A Kalman filtering approach to short-time Fourier analysis

Abstract

The problem of estimating time-varying harmonic components of a signal measured in noise is considered. The approach used is via state estimation. Two methods are proposed, one involving poleplacement of a state observer, the other using quadratic optimization techniques. The result is the development of a new class of filters, akin to recursive frequency-sampling filters, for inclusion in a parallel bank to produce sliding harmonic estimates. Kalman filtering theory is applied to effect the good performance in noise, and the class of filters is parameterized by the design tradeoff between noise rejection and convergence rate. These filters can be seen as generalizing the DFT. I. INTRODUCTION S HORT-TIME Fourier analysis has been a topic of considerable interest in digital signal processing [ 11, [ 2 ] , and is concerned with the evaluation of the slowly timevarying Fourier components of a quasi-periodic signal. This problem arises in many forms in signal processing, notably speech processing and recognition, vibration analysis, power system analysis, and general time series. For example, in a power system, one is frequently interested in studying the "harmonic pollution" of the fundamental mains frequency. The magnitudes and phases of the harmonics are of importance in determining the quality of supply and in identifying pollution sources. Equally, the efficiency of various thyristor motor control devices can best be determined and controlled by harmonic monitoring. The underlying signal model for situations such as these involves regularly sampled values of a continuous time signal. We presume that the period T of this signal is known or is predetermined and that there are 2N + 1 samples per period. The fundamental angular frequency is then wo = 2 r / T and we seek to determine all or part of the 2N + 1 harmonic components. The discrete-time signal model for the sampled data is

Improved Receding Horizon Fourier Analysis for Quasi-periodic Signals

“…k v is the measurement noise, which is a zeromean white Gaussian random process with the covariance R . Then, the quasi-periodic signal model in (1) can be represented with the following state space model [5,6]:…”

“…In order to avoid the poor performance due to noises, Kalman filtering approaches were suggested in [5,17]. In these approaches, FCs are included in parts of the state variables, which are estimated from Kalman filter using the stochastic information on noises.…”

Online Identification of a Two-Mass System in Frequency Domain using a Kalman Filter

“…For real-time implementation, a recursive Kalman filter can be configured to perform like a sample-by-samplebased Fourier transform (Bitmead et al, 1986). The short-time DFT can be obtained by considering the following simplified state-space representation…”

“…(11), the optimal Kalman gain K(k) has a time-varying nature. As proposed in (Bitmead et al, 1986), fixing the covariance matrix at P(k) = αI and choosing R 1x1 = r gives the steady-state values of the Kalman gain vector…”

A Simple Spectral Observer

“…Since the presentation of such a work, several designs of spectral observers with improved features have been proposed either to deal with noise [2], disturbances, lack of data [3] or to estimate other parameters such as frequency [4]. The main goals of a spectral observer are both the estimation of a given signal and the transformation of such a signal to the frequency domain by means of the recursive identification of the coefficients of a Fourier series [5].…”