Short-time harmonic analysis via the state-space optimal FIR filter

We suggest a statistically optimal FIR filter which estimates the time-varying Fourier coefficients of the quasi-periodic signal. It is shown that when the time-varying Fourier coefficients are in the random-walk motion, the quasi-periodic signal is represented by the stochastic state model whose state noise reflects the random increment of the Fourier coefficients. The state-space optimal FIR filter can then be applied to the state model to get the optimal short-time estimate. For the signals whose Fourier coefficients are nearly constant in the short-time interval, a simplified solution of the optimal FIR filter is also suggested. It is shown that the optimal FIR filter in the harmonic analysis problem is a stochastic and structural generalization of the DFT, and the optimal FIR filter gives exact harmonic estimate when the signal is periodic and noiseless. It is also shown by examples that the noise suppressing and the ability to resolve changes of the Fourier coefficients can be tuned by adjusting the filter length and the noise covariance setting.