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seismic data analysis, biomedical engineering, and automotive emission, just to ... Dr. Sergio Barbarossa. B. Barkat is with the School of Electrical and Electronic Engineering, ... To further confirm the high performance of the proposed. TFD, we ...
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A High-Resolution Quadratic Time–Frequency Distribution for Multicomponent Signals Analysis Braham Barkat and Boualem Boashash, Fellow, IEEE

Abstract—The paper introduces a new kernel for the design of a high resolution time–frequency distribution (TFD). We show that this distribution can solve problems that the Wigner–Ville distribution (WVD) or the spectrogram cannot. In particular, the proposed distribution can resolve two close signals in the time–frequency domain that the two other distributions cannot. Moreover, we show that the proposed distribution is more accurate than the WVD and the spectrogram in the estimation of the instantaneous frequency of a stepped FM signal embedded in additive Gaussian noise. Synthetic and real data collected from real-world applications are shown to validate the proposed distribution. Index Terms—Instantaneous frequency, mean square error, spectrogram, time–frequency distributions, Wigner–Ville distribution.

I. INTRODUCTION

I

N the last two decades, there has been an intensive research effort to study time–frequency distributions (TFDs) in order to characterize and analyze nonstationary signals [1], [2]. As a result, many classes of TFDs have been proposed; one of these classes is the quadratic (or Cohen’s) class. Many members of the quadratic class have been successfully applied to various real-life problems such as radar, sonar, seismic data analysis, biomedical engineering, and automotive emission, just to name a few [3], [4]. However, no one single quadratic TFD exists that can be used effectively in all possible applications. The choice of a particular TFD depends on the specific application at hand and the representation properties that are desirable for that application. Some members of the quadratic class are the Wigner–Ville distribution (WVD), the spectrogram, the Choi–Williams distribution [5], the Rihaczek distribution [3], and the Zhao–Atlas–Marks distribution [6]. Members of the quadratic class seem to behave differently in the analysis of multicomponent signals. For instance, the WVD presents high amplitude cross terms for such signals, whereas the spectrogram is, in general, cross terms free. The analysis of multicomponent signals is still a major challenge facing the time–frequency analysis community. Indeed, in an application dealing with multicomponent signals, the analyst should be able to know the number of components present in the signal, discriminate between the components and the existing cross terms, Manuscript received December 1, 1999; revised June 22, 2001. The associate editor coordinating the review of this paper and approving it for publication was Dr. Sergio Barbarossa. B. Barkat is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore (e-mail: [email protected]). B. Boashash is with the Signal Processing Research Centre, Queensland University of Technology, Brisbane, Australia (e-mail: [email protected]). Publisher Item Identifier S 1053-587X(01)08246-0.

resolve components that are close in the time–frequency plane, and estimate the instantaneous frequency of each component separately. These points require that the TFD used in the analysis should be cross terms free and should have high-resolution in the time–frequency domain. Unfortunately, these two conditions seem to be conflicting, and an intensive research effort is still being deployed to design a TFD that satisfies both of these conditions simultaneously [7]–[10]. This paper proposes one solution to this problem. In general, real-life signals are multicomponent and differ from one application to another, and since no one TFD can be accurately applied for all situations, the practitioner would prefer to have a variety of tools at hand to use. In a particular situation, the analyst or practitioner may find that all the available tools fail to achieve the predefined design or processing goals. For this reason, the design of a new tool that has a different objective or has a better performance than existing ones is very important. In this context, we propose in this paper a signal-independent kernel for the design of a new quadratic TFD. A main objective in the design of this new quadratic member is to have high resolution in the time–frequency domain in addition to maximum cross term reduction or suppression. Another important design objective is to have accurate instantaneous frequency (IF) estimates for multicomponent signals. For the sake of simplicity in the comparison of the proposed TFD with existing ones, we consider the most two popular quadratic distributions, namely, the WVD and the spectrogram. This comparison is performed with respect to some criteria detailed later in the paper. We show that the new quadratic member outperforms these two distributions in the analysis of many multicomponent signals. In particular, we show that the proposed TFD can resolve two close components in the time–frequency domain that neither the spectrogram nor the WVD can resolve. Furthermore, in the estimation of the IF of a noisy stepped frequency-modulated (FM) signal, we show that the estimator based on the peak of the proposed TFD, compared wiht those based on the peaks of WVD and the spectrogram, is more accurate and has the smallest mean squared error. Both synthetic and real-life signal data are used in the comparisons to demonstrate the superiority of the new TFD. The proposed distribution has not been compared with other TFDs such as the Choi–Williams distribution or the teh ZhaoMarks-Atlas distribution because optimal kernel parameters for these TFDs are unknown, and therefore, a simple comparison cannot be provided. A more complex comparison is outside the scope of this paper and can be addressed in future work. To further confirm the high performance of the proposed TFD, we compare it with the WVD in the representation and

