The vector 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 . In a computational model of a drum, for example, may be changed corresponding to different striking locations on the drumhead.
The vector in a modal representation (Eq.(G.21)) specifies how the modes are to be mixed into the output. In other words, specifies how the output signal is to be created as a linear combination of the mode states:
In a computational model of an electric guitar string, for example, changes whenever a different pick-up is switched in or out (or is moved ).
The modal representation is not unique since and may be scaled in compensating ways to produce the same transfer function. (The diagonal elements of may also be permuted along with and .) Each element of the state vector 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 will oscillate sinusoidally, after the input becomes zero, at some frequency , where
relates the system eigenvalue to the oscillation frequency , with denoting the sampling interval in seconds. More generally, in the damped case, we have
where is the pole (eigenvalue) radius. For stability, we must have
In practice, we often prefer to combine complex-conjugate pole-pairs to form a real, ``block-diagonal'' system; in this case, the transition matrix is block-diagonal with two-by-two real matrices along its diagonal of the form
where is the pole radius, and re . 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.