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A Theory of Polyspectra for Nonstationary Stochastic Processes Alfred Hanssen, Member, IEEE, and Louis L. Scharf, Fellow, IEEE

Abstract—Harmonizable processes constitute an important class of nonstationary stochastic processes. In this paper, we present a theory of polyspectra (higher order moment spectra) for the harmonizable class. We define and discuss four basic th-order moment function, the th-order quantities: the time-frequency polyspectrum, the th-order ambiguity function, and the th-order frequency-frequency polyspectrum. The latter generalizes the conventional polyspectrum to nonstationary stochastic processes. These four functions are related to one another by Fourier transforms. We show that the frequency and time marginals of the time-frequency polyspectrum are the instantaneous th-order moment and the conventional th-order stationary polyspectrum, respectively. All quantities except the th-order ambiguity function allow for insightful interpretations in terms of Hilbert space inner products. The inner product picture leads to two novel and very powerful definitions of polycoherence for a nonstationary stochastic process. The polycoherences are objective measures of stationarity to order , which can be used to construct various statistical tests. Finally, we give some specific examples and apply the theory to linear time-varying systems, which are popular models for fading multipath communication channels. Index Terms—Coherence, harmonizability, nonstationarities, polyspectra, spectral representation, stochastic processes.

Assume now that the stochastic process representation [5]

has the spectral

(1) is the complex-valued increment process (or the where . If time is generalized Fourier transform) of the process , and if time continuous, the integration limits in (1) are , where is the equidistant is discrete, the limits are , the increment sampling interval. Since we assume , process has a useful Hermitian symmetry where the asterisk denotes complex conjugation. has orthogonal increments [6], i.e., If the process (2) is some function of and is Dirac’s delta where function, then it is easy to show that we obtain the well-known is a function of time difstationary result that ference only: (3)

I. INTRODUCTION

L

ET , be a real-valued stochastic process, where denotes some index set for a time-like variable . For , or time example, time could be continuous, for which . could be discrete, for which is Gaussian, it is well known that knowlIf the process edge of the first and second-order moment functions and is sufficient to characterize the process completely [1]. Non-Gaussian processes are, however, abundant in nature and man-made systems alike. To describe the deviation from Gaussianity, one must often consider moment functions of orders higher than two [1]–[4].

Manuscript received June 21, 2002; revised November 4, 2002. The work of A. Hanssen was supported by the Research Council of Norway. L. L. Scharf was supported by the National Science Foundation under Contracts ECS 9979400 and ITR 0112573 and by the Office of Naval Research under Contract N00014-00-1-0033. The associate editor coordinating the review of this paper and approving it for publication was Dr. Xiang-Gen Xia. A. Hanssen is with the Department of Physics, University of Tromsø, Tromsø, Norway (e-mail: [email protected]). L. L. Scharf is with the Departments of Electrical and Computer Engineering and Statistics, Colorado State University, Ft. Collins, CO 80523-1373 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2003.810298

as the power spectral density Then, we identify . Equation (3) is, of course, nothing but the Einof stein–Wiener–Khinchin relation for stationary processes, relating the correlation function to the power spectral density by a Fourier transform. The class of harmonizable nonstationary processes is now defined as the processes with nonorthogonal increments [7]–[9], i.e., (4) is some complex-valued function of and where . The dual-frequency function is often called the Loève spectrum of the process. In this case, the relation between the nonstationary correlation function and the Loève spectrum takes the form [7]

(5) From (5), we understand that in this representation, spectral correlation among different frequency components is responsible for the nonstationarities. Knowing the spectral correlation in detail is therefore useful for characterizing the nature of the non. stationarity of The main purpose of this paper is to introduce a theory of polyspectra for harmonizable stochastic processes. In other

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words, we will define and interpret the higher order moments to arbitrary of the nonorthogonal increment process order. In doing so, we derive a powerful and natural theory of polyspectra and polycoherences for nonstationary harmonizable stochastic processes. We believe that the resulting representation of polyspectra is important from a fundamental modeling point of view. There are numerous applications in acoustics and electromagnetics that would benefit from applying the proposed representation to the analysis of experimental data. As far as we know, this is the first publication dealing with polyspectral generalizations for nonstationary harmonizable processes. Note that some recently published titles [10], [11] apparently suggest that a theory of bispectra for harmonizable processes exists. However, the bispectral density considered in these contributions is actually a bimeasure, which is a second-order moment spectrum of harmonizable processes and not a third-order quantity, as a bispectrum actually is [3], [12]. The structure of this paper is as follows. In Section II, we present and discuss the four basic ways an th-order nonstationary moment can be represented in time-time, time-frequency, frequency-time, and frequency-frequency, respectively. In Section III, we discuss the stationary limit and show that it reproduces the conventional polyspectrum, and in Section IV, we calculate the time and frequency marginals of the basic time-frequency polyspectrum. In Section V, we discuss an important geometrical interpretation in terms of Hilbert space inner products of three of the representations. In Section VI, we show that the inner product formulations of the th-order frequency-frequency and time-frequency polyspectra lead to two very powerful and useful definitions of coherence for nonstationary harmonizable processes. We discuss some statistical tests that may be constructed on the basis of the proposed coherences. In Section VII, we provide some examples, and we show that the coherences may also be applied to construct a test of determinism versus randomness for amplitude-modulated signals. In Section VIII, we illustrate how the theory works for linear time-varying systems, which model multipath communication channels, and in Section IX, we list our main conclusions, along with some thoughts about the continuation of this work.

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II. BASIC REPRESENTATIONS In the remainder of this paper, we will assume that the specare tral representation (1) holds but that the increments nonorthogonal. Note that we will only discuss moment spectra and not cumulant spectra in this paper. It is easy, however, to construct the corresponding cumulant spectra from the moment spectra by means of a formula due to Leonov and Shiryaev (see, e.g., [12, p. 33]). As our basic nonstationary th-order quantity in the time doby main, we define the th-order moment function of (6) Here, is a global (or absolute) time variable, and are local (or relative) time variables, i.e., the ’s are time shifts relative to the global time , and is a vector of local times (superscript denotes a transpose). Expressing the moment function (6) by means of the nonorthogonal spectral representation, we obtain (7)–(10), , shown at the bottom of the page, where , and . We thus understand as a frequency offset or a local frequency . relative to the global frequencies We immediately identify the th-order frequency-frequency polyspectral density (or f-f polyspectrum)

(11) We notice that the th-order moment function and the th-order f-f polyspectrum constitute an -dimensional Fourier transform pair (12) . We may call (10) the spectral representation of In this paper, we strictly follow the convention that global variables are denoted by Latin letters (e.g., and ) and that local variables are denoted by Greek letters (e.g., and ). Note local time variables of must always be treated that the

(7)

(8)

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collectively and likewise for the global frequency variables in . We now understand that we may invoke Fourier transforms with respect to any of the variables , , , and in order to exhaust all possible time and frequency representations to the th order. A very important quantity is now derived by means of an , with respect inverse Fourier transform of to the local frequency . We obtain (13)

Fig. 1. Fourier relations between the basic polyspectral densities of nonstationary harmonizable processes.

