Generalized Instantaneous Parameters And ... - Semantic Scholar

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 5, MAY 1997

Generalized Instantaneous Parameters and Window Matching in the Time-Frequency Plane Graeme Jones, Member, IEEE, and Boualem Boashash, Senior Member, IEEE Abstract—In this paper, the concept of instantaneous parameters, which has previously been associated exclusively with 1-D measures like the instantaneous frequency and the group delay, are extended to the 2-D time-frequency plane. Such generalized instantaneous parameters are associated with the shorttime Fourier transform. They may also be interpreted as local moments of certain time-frequency distributions. It is shown that these measures enable local signal behavior to be characterized in the time-frequency plane for nonstationary deterministic signals. The usefulness of the generalized instantaneous parameters is demonstrated in their application to optimal selection of windows for spectrograms. This is achieved through window matching in the time-frequency plane. An algorithm is provided that illustrates the performance of this window matching. Results based on simulated and real data are presented. Index Terms— Instantaneous parameters, short-time Fourier transform, time-frequency analysis, window matching.

I. INTRODUCTION

G

LOBAL signal power representations in time or frequency (the instantaneous power and energy density spectrum) provide the analyst with only a limited amount of signal information and cannot adequately characterize the behavior of nonstationary deterministic signals. For example, the energy density spectrum of a linear FM signal simply reveals a broadband spectral character and provides no information as to the direction (increasing or decreasing) of the modulation. This information is contained in the signal’s phase (in time or in frequency). By utilizing both the amplitude and phase of a signal in the time or frequency domains, all is revealed about the signal of interest. The signal information is, however, presented in an inconvenient and confusing way since phase is a difficult quantity to interpret. Since a nonstationary signal may be considered as a signal with a spectrum which varies with respect to time, this has lead to an alternate signal representation—a time-frequency distribution (TFD). A TFD attempts to represent the amplitude and phase of a signal together in the 2-D time-frequency plane—it displays the evolution of a time signal with respect to frequency (the spectral phase

function) and, conversely, the evolution of the frequency signal with respect to time (the time phase function). In other words, signal or spectral phase information is combined with amplitude information in the time-frequency plane to create a representation displaying all signal information. Unfortunately, the use of a TFD to represent the nonstationary behavior of deterministic signals leads to several interpretive problems. For cases where the signal phase characteristic is not monotonic (in time and frequency), the TFD is complicated and confusing, indicating the presence of multiple signals. Such nonlinear behavior may also be described in terms of the so-called cross-terms, which manifest themselves in TFD’s through interaction of signals in some region of the time-frequency plane with those in others. Although there are many types of TFD’s, we shall here consider those of Cohen’s class—they fall into (or between) two classes: (energy) density distributions and energy distributions. TFD’s that are of the density type are expressible in the form [21]

(1) , where is the Wigner–Ville distribufor tion (WVD, a member of the class under the limit ) and is expressed as (2) The other limiting member of this class occurs when and is known as the Rihaczek distribution (RD) [32]. All other distributions of this class have thus been generated by allowing the parameter to vary. It is not necessary to consider any other values for due to the cyclic nature of in (1). Density distributions are so named because they satisfy the following properties:

Manuscript received February 10, 1993; revised October 14, 1996. This work was supported by the Australian Research Council and the Defence Science and Technology Organization. This work was completed while both authors were with Signal Processing Research Centre, Queensland University of Technology. The associate editor coordinating the review of this paper and approving it for publication was Dr. Jelena Kovacević. G. Jones is with Raytheon Canada Limited, Waterloo, Ont., Canada N2J 4K6. B. Boashash is with the Signal Processing Research Centre, Queensland University of Technology, Brisbane Q 4001, Australia. Publisher Item Identifier S 1053-587X(97)03332-1. 1053–587X/97$10.00  1997 IEEE

(3) (4) (5)

(6)

JONES AND BOASHASH: GENERALIZED INSTANTANEOUS PARAMETERS AND WINDOW MATCHING

where is the TFD of , which is any valid 1-D signal. These results demonstrate, respectively, that appropriate integration of the functions will yield the total energy, the instantaneous power, energy density spectrum, or local energy in some region [specified by the time-frequency smoothing function in (6)]. Viewed in such a way, it is seen that these distributions satisfy necessary criteria to be called densities. Additionally, they may even be real (the WVD) and are invertible uniquely to within a phase constant [12]. The result of (6) actually specifies the other type of TFD available—the (local) energy distribution. It is often the case in time-frequency analysis that one signal is designated a window and applied to determine the shape and form of the local region of energy distribution of the other signal. The energy distribution is in actual fact the well known spectrogram, which will be defined here as