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IF estimation of a linear FM signal. As it is known, the WVD is optimal, in the sense of maximum energy concentration about the IF, for linear FM signals. The WVD is also efficient in the IF estimation of such signals [11]. Here, we show that in practice, the proposed TFD is equivalent to the WVD in representing and estimating the IF of a noisy linear FM signal. The paper is organized as follows. In the next section, we review the ambiguity domain filtering, we define the new TFD, and we discuss its properties and implementation. In Section III, several examples are considered for the evaluation of the proposed distribution. In Section IV, we draw some conclusions. II. PROPOSED TFD A. Ambiguity Domain Filtering In general, the auto terms of a multicomponent signal are centered around the origin, in the ambiguity domain, and the cross terms are located away from the origin at a distance function of the separation, in time and frequency, of the respective components [5]. This important property is a key idea in the design of quadratic TFDs that can remove or reduce the cross terms in a multicomponent signal analysis. In order to clarify this point, let us consider the general form of the quadratic class given by

(1) is the analytic multicomponent signal under considwhere is a weighting function, called the kernel, eration, and that determines the distribution and its properties. In particular, , (1) yields the well known Wigner-Ville distrifor bution (WVD). Note: It can be shown that any quadratic TFD can be expressed as a function of the WVD [1]. Moreover, it can be shown that any quadratic TFD can be written as a linear function of any other quadratic TFD [1]. be defined as Let the generalized ambiguity function (2) where

is the Sussman ambiguity function given by (3)

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the cross terms in the analysis of a multicomponent signal (recall that the cross terms are located away from the origin). This result explains why some members of the quadratic class are cross terms free while others are not. For example, the kernel of the WVD is equal to unity . This means that the WVD kernel is an allpass filter in the ambiguity domain, and therefore, the cross terms are neither reduced nor removed. Consequently, the WVD presents undesirable high-amplitude cross terms in the analysis of multicomponent signals. Note that the filtering of the cross terms in the ambiguity domain that reduces (or removes) the cross terms in the time–frequency domain unfortunately results in a lower time–frequency resolution. That is, there is tradeoff between cross term suppression and time–frequency resolution in the design of a given quadratic TFD. In the next subsection, we propose a new kernel for the design of a quadratic TFD. We show that this high-resolution distribution can solve problems that the WVD and the spectrogram cannot. B. Proposed Kernel The design of the new TFD is based on the discussion deof the tailed in the previous section. The kernel function proposed TFD is chosen in the ambiguity domain as a 2-D function centered around the origin with sharp cut-off edges. In this way, the kernel would allow us to retain as many auto-terms energy as possible while filtering out as much cross-terms “energy” as possible. The numbers of auto terms and cross terms kept and filtered out are functions of the volume underneath the . This volume can be changed by varying a 2-D function single parameter , as shown later. Since the proposed TFD is implemented in the time-lag do, the kernel is defined and presented in the same domain main. An intuitive justification in the design of the new kernel is based on the discussion above. That is, we begin from a time cosh whose spectrum presents the narrowest function mainlobe (best frequency resolution) compared with many other considered time functions for the same signal duration. By excosh ) and tending this time function to a 2-D quantity then taking it to a power , we can obtain two desirable char, which is cenacteristics. First, its Fourier transform tered around the origin, presents sharp cut-off edges. Second, the volume beneath it can be controlled by varying the value of . Consequently, the proposed time-lag kernel is given by

