### Relation to Finite Difference Approximation

The Finite Difference Approximation (FDA) (§7.3.1) is a special case of the matched transformation applied to the point . To see this, simply set in Eq.(8.5) to obtain

which is the FDA definition in the frequency domain given in Eq.(7.3).

Since the FDA equals the match *z* transformation for the point , it maps
analog dc () to digital dc () exactly. However, that is the
only point on the frequency axis that is perfectly mapped, as shown in
Fig.7.15.

#### State Space Approach to Modal Expansions

The preceding discussion of modal synthesis was based primarily on
fitting a sum of biquads to measured frequency-response peaks. A more
general way of arriving at a modal representation is to first form a
*state space model* of the system [449], and then convert
to the modal representation by *diagonalizing* the state-space
model. This approach has the advantage of preserving system behavior
between the given inputs and outputs. Specifically, the similarity
transform used to diagonalize the system provides new input and output
gain vectors which properly excite and observe the system modes
precisely as in the original system. This procedure is especially
more convenient than the transfer-function based approach above when
there are multiple inputs and outputs. For some mathematical details,
see [449]^{9.7}For a related worked example, see §C.17.6.

#### Delay Loop Expansion

When a subset of the resonating modes are nearly harmonically tuned,
it can be much more computationally efficient to use a *filtered
delay loop* (see §2.6.5) to generate an entire
*quasi-harmonic series of modes* rather than using a biquad for each
modal peak [439]. In this case, the resonator
model becomes

*distributed*1D propagation medium such as a vibrating string or acoustic tube. More abstractly, a superposition of such quasi-harmonic mode series can provide a computationally efficient

*psychoacoustic equivalent approximation*to arbitrary collections of modes in the range of human hearing.

Note that when is close to instead of , primarily only odd harmonic resonances are produced, as has been used in modeling the clarinet [431].

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