(7)

The spectrogram has been used for many years in speech, underwater acoustic, and time-varying spectrum estimation problems. Many TFD’s bear a dichotomous relation to these two limiting TFD types and actually fall in between the two. The Born–Jordan–Cohen distribution (BJC) [6], [12], for example, satisfies (3)–(5) but not (6). Similar comments may be made as regards other common distributions—the reduced interference distribution (RID) [9] and the Zhao–Atlas–Marks distribution (ZAM) [35] [which only satisfies (4) and (5)].

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II. INSTANTANEOUS PARAMETERS AND THEIR APPLICATION TO NON-STATIONARY SIGNAL ANALYSIS There are many parameters that are inevitably used to describe the global behavior/nature of a signal. For example, practical signals are often described in terms of their bandwidth and duration. Such common measures use the algebraic moments (and cumulants) of the time and frequency power signal representations (the instantaneous power, energy density spectrum). The bandwidth, for example, is the (algebraic) variance (second-order cumulant) of the energy density spectrum. Such global signal parameters are most convenient for describing a stationary signal in time and frequency but provide inadequate characterizations of nonstationary signals since they cannot reflect the local behavior. To describe this nonstationary or instantaneous behavior, a set of parameters may be defined that are perceived to be the instantaneous equivalents of such global moment and cumulant measures—the instantaneous parameters (IP’s). The IP’s may be derived through the general forms of the (Fourier) power theorem, which are written as [31]

(8)

(9) By re-expressing the above equations in the form

A. Density Distributions versus Energy Distributions For representational purposes, an energy distribution has many advantages over a density distribution. This is especially true when it is realized that density TFD’s are complex (the WVD may be thought of as a complex function with the phase restricted to 0 or ). A positive energy distribution would show regions where there was significant local signal presence and would be useful for time-frequency filtering, pattern recognition, and template generation. Since an energy distribution is a magnitude squared quantity with an arbitrary analysis window, unique distribution inversion is not possible, and a density function should always be retained if further postprocessing is required. Nevertheless, as a representational and analysis tool, energy distributions are extremely useful, and their positivity satisfies our intuitive expectations of the form of such a function [14]. Since energy distributions (that is, spectrograms) are dependent on the window function, it will be fruitful to look at ways to generate improved spectrograms that are optimal in some sense. With this and other goals in mind, the paper will describe the general theory of instantaneous parameters in time-frequency and the notion of window matching for TFD’s. An example application generates an adaptive energy distribution with certain optimal properties. The first relevant concept to be investigated is that of instantaneous parameters.

(10) and

(11) the technique of “extracting” instantaneous parameters from global moment calculations may be illustrated. The instantaneous contributions to the moment calculations, that is, the IP’s, are simply those quantities contained in the square braces on the right-hand sides of (10) and (11), which correspond to the time and frequency moment operators on the left hand sides. This form ensures that the instantaneous frequency moments are a function of time and vice versa and that equality holds under the integrals. The best (and most successful) example of the application of these IP’s are the instantaneous

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frequency (the instantaneous first frequency moment with respect to time ) and the group delay (the instantaneous first time moment with respect to frequency, [4], [5], [7], [29]. These IP’s are not the only ones derivable. By rotating a signal, it is possible to analyze it at different orientations—between the time and frequency domains. A timefrequency rotation operator is expressible in the form [30]

(12) A. Instantaneous Parameters and Time-Frequency Distributions One of the significant features of TFD’s is that IP’s may be derived from them by determination of conditional moments. This is only universally true of TFD’s that are density functions [see (1)]. In other words, instantaneous frequency moments are expressible as [6], [12]

(13)

with the instantaneous time equivalents of the form

(14)

The point of interest with these results is that, dependent on the TFD employed, the IP’s calculated will be different (and, thus, not in general equal to the IP’s derived classically from the generalized Power theorem). Since each of these TFD’s contain the same information and are invertible (as already mentioned to within a phase constant), the inherent nonuniqueness of IP’s is highlighted. Each IP value generated through the use of density TFD’s is correct under the global moment calculation, that is (15) and (16) In other words, there are many IP’s that satisfy the global integral equalities of (10) and (11). Each of these IP’s essentially yield the same information since these kernels of the density distributions are simply related by a phase change in the ambiguity plane [32].

