1 Introduction Spectral analysis of (wide-sense) stationary random processes by means of the power spectral density (PSD) is a useful concept. owever,

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1 IEEE Trans. Signal Proc., vol. 45, no. 6, pp. 150{1534, June Generalized Evolutionary Spectral Analysis and the Weyl Spectrum of Nonstationary andom Processes SP-865, nal version Gerald Matz y, Franz lawatsch y, and Werner Kozek z y INTFT Vienna University of Technology Gusshausstrasse 5/389, A-1040 Wien Austria z NUAG, Department of Mathematics University of Vienna Strudlhofgasse 4, A-1090 Wien Austria Correspondence should be addressed to F. lawatsch Tel.: Fa: fhlawats@ .tuwien.ac.at Abstract The evolutionary spectrum (ES) is a \time-varying power spectrum" of nonstationary random processes. Starting from an innovations system interpretation of the ES, we introduce the generalized evolutionary spectrum (GES) as a novel family of time-varying power spectra. The GES contains the ES and the recently introduced transitory evolutionary spectrum as special cases. We consider the problem of nding an innovations system for a process characterized by its correlation function, and we discuss the connection between GES analysis and the class of underspread processes. We furthermore show that another special case of the GES, a novel time-varying power spectrum that we call Weyl spectrum, has substantial advantages over all other members of the GES family. The properties of the Weyl spectrum are discussed, and its superior performance is veried eperimentally for synthetic and real-data processes. This work was supported by FWF Grants P1001- OP and S7001-MAT. 1

2 1 Introduction Spectral analysis of (wide-sense) stationary random processes by means of the power spectral density (PSD) is a useful concept. owever, in many applications the signals must be modeled as nonstationary processes. Etensions of the PSD to nonstationary processes result in \time-varying power spectra" such as the Wigner- Ville spectrum [1]{[3], the physical spectrum [4], and the evolutionary spectrum (ES) [5]{[19]. This paper discusses and etends the ES. The original denition of the ES is based on an epansion of the nonstationary random process under analysis into comple eponentials with uncorrelated, random, time-varying amplitudes [5]. Alternatively, the ES can be epressed via the transfer function of a linear time-varying (LTV) innovations system [5, 9, 10]. Following the introduction of the ES by Priestley [5]{[7], numerous researchers have discussed the theoretical framework of the ES [, 3], [8]{[1], applications of the ES to special types of nonstationary processes [13, 14], and the estimation of the ES [15]{[18]. Furthermore, etensions to parametric models have been established [19] and a concept dual to the ES, the transitory evolutionary spectrum (TES), has been introduced recently [0]. In this paper, motivated by the innovations system interpretation of the ES, we dene the generalized evolutionary spectrum (GES) as a family of time-varying power spectra etending both the ES and the TES. Subsequently, we concentrate on a specic member of the GES family, the novel Weyl spectrum that has important advantages over all other GES members [1]. The paper is organized as follows. Section reviews the ES. Section 3 reviews the TES and gives a novel innovations system interpretation of the TES. An important process classication (underspread/overspread) []{[4] is considered in Section 4. The GES is introduced in Section 5 using the generalized Weyl symbol [5], and the properties of the GES are discussed. The construction of an innovations system and, specically, the advantages of the positive semidenite innovations system are considered in Section 6. In Section 7, we introduce the Weyl spectrum [1] as a new member of the GES family with substantial advantages over all other GES members. In Section 8, our theoretical results are veried eperimentally for synthetic and real-data processes. The Evolutionary Spectrum.1 The Stationary Case The ES can be motivated by the stationary case. Therefore, we rst consider a zero-mean, wide-sense stationary random process (t) with autocorrelation function r () = Ef(t + ) (t)g and PSD 1 S (f) = 1 Integrals are from 1 to 1 unless stated otherwise. r () e jf d 0 :

3 Since f S (f) df = Efj(t)j g, the PSD can be interpreted as a spectral distribution of the average power. It is related to an epansion of the process (t) into comple sinusoids e jft [6], (t) = f X(f) e jft df ; (1) where the epansion coecients can be shown to be uncorrelated with the PSD as average intensity, EfX(f 1 ) X (f )g = S (f 1 ) (f 1 f ) : () Let us set X(f) = N(f) A(f) where N(f) denotes stationary white noise with normalized average intensity, E fn(f 1 ) N (f )g = (f 1 f ), and A(f) is a deterministic, comple-valued weighting function. The lefthand side of () then becomes EfX(f 1 ) X (f )g = ja(f 1 )j (f 1 f ), so that the PSD is seen to be S (f) = ja(f)j : (3) A second important interpretation of the PSD is obtained by representing the process (t) as the output of a linear time-invariant (LTI) \innovations system" whose input is stationary white noise n(t) [6], (t) = (n)(t) = t 0 h(t t 0 ) n(t 0 ) dt 0 with E fn(t 1 ) n (t )g = (t 1 t ) : The PSD then equals the magnitude squared of the system's transfer function (f) = h(t) t ejft dt, S (f) = j(f)j : (4) It is easily seen that (f) = A(f) if we take N(f) to be the Fourier transform of n(t). ence, the PSD epressions (3) and (4) are equivalent.. Denition and Interpretation of the Evolutionary Spectrum We net consider a zero-mean, nonstationary random process (t) with correlation function (t 1 ; t ) = E f(t 1 ) (t )g [6]. Motivated by (1), we postulate an epansion of (t) into comple sinusoids e jft, (t) = f X t (f) e jft df ; (5) where the epansion coecients X t (f) are time-varying but again assumed to be uncorrelated, EfX t (f 1 ) X t (f )g = ES (t; f 1 ) (f 1 f ) : (6) This constitutes an implicit denition of the ES. In order to make this denition more precise, we set X t (f) = N(f) A(t; f) where N(f) denotes stationary white noise with normalized average intensity and A(t; f) is a deterministic, comple-valued weighting function. The epansion (5) then becomes (t) = f N(f) A(t; f) e jft df ; (7) This epansion and similar epansions used in the following are to be interpreted in the mean-square sense [6]. 3

