Estimating the number of components of a multicomponent nonstationary signal using the short-term time-frequency Rényi entropy
© Sucic et al; licensee Springer. 2011
Received: 15 June 2011
Accepted: 7 December 2011
Published: 7 December 2011
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
1.1 Nonstationary signals analysis and quadratic class of time-frequency distributions
Practical signals in the various fields of engineering (telecommunications, acoustics, biomedical engineering) are nonstationary, with the instantaneous frequency (IF) being their key parameter . One of the fundamental information when analyzing such signals is the number of components present in the signal. When applied to a time-frequency distribution (TFD), the Rényi entropy measures the signal complexity [2, 3]. Signals of high complexity are composed of a large number of elementary components .
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. Time-frequency representations (TFRs), or TFDs, are two variable functions, C s (t, f), defined over the two-dimensional (t, f) space . 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 , 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 .
Hence, the WVD can be understood as the Fourier transform of the signal kernel K s (t, f), also known as the instantaneous autocorrelation function (IAF) of s(t).
From (7), we can notice that using the IAF as the kernel function brings nonlinearity in the WVD. The effects of this nonlinearity will be most evident in the case of multicomponent signals, as explained below. Note that, in general, a component in the (t, f) domain is a ridge of energy concentration whose peaks follow the component IF law .
It appears in the (t, f) plane in between the signal components, often degrading the quality of signal representation in the (t, f) plane.
The rule of interference construction in the WVD can be summarized as follows. Two points belonging to the signal will interfere to create a third point which will be located on their geometrical midpoint. The amplitude of the interference will be proportional to the double product of the amplitudes of the interfering points. In addition, the interferences oscillate perpendicularly to the line joining the two signal points, assuming both positive and negative values, with the frequency of oscillation being proportional to the distance between these two points [4, 6].
It can be deduced from the general interference rule that interferences will be also present in the case of monocomponent signals with nonlinear FMs, called inner cross-terms in .
1.2 Separable kernel time-frequency distributions
where k = Γ(2β)/(22β-1Γ2(β)) is the normalizing factor, Γ(·) is the gamma function, and β is a real, positive number. It has been shown in  that the MBD outperforms other TFDs in terms of time-frequency resolution and interference suppression for a large class of non-stationary signals, such as newborn and adult EEG signals as well as heart rate variability signals.
2 Time-frequency Rényi entropy and TFDs
2.1 The definition and limitations of the global Rényi entropy
to be an acceptable tool for measuring the complexity and information content of a nonstationary signal in the (t, f) domain. Since in (24) represents a probability density function, it is natural to expect that a multicomponent signal will have larger entropy when compared to a single pulse in the (t, f) plane. As explained in the first section, nonpositivity (due to the presence of interfering terms) is one of the characteristics of the quadratic class of TFDs. As a TFD can be negative in some regions of the (t, f) plane, the Shannon entropy cannot be used in practice as a signal complexity measure, due to the logarithm in (24).
As shown in , when the parameter α (for the Shannon entropy α → 1) is an odd integer value, the oscillatory structure of interferences are annulled under integration. Let us next consider several examples illustrating the use of the Rényi entropy in the (t, f) domain. Note that all TFDs that are used in this article are normalized as per (23).
2.2 Example I: Case of two Gabor logons with same time duration and FM
This example confirms the accuracy of the Rényi entropy counting property when the entropy of one of the components is known in advance.
2.3 Example II: Case of two Gabor logons with same FM and different time duration
As expected, the signal component sl(t) that occupies a larger region of the (t, f) plane exhibits a considerably larger value of the Rényi entropy when compared to the entropy of the component with the shorter-time support, s2(t). Consequently, the estimation of the number of components based on the difference of the Rényi entropies of the entire signal and one of its components fails regardless which of the two components is chosen as the reference signal. The implication is that the global Rényi entropy can correctly detect the number of components only when all components have same time supports, with arbitrary time/frequency shifts in the TF plane.
2.4 Example III: Case of a signal with different FM components of same time duration
Thus, the presented examples indicate that the counting property of the Rényi entropy is restricted to multicomponent signals composed of components with similar time and frequency supports only. In addition, having to know the entropy of one of the components in advance, makes this approach highly impractical. To remedy this disadvantage, the next section presents a novel method for estimating the local number of components in a signal that can be applied independently of the components respective durations in time and frequency. In addition, the proposed automatic procedure does not require the prior knowledge of the complexity of one of the signal components.
3 A novel algorithm for estimating the number of components using the short-term Rényi entropy
3.1 Assumptions and constraints
all components have the similar structure in the time-frequency plane, and
the Rényi entropy of a single component is available.
However, in most practical applications this will not be the case: in general, signals encountered in many engineering and multidisciplinary fields (telecommunications, acoustics, radar, sonar, etc.) are usually mixtures of signals with different durations and frequency bandwidths. This makes the number of components estimation based on the global Rényi entropy ineffective for real-world signals.