We can show that (1) can be expressed as [1] (4) or equivalently (5) This result indicates that a quadratic TFD is obtained by first (using the smoothing the symmetric ambiguity function ) and then by taking a two-dimensional kernel function (2-D) Fourier transform of the result. This operation is equivalent to a 2-D filtering in the ambiguity domain. Thus, by a proper , we can reduce or remove choice of the kernel function

cosh

(6)

In this expression, is a real parameter that controls the in the ambiguity sharpness of cutoff of the 2-D filter domain. The choice of is application dependent; however, we have found that its range should be between zero and unity . We have also found, from extensive simulations and real-life data analysis, that in general, the value gives the best results in terms of cross term suppression and time–frequency resolution, but we should observe that this value is given only as an indication and is not optimal for all situations. The optimization of the parameter is beyond the scope of this paper and can be dealt with elsewhere.

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The new distribution has many desirable properties. These given by properties are deduced directly from the kernel (7)

An interpolation of the analytic signal in (12) is necessary in order to obtain noninteger values of the signal arguments. However, this interpolation can be avoided by using a similar formulation as the one in [13], namely

(8) (13) C. Properties of the Proposed TFD The proposed TFD satisfies many of the important properties sought in a TFD [12]. In particular, we have the following. . • It is real since • It is time-shift invariant since the Doppler-lag kernel is not a function of time . • It is frequency-shift invariant since the Doppler-lag kernel is not a function of frequency . • Its first moment yields the instantaneous frequency of the signal under consideration, that is (9) This property is satisfied because [1] constant for all

The resulting TFD in (13) is alias-free and periodic in with a is the length of period equal to unity [13]. Note that the rectangular window considered in the implementation. We observe from the previous expression that the implementation procedure of the proposed TFD requires three steps. 1) formation of the quadratic quantity ; of with 2) discrete convolution in time ; 3) discrete Fourier transform (DFT), with respect to , of the previous result. Further simplifications in the implementation procedure may be achieved by using the general procedure outlined in [14]. III. COMPARISONS AND DISCUSSIONS

(10)

Note: The above properties are the most important ones needed to achieve the predefined design objectives for the proposed TFD, namely, a high time–frequency resolution and a high performance for the instantaneous frequency estimation, as shown later. D. Implementation of the Proposed TFD Since in real-life analysis the practitioner is often required to deal with discrete-time signals, we consider here a discrete-time formulation of the proposed TFD. Using (7) in the general formula (1), we can show that the proposed TFD can be written as

(11) The discrete-time version of the above expression is given by

In this section, we evaluate the performance of the proposed TFD by comparing it with other existing quadratic TFDs. We limit the comparison to the two most popular quadratic distributions, namely, the spectrogram and the WVD. The rationale behind the choice of these particular distributions is that they constitute the two extremes of the quadratic class in the sense of time–frequency resolution and cross term suppression, that is, the WVD is chosen because of its very high time–frequency resolution (with high amplitude cross terms), and the spectrogram is chosen because it is, in general, cross term free (with a low time–frequency resolution). A comparison with other existing quadratic distributions such as the Choi–Williams distribution and the Zhao–Marks–Atlas distribution can also be considered. However, for the comparison to be effective, an optimization procedure of a parameter in the distributions kernels expressions is necessary. This optimization is beyond this paper topic and can be dealt with in a separate work. The comparison here is two-fold: One is based on the time–frequency resolution, and the second is based on the ability to estimate a signal IF accurately. A. Signal Representations

(12) and are obHere, the discrete-time expressions and at a frequency tained by sampling such that and . The total signal duration , with being the total number of samples. For is . To simplicity and without loss of generality, we assumed derive the previous expression, we used the commutative property of the convolution operation.

Several examples, using both synthetic and real data, are considered here to demonstrate the high performance of the proposed distribution and its superiority over the WVD and the spectrogram in terms of cross- term suppression and time–frequency resolution. Example A1—Two Parallel Linear FM Signals: In this first example, we consider a multicomponent signal composed of two parallel linear FM components. The sampling frequency is . chosen equal to unity and the signal length is equal to The first component frequency range lies between 0.1 Hz and

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=

Fig. 1. Slices taken at the same time instant n 3 (signal start time instant) of the WVD, the spectrogram, and the proposed TFD for a multicomponent signal composed of two parallel linear FM components using (left) a small size window length, (center) a medium size window length, and (right) a large size window length.