B. Use of Instantaneous Parameters to Characterize Signal Nonstationary Behavior The question remains as to how these IP’s may be used to describe a nonstationary signal. By utilizing IP’s from many time-frequency directions, the complete nonstationary behavior of a signal may be documented. IP’s for a given timefrequency orientation characterize the instantaneous behavior along that 1-D “slice” in the plane (for example, the IF characterizes instantaneous behavior in time). The IP’s are not always useful for characterising the nonstationary behavior of a signal since they are only instantaneous with respect to time, frequency, or some other line through the time-frequency plane. It is envisaged that a large number of them would be needed to enable satisfactory analysis of an arbitrary signal. What is indeed required are IP’s that are instantaneous in the plane (that is, with respect to specific timefrequency locations), rather than instantaneous with respect to a line. Such a class [the generalized IP’s (GIP’s)] are introduced and discussed in the next section. III. GENERALIZED INSTANTANEOUS PARAMETERS IN TIME-FREQUENCY In the time-frequency plane, IP’s are required to represent the signal nonstationarities, which cannot be conveniently achieved (as has been discussed), with 1-D measures. The need for parameters that are instantaneous in the plane requires that the IP’s are functions of both time and frequency—that is, 2D. The generation of 2-D GIP’s proceeds in exactly the same way as for the 1-D IP’s, with the notable difference that now, the general Fourier power theorem must be realized in two dimensions, i.e.,

(17) where (18) It should be noted that in the form given above, variables on both sides of the equation may be interpreted as time and frequency quantities. Since both sides are related by integration, arbitrary variables may be assigned since they have no bearing on the equality relation. Such a form allows the derived GIP’s to be functions of time and frequency, and it will be shown that this is a valid form. If we are to derive appropriate definitions of GIP’s through extension of the 1-D instantaneous quantities’ definition, TFD’s would be used in (17). The chosen types of TFD’s would need to be expressible as a product of complex conjugates (as ). A local energy distribution (i.e., a spectrogram) is expressible in an appropriate form. We will


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This result may simply be generalized for any TFD with a phase product kernel [see (1)] to yield

here redefine the short-time Fourier transform as (19) A 2-D relationship may then be expressed as

Utilizing the above relation, the GIP’s may then be expressed in terms of density TFD’s as

(20) (23) The quantity on the left side of (20) STFT with the time-reversed window cross-WVD) and is expressible as

is a scaled (similar to a

(21)

to be derived Equation (20) allows 2-D GIP’s in the same manner as the 1-D IP’s of Section II. It shall be shown that the relations above are not arbitrary but form appropriate definitions of the GIP’s, suitably extending the class of 1-D IP’s. One important interpretation issue is the relation of these GIP’s to conditional moments of TFD’s, which is addressed in the next subsection. A. Generalized Instantaneous Parameters and Their Relation to Time-Frequency Distributions It has been seen in an earlier section how 1-D IP’s are derivable through the conditional moments of certain TFD’s [4], [6], [11], [12]. It was also noted that they may differ from those derived through the power theorem but satisfies the integral relations of (10) and (11). It will now be shown how GIP’s are also derivable through conditional TFD moments, exhibiting the same properties and behavior as their 1-D counterparts. Whereas for the 1-D IP’s the moments were conditional over the time or frequency axes [see (13) and (14)], for the GIP’s, measures are over local areas in the timefrequency plane—ensuring their representation of localized signal characteristics. To determine the relationship of the GIP’s to TFD’s, we will initially employ Nuttall’s results [26], which are of a form expressed here as

(22)

(24) This result is the 2-D analog to the 1-D IP’s derived in (13) and (14). Similar comments apply to their properties. For example, it may be deduced that substitution of the GIP’s of (24) into (20) is valid. If we employ the WVD, which is a real energy density distribution, all GIP’s will be real. From (23), it is noted that these GIP’s may be rendered an alternate interpretation as the local moments of a density TFD defined over a region specified by the TFD of the window. Such a result once again illustrates that the GIP’s are a natural extension of the 1-D results, in fact forming a broader class. For example, if a window of the form (25) was applied, the GIP’s generated are given by