4 and the left-hand side of (6) becomes EfX t (f 1 ) X t (f )g = ja(t; f 1 )j (f 1 f ), so that [5] ES (t; f) = ja(t; f)j : (8) With (7), it is easy to show that the process' average instantaneous power can be written as E j(t)j = f ja(t; f)j df, so that the ES is a spectral distribution of the average instantaneous power, i.e., f ES (t; f) df = Efj(t)j g (\marginal property"). We now ask if the epansion (7) underlying the ES eists and is unique, and how the ES can be derived given the correlation function (t 1 ; t ). Introducing (t; f) = A(t; f) e jft, (7) becomes (t) = f N(f) (t; f) df ; (9) and it is easily shown that the correlation function can be epressed as (t 1 ; t ) = f (t 1; f) (t ; f) df. In operator notation, this reads = +, where is the correlation operator (i.e., the positive denite or semidenite, self-adjoint linear operator whose kernel is (t 1 ; t )), is the linear operator whose kernel is (t; f), and + denotes the adjoint of the operator [7]. ence, for given correlation operator, is a solution to the factorization problem + =. Such a solution always eists since is positive semidenite, but it is not unique: if is a solution, then so is ~ = U where U is an arbitrary unitary operator (i.e., UU + = I with I the identity operator). ence, the ES as dened in (8) is not unique [5]; the specic ES obtained depends on the particular solution to the factorization problem + =. It is important to note that the interpretation of the ES as a time-varying power spectrum is restricted to the case where the \amplitude function" A(t; f) weighting the comple sinusoids e jft in the epansion (7) is slowly time-varying. Indeed, only in this case can the function (t; f) = A(t; f) e jft in the epansion (9) be interpreted (as a function of t) as a narrowband, amplitude-modulated signal (an \oscillatory function" [5]) spectrally localized around f, so that the parameter f in ES (t; f) = ja(t; f)j can be interpreted as \frequency" in a meaningful sense. This restriction amounts to a kind of quasi-stationarity assumption..3 Innovations System Interpretation We can epress (t) as the response of an LTV innovations system to stationary white noise n(t) [8], (t) = (n)(t) = Calculating the correlation function of (t) from (10) yields t 0 (t; t 0 ) n(t 0 ) dt 0 with E fn(t 1 ) n (t )g = (t 1 t ) : (10) (t 1 ; t ) = t 0 (t 1 ; t 0 ) (t ; t 0 ) dt 0 ; (11) so that the innovations system is obtained as (non-unique) solution to the factorization problem + =, i.e., is a \square root" of the correlation operator (cf. Section 6). It is easily shown that (10) 4

5 is equivalent to the epansion (9), (t) = f N(f) (t; f) df, with N(f) the Fourier transform of n(t) and (t; f) = t (t; t 0 ) e jft0 dt 0. We then obtain 0 A(t; f) = (t; f) e jft = (t; t ) e jf d = (t; f); where (t; f) is the time-varying transfer function of as dened by adeh [9]. ence, the ES can be reformulated as the squared magnitude of adeh's transfer function of the innovations system [5, 9, 10], ES (t; f) = j (t; f)j : (1) This result is very intuitive and etends the PSD epression (4) obtained in the stationary case. owever, recall that the interpretation of the ES as a time-varying power spectrum requires A(t; f) to be slowly time-varying. Since A(t; f) = (t; f), this means that (t; f) should be slowly time-varying as well. Furthermore, the epression (1) is intuitively meaningful only if the LTV system acts as a weighting in the time-frequency (TF) domain, i.e., if the TF shifts caused by the innovations system are small. These considerations will lead us to the class of underspread systems and processes discussed in Section 4. 3 The Transitory Evolutionary Spectrum The transitory evolutionary spectrum (TES) has recently been introduced [0] as a time-varying spectrum that is dual to the ES (see Table I). The TES is matched to \quasi-white" processes. 3.1 The Nonstationary White Case We rst consider a zero-mean, nonstationary white random process (t) with correlation function (t 1 ; t ) = q (t 1 ) (t 1 t ), where q (t) 0 is the process' average instantaneous intensity [6]. The process allows a trivial epansion into Dirac impulses, (t 0 ) = t (t) (t 0 t) dt; (13) or, taking the Fourier transform, X(f) = t (t) e jtf dt : (14) ere, the epansion coecients (t) are uncorrelated, Ef(t 1 ) (t )g = q (t 1 ) (t 1 t ) : (15) Let us set (t) = n(t) a(t) where n(t) is stationary white noise with normalized average intensity and a(t) is a deterministic function. The left-hand side of (15) is then Ef(t 1 ) (t )g = ja(t 1 )j (t 1 t ), so that q (t) = ja(t)j : 5