Fig. 2.

(Left) WVD, (center) the spectrogram, and (right) the proposed TFD for a real-life automotive signal.

0.2 Hz, and the second component frequency range lies between 0.12 Hz and 0.22 Hz. Neither the WVD nor the spectrogram are able to resolve the two signal components at the edges (start and end signal points) in the time–frequency plane (as illustrated later by Fig. 1). This situation is, in general, undesirable in IF estimation applications because the IF of a signal must be accurately estimated at every time instant of the signal duration. To resolve the two components at the edges, we suggest the use of the proposed TFD instead of the WVD or the spectrogram, as shown below. Let us examine more closely slices of the WVD, the spectrogram, and the proposed TFD of the signal. The distributions are evaluated using a small size window length (31 samples), a medium size window length (63 samples), and a large size window length (127 samples), respectively. For each window length, we take slices of the distributions at time instant (recall that ). In Fig. 1, we display the normalized amplitudes of the distributions for each window length separately. We observe that for the small size window length (left plot), none of the distributions can resolve the two components. However, for the two other windows (center and right plots), we observe that the proposed TFD is the only distribution able to resolve the two components. The proposed TFD is also able to resolve the components at all other time instants. Example A2—Automotive Signal: In this example, we analyze a real-life automotive signal, namely, the cylinder pressure signal. This signal was collected using an in-cylinder pressure sensor that was mounted in the vicinity of the combustion chamber [4]. For the analysis of the signal, we use the WVD, . These disthe spectrogram, and the proposed TFD tributions are displayed in Fig. 2. We can clearly observe the su-

periority of the proposed TFD over the WVD in terms of cross term suppression. We can also note the superiority of the proposed TFD over the spectrogram in terms of time–frequency resolution. For this particular figure, the window length used was fixed equal to 101 samples (signal length ); however, other window lengths yield similar results. The superiority of the proposed TFD enables the analyst to separate and filter out, using a time–frequency filter, each component of the signal more precisely compared with when either of the two other distributions is used. This superiority can also be used to estimate the IF of each component more accurately. Since the proposed distribution does not satisfy the time support property, the components seem to last longer in time. Example A3—Passive Acoustic Signal: In this example, we consider another real-life signal to demonstrate the high performance of the proposed distribution. The signal considered here is an acoustic signal emitted from an overflying aircraft recorded and collected, under accurately monitored conditions, by a single ground-based microphone. This signal has been used to estimate the aircraft’s flight parameters, namely, constant height, constant ground speed, constant acoustic frequency, and range [15]. The signal is analyzed using the WVD, the spectrogram, and the proposed TFD. The analysis result is displayed in Fig. 3. The superior performance of the proposed distribution in terms of time–frequency resolution (narrower main lobe) and artifact reduction is apparent. Here, the signal is monocomponent but with a nonlinear FM law; it is for this reason that we have artifacts in the distributions, including the WVD. The previous examples clearly showed the high time–frequency resolution of the proposed TFD. Next, we demonstrate the performance of the new distribution in terms of IF estimation.

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Fig. 3.

(Left) WVD, (center) the spectrogram, and (right) the proposed distribution of a real-life acoustic signal.

Fig. 4. (Top left) WVD, (top center) spectrogram, and (top right) proposed distribution of a stepped FM signal using a 21-sample window length. Slices of these distributions taken at time instant n 3 is displayed in the bottom left plot. Slices of these distributions evaluated using another window length (63 samples) taken at the same time instant is displayed on the bottom right.

