Time-Frequency Representation of Signals Using Kalman Filter

Data analysis is a necessary part of practical applications as well as of pure research. The problem of data analysis is of great interest in engineering areas such as signal processing, speech recognition, vibration analysis, time series modelling, etc. Sensing of physical effects transformed to time series of voltage represents measured reality for us. However, the knowledge of time characteristics of measured data is often quite insufficient for description of real signal properties. Therefore, frequency analysis, as an alternative signal description approach, has been engaged in signal processing methods. Unfortunately, data from real applications are mostly non-stationary and represent non-linear processes in general. This fact has leaded to introduction of joint time-frequency representation of measured signals. Former time-frequency methods dealing with non-stationary data analysis, such as the short-time Fourier transform, repeatedly use a block of data processing with the assumption that frequency characteristics are time-invariant and/or a process is stationary within the usage of the data block. The resolution of such methods is then given by the HeisenbergGabor uncertainty principle. This limitation is mainly addressed by Fourier pairs and by the block processing. In this chapter, the definition of instantaneous frequency introduces a new approach to acquire a proper time-frequency representation of a signal. The description of limitations which have to be taken into account in order to achieve meaningful results is also given there. The section that deals with the instantaneous frequency phenomenon is followed by the part describing the construction of the signal model. A signal is modelled as an output of a system that consists of sum of auto-regressive (AR), linear time-invariant (LTI), secondorder subsystems. In fact, such a model corresponds to a system of resonators formed in parallel. The reason for such model selection is simplicity in matrix implementation as well as preservation of a physical meaning of decomposed signal components. The signal model is developed in state space and there is also demonstrated how to select adequately all system matrices. The estimation of particular signal components is then obtained through the adaptive Kalman filtering which is based on the previously defined state space model. The Kalman filter recursively estimates time-varying signal components in a complex form. The complex form is required in order to compute the instantaneous frequency and amplitude of the O pe n A cc es s D at ab as e w w w .ite ch on lin e. co m