The quantity is the th-order ambiguity function, which is a function of local frequency and local time vector . and constitute a Fourier We observe that transform pair in the vector variables and (21)

(14)

(15)

(16)

is the th-order time-frequency polyspectral where . In a recent density (or t-f polyspectrum) for the process conference paper [13], we used the term higher order Rihaczek spectrum for the t-f polyspectrum. We think that the term suggested in the present paper is far better and more descriptive is a function of of the quantity of interest. Note that global time and global frequencies . It is important to note that the t-f and the f-f polyspectra are a Fourier transform pair in the variables and ,

We now understand that any of the four basic quantities , , , and may be used to characterize the higher order behavior of a nonstationary stochastic process. Depending on the focus and actual application, it may be advisable to exploit the global time—local , the global time correlation function given by , the time—global frequency t-f polyspectral density , local frequency—local time ambiguity function and/or the local frequency—global frequency f-f polyspectral . density It is very important to note that these four basic densities are interrelated by Fourier transforms, as illustrated by the diagram in Fig. 1, and may be interpreted as the expectations of Fig. 2. III. STATIONARY LIMIT To obtain the stationary limit of the polyspectra, we first recall that the th-order moment function has the spectral representation

(17) The fourth and last quantity we may construct results )-dimensional inverse Fourier transform of as an ( , with respect to the global frequency vector . This yields

(22) For (22) to be independent of on the manifold by requiring

, we must require that is nonvanishing only . This is accomplished

(18)

(23) (19)

is a function of only. Note that (23) is the where th-order generalization of the familiar second-order orthogonality condition (2). Inserting (23) into (22) and carrying out the integration over , we obtain

(20)

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Fig. 2.

Fourier relations between the basic expectations of nonstationary harmonizable processes.

where

(32) (25) (33)

as the conventional th-order moWe thus recognize as the corment function for stationary processes and responding conventional th-order moment spectrum for staand tionary processes. In the stationary limit, are the only two th-order functions of interest. Note that these two functions constitute a Fourier transform pair (26) Equation (26) is the th-order Einstein–Wiener–Khinchin relation. From (23), we understand that the stationary th-order denand the th-order nonstationary density sity are related by

Thus, the frequency marginal of

is (34)

It is interesting and reassuring to note that the time marginal (28) is in fact the instantaneous th-order moment of the process and that the frequency marginal (29) is the conventional th-order polyspectrum that one would normally associate with a stationary process, as seen from (34). , we obtain As special cases, it is instructive that for (35) and (36)

(27) The significance of the Dirac delta function in (27) is that is picks out the stationary manifold on which the stationary th-order density is nonzero. IV. MARGINALS We are interested in the marginals of the global time—global . The time marginal is frequency t-f polyspectrum readily derived as

The second-order marginals are obviously the instantaneous power and the ordinary power spectral density, respectively. is a complex-valued quantity in Thus, even though the ( , )-plane, its marginals are real and non-negative. , we obtain For (37) and

(28)

(38)

The frequency marginal is

The third-order marginals are obviously the instantaneous skewness and the conventional stationary bispectral density, respectively. (29)

V. GEOMETRICAL INTERPRETATIONS We note that three of the four corners in Figs. 1 and 2 can be expressed conveniently as Hilbert space inner products

(30) (39)

(31)

(40)

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(41) Here, we use the convention that the Hilbert space inner product between two complex valued stochastic variables and is de. Note that the th-order ambiguity fined by function is not expressible as an inner product. Rather, it is a convolutions. sequence of Fig. 3.

VI. POLYCOHERENCES In order to obtain objective measures of polyspectra, it is customary to introduce coherence. In particular, the bicoherence ) and the tricoherence ( ) have been useful for the ( analysis and interpretation of data from various assumed stationary processes [4], [14]. Stringent statistical tests based on bi- and tricoherences have been suggested, e.g., [15]. As has been pointed out by Thomson [16], the spectral correlations induced by nonstationarities may be mistaken for nonlinearities in a bicoherence analysis if conventional stationary bicoherence analysis is applied to nonstationary data. Thus, there is a need for definitions of polycoherences that retain their validity for nonstationary processes. We will now derive two different generalized coherences for polyspectra of arbitrary order, which are indeed valid for any harmonizable nonstationary stochastic process. A. Frequency-Frequency Polycoherence To derive a meaningful and useful measure of coherence, we now turn to the expression for the th-order f-f . Expressed in terms polyspectral density , we see of the nonorthogonal increment process is actually a measure of from (41) that the coherence or the Hilbert space inner product between the one-dimensional complex-valued stochastic process and the complex valued ( )-dimen. We may therefore normalize sional random field by the mean-square of its two conand stituents Then, the magnitude-squared frequency-frequency polycoherence of order is defined by (42), shown at the bottom of the page. By the use of Schwartz’ inequality, it is easy to show . Note that . that This new definition of f-f polycoherence generalizes prior definitions to nonstationary processes, for which (poly)spectral may be nonzero. correlations at

cos

j

Frequency-frequency polycoherence in terms of the angle (; f )j.

(; f ) =

From (42), we understand that is actually the cosine-squared of an angle between the two quantities comprising the magnitude-squared f-f polycoherence: (43) In Fig. 3, we illustrate this important geometrical relation beand . tween , we observe that takes the For orders form of the conventional stationary magnitude-squared polycowe obtain herence of order , e.g., for

(44) which is the conventional magnitude-squared bicoherence associated with stationary processes. B. Time-Frequency Polycoherence Employing the same inner product reasoning as in the preceding section, we may also define a time-frequency coherence . We based on the time-frequency polyspectrum can be understood as the Hilbert recall that , space inner product between the stochastic process itself, )-dimensional random field and the complex valued ( . Normalizing by the mean-squared values of the two elements of its inner product definition, we define the magnitude-squared time-frequency by polycoherence of

(45)

Again, Schwartz’ inequality ensures that

.

(42)

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VII. EXAMPLES A. Single Random Amplitude Signal Let the nonstationary process

be defined by (48)

where is a zero mean stochastic variable with finite th-order is assumed to be harmonizable; moment. The process hence, it has the representation (49) Fig. 4. Time-frequency polycoherence in terms of the angle j (t; f )j. cos

(t; f ) =

From (45), we understand that is the cosinesquared between the two quantities comprising the magnitudesquared t-f polycoherence (46) and Fig. 4 illustrates this geometrical relation between . An important interpretation of the t-f polycoherence is now as a meaevident from (45). We may understand )-dimensional random sure of how easy it is to estimate the ( at time from an instantaneous obfield , or vice versa. This seems to be a servation of the process fundamentally important interpretation of t-f polycoherence. Rewriting (45) slightly, we can cast it into the form of (47), shown at the bottom of the page. From (47), we obtain another important interpretation of the time-frequency polycoherence. First, note that we may identify as the (complex conjugated) evaluinstantaneous estimate of the increment process of . Then, we understand ated at the sum frequency is a measure of how closely the instantaneous that at time one-term frequency estimate of approximates the random field .

is the corresponding nonorthogonal increment where are uncorrelated. Under process. We assume that and is clearly harmonizable, and the correthese assumptions, sponding increment process is given by (50) We now readily find the th-order moment function to be as in (51) and (52), shown at the bottom of the page. Thus, we identify the th-order frequency-frequency polyspectrum as

(53) (54) The time-frequency polyspectrum is likewise found to be

(55) (56) We can now evaluate the magnitude-squared f-f polycoherence to obtain (57) and (58), shown at the bottom of the next

(47)

(51)

(52)