(26)

which has simply reduced to the frequency IP’s, which are functions of time. This illustrates that the (1-D) IP’s are equivalent to the conditional moments of density TFD’s taken over the time or frequency lines (which are the limiting cases of the window being a delta function or a constant in time). This observation reinforces the notion that the GIP’s are a 2-D extension of the 1-D IP’s. The GIP measures that have thus been derived can essentially characterize the instantaneous signal behavior at any point in the time-frequency plane and will be shown to be able to be of more practical use than the 1-D IP’s. An application of these GIP’s will now be illustrated through the development of the general theory of instantaneous window matching.

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such functions will cause a minimal amount of spread. The obvious example of such a window is a Gaussian

IV. GENERATION OF AN ENERGY DISTRIBUTION THROUGH WINDOW MATCHING One of the major drawbacks of application of the spectrogram to time-frequency analysis is its nonuniqueness and arbitrary window. In addition, one choice for the analysis window cannot usually provide a high-resolution distribution across all regions of the time-frequency plane. This is where the concept of window matching may be employed. A matched filter [15] is the classic example of window matching. The best response (for detection in white noise) occurs when the impulse response of the filter is the time reversed incoming signal conjugate—that is, the filter is matched to the signal. We are faced with a slightly more complicated situation here, although the idea behind the method is fundamentally the same. A trivial extension of the matched filter solution is to simply use the signal as the window for matching; this will produce an energy distribution of the form [18]

(29) which has a WVD of the form (30) (The Gaussian function is completely specified by only two parameters— and —and, in the development of window matching that follows, it will be seen how these parameters vary to generate the local signal and window matching in the time frequency plane.) The Gaussian has the smallest possible variance in any time-frequency direction and satisfies, with equality, the uncertainty principles [16] (31) and [27] (32)

(27) which is a squared and scaled WVD. Such a technique is generally unsuitable for most signals due to the nonstationary nature of their WVD’s. For example, if a signal were composed of two linear FM components, application of a window representing their combined form would be inappropriate for matching to each individual component (over their respective time-frequency locations). By effectively performing global matching, as has been done in (27), the local behavior of the signal is not taken into account and thus not reflected in the choice of the window. The example just discussed illustrates the necessity to match the window to the signal at all locations in the time-frequency plane. If we thus employ a window function that matches the signal locally in the time-frequency plane, the general form of such an energy distribution would be

represents the normalized average operator, and where and are the time and frequency (magnitude squared) signal averages. The generalized uncertainty principle of (31) is well known to the radar community [13], [31]. It is these considerations that also serve to illustrate why the technique of matching a signal to itself could not generally yield a well-concentrated distribution. Unless the signal (and thus the window) is of an “optimum” form (for example, is Gaussian), such a method can be expected to introduce excess smoothing due to the fact that it will most likely have poor concentration properties (at least in some parts of the plane). If, however, the signal and window were Gaussian and identical, the best match is achieved, and the resulting energy distribution is optimally concentrated (in terms of secondorder measures). We shall now examine an interesting and novel approach to the concept of window matching before demonstrating how the GIP’s may be used to produce a window-matched spectrogram. A. Window Matching in Terms of Signal and Window Moments and Cumulants

(28) is the WVD of the location-dependent where window function, where is the equivalent timevarying filter. The window will depend on the signal, but the criterion through which such windows may be chosen is as yet unknown—in the next section, the concept of window matching in the time-frequency plane will be further developed.

