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 [99]).

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

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University