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page. These formulas show that an th-order moment description, including the magnitude-squared f-f polycoherence , is a scaled version of the corresponding th-order for . moment description for 1) Deterministic Limit: It is also possible to treat the case is a deterministic function using the above where theory. This is possible by invoking the correspondence

B. Sum of Random Amplitude Signals An important and straightforward generalization of the previous example is given by the multipulse process (61) are uncorrelated random variables, uncorrelated harmonizable processes, and ’s and ’s are mutually uncorrelated. We find the the to be increment process of

where (59) is the Fourier transform of . where For deterministic signals, the magnitude-squared f-f polycoherence (57) reduces to (60) i.e., it is a combination of statistical moments of the random amplitude . It is important to note that the f-f polycoherence becomes independent of both and for the case of a single deterministic function of random amplitude. In fact, there are very few tests available for assessing determinism versus randomness. The above considerations about the polycoherence suggest such a test. 2) Hierarchy of Tests Based on Frequency-Frequency Polycoherence: For the case where the observed process is given , we can summarize the following hierarchy by of tests based on the th-order magnitude-squared f-f polycois herence. If is deterministic; a) independent of and , then is a stob) independent of but dependent on , then chastic process with stationary th-order moments and spectra; is a stochastic process c) dependent on and , then with nonstationary th-order moments and spectra. 3) Hierarchy of Tests Based on Time-Frequency Polycoherence: The tests one can construct for the time-frequency polycoherence parallel those of the f-f polycoherence completely. If is is deterministic; a) independent of and , then is a stob) independent of but dependent on , then chastic process with stationary th-order moments and spectra; is a stochastic process c) dependent on and , then with nonstationary th-order moments and spectra.

are

(62) is the increment process corresponding to . Having this formulation of at hand, all the desired quantities can be derived in closed form. Note that classical analog communication signals like AM, FM, phase modulation, and classical digital communication schemes like ASK, frequency shift keying, phase shift keying, pulse amplitude modulation (PAM), and multipulse-PAM are included in the model (61). Observation of signal(s) (stochastic, deterministic, or both) in additive noise is also included and both the signal(s), and the noise may be assumed to be nonstationary and non-Gaussian if so desired. The simple model (61) captures a surprisingly large class of interesting and important models for communication signals. 1) Nonstationary Random PAM Processes: Nonstationary random PAM processes are included in the model (61) by assigning the random waveforms

where

(63) is a possibly random pulse where is a Baud interval, and may be written as shape. Then, the process (64) is a sequence of random variables (ampliwhere and are uncorrelated. tudes), and we assume that to belong to the We now require the random input process harmonizable class, which immediately guarantees that is harmonizable. That is (65)

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where (66) and

(67) is the discrete time Fourier transform (DTFT) of the fi. If we consider the nite-length random sequence , we must obviously require that asymptotic case or resort to a harmonizable spectral representation . for From these considerations, we see that the role of the random is to (poly)spectrally shape the amplitude sequence higher order moment spectra of the increment process. In this particular case, we can show that the th-order f-f polyspectrum modulates the f-f polyspectrum of the of the sequence to produce the f-f polyspectrum of stochastic process

where is the nonstationary impulse response of the system. Here, denotes the input time, denotes the output is the input stochastic process. time of the system, and Note that since is an input time and is an output time, the behaves quite differently impulse response function with respect to the two arguments. For the linear system to be stable, we must assume that (71) Because of (71), we can express as transform

(72) Hence, the process

where the increment process is given by (74) To ensure that ther require that

(69) is not to be understood as a density since Note that its definition is based on a DTFT of a sequence (67) rather than an increment process. We find it highly interesting that the random PAM example gives rise to polyspectra that are mixed discrete/continuous (i.e., discrete Fourier transforms and continuous spectral representations). We believe that there are many interesting aspects of polyspectral descriptions of nonstationary digital communication signals buried in this representation. Although it turned out to be quite easy to derive the f-f polyspectrum of the nonstationary random PAM process, it has proven to be much more difficult to express its t-f polyspectrum in an informative way. VIII. LINEAR TIME-VARYING SYSTEMS Linear time-varying systems are important models for nonstationary and possibly randomly varying communication channels. The output of such a system may be written as (70)

has the representation (73)

(68) where we define the f-f polyspectrum of the sequence to be

by its partial Fourier

has finite th-order moment, we must fur-

(75) Inserting the increment processes (74) into (75) and collecting terms, we find that this is equivalent to requiring (76), shown at the bottom of the page, where (77) is the th-order moment function for the input process to the is a uniformly bounded function, then a system. If sufficient condition for (76) to hold is clearly that

(78) Thus, under the assumptions of a stable system and that the increment process has finite moments up to order , it is clear that linear time-varying systems fall within the class of processes harmonizable up to order , provided the extra condition of (78) holds. With the increment process at hand, it is now possible to evaluate the polyspectral densities for linear time-varying sysand are uncorrelated, the tems. If we assume that

(76)

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th-order f-f polyspectrum becomes (79) and (80), shown at the bottom of the page. The f-f polyspectral density is thus

and

(87) (81)

A. Stationary and White Input Assume now that the input process is stationary and white to order . Then

where we define (88) Since the integration is carried out over the input time, we see is a measure of the integrated inner product bethat tween the th-order system polyfrequency response at frequenand . Likewise, can be undercies stood as the corresponding integrated inner product between the )-order polyfrequency system response at time and its ( response at .

(82) IX. CONCLUSIONS where tionary input process simplifies to

is the th-order moment of the sta. Then, the f-f polyspectral density

(83) We can readily find the corresponding t-f polyspectral density by an inverse Fourier transform with respect to , which yields (84)

(85) It is interesting to see that the th-order f-f polyspectrum and the t-f polyspectrum for nonstationary linear systems can be expressed in terms of time integrated Hilbert space inner products

(86)

We have derived and presented a theory of polyspectra (higher order moment spectra) of arbitrary order for nonstationary stochastic process belonging to the harmonizable class. The four basic quantities of the theory were shown to be the th-order moment function, the th-order time-frequency polyspectrum, the th-order ambiguity function, and the th-order frequency-frequency polyspectrum. These basic quantites were interrelated through Fourier transforms. We demonstrated that the stationary limit is readily recoverable and that the Fourier transform pair formed by the th-order moment function and the th-order spectral correlation function are the only relevant functions for stationary processes. Importantly, the frequency and time marginals of the th-order timefrequency polyspectrum were shown to be the instantaneous th-order moment and the conventional stationary th-order polyspectrum, respectively. By realizing that three out of the four basic quantities could be understood as Hilbert space inner products, we were led to suggest two novel and powerful definitions of polycoherences. Our polycoherences generalize the familiar polycoherence of stationary theory to harmonizable nonstationary processes. A whole host of important objective statistical tests can be constructed based on the proposed polycoherences. The most obvious test is whether a nonvanishing th-order polyspectrum is caused by nonstationarities or by non-Gaussian random variables with nonvanishing higher order moments. For random amplitude signals, we demonstrated that the polycoherence could be used to construct a test for whether the underlying signal was deterministic or stochastic. We calculated the relevant nonstationary higher order quantities for some specific families of processes. Finally, we illustrated the theory for random linear time-varying systems, which

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are standard models for fading multipath communication channels. We did not discuss the issue of estimators for the various polyspectral representations of nonstationary processes in this paper. It is natural, however, to expect that kernel based estimators involving local smoothing will play a central role when constructing consistent estimators [17]. Alternatively, computer-intensive resampling techniques may be applied to build consistent polyspectral estimates for nonstationary processes. Various estimators for the polyspectral densities will be presented in forthcoming papers. Note that it is possible to generalize the results of this paper to the more general Karhunen and Cramér classes of processes [9]. The theory of higher order moments and their associated transformed quantities for the Karhunen and the Cramér classes will be presented in forthcoming publications.

