm G(z)$ having an amplitude response bounded by 1. In principle, this allows the network to be arbitrarily different at each frequency.

Gerzon's vector Schroeder allpass is used in the IRCAM Spatialisateur [218].

Previous: Example Allpass Filters
Next: Allpass Digital Waveguide Networks

Order a Hardcopy of Physical Audio Signal Processing

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.

Comments

No comments yet for this page

Add a Comment

You need to login before you can post a comment (best way to prevent spam). ( Not a member? )