#### Late impulse-response analysis

All methods useable with inverse filtering can be modified based on the observation that late in the impulse response, the damped modes have died away, and the least-damped modes dominate. Therefore, by discarding initial impulse-response data, the problem in some sense becomes ``easier'' at the price of working closer to the noise floor. This technique is most appropriate in conjunction with the inverse filtering method for mode extraction (discussed below), since for subtraction, the modal impulse response must be extrapolated back to the beginning of the data record. However, methods used to compute the filter numerator in variations on Prony's method can be used to scale and phase-align a mode for subtraction [428,297].

One simple approximate technique based on looking only at the late impulse response is to take a zero-padded FFT of the latest Hanning-windowed data segment. The least-damped modes should give very clearly dominant peaks in the FFT magnitude data. As discussed above, the peak(s) can be interpolated to estimate the mode resonance frequency, and the bandwidth can be measured to determine the time-constant of decay. Alternatively, the time-constant of decay can be measured in the time domain by measuring the decay slope of the log amplitude envelope across the segment (this time using a rectangular window). Since the least-damped mode is assumed to be isolated in the late decay, it is easy to form a pitch-synchronous amplitude envelope.

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Empirical Notes on Inverse Filtering

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Sinusoidal modeling