ACKNOWLEDGMENT This work was carried out while A. Hanssen spent a sabbatical term at Colorado State University (CSU) during the academic year 2001/2002. He would like to thank L. Scharf for being such a great host and collaborator during his stay and extends his thanks to the Department of Electrical and Computer Engineering at CSU for facilitating the practicalities of his sabbatical. The authors wish to thank D. C. Farden for invaluable help with MetaPost and LATEX and for helpful discussions during this project. We also thank Y. Birkelund, Y. Larsen, and P. Schreier for helpful discussions.

[14] P. J. Huber, B. Kleiner, T. Gasser, and G. Dumermuth, “Statistical methods for investigating phase relations in stationary stochastic processes,” IEEE Trans. Audio Electroacoust., vol. AU-19, pp. 78–86, 1971. [15] T. S. Rao and M. M. Gabr, “A test for linearity of stationary time series,” J. Time Series Anal., vol. 1, no. 1, pp. 145–158, 1980. [16] D. J. Thomson, “Multitaper analysis of nonstationary and nonlinear time series data,” in Nonlinear and Nonstationary Signal Processing, W. Fitzgerald, R. Smith, A. Walden, and P. Young, Eds. Cambridge, U.K.: Cambridge Univ. Press, 2000, ch. 11, pp. 317–394. [17] L. L. Scharf and B. Friedlander, “Toeplitz and hankel kernels for estimating time-varying spectra of discrete-time random processes,” IEEE Trans. Signal Processing, vol. 49, pp. 179–189, Jan. 2001.

Alfred Hanssen (M’93) received the Dr.Scient. (Ph.D.) degree in physics (theoretical plasma physics) from the University of Tromsø, Tromsø, Norway, in 1992. He was a post-doctoral researcher with the Institute of Theoretical Astrophyics, University of Oslo, Oslo, Norway, from 1993 to 1994. From 1994 to 1999, he was an Associate Professor in applied physics with the University of Tromsø, and in 1999, he was appointed Professor of physics at the same university. He has spent several long-term visits abroad: MaxPlanck-Institute für Aeronomie, Katlenburg-Lindau, Germany, from 1988 to 1989; Los Alamos National Laboratory, Theoretical Division, Los Alamos, NM, 1991; European Commission—Joint Research Center, Space Applications Institute, Ispra (VA), Italy, 1996; and the Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, 2001 to 2002. His research interests are in the areas of statistical signal processing, nonstationary stochastic processes and fields, array processing, sonar, nonlinear dynamics, wave dynamics, and higher order statistics. He holds an adjunct research position at the Norwegian Defence Research Establishment, where he is involved in research pertaining to synthetic aperture sonar and array processing. Prof. Hanssen was Technical Program Chair for NORSIG-2002 on board the Hurtigruten ship between Tromsø and Trondheim, Norway, and he organized a special session on nonstationary stochastic processes at Asilomar 2002. He serves on the IEEE SP Technical Committee for Sensor Arays and Multichannel Signal Processing.

REFERENCES [1] M. B. Priestley, Non-Linear and Non-Stationary Time Series Analysis. London, U.K.: Academic, 1988. [2] D. R. Brillinger, “An introduction to polyspectra,” Ann. Math. Stat., vol. 36, pp. 1351–1374, 1965. [3] D. R. Brillinger and M. Rosenblatt, “Computation and interpretation of k th-order spectra,” in Spectral Analysis of Time Series: Proceedings, B. Harrison, Ed., New York, 1967, pp. 189–232. [4] E. J. Powers and S. Im, “Introduction to higher-order statistical signal processing and its applications,” in Higher Order Statistical Signal Processing, B. Boashash, E. J. Powers, and A. M. Zoubir, Eds. New York: Wiley, 1995, ch. 1. [5] H. Cramér, “On the theory of stationary random processes,” Ann. Math., vol. 41, pp. 215–230, 1940. [6] A. M. Yaglom, Correlation Theory of Stationary and Related Random Functions. New York: Springer-Verlag, 1987. [7] M. Loève, Probability Theory, 3rd ed. Princeton, NJ: Van Nostrand, 1963. [8] S. Cambanis and B. Liu, “On harmonizable stochastic processes,” Inform. Contr., vol. 17, pp. 183–202, 1970. [9] M. M. Rao, “Harmonizable, Cramér and Karhunen classes of processes,” in Handbook of Statistics, E. J. Hannan, P. R. Krishnaiah, and M. M. Rao, Eds., 1985, vol. 5, ch. 10, pp. 279–310. [10] H. Soedjak, “Consistent estimation of the bispectral density function of a harmonizable process,” J. Statist. Planning Inference, vol. 100, pp. 159–170, 2002. [11] M. M. Rao, Stochastic Processes: Inference Theory. Dordrecht, The Netherlands: Kluwer, 2002, pp. 557–593. [12] M. Rosenblatt, Stationary Sequences and Random Fields. Boston, MA: Birkhäuser, 1985. [13] A. Hanssen and L. L. Scharf, “Theory of higher-order Rihaczek spectra,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., vol. 2, Orlando, FL, May 2002, pp. 1457–1460.

Louis L. Scharf (F’86) received the Ph.D. degree in electrical engineering in 1969 from the University of Washington, Seattle. He is currently Professor of electrical engineering and statistics at Colorado State University (CSU), Fort Collins. From 1969 to 1971, he was with Honeywell’s Marine Systems Center, Seattle. From 1971 to 1982, he was with CSU. He was Professor and Chair of Electrical Engineering at the University of Rhode Island, Kingston, from 1982 to 1985. From 1985 to 2000, he was with the University of Colorado, Boulder. He has held visiting positions at Duke University, Durham, NC; Ecole Superieure d’ Electricité, Gif-sur-Yvette, France; Ecole Superieure des Télécommunications, Paris, France; EURECOM, Sofia-Antipolis, France; University of LaPlata, LaPlata, Argentina; the University of Wisconsin, Madison; and the University of Tromsø, Tromsø, Norway. His interests are in statistical signal processing as it applies to adaptive radar, sonar, and wireless communication. His most important contributions to date are to invariance theories for detection and estimation; matched and adaptive subspace detectors and estimators for radar, sonar, and data communication; and canonical decompositions for reduced dimensional filtering and quantizing. His current interests are in rapidly adaptive receiver design for space-time and frequency-time signal processing in the wireless communication channel. Prof. Scharf was Technical Program Chair for the 1980 ICASSP Denver, CO, Tutorials Chair for ICASSP 2001, Salt Lake City, UT, and Technical Program Chair for Asilomar 2002. He is past chair of the Fellow Committee of the IEEE Signal Processing Society and serves on its Technical Committees for Theory and Methods and for Sensor Arrays and Multichannel Signal Processing. He has received numerous awards for his research contributions to statistical signal processing, including an IEEE Distinguished Lectureship, an IEEE Third Millennium Medal, and the Technical Achievement Award from the IEEE Signal Processing Society.

