Properties of the Modal Representation

The vector ${\tilde B}$ in a modal representation (Eq. (E.21)) specifies how the modes are driven by the input. That is, the $i$th mode receives the input signal $u(n)$ 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. (E.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:

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

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 [99]).

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 $i$ 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

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

relates the system eigenvalue $\lambda_i$ to the oscillation frequency $\omega_i$, with $T$ denoting the sampling interval in seconds. More generally, in the damped case, we have

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

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

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

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

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

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 Tool Box can be used to convert complex diagonal form to real block-diagonal form.