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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2005

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Assessment of Nonlinear Interaction Between Nonlinearly Coupled Modes Using Higher Order Spectra A. R. Messina, Member, IEEE, and Vijay Vittal, Fellow, IEEE

Abstract—Higher order spectral analysis techniques are used to identify nonlinear interaction involving the electromechanical modes of oscillation in complex power systems. First, the presence and extent of nonlinear interactions between frequency components in oscillatory processes following large perturbations are identified using bispectrum and bicoherence analysis methods. Then, the strength and distribution of nonlinear couplings between frequency components is investigated using the phase relationships between spectral components. A case study with a 377-generator model of the Mexican interconnected system is used to illustrate nonlinear aspects arising from the nonlinear interaction of the different low-frequency inter-area modes. The results of the numerical simulations show that low-frequency modes may interact nonlinearly producing intermodulation components at the sum and/or difference frequency of the fundamental modes of oscillation. Such an identification can be used as a benchmark for validation of nonlinear analysis methods, and to reduce the burden associated with these methods. Index Terms—Nonlinear systems, power system dynamic stability, spectral analysis, statistics.

I. INTRODUCTION

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ONLINEAR interaction between the fundamental modes of oscillation in power systems has been recently studied in the context of the method of normal forms [1]–[4]. These techniques provide insight into the origin and role of the interaction process and may be used to design controllers and predict various aspects of nonlinear behavior. The application of these methods to large power system models, however, is very challenging and computationally demanding. Several approaches and models have been proposed to quantify nonlinear coupling. Among these, nonlinear spectral analysis techniques are finding an ever increasing range of applications in diverse areas of physical systems. These include the use of quadratic phase coupling information in plasma fluctuations and surface gravity waves, and the study of modulation processes in speech signals [5]–[8]. Fourier analysis provides a traditional approach to analyze oscillatory signals in time series, allowing dominant frequencies to be identified. Information concerning the phase associ-

ated with the different frequency components, however, is suppressed. This information can be used to indicate relationships between the frequency components as well as to detect nonlinear behavior, especially under heavy stress conditions [9]. This paper investigates the application of higher order spectral (HOS) techniques to detect mechanisms leading to nonlinear behavior of electromechanical modes. First, the presence and extent of nonlinear interactions between frequency components in oscillatory processes following large perturbations are identified using bispectrum and bicoherence analysis methods. Then, the strength and distribution of nonlinear couplings between frequency components is investigated using the phase relationships between spectral components. The proposed approach constitutes a natural extension to linear analysis techniques and may be used to supplement information on nonlinear system behavior. A case study on a 377-generator model of the Mexican interconnected system is used to illustrate nonlinear aspects arising from nonlinear interaction between low-frequency inter-area modes. The results of the numerical simulations show that low-frequency modes may interact nonlinearly producing intermodulation components at the sum and/or difference frequency of the fundamental modes of oscillation. Such identification is of fundamental importance for validation of analytical models based on normal form theory, and would assist in the determination of measurements for prediction and characterization of system nonlinear behavior and improved control design. In addition, accurate determination of the interacting modes will significantly reduce the computational burden associated with the method of normal forms since several of the computations can be restricted to the interacting modes determined by the HOS analysis. II. BISPECTRAL ANALYSIS A. Basic Definitions be a real discrete, zero-mean time-varying signal. Let is given by The Fourier transform of (1)

Manuscript received June 2, 2004. Paper no. TPWRS-00693-2003. A. R. Messina was with the Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 36849 USA. He is now with the Department of Electrical Engineering, Cinvestav, Guadalajara JAL 45090, Mexico (e-mail: [email protected]). V. Vittal is with the Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 36849 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TPWRS.2004.841240

represents the discrete Fourier transform (DFT) of where at frequency . The power spectrum, , can then be defined in terms of the signal’s DFT as

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(2)