In a similar approach as is used for probability density functions, any finite energy signal representation may be expressed through its associated characteristic function. (Although we are here referring to algebraic and not probabilistic measures, we shall retain the statistical nomenclature). For example, the characteristic function of the WVD of a signal (under normalization) would be defined as

(33) V. CONCEPTS

OF

WINDOW MATCHING

The desire to employ the most appropriate window to analyze the local energy at every point in the time-frequency plane requires the selection of windows that are optimum in some sense. It is thus important to choose windows that are tightly bound in time-frequency to ensure that smoothing by

The above characteristic function may be expressed in the form (34)


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(WVD-derived) GIP’s are calculated is

or (35) Since it is known that the global moments of a WVD are equivalent to those of the actual signal and its spectrum [6], [12], the parameters of (34) are the moments of the signal/spectrum, which are known to be defined as

(39) Using earlier results, the instantaneous characteristic function may be written as

(36)

(37)

(38)

The above relations may be equivalently expressed in terms of the cumulants of the signal and its spectrum. The cumulants [ see (35)] are related to the moments but are often preferred as descriptors due to their more convenient representation of characteristics. In essence, the th cumulant gives an indication of the th-order properties by “removing” from the th moment the distorting influences of the lower order moments. Variance measurements, for example, are cumulants of the second order. The general moment-cumulant relations may be expressed in forms given in [24]. It is through using such measures that window matching may be performed. If a Gaussian window is used, it is entirely determined in the time-frequency plane by its second-order cumulants and no others (indeed, all higher order cumulants are zero for a Gaussian [24]). Thus, when applying a Gaussian window, which has the minimum (second-order) spread in time-frequency, the matching could be based on GIP’s—in this case, the instantaneous second-order cumulant measures. Such a window matching approach may now be devised and theorized. The idea is that the window should match some local set of properties associated with the signal in the time-frequency plane. This approach will thus require that the instantaneous moments or cumulants of the signal-window spectrogram at a given time-frequency point be matched to those instantaneous parameters associated with the the window itself (that is, the self-windowed spectrogram). To formulate this, one needs an “instantaneous characteristic function” through which the instantaneous parameters may be expressed. It should be recalled from earlier sections that for both 1-D IP’s and the GIP’s, the actual measures vary depending on the TFD from which they are derived. We shall employ the WVD-derived measures, which are real and have special physical significance (as will be seen later). From (22), the instantaneous characteristic function through which the

By thinking about window matching in such ways, the problem will simply reduce to the requirement that the appropriate (real, or WVD-derived) GIP’s of the signal-window spectrogram be matched to those of the window-window spectrogram. This is equivalent to locally approximating the signal in time-frequency by the window. The window used will be Gaussian, which implies that the relevant GIP’s to match would be measures of the second order. The matching task, however, is not altogether straightforward. One possible algorithm will now be discussed, followed by simulations that illustrate the worth of the method. VI. AN ALGORITHM TO GENERATE AN ADAPTIVE ENERGY DISTRIBUTION (AED) It will now be examined as to how the GIP’s can be employed to generate a window-matched spectrogram, which will be referred to as the adaptive energy distribution (AED). The problem, as has been stated, requires that the relevant instantaneous moments or cumulants of the signal-window spectrogram be matched with those of the window-window spectrogram. For a Gaussian window, only matching of the instantaneous second-order cumulants is required since they completely characterize the Gaussian. Thus, the GIP’s required to be matched are the instantaneous time variance ITV , the instantaneous frequency variance IFV , and the instantaneous time-frequency covariance ITF Referring to the expression of (23) and (24), where the chosen distribution is the (real) WVD , the variance measures may be calculated from the instantaneous moments and expressed in the following succinct forms, i.e., ITV

(40)

IFV

(41)

ITF (42)

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where indicates the real part of the expression. The equations to be solved for the matching then become ITV IFV ITF

ITV IFV ITF

(43) (44) (45)

Satisfaction of these equations ensures that the final solution has locally approximated (in a second-order sense) the signal by the window. Since, in the Gaussian window situation, two parameters [ and ; see (29)] are required to be solved, the above set of three equations is overdetermined. They are, however, satisfied with equality if the signal is itself Gaussian, with the trivial result that the window parameters become identical to those of the Gaussian signal. In general, since both sides of the equations depend on the window parameters, determination of a general solution would necessarily employ an iterative method. The technique requires selection of a Gaussian window (through the parameters and ) such that the resulting instantaneous second-order cumulants approach those that would be generated if the signal was the same as the window (for the specific time-frequency point under analysis). For signals that are not simple Gaussians, interpretive problems may exist with application of the described method. It is possible that the calculated values for the instantaneous time and frequency variances may become zero or even negative. This would seem to render the measurements meaningless since it does not appear possible that a negative variance could ever occur. This conceptual stumbling block may be resolved, however. The theorem given below expresses a relationship between the ITV and IFV that will be shown to allow a sensible interpretation of these quantities: Theorem 1: The second-order (real) GIP’s, the ITV, and the IFV, for an arbitrary signal and any Gaussian window of satisfy the form IFV

