Estimating the number of components of a multicomponent nonstationary signal using the short-term time-frequency Rényi entropy

Abstract

The time-frequency Rényi entropy provides a measure of complexity of a nonstationary multicomponent signal in the time-frequency plane. When the complexity of a signal corresponds to the number of its components, then this information is measured as the Rényi entropy of the time-frequency distribution (TFD) of the signal. This article presents a solution to the problem of detecting the number of components that are present in short-time interval of the signal TFD, using the short-term Rényi entropy. The method is automatic and it does not require a prior information about the signal. The algorithm is applied on both synthetic and real data, using a quadratic separable kernel TFD. The results confirm that the short-term Rényi entropy can be an effective tool for estimating the local number of components present in the signal. The key aspect of selecting a suitable TFD is also discussed.1 Time-frequency distributions and instantaneous frequency estimation When dealing with highly complex signals, such as multicomponent nonstationary signals, several pieces of information are required for their characterization. Classical approaches of the time signal representation, x(t), and the frequency representation, X(f), are not best tools for obtaining those information when dealing with multicomponent signals. These representations define the signal duration, the changes of amplitude in time, as well as the entire signal frequency content. Timefrequency representations (TFRs), or TFDs, are two variable functions, C s (t, f), defined over the two-dimensional (t, f) space [4]. Such a joint TFR shows how the frequency content of a signal changes in time.One of the most popular TFDs, introduced by Wigner and extended by Ville to analytic signals [4], has been treated as a pseudo probability density function in [2,3,5] to which the Rényi entropy has been applied as a measure of signal complexity. The intuitive idea of the Wigner-Ville distribution (WVD) was to obtain a kind of instantaneous signal spectrum by performing the Fourier transform of a function related to the signal, called the kernel function K s (t, τ). The WVD of a signal s(t), denoted as W s (t, f), represents a monocomponent frequency modulated (FM) signal as a knife-edge ridge in the (t, f) plane, whose crest is the IF of the signal [4].Let s(t) be an analytic FM signal of the form [4] s(t) = a(t)e jφ(t) ,where a(t) is the instantaneous signal amplitude (assumed to be equal to one in the rest of the article), and the signal IF is defined as the time derivative of its instantaneous phase j(t) [4] f i (t) = φ (t) 2π .

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