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A Theory of Polyspectra for Nonstationary Stochastic Processes Alfred Hanssen, Member, IEEE, and Louis L. Scharf, Fellow, IEEE

Abstract—Harmonizable processes constitute an important class of nonstationary stochastic processes. In this paper, we present a theory of polyspectra (higher order moment spectra) for the harmonizable class. We define and discuss four basic th-order moment function, the th-order quantities: the time-frequency polyspectrum, the th-order ambiguity function, and the th-order frequency-frequency polyspectrum. The latter generalizes the conventional polyspectrum to nonstationary stochastic processes. These four functions are related to one another by Fourier transforms. We show that the frequency and time marginals of the time-frequency polyspectrum are the instantaneous th-order moment and the conventional th-order stationary polyspectrum, respectively. All quantities except the th-order ambiguity function allow for insightful interpretations in terms of Hilbert space inner products. The inner product picture leads to two novel and very powerful definitions of polycoherence for a nonstationary stochastic process. The polycoherences are objective measures of stationarity to order , which can be used to construct various statistical tests. Finally, we give some specific examples and apply the theory to linear time-varying systems, which are popular models for fading multipath communication channels. Index Terms—Coherence, harmonizability, nonstationarities, polyspectra, spectral representation, stochastic processes.

Assume now that the stochastic process representation [5]

has the spectral

(1) is the complex-valued increment process (or the where . If time is generalized Fourier transform) of the process , and if time continuous, the integration limits in (1) are , where is the equidistant is discrete, the limits are , the increment sampling interval. Since we assume , process has a useful Hermitian symmetry where the asterisk denotes complex conjugation. has orthogonal increments [6], i.e., If the process (2) is some function of and is Dirac’s delta where function, then it is easy to show that we obtain the well-known is a function of time difstationary result that ference only: (3)

I. INTRODUCTION

L

ET , be a real-valued stochastic process, where denotes some index set for a time-like variable . For , or time example, time could be continuous, for which . could be discrete, for which is Gaussian, it is well known that knowlIf the process edge of the first and second-order moment functions and is sufficient to characterize the process completely [1]. Non-Gaussian processes are, however, abundant in nature and man-made systems alike. To describe the deviation from Gaussianity, one must often consider moment functions of orders higher than two [1]–[4].

Manuscript received June 21, 2002; revised November 4, 2002. The work of A. Hanssen was supported by the Research Council of Norway. L. L. Scharf was supported by the National Science Foundation under Contracts ECS 9979400 and ITR 0112573 and by the Office of Naval Research under Contract N00014-00-1-0033. The associate editor coordinating the review of this paper and approving it for publication was Dr. Xiang-Gen Xia. A. Hanssen is with the Department of Physics, University of Tromsø, Tromsø, Norway (e-mail: [email protected]). L. L. Scharf is with the Departments of Electrical and Computer Engineering and Statistics, Colorado State University, Ft. Collins, CO 80523-1373 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2003.810298

as the power spectral density Then, we identify . Equation (3) is, of course, nothing but the Einof stein–Wiener–Khinchin relation for stationary processes, relating the correlation function to the power spectral density by a Fourier transform. The class of harmonizable nonstationary processes is now defined as the processes with nonorthogonal increments [7]–[9], i.e., (4) is some complex-valued function of and where . The dual-frequency function is often called the Loève spectrum of the process. In this case, the relation between the nonstationary correlation function and the Loève spectrum takes the form [7]

(5) From (5), we understand that in this representation, spectral correlation among different frequency components is responsible for the nonstationarities. Knowing the spectral correlation in detail is therefore useful for characterizing the nature of the non. stationarity of The main purpose of this paper is to introduce a theory of polyspectra for harmonizable stochastic processes. In other

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words, we will define and interpret the higher order moments to arbitrary of the nonorthogonal increment process order. In doing so, we derive a powerful and natural theory of polyspectra and polycoherences for nonstationary harmonizable stochastic processes. We believe that the resulting representation of polyspectra is important from a fundamental modeling point of view. There are numerous applications in acoustics and electromagnetics that would benefit from applying the proposed representation to the analysis of experimental data. As far as we know, this is the first publication dealing with polyspectral generalizations for nonstationary harmonizable processes. Note that some recently published titles [10], [11] apparently suggest that a theory of bispectra for harmonizable processes exists. However, the bispectral density considered in these contributions is actually a bimeasure, which is a second-order moment spectrum of harmonizable processes and not a third-order quantity, as a bispectrum actually is [3], [12]. The structure of this paper is as follows. In Section II, we present and discuss the four basic ways an th-order nonstationary moment can be represented in time-time, time-frequency, frequency-time, and frequency-frequency, respectively. In Section III, we discuss the stationary limit and show that it reproduces the conventional polyspectrum, and in Section IV, we calculate the time and frequency marginals of the basic time-frequency polyspectrum. In Section V, we discuss an important geometrical interpretation in terms of Hilbert space inner products of three of the representations. In Section VI, we show that the inner product formulations of the th-order frequency-frequency and time-frequency polyspectra lead to two very powerful and useful definitions of coherence for nonstationary harmonizable processes. We discuss some statistical tests that may be constructed on the basis of the proposed coherences. In Section VII, we provide some examples, and we show that the coherences may also be applied to construct a test of determinism versus randomness for amplitude-modulated signals. In Section VIII, we illustrate how the theory works for linear time-varying systems, which model multipath communication channels, and in Section IX, we list our main conclusions, along with some thoughts about the continuation of this work.

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II. BASIC REPRESENTATIONS In the remainder of this paper, we will assume that the specare tral representation (1) holds but that the increments nonorthogonal. Note that we will only discuss moment spectra and not cumulant spectra in this paper. It is easy, however, to construct the corresponding cumulant spectra from the moment spectra by means of a formula due to Leonov and Shiryaev (see, e.g., [12, p. 33]). As our basic nonstationary th-order quantity in the time doby main, we define the th-order moment function of (6) Here, is a global (or absolute) time variable, and are local (or relative) time variables, i.e., the ’s are time shifts relative to the global time , and is a vector of local times (superscript denotes a transpose). Expressing the moment function (6) by means of the nonorthogonal spectral representation, we obtain (7)–(10), , shown at the bottom of the page, where , and . We thus understand as a frequency offset or a local frequency . relative to the global frequencies We immediately identify the th-order frequency-frequency polyspectral density (or f-f polyspectrum)

(11) We notice that the th-order moment function and the th-order f-f polyspectrum constitute an -dimensional Fourier transform pair (12) . We may call (10) the spectral representation of In this paper, we strictly follow the convention that global variables are denoted by Latin letters (e.g., and ) and that local variables are denoted by Greek letters (e.g., and ). Note local time variables of must always be treated that the

(7)

(8)

(9) (10)

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collectively and likewise for the global frequency variables in . We now understand that we may invoke Fourier transforms with respect to any of the variables , , , and in order to exhaust all possible time and frequency representations to the th order. A very important quantity is now derived by means of an , with respect inverse Fourier transform of to the local frequency . We obtain (13)

Fig. 1. Fourier relations between the basic polyspectral densities of nonstationary harmonizable processes.