=

B. Instantaneous Frequency Estimation Example B1—Stepped FM Signal: In this example, we focus on the IF estimation of a stepped FM signal. The respective frequencies of the signal are equal to 0.1 Hz, 0.25 Hz, and 0.4 Hz. The sampling frequency is equal to unity and the signal length . is equal to The WVD, the spectrogram, and the proposed TFD for this signal are displayed in Fig. 4 (top plots, respectively). Slices of are also plotted in the these distributions at time instant same figure (bottom left). The window length considered in the evaluation of these distributions was fixed equal to 21 samples. However, larger window lengths give similar results as shown, for example, for a window length equal to 63 samples (see the bottom right of Fig. 4). From this figure, we can observe that the proposed TFD is the best distribution in terms of narrowest lobe around the signal IF (0.1 Hz) in the analysis of the noiseless signal under consideration. Let us now consider the case where additive white Gaussian noise distorts the signal. The signal-to-noise ratio (SNR), which is defined as the power of the signal over the power of the noise,

Fig. 5. Slices of the WVD, the spectrogram, and the proposed distribution of a stepped FM signal in additive Gaussian noise. The SNR is equal to 5 dB.

0

is chosen equal to 5 dB. Once again, we use the previous distributions, with a window length equal to 21, in the analysis of , of these distrithe noisy signal. Slices, taken at time butions are displayed in Fig. 5. We can again observe that the proposed distribution not only has the narrowest lobe around the true signal IF but also yields the smallest shift from it (minimum bias). In order to quantify the superior performance of the

BARKAT AND BOASHASH: HIGH-RESOLUTION QUADRATIC TIME–FREQUENCY DISTRIBUTION

Fig. 6.

IF estimator mean squared errors (estimated) for different window lengths for a stepped FM signal at time instant n

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= 3.

Fig. 7. (Left plot) Slices of the WVD and the proposed TFD ( = 1) at the middle of the time interval for a linear FM signal. (Right plot) Estimated mean squared errors in the IF estimation of the same signal at the same time instant.

proposed TFD over the WVD and the spectrogram in a noisy environment, a statistical performance comparison of the distributions is considered. In what follows, we consider the IF estimation of the noisy (true IF equal to 0.1 Hz). The SNR signal at time instant , defined earlier can be expressed as SNR is the noise variance. where is the signal amplitude, and The peaks of the WVD, the spectrogram, and the proposed TFD are used as IF estimators of the noisy signal. The SNR is varied dB to dB by a 1-dB step. For each SNR value, we from run 1000 Monte Carlo simulations. The experiment is performed for a small, a medium, and a large size window length, respecin the protively. For the two first windows, we chose . posed TFD, and for the largest window, we chose Simulation results, for each window length are displayed in Fig. 6. The axis represents the SNR (in decibels), and the axis represents the IF estimated mean squared errors (in decibels). The simulation results displayed here show that for each window length, the proposed TFD yields the minimum mean squared error for the IF estimators and, consequently, is more accurate in estimating the IF of the noisy signal. Example B2—Linear FM Signal: In this example, we confirm the high performance of the proposed TFD by comparing it with the WVD in the analysis and IF estimation of a linear FM signal. It is well known that the WVD is optimal in the sense of maximum concentration of the signal energy around its IF for a monocomponent linear FM signal [1]. It is also shown that the IF estimator based on the peak of WVD is sufficient for a linear FM signal in additive white Gaussian noise [11]. These

two properties of the WVD makes it a very powerful tool in the analysis of noisy linear FM signals. In this example, we show that the WVD and the proposed TFD are, in practice, equivalent for the analysis of such signals. For that purpose, let us consider a linear FM signal of length equal to sampled at a frequency equal to unity. The signal is analyzed using both the WVD and the proposed TFD . In the left plot of Fig. 7, we display slices of both with . It is clear distributions at the middle of the time interval from the figure that the two distributions are equivalent in terms of spectral resolution (main lobe width around the signal IF) for the noiseless signal. For statistical comparison of the two distributions, white Gaussian noise is added to the signal. The peak of the WVD and the peak of the proposed TFD are used as IF estimators of the noisy linear FM signal at the middle of the time interval. For each SNR value defined as in the previous example and varied from 0 to 20 dB, the IF is estimated 1000 times. Then, its variance is evaluated at that particular SNR value. Simulations results are plotted in the right plot of Fig. 7. Here again, we can clearly observe that the two distributions are equivalent. Their respective based estimators variances reach the Cramér-Rao bound [16] (represented by the continuous line), and both estimators have an equivalent threshold. Larger signal lengths were also considered, and they yield similar results. C. Summary of Main Results Five different examples were provided to illustrate the effectiveness of the proposed TFD. The first example showed that