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where is the expected value or statistical average of the denotes the complex conjugate of . The ensemble and , power spectrum is real valued, symmetric, i.e., and contains no information about the phase of oscillations; this prevents the analysis of phase relations between oscillations at different frequencies. B. HOS Analysis HOS are the extension to higher orders of the concept of the power spectrum. In cases where the process is non-Gaussian or is generated by nonlinear mechanisms, HOS provide information which can not be obtained from the conventional spectrum. For a discrete, zero-mean process, the bispectrum is defined as [5], [10] (3) , where is the third-order cumufor lant of the process . Alternatively, the bispectrum may be written in the more familiar form (4) where is the time duration of the signal. The bispectrum is a complex, two-dimensional doubly periodic quantity that measures the magnitude and the phase of the correlation of a signal at different Fourier frequencies and plane. Knowlhas several symmetry regions in the edge of the bispectrum in the nonredundant triangular region is sufficient for a complete description of the bispectrum [9]. It can then be readily seen that

for all contrast to this, the system is linear for . The ability to separate the magnitude and distribution of the nonlinearity as a function of frequency makes higher order analysis a particularly attractive technique for the study of oscillatory processes. The amplitude of the bicoherence can be interpreted as the contribution of energy of nonlinear interaction to the wave en. It is important to emphasize, ergy with a frequency however, that the bispectrum does not indicate which of the three or frequencies interact, i.e., . The nature of this interaction has to be determined using other techniques such as Fourier analysis. C. Quadratic Phase Coupling (QPC) Second-order nonlinear interactions between frequency components may give rise to a phase relationship known as QPC [6]. Following [9], three harmonic components with frequencies and phase , are said to be quadratically phase coupled if and . When such a relationship exists, the phase coupled components contribute to the third moment sequence of a process; the degree of coupling between the components may be measured by the bispectrum. To define more precisely these concepts, consider a simple quadratic system of the form where represents the input to the system, and denotes the output; the parameter represents the coefficient of nonlinearity. Let (8) , and denote, respectively, the amplitude, angular where frequency, and phase of the waves. Then

(5) (9)

where where (6) is known as the biphase of the bispectrum, and ; the triplet is called a bifrequency. An immediate consequence of the definition of the bispecand are trum in (4) is that when the frequencies independent of one another, or there is a random phase relationship among them, the bispectrum will average to a very small value. A quantitative tool is therefore needed to determine the statistical significance of this correlation. Given estimates of the spectrum, and the bispectrum, the bicoherence is defined [5] (7) is the power spectrum of in (2). Since bicoherwhere close ence is the normalized bispectrum, a value of to unity indicates a nonlinear signal production mechanism. In

, and . From (9), it can easily be verified that , contains components at the input frequencies, in , addition to harmonic terms of the form , arising from and modulation components at . nonlinear interaction between the primary frequencies The form of the bispectrum in (4) suggests that, if the energy at the sum, or difference, frequency is generated by a nonlinear process, phase coherence among the bifrequency components exists, and therefore the statistical average will lead to a nonzero value of the bispectrum. Also, the bicoherence, , will be close to unity and the biphase at that frequency will be zero. For a harmonically related processes, the , and bispectrum consists of a peak at the bifrequency is theoretically zero for other uncoupled harmonic processes, having no phase-coupled interactions. Fig. 1 illustrates the use of the bispectrum for QPC detection Hz, Hz, . of the process (9) with Note the generation of second harmonics at 0.50 Hz, 2.0 Hz and

MESSINA AND VITTAL: NONLINEAR INTERACTION BETWEEN NONLINEARLY COUPLED MODES USING HOS

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Fig. 1. Process with quadratic phase coupling. (a) Contour plot of bispectrum. (b) Power spectrum.

phase-coupled intermodulation components at Hz and Hz in the power spectrum. The corresponding bispectrum estimate resulting from (3) shows a peak at the bifrequency (1.00 Hz, 0.25 Hz) with the intermodulation components being implicit, and self coupling of the 0.25- and 1.00-Hz primary frequencies1. In practical applications, spurious peaks may appear in the bispectrum at locations without significant QPC due to various aspects such as finite data length. Furthermore, one of the frequency components may be equal to the sum of the other two harmonics without nonlinear relation among them. The use of HOS may provide further verification of phase coherence. Conventional techniques for QPC detection based on fast Fourier transform (FFT) approaches suffer from low-frequency resolution, especially when there is a limited amount of data. This has provided the thrust to develop alternate bispectrum estimation methods.

and satisfies the third-order recursion (TOR) equations (11) is the third moment sequence of the AR in which is the impulse function. The bispectrum of process and the AR process of (10) is given by

(12) where

is the transfer function of the process (13)

D. Parametric Methods for Detection of QPC Parametric techniques based on Auto-Regressive (AR) modeling of the third-order cumulants have been shown to provide a more accurate detection of QPC, especially for short-length, non-Gaussian data [9], [10]. These techniques may be summarized briefly as follows. -order AR process, , described by Consider a real

and the are the AR-model parameters. Once the bispectrum is computed, expressions for the bicoherence can then be obtained using (7). Since in this case, the power spectrum and bispectrum involve the use of parameter estimates of different models, parametric models are often considered more appropriate to detect rather than to quantify QPC [9]. In the following sections, we provide guidelines for the investigation of QPC in the context of the analysis of power system dynamic behavior.