6 Table I Duality of evolutionary spectrum and transitory evolutionary spectrum (F denotes the Fourier transform operator). Evolutionary Spectrum (t) = f N(f) (t; f) df X(f) (t 1 ; t ) = (t f 1; f) (t ; f) df Transitory Evolutionary Spectrum = t n(t) (t; f) dt X (f 1 ; f ) = t (t; f 1) (t; f ) dt (t; f) = A(t; f) e jft (t; f) = a(t; f) e jft (t) = (n)(t) = t 0 (t; t0 ) n(t 0 ) dt 0 (t; f) = F 1 t 0!f f(t; t0 )g (t 0 ; f) = F t!f f(t; t 0 )g A(t; f) = (t; f) = L (1=) (t; f) a(t; f) = e (t; f) = L (1=) (t; f) The relation (t) = n(t) a(t) can be interpreted in the sense that stationary white noise n(t) is passed through a linear \frequency invariant" (LFI) system, i.e., an LTV system acting as a multiplier. This is dual to the interpretation of stationary processes in terms of an LTI innovations system (cf. Section.1). 3. Denition and Interpretation of the Transitory Evolutionary Spectrum We net consider a zero-mean, nonstationary random process (t) with correlation function (t 1 ; t ). Motivated by the epansion (14) in the nonstationary white case, let us postulate an epansion X(f) = = t t f (t) e jtf dt = t n(t) a(t; f) e jft dt (16) n(t) (t; f) dt : (17) ere, n(t) is stationary white noise with average intensity one, a(t; f) is a deterministic, comple-valued weighting function dened by f (t) = n(t) a(t; f), and (t; f) = a(t; f) e jft. The epansion coecients f (t) are frequency-varying but still assumed uncorrelated, Ef f (t 1 ) f(t )g = TES (t 1 ; f) (t 1 t ) : (18) The frequency variation in f (t) will be seen to imply a broadening of the Dirac impulses in (13). Eq. (18) constitutes an implicit denition of the TES. With f (t) = n(t) a(t; f), (18) can be rewritten as Ef f (t 1 ) f (t )g = ja(t 1 ; f)j (t 1 t ), so that TES (t; f) = ja(t; f)j : (19) With (16), one can show that the TES is a temporal distribution of the average spectral energy density, i.e., the TES satises the \marginal property" TES t (t; f) dt = EfjX(f)j g. 6

7 Taking the inverse Fourier transform of (16) or (17), the process (t 0 ) is represented as (t 0 ) = t n(t) (t 0 ; t) dt where (t 0 ; t) = f (t; f) ejt0f df = f [a(t; f) ejtf ] e jt0f df. If a(t; f) is slowly varying with respect to f, then (t 0 ; t), as a function of t 0, is localized about t 0 = t (in the limiting case where a(t; f) = a(t), we get (t 0 ; t) = a(t) (t 0 t)). The TES is then based on an epansion of (t 0 ) into narrow pulses with uncorrelated coecients, which justies the interpretation of the TES parameter t as \time." Note that the assumption that a(t; f) is slowly varying with respect to f amounts to a \quasi-whiteness" property of (t 0 ). With (17), the spectral correlation function X (f 1 ; f ) = E fx(f 1 ) X (f )g (which is related to the temporal correlation function (t 1 ; t ) by a -D Fourier transform) can be epressed as X (f 1 ; f ) = (t; f t 1) (t; f ) dt. ence, calculation of (t; f) given (t 1 ; t ) amounts to solving the factorization problem + = X where is the linear operator whose kernel is (f; t) = (t; f). Again, always eists but it is not unique, so that the TES as given by (19) is not unique either. 3.3 Innovations System Interpretation We now establish a novel reformulation of the TES in terms of LTV innovations systems. In (10), we have modeled (t) as the response of an LTV system to stationary white noise n(t). Taking the Fourier transform of (10) with respect to t, one re-obtains the epansion (17) with (t 0 ; f) = t (t; t0 ) e jft dt, so that a(t; f) = (t; f) e jft = (t + ; t) e jf d = e (t; f) : ence, the amplitude function a(t; f) can be interpreted as a time-varying transfer function e (t; f) of which, however, is dierent from adeh's function (t; f) arising in the case of the ES. With (19), the TES can be epressed as the squared magnitude of the new transfer function of the innovations system, TES (t; f) = e (t; f) Comparing with (1), we see that the only dierence between the ES and the TES is in the denition of the time-varying transfer function. This viewpoint will motivate the denition of the GES in Section 5. The identity a(t; f) = e (t; f) suggests that e (t; f) should be a smooth function of f as it was required for a(t; f). In addition, the interpretation of TES (t; f) via the transfer function of is meaningful only if acts as a weighting in the TF domain (i.e. the TF shifts caused by are negligible). This restriction was already encountered in the contet of the ES and will be further considered in the net section. : 7

8 4 Underspread Systems and Processes We have argued above that a meaningful interpretation of the ES in terms of a process epansion into uncorrelated narrowband signals is restricted to quasi-stationary processes, while the interpretation of the TES in terms of an epansion into uncorrelated short pulses is restricted to quasi-white processes. These restrictions correspond to the requirement that the transfer functions (t; f) and e (t; f) be smooth with respect to t and f, respectively, and that the innovations system acts as a pure TF weighting in the sense that it introduces negligible TF displacements. For later use, we shall now discuss characterizations of the TF displacements of the innovations system and of the correlation structure of the resulting process. 4.1 Underspread Systems The TF shifts caused by an LTV system are characterized by the generalized spreading function [3, 5] 1 1 S () t (; ) = () (t; ) e jt dt with () (t; ) = t + ; t + : (0) ere, and denote time lag and frequency lag, respectively, and is a real parameter. Since the magnitude of S () (; ) is independent of, we shall use the simplied notation js (; )j = S () (; ). For given (; ), js (; )j indicates how much the TF-shifted input signal (S ; )(t) = (t ) e jt contributes to the output signal [5]. It follows that the TF shifts caused by an LTV system are crudely characterized by the eective support of js (; )j. Let us dene the \displacement spread" [3, 4] as the area of the smallest rectangle (centered about the origin of the (; )-plane) containing the eective support of js (; )j, as shown in Fig. 1. A system is called underspread if 1, meaning that js (; )j is concentrated about the origin of the (; )-plane, so that causes only small TF shifts. Furthermore, an underspread system will be called strictly underspread if is oriented parallel to the and aes, such that = and = where ; ;, and are dened in Fig. 1. ere, = 4 so that a system is strictly underspread if 4 1. Quasi-LTI systems with small frequency shifts (small ) and quasi-lfi systems with small time shifts (small ) are potentially strictly underspread systems. 4. Underspread Processes Quasi-stationary processes have small spectral correlation, while quasi-white processes have small temporal correlation. These two situations are generalized by the concept of underspread processes. We rst dene the epected ambiguity function (EAF) [, 3] of a nonstationary process (t) as nd A (; ) = E S- ;- ; S Eo ; = t + ; t e jt dt ; t 8

