On the time-frequency analysis based filtering

Efficient processing of nonstationary signals requires time-varying approach. An interesting research area within this approach is time-varying filtering. Since there is a certain amount of freedom in the definition of timevarying spectra, several definitions and solutions for the time-varying filtering have been proposed so far. Here we will consider the Wigner distribution based time-varying filtering form defined by using the Weyl correspondence. Its slight modification will be proposed and justified in the processing of noisy frequency modulated signals based on a single signal realization. An algorithm for the efficient determination of the filters’ region of support in the time-frequency plane, in the case of noisy signals, will be presented. In the second part of the paper, the theory is applied on the filtering of multicomponent noisy signals. The Smethod is used as a tool for the filters’ region of support estimation in this case. This method, combined with the presented algorithm, enables very efficient time-varying filtering of the multicomponent noisy signals based on a single realization of the signal and noise. Theory is illustrated by examples. I. I Analysis and processing of stationary signals is usually performed either in time or in frequency domain. However, when the signals exhibit nonstationary characteristics more efficient processing can be done using joint timefrequency domain tools. They are based on the time-frequency representations of signal. Time-varying filtering is one of the challenging areas where one can benefit from the joint time-frequency representations. This kind of filtering can produce better results, in the nonstationary signal cases, than the processing of signals in either time or frequency domain separately. However, in the definition of time-varying spectra there is a certain amount of freedom [3], [5], [6], [8], [16], [26], what has resulted in several solutions for this apAnnales des Télécommunications, Vol.55, No.5-6, May 2000. proach. The first, classical solution, has been presented by Zadeh [25]. It can be related to the Richaczek distribution [16]. However, this distribution exhibits some serious drawbacks as a possible tool for the time-frequency representation of signals [6], [8]. This fact was the reason for a redefinition of the time-varying filtering relations into the Wigner distribution framework, using the Weyl correspondence [3], [9], [13], [14], [18]. In this paper we will consider the Wigner distribution approach. Its slight modification will be proposed and justified in the treatment of noisy frequency modulated signals when only one signal realization is available. The algorithm proposed in [23] has been used for determination of the filter’s region of support. This algorithm is based on the optimal window length relation in the Wigner distribution. It uses a specific statistics approach of comparing the bias and variance [11], [23]. Theory is extended to the multicomponent noisy signals. The S-method is used as a basic tool for the time-varying filter support estimation [20], [21]. This method combined with the described algorithm, enables very efficient time-varying filtering of the multicomponent noisy signals, based on a single signal realization. Examples illustrate the presented theory. The paper is organized as follows. The basic theory, including the definitions of timevarying filtering, its discrete and pseudo forms, along with the illustrations, are given in Section II. Filtering of a monocomponent signals is considered in the next Section. In Section III the algorithm for the region of support estimation is presented, as well. The algorithm efficiency in the time-varying filtering applications is illustrated on examples. Time-varying filtering of multicomponent signals is studied and illustrated in Section IV. 756 TIME-FREQUENCY SIGNAL ANALYSIS