(10) III. CASE STUDY is a non-Gaussian process with , and third moments of the process is given by

A. Description of the Study System

where

1Due

to symmetry properties, the bispectrum is symmetric about f

. The

=f

System studies are based on a 6-area dynamic model of the Mexican interconnected system (MIS) that includes the detailed representation of: 377 generators, 3759 buses, 2936 branches, 10 large static VAR compensators, and 1986 transformers. Fig. 2 shows a schematic representation of the MIS illustrating the

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Fig. 2.


Geographical representation of the study system.

location of major transmission and generation facilities, along with selected areas of study. The base case condition is the summer peak-load 2002; for the base case, the overall state-space model of the system has 2256 states for the base case condition. The six-area model of the MIS exhibits three critical inter-area modes involving the interaction of machines in several parts of the system: two critical inter-area modes at 0.32 and 0.52 Hz involving the interaction of machines in the north and south systems, and east and west systems, respectively, and a mode at 0.62 Hz involving the interaction of machines in the northern and northeastern systems. Also of relevance, the system exhibits two higher frequency modes at 0.78 and 0.91 Hz involving localized phenomena within the geographical areas. Table I displays the main characteristics of these modes showing their modal damping ratios and swing frequency. Previous work has shown that these modes are dynamically coupled and may interact to produce a complex system behavior [4]. This has provided motivation for the analysis of this phenomenon. B. Spectral Characteristics Detailed transient stability studies were performed to examine the presence of nonlinear behavior arising from the interaction of critical inter-area modes. Cases of particular interest considered in the analysis included the following. Case A. Outage of the unit #1 (650 MW) of the Laguna Verde (LGV) nuclear power station in the southeastern network of the system. This contingency is known from previous studies to strongly excite the 0.32-Hz north-south inter-area mode 1 Case B. Simultaneous tie-line tripping without fault of one circuit of the MMT-JUI 400 kV line and the

TABLE I SLOWEST OSCILLATION MODES OF THE SYSTEM

TMD-PBD 400-kV line in the southeastern network of the system. This scenario is known to excite modes associated with the southern regions of the system. Case A is of interest since it involves the presence of a dominant mode at 0.32 Hz and provides the opportunity to assess nonlinear self coupling of the dominant mode. Case B results in nonlinear modal interaction involving the nonlinear interaction of inter-area modes 1–4 and permits the study of the strength and distribution of nonlinearity and the study of intermodulation generation. For each contingency case, several tie-lines were selected representing major interconnection in the system. Table II lists the tie-lines on which the real power flows are the signals selected for evaluating nonlinear performance (refer to Fig. 2) whilst Fig. 3 shows the variation of the real power flow at selected system locations following selected system disturbances.


TABLE II MAJOR INTERTIES SELECTED FOR ANALYSIS OF NONLINEAR BEHAVIOR

Fig. 3.

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TABLE III EIGENVALUES OF THE SYSTEM FOR THE POST-CONTINGENCY CONDITION

Real power flow at selected transmission lines. (a) Case A. (b) Case B.

Table III shows the corresponding eigenvalues for the post-contingency condition (PCC) along with the swing frequency and . damping ratios The north-south interface has a vital role in interconnecting the north and south systems and has a decisive effect on the 0.32-Hz inter-area mode 1. Further, the two 400-kV lines, MMT-JUI and TEC-TOP are representative of a complex transmission system in the southeastern network enabling the transmission of nearly 4000 MW of hydro generation from the southeastern network of the MIS to the Mexico City metropolitan area. Finally, the 400-kV QRO-SLM line constitutes the West-East interface of the system. For case A, the critical contingency results in growing power oscillations for both the north–south interface and major interconnections linking the southeastern system with the central

Fig. 4.

