 Research Article
 Open Access
Robust TimeFrequency Distributions with ComplexLag Argument
 Nikola Žarić^{1}Email author,
 Irena Orović^{1} and
 Srdjan Stanković^{1}
https://doi.org/10.1155/2010/879874
© Nikola Žarić et al. 2010
 Received: 30 December 2009
 Accepted: 1 March 2010
 Published: 13 April 2010
Abstract
The robust timefrequency distributions with complexlag argument are proposed. They can provide an accurate estimation of fast varying instantaneous frequency in the presence of noise with heavytailed probability density function. The Lestimate form of this distribution is defined and it includes the Lestimate form of Wigner distribution as a special case. A modification for multicomponent signal representation is proposed, as well. Theoretical considerations are illustrated by the examples.
Keywords
 Mean Square Error
 Loss Function
 Instantaneous Frequency
 Impulse Noise
 Wigner Distribution
1. Introduction
Nonstationary signals such as speech, radar, seismic, sonar, and biomedical signals can be found in many practical applications. Due to timevarying spectra of these signals, timefrequency analysis has been used in their analysis. For different types of signals, various timefrequency distributions (TFDs) have been proposed [1–5].
In real applications we deal with signals corrupted by noise. If noise is additive with Gaussian probability density function (pdf), the standard timefrequency distributions represent a maximum likelihood (ML) estimate [6]. However, if the signal is corrupted by noise with heavytailed pdf (usually caused by environmental or humanmade activities), the standard TFDs produce poor results. Consequently, the robust timefrequency distributions have been introduced [6–12]. The simplest and the most commonly used robust timefrequency representation is the robust shorttime Fourier transform (STFT). The marginal median robust STFT has been introduced as an ML estimate of signals with Laplacian noise [8]. This form can also be successfully used for other types of heavytailed noises. The Lestimate robust STFT is introduced for signals with a mixture of Gaussian and impulse noises [9]. As in the case of the standard STFT, the main drawback of the robust STFT is a poor timefrequency resolution. In order to improve the timefrequency concentration, the robust forms of the Wigner distribution (WD) have been introduced [9–12]. They can provide an ideal concentration for signals with a linear instantaneous frequency (IF). However, for multicomponent signals the crossterms appear. The robust Smethod that combines good properties of the STFT and the WD has been introduced to provide a crossterms free representation [12]. However, it cannot provide good concentration for signals with fast varying IF. Thus, the timefrequency distributions with complexlag argument have been used to estimate nonlinear and fast IF variations [13–20]. Similarly as other TFDs, these distributions provide poor signal representation in the presence of heavytailed noise.
In this paper we propose a robust form of the N thorder complexlag timefrequency distribution (CTD). An arbitrary high concentration can be achieved by increasing the distribution order N. The standard CTD has been defined as convolution of the WD and the Fourier transform of the higher order complexlag moment, called concentration function (CF) [17, 18]. Similarly, the robust CTD can be obtained as convolution of the robust WD and CF forms. Additionally, a crossterms free robust complexlag timefrequency distribution is proposed for multicomponent signals.
The paper is organized as follows. The elementary theory behind linear and quadratic robust timefrequency representations is presented in Section 2. The N th order robust timefrequency distributions with complexlag argument are proposed in Section 3. The advantages of the proposed distributions are proven through various examples in Section 4. Concluding remarks are given in Section 5.
2. Theoretical Background
2.1. Robust ShortTime Fourier Transform
where N _{ s } is even, while the parameter α takes values within the range For and the standard STFT and the marginal median STFT are obtained, respectively. Higher value of provides an enhanced reduction of heavytailed noise, while smaller value of improves spectral characteristics. Thus, depending on the application, the value of parameter should be chosen to provide good tradeoff between these requirements.
where STFT_{ h } may be STFT_{ M } or STFT_{ L }.
2.2. Robust Quadratic TimeFrequency Distributions
where the elements are sorted in nondecreasing order as , while the coefficientsa _{ i } are defined by (6). The marginal median WD follows for =1/2.
where STFT_{ h } can be the medianbased STFT or the Lestimate STFT.
where P(l) is a frequency domain window with its width equal to For and the robust spectrogram and the robust WD are obtained, respectively. More details about parameter selection can be found in [5].
3. Robust ComplexLag TimeFrequency Distributions
Spread factors for some timefrequency distributions.
Distribution  Spread factor 

