- Research Article
- Open Access
Robust Time-Frequency Distributions with Complex-Lag Argument
© Nikola Žarić et al. 2010
- Received: 30 December 2009
- Accepted: 1 March 2010
- Published: 13 April 2010
The robust time-frequency distributions with complex-lag argument are proposed. They can provide an accurate estimation of fast varying instantaneous frequency in the presence of noise with heavy-tailed probability density function. The L-estimate form of this distribution is defined and it includes the L-estimate 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.
- Mean Square Error
- Loss Function
- Instantaneous Frequency
- Impulse Noise
- Wigner Distribution
Nonstationary signals such as speech, radar, seismic, sonar, and biomedical signals can be found in many practical applications. Due to time-varying spectra of these signals, time-frequency analysis has been used in their analysis. For different types of signals, various time-frequency 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 time-frequency distributions represent a maximum likelihood (ML) estimate . However, if the signal is corrupted by noise with heavy-tailed pdf (usually caused by environmental or human-made activities), the standard TFDs produce poor results. Consequently, the robust time-frequency distributions have been introduced [6–12]. The simplest and the most commonly used robust time-frequency representation is the robust short-time Fourier transform (STFT). The marginal median robust STFT has been introduced as an ML estimate of signals with Laplacian noise . This form can also be successfully used for other types of heavy-tailed noises. The L-estimate robust STFT is introduced for signals with a mixture of Gaussian and impulse noises . As in the case of the standard STFT, the main drawback of the robust STFT is a poor time-frequency resolution. In order to improve the time-frequency 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 cross-terms appear. The robust S-method that combines good properties of the STFT and the WD has been introduced to provide a cross-terms free representation . However, it cannot provide good concentration for signals with fast varying IF. Thus, the time-frequency distributions with complex-lag 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 heavy-tailed noise.
In this paper we propose a robust form of the N th-order complex-lag time-frequency 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 complex-lag 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 cross-terms free robust complex-lag time-frequency distribution is proposed for multicomponent signals.
The paper is organized as follows. The elementary theory behind linear and quadratic robust time-frequency representations is presented in Section 2. The N th order robust time-frequency distributions with complex-lag 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.
2.1. Robust Short-Time 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 heavy-tailed noise, while smaller value of improves spectral characteristics. Thus, depending on the application, the value of parameter should be chosen to provide good trade-off between these requirements.
where STFT h may be STFT M or STFT L .
2.2. Robust Quadratic Time-Frequency 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 median-based STFT or the L-estimate 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 .
Spread factors for some time-frequency distributions.
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. L-Estimate Form of the Robust CTD
The marginal median and the L-estimate approach can be used to overcome disadvantages of iterative procedure for the robust CTD calculation. In the sequel, only the L-estimate approach is considered, since the marginal median follows as a special case of the L-estimate forms for =1/2. Also, the L-estimates exhibit enhanced performance in the presence of mixture of Gaussian and impulse noise, common to real applications. Thus, the L-estimate approach is used to define the robust CTD.
where and are such that and .
where WDL represents the L-estimate robust WD.
3.2. Robust CTD Form for Multicomponent Signals
where is the position of the q th signal component maximum in the L-estimate robust STFT. It is assumed that the q-th 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 cross-terms will be avoided, if the distance between signal components is higher than 2W q (see  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 S-method.
Highly nonstationary signals with fast varying instantaneous frequencies are considered. The signals are corrupted with heavy-tailed noise. The standard and the robust forms of CTD are considered and compared with corresponding forms of the Wigner distribution.
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 L-estimate forms are calculated by using parameter for all distributions. Namely, this value provides satisfying trade-off between noise reduction and distribution concentration. For a given signal, the standard WD, the L-estimate WD, the standard the marginal median (obtained according to (27) with α=1/2), and the L-estimate are shown in Figure 1. Since the signal has a fast varying IF, both the standard WD (Figure 1(a)) and the L-estimate WD (Figure 1(b)) yield poor results. The fourth-order CTD is introduced to improve concentration. However, as expected, the standard (Figure 1(c)) contains significant disturbances, due to the presence of strong heavy-tailed noise. Furthermore, the robust complex-lag 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 L-estimate (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 L-estimate provides the lowest MSE.
The MSE of instantaneous frequency estimation in the presence of Gaussian noise.
the fourth-order distribution is no longer optimal. Thus, the concentration can be improved by using the sixth-order distribution. The noise is the same as in the previous example. The L-estimate RWD, the L-estimate and L-estimate are shown in Figure 3. The parameter is used in all cases.
The MSE of instantaneous frequency estimation.
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 L-estimate SM, the standard CTDN=4, and the L-estimate RCTDN=4 are shown in Figure 4.
The L-estimate-based robust N th-order complex-lag time-frequency distribution has been proposed. It provides an efficient estimation for nonstationary signals corrupted with a mixture of Gaussian and heavy-tailed impulse noise. Additionally, we proposed the modified L-estimate robust CTD form that provides a cross-terms free representation for multicomponent signals.
The L-estimate 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 complex-time distributions based on the ambiguity domain .
This work is supported by the Ministry of Education and Science of Montenegro.
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