DFT spectrum of the signals of interest. (a) Case A. (b) Case B.

system. Furthermore, the analysis of real power-flow deviations for case B shows sustained oscillations in the western and southwestern systems, suggesting the presence of various interacting modes. Referring to Fig. 3(b), it can also be seen that the real power flow on major interconnections in the western and southeastern systems swings in phase suggesting coupling between the fundamental modes. The analysis of the Fourier spectra for cases A and B in Fig. 4 indicates that the critical contingency stimulates different dynamic patterns. For interconnections along the north-south in-

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terface (namely the LAJ-HUE and the ALT-TMO 400-kV lines) the analysis for case A discloses the presence of the 0.32 (0.291) Hz inter-area mode 12, and to a less extent the 0.62 Hz inter-area mode 3 local to machines in the north systems. The power level peak for mode 1 is significantly larger than the ones at higher frequency indicating an essentially north-south interaction. In contrast, the analysis of the TEC-TOP signal reveals the presence of the 0.52-Hz East-west inter-area mode 2, and the 0.77-Hz inter-area mode 4, in addition to the 0.28-Hz inter-area mode 1. For case B, on the other hand, spectral analysis confirms that power-flow signals in the western and southeastern systems of the MIS have a similar dynamic pattern [refer to Fig. 3(b)] involving the interaction of three major inter-area modes. Coupling, in this case, is dominated by inter-area modes 1, 2, and 4 in Table I. Simulation results agree well with observed system behavior and conventional eigen-analysis studies. Although Fourier-based methods properly identify the critically interacting modes, no direct evidence of nonlinear coupling is provided. In the following, HOS techniques are used to determine evidence of nonlinear effects arising from interactions between the different frequency components. IV. BISPECTRUM ANALYSIS A. Bispectrum Analysis To examine the potential for nonlinear coupling in the system response, the magnitude squared bispectrum was computed for each signal. In this analysis, the mean value was removed from each signal and averaging was used to obtain a satisfactory signal-to-noise ratio for the signals. The bispectrum of the records was then computed via (4) using the Gabr–Rao algorithm [10], [11]. Fig. 5 shows a contour plot of the bispectrum of the real power-flow signals for case A. The contour lines indicate the bispectrum components with maximum activity; in this plot, light shades represent low bispectrum values, while dark shades indicate high bispectrum magnitudes. For clarity of presentation, only the right upper quadrant of the plots is presented since this is the part of physical interest here. Further, the frequencies plotted are limited to those within the low-frequency range of concern. For the 400-kV line, LAJ-HUE, examination of the bispectrum in Fig. 5 shows essentially the presence of nonlinear coupling at the frequency pair (0.26 Hz, 0.26 Hz) arising from nonlinear self interaction of inter-area mode 1. In contrast to this case, the analysis of the TEC-TOP signal shows a weak nonlinear interaction between the 0.52-Hz inter-area mode 2 and the 0.26-Hz mode in addition to self-interaction at 0.52 Hz. The analysis also shows small peaks at about 0.42 and 0.18 Hz, but the frequency resolution of the bispectrum for this case is insufficient to provide a more in-depth analysis of the phenomenon. The strength of the nonlinear interaction originating from these components is demonstrated in Fig. 6 that shows the bispectrum

2Note that for case A, the frequency of inter-area mode 1 for the PCC reduces

to 0.291 Hz; the frequency identified by Fourier analysis (see Fig. 4) is about 0.267 Hz. For case B, the frequency of mode 2 reduces to 0.51 Hz, whilst that of mode 4 is 0.749 Hz. Other changes are not significant.

Fig. 5.

Contour plots of the bispectrum of the power-flow signals for Case A.

Fig. 6.

Diagonal slice of the bispectrum of the power-flow signals for Case A.

slice as a function of frequency. It should be emphasized that, since the bispectrum is estimated directly from the nonlinear


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Fig. 8. Diagonal slice of bispectrum of power signals for Case B.