9 js (; )j Figure 1. Eective support of the spreading function of a self-adjoint LTV system. where again (S ; )(t) = (t ) e jt. The EAF describes the average correlation of all TF locations separated by in time and by in frequency. Noting that the EAF is the spreading function (with = 0) of the correlation operator, A (; ) = S (0) (; ), we are led to dene the TF correlation spread of 4 (t) as the displacement spread of the correlation operator, =. Then, a process (t) is underspread if 1 [, 3], which means that the EAF is concentrated about the origin of the (; )-plane, and hence, that components of (t) that are suciently separated in the TF plane will be nearly uncorrelated. 4 4 The eective support of the EAF can also be described by the quantities = and = (see Fig. 1), and (t) will be called strictly underspread if the correlation operator is strictly underspread, i.e., 4 1. Two potential special cases of strictly underspread processes are quasi-stationary processes (with small ) and quasi-white processes (with small ). The TF shifts caused by the innovations system are related to the TF correlation structure of the associated process (t). It can be shown [3] that the correlation spread of (t) is bounded in terms of the displacement spread of as 4. ence, an underspread innovations system implies an underspread process. Conversely, if (t) is (strictly) underspread, then it is not true that every innovations system is (strictly) underspread even though a (strictly) underspread can always be found (see Section 6). 5 The Generalized Evolutionary Spectrum 5.1 Denition and Interpretation The epressions ES (t; f) = j (t; f)j and TES (t; f) = j e (t; f)j suggest an etension of the ES and TES. Indeed, the time-varying transfer functions (t; f) and e (t; f) are just two special cases of a family 9

10 of time-varying transfer functions known as generalized Weyl symbol [3, 5] and dened as L () (t; f) = () (t; ) e jf d ; where () (t; ) has been dened in (0). The transfer functions underlying the ES and the TES are reobtained with = 1= and = 1=, respectively, i.e., (t; f) = L (1=) (t; f) and e (t; f) = L (1=) (t; f). We now introduce the generalized evolutionary spectrum (GES) as [1] GES () (t; f) 4 = L () (t; f) : (1) The GES comprises the ES and TES as special cases with = 1= and = 1=, respectively: ES (t; f) = GES (1=) (t; f) ; TES (t; f) = GES (1=) (t; f) : The denition of the GES in (1) contains a twofold ambiguity corresponding to the choice of the innovations system and of the parameter. This will be discussed in Sections 6 and 7. For a strictly underspread process, one can always nd an innovations system that is strictly underspread (see Section 6); here, the primary eect of is a TF weighting or, equivalently, the TF shifts caused by are small. Let us for the moment assume that the GES is based on this. The average energy content of a process (t) around a TF analysis point (t; f) can be measured by the physical spectrum [4] P (t; f) = Efjh; g (t;f ) ij g, where g (t;f ) (t 0 ) = g(t 0 t) e jft0 with g(t 0 ) being a normalized \window" or \test function" that is real-valued, even, and concentrated about the origin of the TF plane (note that g (t;f ) (t 0 ) is then normalized too and is properly concentrated about the analysis TF point (t; f)). For a strictly underspread innovations system, it is proved in Appendi A that GES () (t; f) P (t; f) ; () which shows that the GES is here physically meaningful. For = 0, () holds even when is (weakly) underspread. ence, the GES with = 0 is meaningful for a wider class of processes (see Section 7). From the fact that the right-hand side of the approimation () is independent of, it follows that the GES based on a strictly underspread innovations system is approimately independent of, i.e., GES ( 1) (t; f) GES ( ) (t; f). Let us compare two GES that are based on the same (strictly underspread) innovations system but have dierent values. It can then be shown (see Appendi B) that the dierence (t; f) = GES ( 1) (t; f) GES ( ) (t; f) is bounded both in a pointwise and an L -norm sense: j(t; f)j jj (4 ) 3 tr n 1= o ; kk 8 jj (4 ) 3 trf g ; (3) where = 1, and have been dened in Fig. 1, trfg denotes the trace of an operator [7], and 1= is the positive semidenite, self-adjoint operator square root of. ence, for a strictly underspread 10

11 innovations system the choice of the GES parameter is not critical. This is not true for weakly underspread processes since here no innovations system with small eists. The generalized spreading function introduced in Section 4 can be shown [3, 5] to be the -D Fourier transform of the generalized Weyl symbol underlying the denition of the GES, S () (; ) = t f L () (t; f) ej(t f ) dt df : (4) For an underspread, S () (; ) is concentrated about the origin of the (; )-plane, and thus (4) implies that L () (t; f), and hence GES() (t; f), is a -D lowpass function (i.e., a smooth function). In particular, a small spread of in the direction implies that L () (t; f) and GES() (t; f) are smooth with respect to t, while a small spread in the direction implies that L () (t; f) and GES() (t; f) are smooth with respect to f. These are the two situations allowing a meaningful interpretation of the ES and TES, respectively. This is consistent, as the response of a quasi-lti system to stationary white noise is a quasi-stationary process, and the response of a quasi-lfi system to stationary white noise is a quasi-white process. 5. Properties In the following, we discuss some important properties of the GES. Consistency. For a stationary process, the GES can be shown to reduce to the PSD and to be independent of t, GES () (t; f) S (f). For a nonstationary white process, the GES reduces to the average instantaneous intensity and is independent of f, GES () (t; f) q (t). For a stationary white process with S (f) = q (t) N 0, the GES is constant over the entire TF plane, GES () (t; f) N 0. Positivity. The GES is real-valued and nonnegative, GES () (t; f) 0. Self-Adjoint Innovations System. It can be shown that L () (t; f) = L()(t; f), where + is the adjoint + of. ence, the GES with parameters and are identical for a self-adjoint innovations system : = + =) GES () (t; f) = GES () (t; f): In particular, = + implies ES (t; f) = TES (t; f). Thus, even though the ES and TES are based on dierent epansion models, they will produce identical results if they are based on a self-adjoint innovations system (which can always be found, see Section 6). Marginals. The ES has correct time marginals, f GES(1=) (t; f) df = Efj(t)j g, while the TES has correct frequency marginals, t GES(1=) (t; f) dt = EfjX(f)j g. If (and only if) the innovations system is normal, + = +, then both the ES and the TES will satisfy both marginal properties, i.e. we have also t GES(1=) (t; f) dt = EfjX(f)j g and f GES(1=) (t; f) df = Efj(t)j g. It can be shown that underspread systems are approimately normal [3]. 11