ITF

(46)

The proof of this theorem is given in the Appendix. Lemma 1: The above theorem may be generalized such that (46) is true for any Gaussian window of the form by defining the measures IFV and ITV over a new coordinate basis

a matter of shifting and rotating the time-frequency axes such that this Gaussian window has rotational symmetry (which ensures the new time and frequency axes have the same scale). By measuring the spread of the Gaussian in time and in frequency, as well as the time-frequency covariance, a straightforward plane rotation and then axis scaling achieves the desired affect (that is, ). The special results given above are true even if one of the ITV or IFV is negative. It may also be observed that it is of the form of the addition uncertainty principle of (32)—it thus may be regarded as a type of local uncertainty principle, which is independent of the signal. This result allows effective use to be made of the GIP’s. It specifically illustrates that although one of the ITV or IFV may be negative, they are never both so. As a result, no problems will occur with application of the GIP’s. A negative value for one of the instantaneous variance measures means that the complementary variance is very large and positive. This implies a bad signal-window match and, as will be shown, prompts the appropriate change in the window. The iterative window matching algorithm may now be formulated. It was noted that the two-parameter problem required the solution of an overdetermined set of equations. This is circumvented by using only two of the second-order measures to derive the parameters for each iteration. Thus, the equation involving the ITF is used, and either of the IFV or ITV yield a positive (or the largest) value (they can never both be negative). As a result, after measuring the IP’s for the th iteration, the Gaussian parameters for the next iteration are directly calculated, assuming that the Gaussian window is matched to itself. This leaves the iterative equations, and the algorithm, in a very simple form. The algorithm may be succinctly stated: 1) Form the spectrogram [see (28)] at the desired point in the time-frequency plane (with no a priori information, initial conditions are , and 2) Determine the GIPS—ITV, IFV, and ITF [see (40)–(42)]. 3) At the th iteration, update the window parameters and through ITF

(50)

IFV

(51)

(47) or, if ITV

where (48) and (49) The theorem above is valid for a Gaussian window that possesses rotational symmetry in the time-frequency plane (that is, any TFD appears circular on the scale). To show that it may be generalized for any Gaussian, it is simply

IFV ITV

, then replace (51) by (52)

4) Repeat the above procedure to the desired level of matching. This iterative method continually reduces the error between the window and the best fit matched condition. If, for example, at one stage the ITV was very large (and the IFV small or even negative), this would indicate that the window has too great a time spread compared with the signal. Equation (52) would then reduce this spread (since ITV This


Fig. 1.

Wigner–ville distribution of simulated signal.

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Fig. 2. Spectrogram of simulated signal.

occurs since the Gaussian parameters derived assume that the window is matched to itself. As a result, the action of the signal on the window at the next iteration further adjusts the estimate and selects an improved window. Through the use of derivatives [see (40)–(42)], rather than moment calculations in the time-frequency plane, the full (discrete) algorithm is computations [20], [21], where is the signal length, and is typically The iterations are stopped when there is less than a 3-bin-width change in window (3 dB) spread—it has been found that four iterations are generally adequate. VII. AED APPLIED TO SYNTHETIC AND REAL DATA

Fig. 3. Adaptive energy distribution of simulated signal.

The performance and behavior of the AED will now be illustrated through its application to the analysis of simulated and real signals. Many TFD’s have recently been formulated [2], [3], [19], [33], [34] that employ adaptivity to improve resolution. The AED is also of such a type, although it is presented here mainly to demonstrate the applicability of the GIP’s. Further investigation of its performance and properties will not be undertaken here.

The AED is unlike the WVD, which displays large crossterms midway between the signal autoterms. In this case, the AED seems to provide the most intuitive representation. The effect of the window matching allows a sharper picture of the local energy in the time-frequency plane than is afforded by the spectrogram.

