In this letter, we have proposed a signal denoising method based on a modification of the intersection of confidence intervals (ICI) rule. The ICI rule is complemented by the relative intersection of confidence intervals length which is used as an additional criterion for adaptive filter support selection. It is shown that the proposed method outperforms the original ICI method equipped with the local polynomial approximation (LPA), as well as various conventional wavelet shrinkage methods.Index Terms-Adaptive filter support, adaptive filtering, edge preserving, intersection of confidence intervals (ICI) rule.

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A method for components instantaneous frequency (IF) estimation of multicomponent signals in low signal-to-noise ratio (SNR) is proposed. The method combines a new proposed modification of a blind source separation (BSS) algorithm for components separation, with the improved adaptive IF estimation procedure based on the modified sliding pairwise intersection of confidence intervals (ICI) rule. The obtained results are compared to the multicomponent signal ICI-based IF estimation method for various window types and SNRs, showing the estimation accuracy improvement in terms of the mean squared error (MSE) by up to 23%. Furthermore, the highest improvement is achieved for low SNRs values, when many of the existing methods fail. Signal Model and Problem FormulationMany signals in practice, such as those found in speech processing, biomedical applications, seismology, machine condition monitoring, radar, sonar, telecommunication, and many other applications are nonstationary [1]. Those signals can be categorized as either monocomponent or multicomponent signals, where the monocomponent signal, unlike the multicomponent one, is characterized in the timefrequency domain by a single "ridge" corresponding to an elongated region of energy concentration [1].For a real signal s(t), its analytic equivalent z(t) is defined aswhere H {s(t)} is the Hilbert transformation of s(t), a(t) is the signal instantaneous amplitude, and φ(t) is the signal instantaneous phase. The instantaneous frequency (IF) describes the variations of the signal frequency contents with time; in the case of a frequency-modulated (FM) signal, the IF represents the FM modulation law and is often referred to as simply the IF law [2,3]. The IF of the monocomponent signal z(t) is the first derivative of its instantaneous phase, that is, ω(t) = φ (t) [1]. Furthermore, the crest of the "ridge" is often used to estimate the IF of the signal z(t) as [1]whereOn the other hand, the analytical multicomponent signal x(t) can be modeled as a sum of two or more monocomponent signals (each with its own IF ω m (t))where M is the number of signal components, a m (t) is the mth component instantaneous amplitude, and φ m (t) is its instantaneous phase. When calculating the Hilbert transform of the signal s(t) in (1), the conditions of Bedrosian's theorem need to be satisfied, that is, a(t) has to be a low frequency function with the spectrum which does not overlap with the e jφ(t) spectrum [2-5].

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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