The quantity is the th-order ambiguity function, which is a function of local frequency and local time vector . and constitute a Fourier We observe that transform pair in the vector variables and (21)

(14)

(15)

(16)

is the th-order time-frequency polyspectral where . In a recent density (or t-f polyspectrum) for the process conference paper [13], we used the term higher order Rihaczek spectrum for the t-f polyspectrum. We think that the term suggested in the present paper is far better and more descriptive is a function of of the quantity of interest. Note that global time and global frequencies . It is important to note that the t-f and the f-f polyspectra are a Fourier transform pair in the variables and ,

We now understand that any of the four basic quantities , , , and may be used to characterize the higher order behavior of a nonstationary stochastic process. Depending on the focus and actual application, it may be advisable to exploit the global time—local , the global time correlation function given by , the time—global frequency t-f polyspectral density , local frequency—local time ambiguity function and/or the local frequency—global frequency f-f polyspectral . density It is very important to note that these four basic densities are interrelated by Fourier transforms, as illustrated by the diagram in Fig. 1, and may be interpreted as the expectations of Fig. 2. III. STATIONARY LIMIT To obtain the stationary limit of the polyspectra, we first recall that the th-order moment function has the spectral representation

(17) The fourth and last quantity we may construct results )-dimensional inverse Fourier transform of as an ( , with respect to the global frequency vector . This yields

(22) For (22) to be independent of on the manifold by requiring

, we must require that is nonvanishing only . This is accomplished

(18)

(23) (19)

is a function of only. Note that (23) is the where th-order generalization of the familiar second-order orthogonality condition (2). Inserting (23) into (22) and carrying out the integration over , we obtain

(20)

(24)

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Fig. 2.

Fourier relations between the basic expectations of nonstationary harmonizable processes.

where

(32) (25) (33)

as the conventional th-order moWe thus recognize as the corment function for stationary processes and responding conventional th-order moment spectrum for staand tionary processes. In the stationary limit, are the only two th-order functions of interest. Note that these two functions constitute a Fourier transform pair (26) Equation (26) is the th-order Einstein–Wiener–Khinchin relation. From (23), we understand that the stationary th-order denand the th-order nonstationary density sity are related by

Thus, the frequency marginal of

is (34)

It is interesting and reassuring to note that the time marginal (28) is in fact the instantaneous th-order moment of the process and that the frequency marginal (29) is the conventional th-order polyspectrum that one would normally associate with a stationary process, as seen from (34). , we obtain As special cases, it is instructive that for (35) and (36)

(27) The significance of the Dirac delta function in (27) is that is picks out the stationary manifold on which the stationary th-order density is nonzero. IV. MARGINALS We are interested in the marginals of the global time—global . The time marginal is frequency t-f polyspectrum readily derived as

The second-order marginals are obviously the instantaneous power and the ordinary power spectral density, respectively. is a complex-valued quantity in Thus, even though the ( , )-plane, its marginals are real and non-negative. , we obtain For (37) and

(28)

(38)

The frequency marginal is

The third-order marginals are obviously the instantaneous skewness and the conventional stationary bispectral density, respectively. (29)

V. GEOMETRICAL INTERPRETATIONS We note that three of the four corners in Figs. 1 and 2 can be expressed conveniently as Hilbert space inner products

(30) (39)

(31)

(40)

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(41) Here, we use the convention that the Hilbert space inner product between two complex valued stochastic variables and is de. Note that the th-order ambiguity fined by function is not expressible as an inner product. Rather, it is a convolutions. sequence of Fig. 3.

VI. POLYCOHERENCES In order to obtain objective measures of polyspectra, it is customary to introduce coherence. In particular, the bicoherence ) and the tricoherence ( ) have been useful for the ( analysis and interpretation of data from various assumed stationary processes [4], [14]. Stringent statistical tests based on bi- and tricoherences have been suggested, e.g., [15]. As has been pointed out by Thomson [16], the spectral correlations induced by nonstationarities may be mistaken for nonlinearities in a bicoherence analysis if conventional stationary bicoherence analysis is applied to nonstationary data. Thus, there is a need for definitions of polycoherences that retain their validity for nonstationary processes. We will now derive two different generalized coherences for polyspectra of arbitrary order, which are indeed valid for any harmonizable nonstationary stochastic process. A. Frequency-Frequency Polycoherence To derive a meaningful and useful measure of coherence, we now turn to the expression for the th-order f-f . Expressed in terms polyspectral density , we see of the nonorthogonal increment process is actually a measure of from (41) that the coherence or the Hilbert space inner product between the one-dimensional complex-valued stochastic process and the complex valued ( )-dimen. We may therefore normalize sional random field by the mean-square of its two conand stituents Then, the magnitude-squared frequency-frequency polycoherence of order is defined by (42), shown at the bottom of the page. By the use of Schwartz’ inequality, it is easy to show . Note that . that This new definition of f-f polycoherence generalizes prior definitions to nonstationary processes, for which (poly)spectral may be nonzero. correlations at

cos

j

Frequency-frequency polycoherence in terms of the angle (; f )j.

(; f ) =

From (42), we understand that is actually the cosine-squared of an angle between the two quantities comprising the magnitude-squared f-f polycoherence: (43) In Fig. 3, we illustrate this important geometrical relation beand . tween , we observe that takes the For orders form of the conventional stationary magnitude-squared polycowe obtain herence of order , e.g., for

(44) which is the conventional magnitude-squared bicoherence associated with stationary processes. B. Time-Frequency Polycoherence Employing the same inner product reasoning as in the preceding section, we may also define a time-frequency coherence . We based on the time-frequency polyspectrum can be understood as the Hilbert recall that , space inner product between the stochastic process itself, )-dimensional random field and the complex valued ( . Normalizing by the mean-squared values of the two elements of its inner product definition, we define the magnitude-squared time-frequency by polycoherence of

(45)

Again, Schwartz’ inequality ensures that

.

(42)

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VII. EXAMPLES A. Single Random Amplitude Signal Let the nonstationary process

be defined by (48)

where is a zero mean stochastic variable with finite th-order is assumed to be harmonizable; moment. The process hence, it has the representation (49) Fig. 4. Time-frequency polycoherence in terms of the angle j (t; f )j. cos

(t; f ) =

From (45), we understand that is the cosinesquared between the two quantities comprising the magnitudesquared t-f polycoherence (46) and Fig. 4 illustrates this geometrical relation between . An important interpretation of the t-f polycoherence is now as a meaevident from (45). We may understand )-dimensional random sure of how easy it is to estimate the ( at time from an instantaneous obfield , or vice versa. This seems to be a servation of the process fundamentally important interpretation of t-f polycoherence. Rewriting (45) slightly, we can cast it into the form of (47), shown at the bottom of the page. From (47), we obtain another important interpretation of the time-frequency polycoherence. First, note that we may identify as the (complex conjugated) evaluinstantaneous estimate of the increment process of . Then, we understand ated at the sum frequency is a measure of how closely the instantaneous that at time one-term frequency estimate of approximates the random field .

is the corresponding nonorthogonal increment where are uncorrelated. Under process. We assume that and is clearly harmonizable, and the correthese assumptions, sponding increment process is given by (50) We now readily find the th-order moment function to be as in (51) and (52), shown at the bottom of the page. Thus, we identify the th-order frequency-frequency polyspectrum as

(53) (54) The time-frequency polyspectrum is likewise found to be

(55) (56) We can now evaluate the magnitude-squared f-f polycoherence to obtain (57) and (58), shown at the bottom of the next

(47)

(51)

(52)

