Elsevier

Signal Processing

Volume 77, Issue 3, September 1999, Pages 343-347
Signal Processing

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Time-varying signal frequency estimation by VFF Kalman filtering

https://doi.org/10.1016/S0165-1684(99)00085-7Get rights and content

Abstract

In this paper, a new algorithm is proposed for estimating a time-varying signal frequency. The proposed method introduces a variable forgetting factor (VFF) into Kalman filter so that this approach does not require any pre-determined forgetting factor. The proposed method is tested on two nonstationary signals. Results show that the proposed algorithm offers a more accurate estimation of signal frequency and faster convergence speed than conventional Kalman filtering algorithm.

Introduction

Kalman filter is an effective means to estimate the signal frequency [3]. However, the performance of Kalman filter in estimating a time-varying signal frequency is degraded since it refers to the entire history of the past observations. For this reason, a time-weighted-error Kalman filter was proposed which reduces the observation window size by introducing a forgetting factor [4]. The time-weighted-error Kalman filtering, however, has the drawback that it requires a pre-determined optimal forgetting factor according to the signal condition. Recently, a speech analysis method via VFF-RLS algorithm has been reported, which generates a VFF according to the glottal pulse [1].

In this paper, we present a VFF Kalman filtering algorithm. The update equation of a VFF for the Kalman filter is derived. In comparison with the time-weighted-error Kalman filtering algorithm, the proposed algorithm does not require a pre-determined forgetting factor since it calculates the forgetting factor adaptively. The proposed method provides better frequency estimation when the signal experiences nonstationarity.

Section snippets

VFF Kalman filtering

The system for signal frequency estimation can be represented by a state–space model:a(k)=Φ(k,k−1)a(k−1),s(k)=ST(k−1)a(k)+v(k),where a(k) is an n-dimensional parameter vector, Φ(k,k−1) is a time-varying parameter transition matrix, and S(k−1) is an n-dimensional vector of past observation represented by [s(k−1),…,s(k−n)]T. v(k) is a white Gaussian noise with the normal density N(0,R(k)), where R(k) is assumed to be known (or estimated). The cost function of Kalman filter with fixed forgetting

Experimental results

The proposed Kalman filter has been tested on two signals whose frequency vary with time. In each test, the frequency is estimated using second-order AR model which is updated at each sample. Fig. 1(a) and Fig. 2(a) show the estimated frequency plot using the fixed forgetting factor Kalman filter for three different forgetting factors and Fig. 1(b) and Fig. 2(b) show the result using the VFF Kalman filter. The initial values are as follows:Φ(k,k−1)=I,a(0|0)=0,P(0|0)=diag[100…100].

Test signal 1

Conclusion

In this paper, we have proposed a VFF Kalman filter to estimate the frequency of a time-varying signal. The proposed algorithm calculates the forgetting factor adaptively and reflects it to the error covariance so that the algorithm can efficiently estimate the time-varying signal frequency. The performance of the VFF Kalman filter has been verified through the simulations on two nonstationary signals. The results of the experiment with a chirp signal show that the tracking performance of the

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