The longest ringing modes are associated with the narrowest bandwidths. When they are important resonances in the frequency response, they also tend to be the tallest peaks in the frequency response magnitude. (If they are not the tallest near the beginning of the impulse response, they will be the tallest near the end.) Therefore, one effective technique for measuring the least-damped resonances is simply to find the precise location and width of the narrowest and tallest spectral peaks in the measured amplitude response of the resonator. The center-frequency and bandwidth of a narrow frequency-response peak determine two poles in the resonator to be factored out. Expressing a filter in terms of its poles and zeros is one type of ``parametric'' filter representation, as opposed to ``nonparametric'' representations such as the impulse response or frequency response. Prony's method [449,297,273] is one well known technique for estimating the frequencies and bandwidths of sums of exponentially decaying sinusoids (two-pole resonator impulse responses).
In the factoring example presented in §8.8.6, the frequency and bandwidth of the main Helmholtz air mode are measured manually using an interactive spectrum analysis tool. However, it is a simple matter to automate peak-finding in FFT magnitude data. (See, for example, the peak finders used in sinusoidal modeling, discussed a bit further in §8.8.1 below.)
Weighted digital filter design
Converting the Desired Amplitude Response to Minimum Phase