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page. These formulas show that an th-order moment description, including the magnitude-squared f-f polycoherence , is a scaled version of the corresponding th-order for . moment description for 1) Deterministic Limit: It is also possible to treat the case is a deterministic function using the above where theory. This is possible by invoking the correspondence

B. Sum of Random Amplitude Signals An important and straightforward generalization of the previous example is given by the multipulse process (61) are uncorrelated random variables, uncorrelated harmonizable processes, and ’s and ’s are mutually uncorrelated. We find the the to be increment process of

where (59) is the Fourier transform of . where For deterministic signals, the magnitude-squared f-f polycoherence (57) reduces to (60) i.e., it is a combination of statistical moments of the random amplitude . It is important to note that the f-f polycoherence becomes independent of both and for the case of a single deterministic function of random amplitude. In fact, there are very few tests available for assessing determinism versus randomness. The above considerations about the polycoherence suggest such a test. 2) Hierarchy of Tests Based on Frequency-Frequency Polycoherence: For the case where the observed process is given , we can summarize the following hierarchy by of tests based on the th-order magnitude-squared f-f polycois herence. If is deterministic; a) independent of and , then is a stob) independent of but dependent on , then chastic process with stationary th-order moments and spectra; is a stochastic process c) dependent on and , then with nonstationary th-order moments and spectra. 3) Hierarchy of Tests Based on Time-Frequency Polycoherence: The tests one can construct for the time-frequency polycoherence parallel those of the f-f polycoherence completely. If is is deterministic; a) independent of and , then is a stob) independent of but dependent on , then chastic process with stationary th-order moments and spectra; is a stochastic process c) dependent on and , then with nonstationary th-order moments and spectra.

are

(62) is the increment process corresponding to . Having this formulation of at hand, all the desired quantities can be derived in closed form. Note that classical analog communication signals like AM, FM, phase modulation, and classical digital communication schemes like ASK, frequency shift keying, phase shift keying, pulse amplitude modulation (PAM), and multipulse-PAM are included in the model (61). Observation of signal(s) (stochastic, deterministic, or both) in additive noise is also included and both the signal(s), and the noise may be assumed to be nonstationary and non-Gaussian if so desired. The simple model (61) captures a surprisingly large class of interesting and important models for communication signals. 1) Nonstationary Random PAM Processes: Nonstationary random PAM processes are included in the model (61) by assigning the random waveforms

where

(63) is a possibly random pulse where is a Baud interval, and may be written as shape. Then, the process (64) is a sequence of random variables (ampliwhere and are uncorrelated. tudes), and we assume that to belong to the We now require the random input process harmonizable class, which immediately guarantees that is harmonizable. That is (65)

(57)

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where (66) and

(67) is the discrete time Fourier transform (DTFT) of the fi. If we consider the nite-length random sequence , we must obviously require that asymptotic case or resort to a harmonizable spectral representation . for From these considerations, we see that the role of the random is to (poly)spectrally shape the amplitude sequence higher order moment spectra of the increment process. In this particular case, we can show that the th-order f-f polyspectrum modulates the f-f polyspectrum of the of the sequence to produce the f-f polyspectrum of stochastic process

where is the nonstationary impulse response of the system. Here, denotes the input time, denotes the output is the input stochastic process. time of the system, and Note that since is an input time and is an output time, the behaves quite differently impulse response function with respect to the two arguments. For the linear system to be stable, we must assume that (71) Because of (71), we can express as transform

(72) Hence, the process

where the increment process is given by (74) To ensure that ther require that

(69) is not to be understood as a density since Note that its definition is based on a DTFT of a sequence (67) rather than an increment process. We find it highly interesting that the random PAM example gives rise to polyspectra that are mixed discrete/continuous (i.e., discrete Fourier transforms and continuous spectral representations). We believe that there are many interesting aspects of polyspectral descriptions of nonstationary digital communication signals buried in this representation. Although it turned out to be quite easy to derive the f-f polyspectrum of the nonstationary random PAM process, it has proven to be much more difficult to express its t-f polyspectrum in an informative way. VIII. LINEAR TIME-VARYING SYSTEMS Linear time-varying systems are important models for nonstationary and possibly randomly varying communication channels. The output of such a system may be written as (70)

has the representation (73)

(68) where we define the f-f polyspectrum of the sequence to be

by its partial Fourier

has finite th-order moment, we must fur-

(75) Inserting the increment processes (74) into (75) and collecting terms, we find that this is equivalent to requiring (76), shown at the bottom of the page, where (77) is the th-order moment function for the input process to the is a uniformly bounded function, then a system. If sufficient condition for (76) to hold is clearly that

(78) Thus, under the assumptions of a stable system and that the increment process has finite moments up to order , it is clear that linear time-varying systems fall within the class of processes harmonizable up to order , provided the extra condition of (78) holds. With the increment process at hand, it is now possible to evaluate the polyspectral densities for linear time-varying sysand are uncorrelated, the tems. If we assume that

(76)

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th-order f-f polyspectrum becomes (79) and (80), shown at the bottom of the page. The f-f polyspectral density is thus

and

(87) (81)

A. Stationary and White Input Assume now that the input process is stationary and white to order . Then

where we define (88) Since the integration is carried out over the input time, we see is a measure of the integrated inner product bethat tween the th-order system polyfrequency response at frequenand . Likewise, can be undercies stood as the corresponding integrated inner product between the )-order polyfrequency system response at time and its ( response at .

(82) IX. CONCLUSIONS where tionary input process simplifies to

is the th-order moment of the sta. Then, the f-f polyspectral density

(83) We can readily find the corresponding t-f polyspectral density by an inverse Fourier transform with respect to , which yields (84)

(85) It is interesting to see that the th-order f-f polyspectrum and the t-f polyspectrum for nonstationary linear systems can be expressed in terms of time integrated Hilbert space inner products

(86)

We have derived and presented a theory of polyspectra (higher order moment spectra) of arbitrary order for nonstationary stochastic process belonging to the harmonizable class. The four basic quantities of the theory were shown to be the th-order moment function, the th-order time-frequency polyspectrum, the th-order ambiguity function, and the th-order frequency-frequency polyspectrum. These basic quantites were interrelated through Fourier transforms. We demonstrated that the stationary limit is readily recoverable and that the Fourier transform pair formed by the th-order moment function and the th-order spectral correlation function are the only relevant functions for stationary processes. Importantly, the frequency and time marginals of the th-order timefrequency polyspectrum were shown to be the instantaneous th-order moment and the conventional stationary th-order polyspectrum, respectively. By realizing that three out of the four basic quantities could be understood as Hilbert space inner products, we were led to suggest two novel and powerful definitions of polycoherences. Our polycoherences generalize the familiar polycoherence of stationary theory to harmonizable nonstationary processes. A whole host of important objective statistical tests can be constructed based on the proposed polycoherences. The most obvious test is whether a nonvanishing th-order polyspectrum is caused by nonstationarities or by non-Gaussian random variables with nonvanishing higher order moments. For random amplitude signals, we demonstrated that the polycoherence could be used to construct a test for whether the underlying signal was deterministic or stochastic. We calculated the relevant nonstationary higher order quantities for some specific families of processes. Finally, we illustrated the theory for random linear time-varying systems, which

(79)

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are standard models for fading multipath communication channels. We did not discuss the issue of estimators for the various polyspectral representations of nonstationary processes in this paper. It is natural, however, to expect that kernel based estimators involving local smoothing will play a central role when constructing consistent estimators [17]. Alternatively, computer-intensive resampling techniques may be applied to build consistent polyspectral estimates for nonstationary processes. Various estimators for the polyspectral densities will be presented in forthcoming papers. Note that it is possible to generalize the results of this paper to the more general Karhunen and Cramér classes of processes [9]. The theory of higher order moments and their associated transformed quantities for the Karhunen and the Cramér classes will be presented in forthcoming publications.