12 For 6= 1=, the marginal properties will not be satised eactly, but they will be approimately satised for strictly underspread innovations systems. Indeed, the deviation between the time marginal of the GES and the epected instantaneous power, 1 (t) = 4 f GES() (t; f) df Efj(t)j g, can be bounded as j 1 (t)j 4 1 n o (4 ) tr 1= ; k 1 k 1 p (4 ) trf g ; where and have been dened in Fig. 1. Similarly, the deviation between the frequency marginal of the GES and the epected spectral energy density, (f) = 4 t GES() (t; f) dt EfjX(f)j g, can be bounded as j (f)j n o (4 ) tr 1= ; k k + 1 p (4 ) trf g : Note that the above bounds correctly reect the fact that the ES ( = 1=) has correct time marginals, while the TES ( = 1=) has correct frequency marginals. Finite Support. From the marginal and positivity properties, it follows that the GES with = 1= and normal satises the following \strong" nite support properties: Efj(t 0 )j g = 0 =) GES (1=) (t 0 ; f) = 0; EfjX(f 0 )j g = 0 =) GES (1=) (t; f 0 ) = 0: For jj < 1= and normal, the GES satises the following \weak" nite support properties: Efj(t)j g = 0; t = [t 1 ; t ] =) GES () (t; f) = 0; t = [t 1 ; t ] EfjX(f)j g = 0; f = [f 1 ; f ] =) GES () (t; f) = 0; f = [f 1 ; f ]: TF Shift and Scaling Covariance. Let GES () (t; f) be based on an innovations system. If (t) is shifted in time by and in frequency by, ~(t) = (S ; )(t) = (t ) e jt, then the correlation operator of the shifted process ~(t) is ~ = S ; S + ;, and a specic innovations system of ~(t) is b ~ = S ; S + ;. If (and only if) the innovations system used for calculating the GES of ~(t) is chosen as ~ = b ~, then the GES of the shifted process is an appropriately shifted GES, GES () ~ (t; f) = GES() (t ; f ) : By choosing the respective positive semidenite innovations systems for both (t) and ~(t) (see Section 6), it is guaranteed that ~ = b ~ so that the above \covariance property" will always be satised. In a similar manner, it can be shown that the GES will satisfy the covariance property with respect to a TF scaling, q ~(t) = jaj (at) =) GES () ~ (t; f) = GES() at; f ; a if it is based on the positive semidenite innovations system. 1

13 LTV System. If (t) is transformed by a positive semidenite LTV system K with kernel K(t; t 0 ), y(t) = (K)(t) = t 0 K(t; t 0 ) (t 0 ) dt 0 ; then the correlation operator of the transformed process y(t) is y = K K = K + K, where is an innovations system of (t). For reasons to be eplained in Section 6, we choose to be the positive semidenite innovations system, and we look for the positive semidenite innovations system y of y(t). Let us assume that K and are jointly strictly underspread in the sense that the eective supports of their spreading functions are both bounded by the same rectangular region that is parallel to the and aes and whose area is much less then 1 (this requires that both systems K and are individually strictly underspread, i.e. 4 L () K 1= K 1= (t; f) L () K 1= (t; f) L () (t; f) L () GES () y (t; f) L () 1 and 4 K K 1). It can then be shown that y K 1= K 1= and K 1= K 1= (t; f) (t; f) L () K 1= K L () K (t; f) L() (t; f) [3], so that the GES of y(t) is (t; f) L() (t; f) = L () K (t; f) GES () (t; f) : (5) Thus, the GES of the output process is approimately equal to the GES of the input process multiplied by the squared magnitude of the generalized Weyl symbol of the LTV system K. This relation suggests to interpret the eect of K as a TF weighting characterized by L () K (t; f). It generalizes the relation Sy (f) = jg(f)j S (f) obtained when a stationary process is transformed by an LTI system with transfer function G(f), and the relation q y (t) = jm(t)j q (t) obtained when a nonstationary white process is transformed by an LFI system with multiplier function m(t). 6 The Factorization Problem For given correlation operator, the innovations systems are dened by (11) or equivalently + = : (6) The solution to this factorization problem is not unique. Indeed, if is a valid innovations system satisfying (6), and if U is an arbitrary unitary operator (satisfying UU + = I), then e = U is an innovations system as well: e + e = UU + + = + =. In the stationary case, the innovations systems are time-invariant and have identical transfer function magnitude. A similar situation eists in the nonstationary white case. owever, in the general nonstationary case, dierent choices of will lead to dierent generalized Weyl symbol magnitudes and hence to dierent GES results. This ambiguity of the GES denition can be resolved by imposing certain constraints on. For eample, the Wold-Cramer ES [9, 10] is obtained with a causal. In this section, however, we shall discuss the advantages of the positive semidenite. 13