modal components is significantly greater indicating the presence of both, frequency mixing between various modal components and nonlinear self coupling. Thus for instance, the analysis of the MMT-JUI power-flow signal in Fig. 7 identifies nonlinear self-interaction at the frequency pairs, (0.74 Hz, 0.74 Hz) and frequency mixing at (0.74 Hz, 0.32 Hz). Again, the low-frequency resolution of the method fails to precisely identify the exact location of the peaks. Of particular interest for this analysis, the results show that the inter-area mode 4 associated with machines in the western and southeastern systems (refer to Table I) is seen to interact nonlinearly with the 0.32-Hz inter-area mode 1. Moreover the analysis of the QRO-SLM real power signal suggests the presence of strong nonlinear interaction between inter-area modes 1 and 2 at the bifrequency (0.51 Hz, 0.32 Hz) in addition to nonlinear coupling at (0.78 Hz, 0.32 Hz) and self coupling at about (0.74 Hz, 0.74 Hz). This is illustrated in Fig. 8, showing the diagonal slice of the bispectrum as a function of frequency. Comparison of Figs. 7 and 8, shows that the magnitude of nonlinear self coupling is in the same order of magnitude as the coupling for frequency mixing. This provides additional information regarding the strength and distribution of nonlinearity in the system which is complementary to that obtained from conventional linear analysis. B. Quadratic Phase Coupling (QPC)

Fig. 7.

Contour plots of the bispectrum of power-flow signals for Case B.

time-domain records, the frequencies detected by the HOS techniques are expected to be in close agreement with those of the Fourier spectra in Fig. 4 (refer to footnote 2). For Case B, on the other hand, the analysis of the bispectrum contours in Fig. 7 demonstrates a more complex dynamic pattern suggesting the strongest evidence for nonlinear interactions. In particular, the extent of nonlinear coupling between

In order to obtain a more accurate assessment of QPC between system quantities, the time series were analyzed to produce the squared bicoherence using both, the bispectrum estimated from the direct (FFT) method, and a parametric approach based on the TOR method in Section II. Fig. 9(a) and (b) shows the direct-method estimate of the bispectrum for selected power-flow signals. In interpreting these results, it is important to note that a peak in the bicoherence magnitude of the record at the frequency pair indicates QPC between frequency and ; the value of components at the frequencies can be interpreted as the proportion of the signal power at that frequency due to QPC.

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Fig. 9. Bicoherence estimate of power signals using the direct method. (a) Case A. (b) Case B.

Application of bicoherence analysis to data set for case A [see Fig. 9(a)]suggests that QPC is present and that it manifest itself as nonlinear self-coupling of the inter-area mode 1 at Hz but, the frequency resolution of the FFT-based methods is too low to allow detailed analysis of other energy transfers.

Fig. 10.

Parametric bispectrum of power-flow signals. (a) Case A. (b) Case B.

By contrast, the analysis of power-flow signals for Case B in Fig. 9(b) reveals a rather more complex dynamic pattern


involving intermodulation components between fundamental modes at (0.74 Hz, 0.32 Hz), (0.51 Hz, 0.32 Hz) and self interaction at (0.32 Hz, 0.32 Hz) as suggested by the bispectrum estimates. To further estimate the significance and accuracy of the QPC estimates, the bicoherence was computed using the TOR method. This technique may provide a closer resolution of closely spaced frequencies, as well as to detect the presence or absence of QPC and is used as a benchmark to validate previous findings. Fig. 10 shows the bispectrum estimate from the parametric method for some selected power-flow signals together with the conventional power spectrum. For case A, the analysis discloses the presence of a dominant component at (0.26 Hz, 0.26 Hz) confirming the presence of self interaction of inter-area mode 1 and weak coupling at (0.52 Hz, 0.26 Hz) suggesting interaction between inter-area modes 1 and 2. Simulation results for the MMT-JUI power-flow signal in Fig. 10(b) indicate that inter-area mode 4 interacts nonlinearly with the inter-area mode 1 to produce a new mode at 0.42 Hz, as shown in the power spectrum. In addition, the analysis of the SLM-QRO power-flow signal confirms the presence of frequency mixing at (0.51 Hz, 0.32 Hz) with a corresponding intermodulation component at 0.83 Hz. Furthermore, the additional peak in Fig. 10(b) appears to indicate that self interaction of the primary frequency at 0.83 Hz transfer energy to oscillations at the second harmonics at 1.66 Hz. These results are consistent with previous findings but provide a more accurate characterization of QPC. Simulation results show new insights and have important implications for the analysis of power system dynamic behavior. Firstly, study experience with complex systems suggests that bispectral analysis can aid in identifying the distribution and strength of nonlinearity. This information, in turn, can be used to validate nonlinear analysis methods such as normal form analysis. Secondly, knowledge of nonlinear interaction can provide the basis for a better appraisal of control capabilities in the system and for reduced computational burden in the normal form calculations. These aspects are to be addressed in futures stages of this research. V. CONCLUSION Bispectral analysis of power system records has identified the existence of nonlinear interaction between the fundamental modes of oscillation in the dynamic model of the system. This interaction manifests in the form of nonlinearly interacting components at the sum and/or frequency difference of the original major electromechanical modes of the system. Understanding the origin and role of the spectral components interacting nonlinearly requires a thorough understanding of the system. The use of HOS is seen to add important information to system analysis which can be of interest in the modeling and control of large stressed power systems and provides direct comparison to nonlinear analytical techniques. In particular, system

