Modeling, Estimation and Optimal Filtration in Signal Processing

Chapter 6

Application of the Kalman Filter to Signal Enhancement

6.1. Introduction

The Kalman filter, introduced in the previous chapter, is extensively used in signal analysis, for a variety of applications such as biomedical, navigation, guidance, econometrics, etc. It has been the topic of a large amount of research. The reader is referred to [2] [5] [20] [25] and [32]. This list is by no means an exhaustive one.

This chapter will mainly be concerned with the use of Kalman filtering in the following case: given a signal disturbed by an additive noise, how can we enhance the signal when only a single sequence of the noisy signal is available, and when there is no a priori information either on the signal or on the noise? This formulation applies, for example, to the case of the single-channel enhancement of a speech signal disturbed by an additive noise.

We will start out with the case of a signal disturbed by a white noise. Assuming that the speech signal can be modeled by a p^th-order autoregressive process, we will look at the state space representation of the system. Thereafter, we will present the Kalman filter whose conventional form requires prior knowledge of the state space's dynamic parameters and of the variances of both the driving process and the additive noise. We will then review the single-channel enhancement methods based on the Kalman filter. Then, we will propose several alternative approaches which forego the variances of the driving process and the measurement ...