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the proposed distribution could resolve two close linear FM signals at some time instant where the WVD and the spectrogram cannot. In the second and third examples, we considered real-life data collected from automotive and acoustic experiments. The analysis of these signals showed the superiority, in terms of time–frequency resolution and artifacts (or cross term) suppression, of the proposed TFD compared with the two other distributions. In the last two examples, we quantified the high performance of the proposed TFD using a statistical performance evaluation. In these two examples, we focused on the IF estimation of a noisy stepped FM signal as well as a noisy linear FM signal. The fourth example clearly showed that the proposed distribution yields the smallest estimator mean squared error (for different window lengths) in the estimation of the noisy signal IF. In the last example, we compared the proposed distribution to the WVD, which is optimal and efficient for representing and estimating the IF of a linear FM signal, and showed that both distributions are practically equivalent in this case. These examples have shown that the proposed TFD outperforms the WVD and the spectrogram in terms of time–frequency resolution and cross term suppression in the analysis of multicomponent signals. They have also shown that the proposed TFD is more accurate in IF estimation of various noisy signals. In addition to the WVD and the spectrogram, there exist other quadratic TFDs such as the Choi–Williams distribution or the Zhao–Marks–Atlas distribution. The comparison of these TFDs to the proposed one could also be considered. However, as stated earlier, the comparison would not be effective without an optimization procedure of the varying parameter that appears in their respective kernels. This optimization is beyond the scope of this paper and could be addressed elsewhere. IV. CONCLUSION In this paper, we presented a kernel for a new quadratic time–frequency distribution. The proposed distribution outperforms the WVD and the spectrogram in terms of time–frequency resolution and cross term suppression. We have shown that the proposed distribution can resolve two close signals in the time–frequency plane that the two other distributions cannot. Synthetic and real data examples have been used to demonstrate the high performance of the proposed distribution. We have also shown that the proposed distribution is more accurate in IF estimation of a stepped FM signal embedded in additive white Gaussian noise. Moreover, the proposed distribution was shown to be equivalent to the WVD in representing and estimating a noisy linear FM signal. ACKNOWLEDGMENT The authors would like to thank Prof. J. F. Böhme from Ruhr University Bochum and Volkswagen AG in Germany for providing the data for the automotive signal example. The authors would also like to express their gratitude to the anonymous reviewers, whose valuable comments helped improve the quality of the paper.

