Relation to Finite Difference ApproximationThe 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 ExpansionsThe 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 , 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 9.7For a related worked example, see §C.17.6.
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 . In this case, the resonator model becomes
Ideal Differentiator (Spring Admittance)
Further Reading in Nonlinear Methods