A. Simulated Test Signal Results

B. Results for the Analysis of Humpback Whale Sounds

The simulated signal to be analyzed is of the form

where ms, Hz, Hz, , and The performance of the AED will now be compared with the WVD and the spectrogram. The spectrogram employs, as its window function, the window used for the first iteration of the AED algorithm. This window is Gaussian, with an associated WVD that has circular symmetry over the timefrequency scale with ms, and Hz. The AED is calculated using the algorithm described earlier—four iterations of the method are used, which have been shown through simulation to produce good results. All plots shown have been normalized. The TFD’s for this signal are displayed in Figs. 1–3. The AED is similar to the spectrogram but has improved resolution since the window function is locally matched to the signal.

The sounds of the humpback whales are analyzed in this section. These sounds are nonstationary deterministic (but subject to various noise interferences) and are generally nontransient and multicomponent—this makes them well suited to a time-frequency analysis. The humpback whale data has been supplied by the Defence Science and Technology Organization (Australia). The data has been decimated to yield an effective sampling rate of 5000 Hz. One of the fundamental sounds in humpback whale songs will be investigated here—this sound has been described by biologists as a deep throat growl or a moan [28]. The moans are generally low frequency, relatively broadband, and extremely loud. Two such sounds were isolated and placed into separate 64-point data files. They will now be analyzed by four TFD’s—the WVD (fixed window) spectrogram, ZAM (with Gaussian smoothing parameter [35]) and the AED. The results for the first test signal are displayed in Figs. 4–7. The WVD is not easily interpretable due to its positivenegative (density) nature and the large number and size of

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Fig. 4. Wigner–Ville distribution of humpback whale sound 1.

Fig. 5.

Fig. 7.

Adaptive energy distribution of humpback whale sound 1.

Spectrogram of humpback whale sound 1. Fig. 8.

Fig. 6.

Wigner–Ville distribution of humpback whale sound 2.

ZAM distribution of humpback whale sound 1.

the cross-terms that are inherently generated. The spectrogram is well smoothed, but it does provide some indication as to the nature of the moan—it is composed of a series of linear FM pulses, which occur at various times and frequency locations. The ZAM provides excellent resolution for signals aligned with the time or frequency axes but degrades for signals with larger FM gradients, as is evidenced in Fig. 6. The signal components appear sharp due to the use of a lateral inhibition window function [35]. It is also of significance to note that the broadband component at about 5.5 ms is well represented by the ZAM but subject to some frequency smoothing introduced by the distribution. The AED (Fig. 7) in this case has yielded

Fig. 9. Spectrogram of humpback whale sound 2.

what may be considered the best result; it is a better resolved version of the spectrogram since no assumptions (as regards window selection) have been made on the signal. This has the benefit that no direction of signal evolution is attenuated or favored over another. The second signal is a more complicated moan. These results (Figs. 8–11) vividly illustrate the fundamental limitations of nonadaptive distributions. The WVD is cluttered with cross-terms to the point of incomprehensibility, whereas the spectrogram shows many characteristics of the signal but is


Fig. 10.

Fig. 11.

ZAM distribution of humpback whale sound 2.

Adaptive energy distribution of humpback whale sound 2.

smoothed to the point that much fine detail is lost. The ZAM has failed to characterize the signal—the linear FM’s with significant gradients have been poorly resolved. Once again, the AED has well-represented the signal in time-frequency—it appears to be composed of multiple components, with linear FM behavior, occurring at various regions throughout the plane. There is much detail present that has not been resolved by the other TFD’s. Most of the modulated signals collapse into one high-energy sinusoid (at about 9 ms), which may be a characteristic of some resonant oscillation associated with the humpback whale’s vocal tract. The linear FM signals may reflect some intrinsic modulating behavior of the animal’s vocal system, especially considering they are of similar (positive or negative) gradient. This section has provided some results that have demonstrated the successful application of the theory of generalized instantaneous parameters and window matching to generation of the adaptive energy distribution.

VIII. CONCLUSIONS This paper has attempted to explain how signal representations of 1-D signals in the time-frequency plane may be interpreted. All such TFD’s are either energy density distributions, (local) energy distributions, or lie somewhere between those two limiting classes.