ACKNOWLEDGMENT This work was carried out while A. Hanssen spent a sabbatical term at Colorado State University (CSU) during the academic year 2001/2002. He would like to thank L. Scharf for being such a great host and collaborator during his stay and extends his thanks to the Department of Electrical and Computer Engineering at CSU for facilitating the practicalities of his sabbatical. The authors wish to thank D. C. Farden for invaluable help with MetaPost and LATEX and for helpful discussions during this project. We also thank Y. Birkelund, Y. Larsen, and P. Schreier for helpful discussions.

[14] P. J. Huber, B. Kleiner, T. Gasser, and G. Dumermuth, “Statistical methods for investigating phase relations in stationary stochastic processes,” IEEE Trans. Audio Electroacoust., vol. AU-19, pp. 78–86, 1971. [15] T. S. Rao and M. M. Gabr, “A test for linearity of stationary time series,” J. Time Series Anal., vol. 1, no. 1, pp. 145–158, 1980. [16] D. J. Thomson, “Multitaper analysis of nonstationary and nonlinear time series data,” in Nonlinear and Nonstationary Signal Processing, W. Fitzgerald, R. Smith, A. Walden, and P. Young, Eds. Cambridge, U.K.: Cambridge Univ. Press, 2000, ch. 11, pp. 317–394. [17] L. L. Scharf and B. Friedlander, “Toeplitz and hankel kernels for estimating time-varying spectra of discrete-time random processes,” IEEE Trans. Signal Processing, vol. 49, pp. 179–189, Jan. 2001.

Alfred Hanssen (M’93) received the Dr.Scient. (Ph.D.) degree in physics (theoretical plasma physics) from the University of Tromsø, Tromsø, Norway, in 1992. He was a post-doctoral researcher with the Institute of Theoretical Astrophyics, University of Oslo, Oslo, Norway, from 1993 to 1994. From 1994 to 1999, he was an Associate Professor in applied physics with the University of Tromsø, and in 1999, he was appointed Professor of physics at the same university. He has spent several long-term visits abroad: MaxPlanck-Institute für Aeronomie, Katlenburg-Lindau, Germany, from 1988 to 1989; Los Alamos National Laboratory, Theoretical Division, Los Alamos, NM, 1991; European Commission—Joint Research Center, Space Applications Institute, Ispra (VA), Italy, 1996; and the Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, 2001 to 2002. His research interests are in the areas of statistical signal processing, nonstationary stochastic processes and fields, array processing, sonar, nonlinear dynamics, wave dynamics, and higher order statistics. He holds an adjunct research position at the Norwegian Defence Research Establishment, where he is involved in research pertaining to synthetic aperture sonar and array processing. Prof. Hanssen was Technical Program Chair for NORSIG-2002 on board the Hurtigruten ship between Tromsø and Trondheim, Norway, and he organized a special session on nonstationary stochastic processes at Asilomar 2002. He serves on the IEEE SP Technical Committee for Sensor Arays and Multichannel Signal Processing.

REFERENCES [1] M. B. Priestley, Non-Linear and Non-Stationary Time Series Analysis. London, U.K.: Academic, 1988. [2] D. R. Brillinger, “An introduction to polyspectra,” Ann. Math. Stat., vol. 36, pp. 1351–1374, 1965. [3] D. R. Brillinger and M. Rosenblatt, “Computation and interpretation of k th-order spectra,” in Spectral Analysis of Time Series: Proceedings, B. Harrison, Ed., New York, 1967, pp. 189–232. [4] E. J. Powers and S. Im, “Introduction to higher-order statistical signal processing and its applications,” in Higher Order Statistical Signal Processing, B. Boashash, E. J. Powers, and A. M. Zoubir, Eds. New York: Wiley, 1995, ch. 1. [5] H. Cramér, “On the theory of stationary random processes,” Ann. Math., vol. 41, pp. 215–230, 1940. [6] A. M. Yaglom, Correlation Theory of Stationary and Related Random Functions. New York: Springer-Verlag, 1987. [7] M. Loève, Probability Theory, 3rd ed. Princeton, NJ: Van Nostrand, 1963. [8] S. Cambanis and B. Liu, “On harmonizable stochastic processes,” Inform. Contr., vol. 17, pp. 183–202, 1970. [9] M. M. Rao, “Harmonizable, Cramér and Karhunen classes of processes,” in Handbook of Statistics, E. J. Hannan, P. R. Krishnaiah, and M. M. Rao, Eds., 1985, vol. 5, ch. 10, pp. 279–310. [10] H. Soedjak, “Consistent estimation of the bispectral density function of a harmonizable process,” J. Statist. Planning Inference, vol. 100, pp. 159–170, 2002. [11] M. M. Rao, Stochastic Processes: Inference Theory. Dordrecht, The Netherlands: Kluwer, 2002, pp. 557–593. [12] M. Rosenblatt, Stationary Sequences and Random Fields. Boston, MA: Birkhäuser, 1985. [13] A. Hanssen and L. L. Scharf, “Theory of higher-order Rihaczek spectra,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., vol. 2, Orlando, FL, May 2002, pp. 1457–1460.

Louis L. Scharf (F’86) received the Ph.D. degree in electrical engineering in 1969 from the University of Washington, Seattle. He is currently Professor of electrical engineering and statistics at Colorado State University (CSU), Fort Collins. From 1969 to 1971, he was with Honeywell’s Marine Systems Center, Seattle. From 1971 to 1982, he was with CSU. He was Professor and Chair of Electrical Engineering at the University of Rhode Island, Kingston, from 1982 to 1985. From 1985 to 2000, he was with the University of Colorado, Boulder. He has held visiting positions at Duke University, Durham, NC; Ecole Superieure d’ Electricité, Gif-sur-Yvette, France; Ecole Superieure des Télécommunications, Paris, France; EURECOM, Sofia-Antipolis, France; University of LaPlata, LaPlata, Argentina; the University of Wisconsin, Madison; and the University of Tromsø, Tromsø, Norway. His interests are in statistical signal processing as it applies to adaptive radar, sonar, and wireless communication. His most important contributions to date are to invariance theories for detection and estimation; matched and adaptive subspace detectors and estimators for radar, sonar, and data communication; and canonical decompositions for reduced dimensional filtering and quantizing. His current interests are in rapidly adaptive receiver design for space-time and frequency-time signal processing in the wireless communication channel. Prof. Scharf was Technical Program Chair for the 1980 ICASSP Denver, CO, Tutorials Chair for ICASSP 2001, Salt Lake City, UT, and Technical Program Chair for Asilomar 2002. He is past chair of the Fellow Committee of the IEEE Signal Processing Society and serves on its Technical Committees for Theory and Methods and for Sensor Arrays and Multichannel Signal Processing. He has received numerous awards for his research contributions to statistical signal processing, including an IEEE Distinguished Lectureship, an IEEE Third Millennium Medal, and the Technical Achievement Award from the IEEE Signal Processing Society.