What is required is a solution for determining the number of components present in a signal in a short-time interval. Such a method is presented here, based on the estimation of the short-time Rényi entropy. It exploits the fact that signals with similar time durations and frequency supports have similar Rényi entropies. Since the Rényi entropy of a signal is invariant to time and frequency shifts , it is expected that a signal represented in the time-frequency domain by two short-energy impulses will have twice the energy in the (t, f) plane of a signal represented by a single energy impulse of the same duration, and its Rényi entropy should be larger by one bit. This suggests that instead of observing the entire (t, f) plane when detecting the number of signal components, we should focus on a finite time interval of the (t, f) plane, and compare the Rényi entropy of this short-time segment with the Rényi entropy of a reference test signal with the same time support. In this way, the number of components present in the chosen time interval will be automatically estimated.
3.2 The proposed algorithm
where t0 controls the length of the observed time interval. We next compare the short-term Rényi entropy of this TFD, denoted as , with the Rényi entropy of the TFD of the reference cosine signal (an arbitrary chosen stationary cosine signal), , for the same Δt. In the proposed algorithm, the reference TFD, C ref (t, f), must be a TFD of the same type and with the same set of parameters as the signal TFD, C s (t, f). C ref (t, f) must also have same dimensions as C s (t, f). The selected synthetic reference signal is a cosine signal of arbitrary amplitude and arbitrary constant frequency. Since the FM affects the bandwidth of a component in the TF domain (signals with fast changes in the IF may present relatively larger bandwidths in the TF domain), a synthetic signal with constant or linear FM is used as a reference signal in order to maintain a constant bandwidth. The amplitude of the reference signal can also be arbitrary chosen since the Rényi entropy of a signal is amplitude invariant.
For each time instant p, a different time portion of the (t, f) plane is extracted, and thus a different value of the Rényi entropy is obtained. After comparing the obtained values of the Rényi entropy of the observed multicomponent signal and the reference one, a function of the instantaneous number of components, n(p), is obtained.
The first step of the algorithm consists in the thresholding of the TFD to remove noise and interference low energy peaks that may locally affect the entropy of the signal. The same must be done with the TFD of the reference signal.
The fundamental step in determining the local number of a signal components using the short-term Time-Frequency Rényi entropy is the choice of the TFD. As it has been shown in , the Rényi entropy is invariant to cross-term since they annulate under integration over the entire (t, f) plane with odd powers of α (25). Naturally, this is not the case in our algorithm, where only a short-time portion of the (t, f) plane is observed, and consequently the eventual presence of cross-terms would cause inaccurate results. Thus, the key requirement we put on the TFD is the minimization of the interference terms spreading in the sense of both the time and frequency axis. TFDs with separable kernels successfully reduce the interferences, by an independent smoothing of the WVD in time and frequency . They have also been shown to outperform other popular TFDs in terms of the time-frequency resolution .
4 Results and discussion
4.1 The experiment
The algorithm presented in the previous section exploits the short-term Rényi entropy to detect the number of components present in a short-time interval of the analyzed signal. In this section, we test its performance on both synthetic and real-life signals examples.
Figure 9a shows a three-compound signal, whose components have parabolic frequency modulations and different time durations. As seen from Figure 9b, the proposed algorithm has correctly detected the number of components present at each time instant for three different values of the parameter α. Figure 9c shows the performance of the algorithm for three different values of Δt, confirming that Δt = 7 s gives the most accurate results.
4.2 Performance on real data
4.3 Strengths and limitations of the algorithm
The results in this section illustrate a high accuracy of the presented algorithm in detecting the local number of signal components. Even in the case of noisy signals or low energy components, the algorithm has correctly estimated the components number. The presented method provides the signal analyst with information on the minimum number of components present in the signal. All this can be useful in various applications that require components separation and extraction [12–14].
The choice of the TFD is crucial for the successful performance of the algorithm. TFDs with separable kernels are recommended in order to avoid the undesirable influence of both inner and outer artifacts of the signal TFD on the results. In this article, the MBD was shown to be a good choice for the real data local component number estimation.
This article proposes a method for estimating the local number of signals components. It is based on the short-term Rényi entropy of signals in the time-frequency plane. Using the Rényi entropy of a short-term segment of a TFD of a multicomponent nonstationary signal, relative to the short-term Rényi entropy of a reference signal, the number of components present in the signal can be accurately estimated. The proposed method does not require any a priori information about the analyzed signal, nor the knowledge of the Rényi entropy of one of the signal components. The method was tested on various synthetic signals, including signals embedded in additive white Gaussian noise, and its use in practice was illustrated on a real-life signal. The method is sensitive to the selection of the TFD. The presented results indicate that the MBD, being an example of Separable kernel TFDs, is a good choice of a TFD when the proposed method is applied in practical situations . These results show that the proposed algorithm can be useful in many applications that require component count and component separation; and, it can be a preferred alternative to other methods such as the Empirical Mode Decomposition .
This study is a part of the research project "optimization and Design of Time-Frequency Distributions" (No. 069-0362214-1575), which was financially supported by the Ministry of Science, Education and Sports of the Republic of Croatia. This article was also partly funded by the Australian Research Council (Grant ID: ARC DP1094498).
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