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studies suggest that control capabilities should be coordinated to enhance damping of the interacting modes. Accurate determination of the interacting modes is also expected to reduce the computational burden associated with the analysis of nonlinear modal interaction using other nonlinear analysis techniques. Future work including improved analytical models and validation under more general operating conditions is needed. In addition, parametric investigations have to be performed to identify nonlinear aspects that lead to nonlinear coupling. REFERENCES [1] N. Yorino, H. Sasaki, Y. Tamura, and R. Yokoyama, “A generalized analysis method of auto-parametric resonances in power systems,” IEEE Trans. Power Syst., vol. 4, no. 3, pp. 1057–1064, Aug. 1989. [2] V. Vittal, N. Bathia, and A. A. Fouad, “Analysis of the interarea mode phenomena in power systems following large disturbances,” IEEE Trans. Power Syst., vol. PS-6, no. 2, pp. 1515–1521, Mayt 1991. [3] C.-M. Lin, V. Vittal, W. Kliemann, and A. A. Fouad, “Investigation of modal interaction and its effects on control performance in stressed power systems using normal forms of vector fields,” IEEE Trans. Power Syst., vol. 11, no. 2, pp. 781–787, May 2003. [4] A. R. Messina, E. Barocio, and J. Arroyo, “Analysis of modal interaction in power systems with FACTS controllers using normal forms,” in Proc. 2003 IEEE Power Engineering Society Winter Meeting, pp. 2111–2117. [5] Y. C. Kim and E. J. Powers, “Digital bispectral analysis and its application to nonlinear wave interactions,” IEEE Trans. Plasma Sci., vol. PS-7, no. 2, pp. 120–131, Jun. 1979. [6] A. G. Beard, N. J. Mitchell, P. J. S. Williams, and M. Kunitake, “Nonlinear interactions between tides and planetary waves resulting in periodic tidal variability,” J. Atmospher. Solar Terrestrial Phys., vol. 61, pp. 363–376, 1999. [7] R. R. Clark and J. S. Bergin, “Bispectral analysis of mesosphere winds,” J. Atmospher. Solar Terrestrial Phy., vol. 59, no. 6, pp. 629–639, 1997. [8] J. W. A. Fackrell and S. McLaughlin, “The higher order statistics of speech signals,” in Proc. 1994 IEE Colloq. Techniques for Speech Processing and Their Application, London, U.K., Jun. 1994. [9] C. L. Nikias and A. P. Petropulu, Higher-Order Spectra Analysis—A Nonlinear Signal Processing Framework. Englewood Cliffs, NJ: Prentice-Hall, 1993. [10] M. R. Raghuveer and C. L. Nikias, “Bispectrum estimation: A parametric approach,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, no. 4, pp. 1213–1230, Oct. 1985. [11] Higher-Order Spectral AnalysisToolbox User’s Guide, E. B. Rao, Ed., The MathWorks, Inc., Version 2, 1998.

A. R. Messina (M’85) received the M.Sc. degree (Hons.) in electrical engineering from the National Polytechnic Institute of Mexico in 1987 and the Ph.D. degree from Imperial College, London, U.K., in 1991. Since 1997, he has been an Associate Professor at the Center for Research and Advanced Studies, Guadalajara, Mexico. He was a Visiting Professor at Iowa State University, Ames.

Vijay Vittal (M’82–SM’87–F’97) received the B.E. degree in electrical engineering from B.M.S. College of Engineering, Bangalore, India, in 1977, the M.Tech. degree from the Indian Institute of Technology, Kanpur, India, in 1979, and the Ph.D. degree from Iowa State University, Ames, in 1982. He is a Professor in the Electrical Engineering and Computer Engineering Department, Iowa State University. Dr. Vittal was the recipient of the 1985 Presidential Young Investigator Award.