REFERENCES [1] B. Boashash, “Time-frequency signal analysis,” in Advances in Spectrum Analysis and Array Processing, S. Haykin, Ed. Englewood Cliffs, NJ: Prentice-Hall, 1991, vol. 1, ch. 9, pp. 418–517. [2] M. G. Amin and W. J. Williams, “High spectral resolution time-frequency distribution kernels,” IEEE Trans. Signal Processing, vol. 46, pp. 2796–2804, Oct. 1998. [3] B. Boashash, Ed., Time Frequency Signal Analysis, Melbourne, Australia: Longman Cheshire, 1992. [4] D. König and J. F. Böhme, “Wigner-Ville spectral analysis of automotive signals captured at knock,” Appl. Signal Process., vol. 3, pp. 54–64, 1996. [5] H. I. Choi and W. J. Williams, “Improved time-frequency representation of multicomponent signals using the exponential kernels,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, pp. 862–871, June 1989. [6] Y. Zhao, L. E. Atlas, and R. J. Marks II, “The use of cone shaped kernels for generalized time-frequency representation of nonstationary signals,” IEEE Tran. Acoust., Speech, Signal Processing, vol. 38, pp. 1084–1091, June 1990. [7] P. Zavarsky and N. Fujii, “Introduction of cross ambiguity function for elimination of crossterms in wigner distribution of the third order,” Electron. Lett., vol. 32, no. 2, pp. 94–95, Jan. 1996. [8] L. Stankovic`, “S-class of time-frequency distributions,” in Proc. Inst. Elect. Eng., Vis. Image Signal Process., vol. 144, Apr. 1997, pp. 57–64. [9] B. Ristic and B. Boashash, “Kernel design for time-frequency analysis using the Radon transform,” IEEE Trans. Signal Processing, vol. 41, pp. 1996–2008, May 1993. [10] G. Jones and B. Boashash, “Generalized instantaneous parameters and window matching in the time-frequency plane,” IEEE Trans. Signal Processing, vol. 45, pp. 1264–1275, May 1997. [11] P. Rao and F. J. Taylor, “Estimation of IF using the discrete Wigner-Ville distribution,” Electron. Lett., vol. 26, pp. 246–248, 1990. [12] J. Jeong and W. J. Williams, “Kernel design for reduced interference distributions,” IEEE Trans. Signal Processing, vol. 40, pp. 402–412, Feb. 1992. [13] , “Alias-free generalized discrete-time time-frequency distributions,” IEEE Trans. Signal Processing, vol. 40, pp. 2757–2765, Nov. 1992. [14] B. Boashash and P. J. Black, “An efficient real-time implementation of the Wigner-Ville distribution,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, pp. 1611–1618, Nov. 1987. [15] D. C. Reid, A. M. Zoubir, and B. Boashash, “Aircraft flight parameter estimation based on passive acoustic techniques using the polynomial Wigner-Ville distribution,” J. Acoust. Soc. Amer., vol. 102, no. 1, pp. 207–223, 1997. [16] B. Barkat and B. Boashash, “Instantaneous frequency estimation of polynomial FM signals using the peak of the PWVD: Statistical performance in the presence of additive Gaussian noise,” IEEE Trans. Signal Processing, vol. 47, pp. 2480–2490, Sept. 1999.

Braham Barkat received the “Ingenieur d’Etat” degree in electronics in 1985 from the National Polytechnic Institute of Algiers (ENPA), Algiers, Algeria, the M.S. degree in control systems in 1988 from the University of Colorado, Boulder, and the Ph.D. degree from the Signal Processing Research Centre, Queensland University of Technology (QUT), Brisbane, Australia. In 1989, he joined University of Blida, Blida, Algeria, where he held a lecturer position in digital and advanced control systems. In 1996, he joined the Signal Processing Research Centre at Queensland University of Technology (QUT), Brisbane, Australia as a Senior Research Assistant and then as a Ph.D. candidate in signal processing. From September 1999 to November 2000, he had been a Postdoctoral Research Fellow, first at QUT and then at Curtin University of Technology, Perth, Australia. In November 2000, he joined the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, as an Assistant Professor. His research interests include time–frequency signal analysis, estimation and detection, statistical array processing, and signal processing in telecommunications.

BARKAT AND BOASHASH: HIGH-RESOLUTION QUADRATIC TIME–FREQUENCY DISTRIBUTION

Boualem Boashash (F’99) received the Dipl.-Ing. degree in physics and electronics from the ICIP, University of Lyon, Lyon, France, in 1978, the M.S. degree from the Institut National Polytechnique de Grenoble (INPG), Grenoble, France, in 1979, and the Doc.-Ing. degree from INPG in May 1982. In 1979, he joined Elf-Aquitaine Geophysical Research Centre, Pau, France. In May 1982, he joined the Institut National des Sciences Appliqées de Lyon. In 1984, he joined the Electrical Engineering Department, the University of Queensland, Brisbane, Australia, as a Lecturer, Senior Lecturer in 1986, and Reader in 1989. In 1990, he joined the Graduate School of Science and Technology, Bond University, Gold Coast, Australia, as Professor of electronics. In 1991, he joined the Queensland University of Technology, Brisbane, as the Foundation Professor of Signal Processing and Director of the Signal Processing Research Centre. He is the editor of two books and has written over 200 technical publications and has supervised more than 20 Ph.D. students and five Masters students. His research interests are time–frequency signal analysis, spectral estimation, signal detection and classification, and higher order spectra. He is also interested in wider issues such as the effect of engineering developments on society. Dr. Boashash was Technical Chairman of ICASSP 1994, which is the premium conference in signal processing. He is a Fellow of IE Australia and of IREE.

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