that is, WD 







where , while X _{ S }(k) is the standard Fourier transform. The coordinate is multiplied by . The influence of this term can be such that an additional oversampling (or interpolations) of signal is required.
The iterative procedure for the nonlinear equation (21) is even more demanding than in the cases of robust STFT and robust WD calculations [6, 11]. Namely, to calculate the robust form of (18) has to be used. It requires an additional iterative procedure. Hence, the CF calculation requires nested iterative procedures, inappropriate for practical realization.
3.1. LEstimate Form of the Robust CTD
The marginal median and the Lestimate approach can be used to overcome disadvantages of iterative procedure for the robust CTD calculation. In the sequel, only the Lestimate approach is considered, since the marginal median follows as a special case of the Lestimate forms for =1/2. Also, the Lestimates exhibit enhanced performance in the presence of mixture of Gaussian and impulse noise, common to real applications. Thus, the Lestimate approach is used to define the robust CTD.
where and are such that and .
where WD_{L} represents the Lestimate robust WD.
3.2. Robust CTD Form for Multicomponent Signals
where is the position of the q th signal component maximum in the Lestimate robust STFT. It is assumed that the qth signal component is of 2W _{ q }+1 width; that is, it is within the region [k _{ q }(n)W _{ q }, k _{ q }(n)+W _{ q }]. Observe that the crossterms will be avoided, if the distance between signal components is higher than 2W _{ q } (see [14] for details). After the first signal component is obtained, the values of STFT_{ L }(n,k) within the region [k _{ q }(n)W _{ q }, k _{ q }(n)+W _{ q }] will be set to 0. Then, this procedure is repeated for other components.
where and .
The role of window P(l) is same as in the Smethod.
4. Examples
Highly nonstationary signals with fast varying instantaneous frequencies are considered. The signals are corrupted with heavytailed noise. The standard and the robust forms of CTD are considered and compared with corresponding forms of the Wigner distribution.
Example 1.
where and are mutually independent Gaussian noises (zero mean with variance equal to 1). The time interval with sampling rate is used. The Gaussian window of width is applied in all cases.
The Lestimate forms are calculated by using parameter for all distributions. Namely, this value provides satisfying tradeoff between noise reduction and distribution concentration. For a given signal, the standard WD, the Lestimate WD, the standard the marginal median (obtained according to (27) with α=1/2), and the Lestimate are shown in Figure 1. Since the signal has a fast varying IF, both the standard WD (Figure 1(a)) and the Lestimate WD (Figure 1(b)) yield poor results. The fourthorder CTD is introduced to improve concentration. However, as expected, the standard (Figure 1(c)) contains significant disturbances, due to the presence of strong heavytailed noise. Furthermore, the robust complexlag distribution forms are considered. Thus, the results are improved by using the median (Figure 1(d)), but the accuracy of estimation is still not satisfactory. The best result in this example, regarding representation and IF estimation precision, is achieved by using the Lestimate (Figure 1(e)).
where is the true IF, while is the estimated IF: . The mean values of MSEs are given in Figure 2 for 100 realizationsof noises. Note that the Lestimate provides the lowest MSE.
The MSE of instantaneous frequency estimation in the presence of Gaussian noise.
Distribution  MSE 

Standard  2.45 
Lestimate  2.56 
Example 2.
the fourthorder distribution is no longer optimal. Thus, the concentration can be improved by using the sixthorder distribution. The noise is the same as in the previous example. The Lestimate RWD, the Lestimate and Lestimate are shown in Figure 3. The parameter is used in all cases.
The MSE of instantaneous frequency estimation.
Distribution  MSE 

Lestimate WD  51.83 
Lestimate  8.52 
Lestimate  3.45 
Example 3.
where and represent Gaussian noises. The same parameters for time interval, window, and noise strength are used as in the Example 2. The results for the standard SM, the Lestimate SM, the standard CTD^{N=4}, and the Lestimate RCTD^{N=4} are shown in Figure 4.
5. Conclusion
The Lestimatebased robust N thorder complexlag timefrequency distribution has been proposed. It provides an efficient estimation for nonstationary signals corrupted with a mixture of Gaussian and heavytailed impulse noise. Additionally, we proposed the modified Lestimate robust CTD form that provides a crossterms free representation for multicomponent signals.
The Lestimate and standard distribution approaches could be combined in some future work to reduce the calculation complexity. Also, the future research could be focused to generalize the proposed approach to the class of complextime distributions based on the ambiguity domain [20].
Declarations
Acknowledgment
This work is supported by the Ministry of Education and Science of Montenegro.
Authors’ Affiliations
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