14 It is reasonable to adopt the \maimally underspread" innovations system for which TF displacement eects are minimized (see Section 4). This system primarily produces a TF weighting that can be described by the squared magnitude of the generalized Weyl symbol, which is the GES. This permits the interpretation of the GES as an average TF energy distribution and is also consistent with the conditions that (t; f) and e (t; f) be smooth with respect to t and f, respectively. In the following, the maimally underspread will be dened as the minimizing the TF displacement radius 4 = T + T where T is an arbitrary normalization time constant and 4 = js (; )j d d js (; )j d d ; 4 = js (; )j d d js (; )j d d (7) measure the etension of the spreading function in the and direction, respectively. The minimization of will be performed within the class of normal (satisfying + = + ). A normal system is advantageous since here both marginal properties are satised by both the ES and the TES (see Section 5.). Any normal innovations system allows a polar decomposition [7] = p U ; (8) where p (satisfying p + p = ) is the positive semidenite, self-adjoint operator square root of, p = 1=, and U is a unitary operator constrained by the normality of. The kernel of p is 1X p p (t; t 0 ) = k u k (t) u k(t 0 ) ; where k 0 are the eigenvalues and u k (t) are the eigenfunctions of the correlation operator. k=1 Since p is ed, it remains to choose the unitary factor U such that the TF displacement radius is minimized. It is shown in Appendi C that the solution to this problem is the identity operator up to a trivial phase factor that will be set to 1 in the following, i.e., U opt = I [30]. Thus, the innovations system with minimum TF displacement radius is the positive semidenite root of, opt = p = 1= : We note that this maimally underspread innovations system will lead to a generalized Weyl symbol, and in turn a GES, that is maimally smooth (cf. Section 5.1). Using the positive semidenite root of as innovations system has another important advantage. Consider a unitary transformation of the process, ~(t) = (U)(t) where UU + = I. The correlation operator of the new process ~(t) is ~ = U U +, and its positive semidenite root is given by e p = U p U + where 14

15 p is the positive semidenite root of. This relation between the innovations systems of (t) and ~(t) was seen in Section 5. to guarantee the shift covariance and scaling covariance properties of the GES when U is a TF shift and TF scaling operator, respectively. Positive semidenite factorization is thus consistent with unitary signal transformations. This is not generally true for other types of factorizations. 7 The Weyl Spectrum 7.1 Denition and Interpretation The denition of the GES in Section 5 contained a twofold ambiguity, namely, the choice of the parameter and that of the innovations system. We now consider the case = 0, where the generalized Weyl symbol reduces to the Weyl symbol, L (t; f) = L (0) (t; f) [31]-[34]. Furthermore, for reasons eplained in the previous section, we adopt the positive semidenite root p = 1= innovations system. These two choices result in a member of the GES family given by 3 WS (t; f) = 4 GES (0) (t; f) = L p (t; f) = =p of the correlation operator as p t + ; t e jf d : This time-varying power spectrum will be called Weyl spectrum (WS) [1]. Being the GES with = 0 using the positive semidenite innovations system, the WS is uniquely dened for given. It has important advantages over all other GES members obtained for 6= 0 and/or 6= 1= : The approimation GES () (t; f) Efjh; g (t;f ) ij g (see Section 5.1) imparting an energetic interpretation to the GES holds for 6= 0 only if the process (t) is strictly underspread. In contrast, the WS will satisfy the above approimation even if (t) is merely weakly underspread, i.e. if 1 but not 4 1. ence, the WS is physically meaningful for a broader class of processes. In particular, the WS is much better suited to processes with \chirp components" (appearing as slanted structures in the TF plane) than is the ES or the TES. Some eamples will be shown in Section 8. The WS is based on the positive semidenite innovations system which introduces minimal TF displacement eects. This favors the interpretation of the WS as a proper time-varying power spectrum. Also, the use of the positive semidenite innovations system is a prerequisite for covariance properties with respect to unitary signal transformations such as TF shifts or TF scalings (see Section 6). The WS is based on the Weyl symbol (generalized Weyl symbol with = 0) whose symmetric structure leads to important advantages over generalized Weyl symbols with 6= 0. This entails corresponding 3 Note that the Weyl symbol of self-adjoint operators is real-valued so that L p (t; f) = L p (t; f). 15

16 advantages of the WS over other members of the GES family. In particular, the WS satises certain covariance properties that are not satised by the GES with 6= 0, as detailed further below. 7. Properties We now discuss the properties of the WS in more detail. Since the general properties of the GES have been discussed in Section 5., we concentrate on WS properties that are not satised by other GES members. TF Coordinate Transforms. There eist a class of unitary signal transformations U corresponding to area-preserving, ane TF coordinate transforms (t; f)! (at+bf ; ct+df ) with det( a b ) = adbc = 1. c d Specic signal transformations U depend on the TF coordinate transform parameters a; b; c; d;, and [35]. The WS satises the following general covariance property with respect to the unitary transformations U: ~(t) = (U)(t) =) WS ~ (t; f) = WS (at + bf ; ct + df ) : (9) Important special cases are listed in the following. For any set of parameters a; b; c; d;, and with ad bc = 1, the corresponding signal transformation U can be composed of some of these special transformations. TF shifts: ~(t) = (S ; )(t) = (t ) e jt =) WS ~ (t; f) = WS (t ; f ) TF scalings: ~(t) = qjaj (at) =) WS ~ (t; f) = WS at; f a Chirp multiplication: ~(t) = e jct (t) =) WS ~ (t; f) = WS (t; f ct) Chirp convolution: ~(t) = qjcj e jct (t) =) WS ~ (t; f) = WS t f c ; f Fourier transform: ~(t) = qjcj X(ct) =) WS ~ (t; f) = WS f c ; ct : We emphasize that the GES with 6= 0 satises only the covariance properties with respect to TF shifts and TF scalings provided that the innovations systems of (t) and ~(t) are related as ~ = U U + (see Section 5.). The general covariance property (9) will not be satised for 6= 0. Marginals. For the important class of (weakly) underspread processes, the marginal properties will be satised by the WS in an approimate manner. Specically, the deviation between the time marginal of the WS and the epected instantaneous power, 1 (t) 4 = f WS (t; f) df Efj(t)j g, can be bounded as j 1 (t)j 8 1= n o tr 1= ; k 1 k p 1= trf g ; and the deviation between the frequency marginal of the WS and the epected spectral energy density, (f) 4 = t WS (t; f) dt EfjX(f)j g, can be bounded as j (f)j 8 1= n o tr 1= ; k k p 1= trf g : 16