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Since a TFD is expected to represent nonstationary behavior in the time-frequency plane, there is an inherent notion of instantaneous measures. Such quantities, as functions of time or frequency, may describe the instantaneous or local behavior of a signal in the same manner that global moment measures describe the overall signal characteristics. Unfortunately, such instantaneous measures, as currently defined, are only instantaneous with respect to time or frequency but never both. A new class of instantaneous parameters—the generalized instantaneous parameters—have been defined in this paper. They are functions of both time and frequency and are associated with the short-time Fourier transform in an analogous manner in that the 1-D parameters are associated with the signal or its Fourier transform. Their particular advantage is that they provide measures with localization in the time-frequency plane. These generalized instantaneous parameters may be employed to characterize local behavior in the time-frequency plane. An example application is given in this paper—window matching. Generation of a time-frequency energy distribution (a spectrogram) requires use of an analysis window. If the window does not match the signal well at the given timefrequency point under analysis, the representation is smeared as a result. By applying the ideas of matched filtering on the fine time-frequency scale, better spectrogram representations may be achieved, which are sharper than any using an arbitrary window. The generalized instantaneous parameters are used for this purpose (with a Gaussian window) to produce an adaptive energy distribution. The simulated and real data results provided demonstrate both window matching in the time-frequency plane as well as the usefulness of the newly defined parameters. APPENDIX The theorem of (46) shall be proved in this Appendix. The proof requires the decomposition of the arbitrary signal into a complete series of orthonormal Hermite functions [17] of the form (53) where

is a Hermite polynomial of the form [17] (54)

The signal could then be expressed as (55) where are the resulting coefficients of the orthonormal Hermite decomposition. To prove the theorem, the ITV and IFV must be calculated. Let us commence with a Gaussian window, which has equal “spread” in time and in frequency and Introducing a normalizing scalar, this window may be expressed as a zeroth-order Hermite function, that is, (56)

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By first rewriting the STFT defined in (19) in the equivalent form

as IFV

(57)

an STFT using the window of (56), and the form of (55), may be expressed as

in

(58)

where

By making the variable substitution equation may be expressed as

the above Similarly, the ITV [using (40)] becomes ITV

which can be simplified to yield Combining these quantities, one obtains IFV

ITV

By realizing that , the STFT of the window and one Hermite polynomial of the orthogonal decomposition of can be written as

where

is a Laguerre polynomial [17] such that As a result, the signal-window IFV calculated for a WVD, the expression for which is given in (41), is expressible

which proves the theorem.


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Graeme Jones (M’91) was born on the Gold Coast, Queensland, Australia, on November 29, 1966. He received the B.E. and Ph.D degrees from the University of Queensland and the Queensland University of Technology (QUT), Brisbane, in 1987 and 1992, respectively. He remained at QUT as a lecturer in signal processing in 1992. From 1992 to 1994, he was a postdoctoral fellow at McMaster University, Hamilton, Ont., Canada, where he received an NSERC International Fellowship. He then returned to QUT, remaining for a year as a research fellow and lecturer in the Aerospace and Aeronautical Division of the School of Electrical and Electronic Systems Engineering. Since September 1995, he has been a Senior Systems Engineer in the Advanced Systems Department of Raytheon Canada Limited, Waterloo, Ont. His research interests include time-frequency and general higher dimensional signal analysis and radar signal processing, detection, and tracking.

Boualem Boashash (SM’89) received the Diplôme d’ingénieur–Physique–Eléctronique from the ICPI University of Lyon, France, in 1978, the M.S. degree from the Institut National Polytechnique de Grenoble, France, in 1979, and Doctorate (Docteur–Ingenieur) from the same university in May 1982. In 1979, he joined the Elf-Aquitaine Geophysical Research Centre, Pau, France. In May 1982, he joined the Institut National des Sciences Appliquées de Lyon, Lyon, France, where he was a MaitreAssistant associé. In January 1984, he joined the Electrical Engineering Department of the University of Queensland, Brisbane, Australia, as a lecturer, Senior Lecturer (1986), and Reader (1989). In 1990, he joined Bond University, Graduate School of Science and Technology, as a Professor of Electronics. In 1991, he joined the Queensland University of Technology as the foundation Professor of Signal Processing and Director of the Signal Processing Research Centre. His research interests are time-frequency signal analysis, spectral estimation, signal detection and classification, and higher order spectra. He is also interested in wider issues such as the effect of engineering developments on society. Dr. Boashash was technical chairman of ICASSP 1994. He is the editor of two books, has written over 200 technical publications, and has supervised 20 Ph.D. students and five Masters students. He is fellow of both IE Australia and of the IREE.