Method 2

The second method is based on constructing a partial fraction expansion of the admittance directly:

$$\Gamma (z) = \Gamma _0 (1-z^{-1}) \sum_{i=1}^{M/2} \frac{1}{1 + a_1(i) z^{-1} + a_2(i) z^{-2}}$$

with $a_m(i)$ as above. While such a construction is not guaranteed to be positive real, (please feel free to find general conditions for which the positive-real condition can be guaranteed), we now have direct control over the bandwidths and modal gains (pole residues in the admittance). The reason the construction tends to be positive real is that by using the same phase for each section (the 1 in all the section numerators), we are sure to get a zero forming at some frequency near the middle between the resonance frequencies, and at a similar distance from the unit circle. This means we are constructing interlacing poles and zeros by simply adding the resonators in parallel. The extra zero near dc is to ensure that the admittance looks like a lightly damped spring at zero frequency. Since half the sampling rate merely ``cuts off'' the frequency response, there is no corresponding requirement to add a pole near $z=-1$ as if it were the point at infinity. However, the phase should be checked not to exceed plus or minus $90$ degrees there (or at any other frequency), and a pole or zero added if necessary to pull it into the positive-real range.

A simple example of a synthetic bridge constructed using this method with is shown in Fig. K.7.

Figure K.7: Synthetic guitar-bridge admittance using method 2.