17 ere, and have been dened in Fig. 1. Since 1= 1 for underspread processes, these bounds imply the approimate validity of the marginal properties. Note that for the GES with 6= 0, approimate validity of the marginal properties required (t) to be strictly underspread in general (see Section 5.). Superposition Law. Let (t) = P N k=1 k(t) be the sum of N uncorrelated, zero-mean processes k (t). Since k ; l (t; t 0 ) = Ef k (t) l (t0 )g = 0 for k 6= l, one has = P N k=1 k. In general, there is no simple way to epress an innovations system of (t) in terms of innovations systems of the component processes k (t). owever, if the realizations of the k (t) belong to orthogonal signal spaces, then it can be shown that the positive semidenite root of (cf. Section 6) is equal to to the sum of the positive semidenite roots of the k, i.e., p; = P N k=1 p;k. By the linearity of the Weyl symbol, we then obtain " NX NX NX NX WS (t; f) = L p;k (t; f)# = WS k (t; f) + L p;k (t; f) L p;l (t; f) : k=1 k=1 k=1 l=1 k6=l With the assumption that the realizations of the processes k (t) are TF disjoint (which then also implies that they belong to orthogonal signal spaces [36]), the cross terms L p;k (t; f) L p;l (t; f) with k 6= l vanish since the L p;k (t; f) do not overlap. We then obtain the superposition law WS (t; f) = NX k=1 WS k (t; f) : (30) In practice, the k (t) will typically be eectively TF disjoint rather than eactly TF disjoint, in which case (30) is valid in an approimate sense. Deterministic Signal Components. Let us now assume that k (t) = k s k (t), i.e. (t) = NX k=1 k s k (t); (31) with deterministic signals s k (t) and uncorrelated, zero-mean random factors k with powers k = Efj kj g. The positive semidenite innovations system of k (t) is given by p;k (t; t 0 ) = k s k (t) s k (t0 ). If the s k (t) are TF disjoint in the sense that their Wigner distributions W sk (t; f) [36]{[40] do not overlap, then we obtain with (30) WS (t; f) = NX k=1 k W s k (t; f) : (3) Chirp Processes. The WS features superior TF concentration for \chirp processes" corresponding to slanted structures in the TF plane. Let us consider a chirp process (t) = w(t) e jct with zero-mean random factor and deterministic envelope w(t). The WS is obtained as WS (t; f) = W w(t; f ct), where W w (t; f) is the Wigner distribution of w(t). This result shows that the WS of a chirp process is well concentrated along the instantaneous frequency f (t) = ct. This can be generalized to multicomponent chirp signals whose components are approimately nonoverlapping in the TF plane (see (31)). Numerical eamples illustrating these results are provided in Section 8. 17

18 7.3 Comparison with the Wigner-Ville Spectrum The Wigner-Ville spectrum (WVS) [1]-[3] is an important time-varying spectrum dened as the Weyl symbol of the correlation operator, W (t; f) = L (t; f) = t + ; t e jf d : The WVS satises many desirable properties; in particular, it is unitarily related to the correlation function (t 1 ; t ) and it satises both marginal properties. owever, it may assume negative values [41]. In the case of an underspread process (t), the WS and WVS yield very similar results. Indeed, for underspread it can be shown (using techniques similar to those used in Appendices A and B) that L (t; f) L 1= (t; f), i.e., taking the square root of the correlation operator is approimately compensated by taking the square of the resulting Weyl symbol. ence, WS (t; f) = L (t; f) L 1= (t; f) = W (t; f) : Note that the approimate equivalence of the WS and WVS for an underspread process implies that the WVS of an underspread process is approimately nonnegative. While the WS and WVS are similar for underspread processes, they may be quite dierent otherwise. Let us reconsider the multicomponent process in (31), consisting of deterministic, TF disjoint signals s k (t) with statistically independent random factors k. WS and WVS are here obtained as (cf. (3)) WS (t; f) = NX k=1 k W s k (t; f) ; W (t; f) = NX k=1 k W s k (t; f) : We see that the WVS is given by a weighted superposition of the Wigner distributions of the individual components, whereas in the WS these Wigner distributions are additionally squared. This squaring entails a sharper representation of the process components in the WS (see net section). 8 Numerical Simulations We now apply the WS and GES to the TF analysis of synthetic and real-data processes. The duration of all processes considered is 18 samples. Our rst eample, shown in Fig., illustrates the superiority of the WS over other members of the GES family in the case of chirp processes. The (synthetic) random process under analysis is of the type (31); it consists of three time-frequency-shifted windowed \parallel" chirp signals s k (t) = w(t t k ) e jc(tt k) e jf kt with identical chirp rates c and statistically independent amplitude factors k with equal average powers. The EAF in part (d) shows that this process is reasonably underspread but not strictly underspread. As a consequence, the WS performs satisfactorily whereas the 18

