Properties of the Modal Representation

The vector ${\tilde B}$ in a modal representation (Eq. $\,$ (G.21)) specifies how the modes are driven by the input. That is, the th mode receives the input signal weighted by ${\tilde b}_i$ . In a computational model of a drum, for example, ${\tilde B}$ may be changed corresponding to different striking locations on the drumhead.

The vector ${\tilde C}$ in a modal representation (Eq. $\,$ (G.21)) specifies how the modes are to be mixed into the output. In other words, ${\tilde C}$ specifies how the output signal is to be created as a linear combination of the mode states:

$\displaystyle y(n) = {\tilde C}\underline{{\tilde x}}(n) = {\tilde c}_1 {\tilde x}_1(n) + {\tilde c}_2 {\tilde x}_2(n) + \cdots + {\tilde c}_N {\tilde x}_N(n)$

In a computational model of an electric guitar string, for example, ${\tilde C}$ changes whenever a different pick-up is switched in or out (or is moved [98]).

The modal representation is not unique since ${\tilde B}$ and ${\tilde C}$ may be scaled in compensating ways to produce the same transfer function. (The diagonal elements of $\tilde{A}$ may also be permuted along with ${\tilde B}$ and ${\tilde C}$ .) Each element of the state vector $\underline{{\tilde x}}(n)$ holds the state of a single first-order mode of the system.

For oscillatory systems, the diagonalized state transition matrix must contain complex elements. In particular, if mode is both oscillatory and undamped (lossless), then an excited state-variable ${\tilde x}_i(n)$ will oscillate sinusoidally, after the input becomes zero, at some frequency $\omega_i$ , where

$\displaystyle \lambda _i = e^{j\omega_iT}$

relates the system eigenvalue $\lambda_i$ to the oscillation frequency $\omega_i$ , with

denoting the sampling interval in seconds. More generally, in the damped case, we have

$\displaystyle \lambda _i = R_i e^{j\omega_iT}$

where

is the pole (eigenvalue) radius. For stability, we must have

$\displaystyle \left\vert R_i\right\vert<1.$

In practice, we often prefer to combine complex-conjugate pole-pairs to form a real, ``block-diagonal'' system; in this case, the transition matrix $\tilde{A}$ is block-diagonal with two-by-two real matrices along its diagonal of the form

$\displaystyle \mathbf{A}_i = \left[\begin{array}{cc} 2R_iC_i & -R_i^2 \\ [2pt] 1 & 0 \end{array}\right]$

where $R_i=\left\vert\lambda_i\right\vert$ is the pole radius, and $2R_iC_i \isdef 2R_i\cos(\omega_i T) = \lambda_i + \overline{\lambda_i} = 2$ re $\left\{\lambda_i\right\}$ . Note that, for real systems, a real second order block requires only two multiplies (one in the lossless case) per time update, while a complex second-order system requires two complex multiplies. The function cdf2rdf() in the Matlab Control Toolbox can be used to convert complex diagonal form to real block-diagonal form.

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written by 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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Introduction to Digital Filters
    State Space Filters
       Modal Representation
          Properties of the Modal Representation