19 f (a) f (b) f (c) (d) t t t Area=1 Figure. Synthetic process consisting of three \parallel" chirp signals: (a) WS, (b) ES/TES with positive semidenite innovations system, (c) WVS, and (d) magnitude of EAF (a hatched square of area 1 is included in this and subsequent plots to allow an assessment of the process' underspread property). f (a) f (b) f (c) (d) t t t Area=1 Figure 3. Synthetic process consisting of three \non-parallel" chirp signals and a Gaussian signal: (a) WS, (b) ES/TES with positive semidenite innovations system, (c) WVS, and (d) EAF magnitude. ES (simultaneously the TES due to the use of the positive semidenite innovations system) totally fails to resolve the three chirp components. The WVS, shown for comparison, performs satisfactorily as well. Fig. 3 shows that the good performance of the WS etends to the case where the overall process is not underspread but all process components are TF disjoint and individually underspread. The process underlying Fig. 3 consists of three windowed \non-parallel" chirp signals s k (t) = w(t) e jc kt (with dierent chirp rates c k ) and a Gaussian signal, again with statistically independent amplitude factors k. Note that the ES/TES does not correctly indicate the frequency modulation of the three chirp components. While the WS and the ES/TES yielded dramatically dierent results in Figs. and 3, Fig. 4 shows that these spectra become very similar for strictly underspread processes. The process under analysis, whose correlation function was constructed using the TF synthesis method proposed in [4], consists of three uncorrelated random components appearing as smooth structures in the TF plane. The EAF shows that the process is indeed strictly underspread. The strong similarity of the WS and the ES/TES corroborates the approimate -invariance of the GES in the case of strictly underspread processes (see Section 5.1). The WS and ES/TES are also very similar to the WVS, as predicted in Section

20 f (a) f (b) f (c) (d) t t t Area=1 Figure 4. Strictly underspread synthetic process: (a) WS, (b) ES/TES with positive semidenite innovations system, (c) WVS, and (d) EAF magnitude. f (a) f (b) f (c) (d) t t t Area=1 Figure 5. LTV ltering of the strictly underspread process from Fig. 4: (a) squared Weyl symbol of LTV lter K, (b) WS of input process (see Fig. 4(a)) multiplied by squared Weyl symbol of K (approimation to WS of lter output process), (c) eact WS of lter output process, and (d) spreading function of K. Fig. 5 corroborates the approimation (5) for the GES of a ltered process. The three-component process (t) from Fig. 4 was ltered by an LTV system K in order to isolate the middle component. Comparing the spreading function of K with the EAF of (t) (shown in Fig. 4(d)), we see that (t) and K are jointly underspread, which is the condition for the approimation (5). Fig. 5 shows that the WS of the output process (K)(t) is indeed approimately equal to the WS of the input process (t) multiplied by the squared Weyl symbol of the LTV system K. Similar results (not shown) are obtained for the ES/TES. We nally applied the WS, ES, and TES to cylinder pressure signals measured in the course of combustion cycles in a car engine 4 [43, 44]. This process is well described by the multicomponent process model discussed in Section 7. (see (31)). The signal corresponding to a given combustion cycle consists of several resonant components (due to knocking). Within one cycle, the resonance frequencies decrease with time due to the decreasing gas temperature. All spectra shown are based on an estimate of the process' correlation function that was derived from 149 realizations corresponding to 149 dierent combustion cycles. Fig. 6 shows that the resulting WS is considerably more concentrated than the ES/TES. In particular, the ES/TES does not 4 We are grateful to D. Konig and J. F. Bohme and to Volkswagen for making these data accessible to us. 0

21 (a) (b) (c) (d) f f f t t t t (e) (f) (g) (h) f f f t t t Area=1 Figure 6. Cylinder pressure process: (a) typical process realization, (b) WS, (c) ES/TES with positive semidenite innovations system, (d)-(f) GES with causal innovations system with (d) = 0, (e) = 1=, (f) = 1=, (g) WVS, and (h) EAF magnitude. clearly indicate the decrease of the resonance frequencies. Fig. 6 also shows that the results obtained with the positive semidenite innovations system are much better than those obtained with the causal innovations system. Finally, it is seen that the WS shows better TF concentration and contains less interference terms than the WVS. 9 Conclusions We have introduced and studied a family of time-varying spectra called generalized evolutionary spectrum (GES). While two prominent special cases of the GES are the classical evolutionary spectrum and the recently introduced transitory evolutionary spectrum, we have shown that another special case of the GES, the novel Weyl spectrum (WS), features signicant advantages over all other GES members. Based on the denition of the GES in terms of an innovations system of the process under analysis, we have furthermore shown the importance of an underspread property for a satisfactory interpretation of the GES as a time-varying spectrum. ere again, the WS is advantageous since it merely requires the process to be underspread whereas the other GES members require the process to be strictly underspread. We have also shown and veried by simulations that in the underspread case the WS is approimately identical to the Wigner-Ville spectrum; for deterministic signal components, however, it is more concentrated than the Wigner-Ville spectrum. 1

22 Appendi A: Proof of Approimation () In order to prove 5 (), we rst note that Efjh; g (t;f ) ij g = Efjhn; g (t;f ) ij g with g (t;f ) (t 0 ) = g(t 0 t) e jft0 can be reformulated as Efjhn; g (t;f ) ij g = 0 S (; ) S( 0 ; 0 ) A g ( 0 ; 0 ) e j(( 0 )f( 0)t) e j( 0 0 ) d d d 0 d 0 ; 0 where A g (; ) is the ambiguity function [40, 45] of g(t). Similarly, using (4) the GES can be written as GES () (t; f) = L () (t; f) = The dierence 0 (t; f) = Efjh; g (t;f ) ij g GES () 0 (t; f) = S () (; ) S() ( 0 ; 0 ) e j(( 0 )f( 0 )t) e j( 0 0 ) d d d 0 d 0 : (t; f) can hence be epressed as 0 S (; ) S ( + 0 ; + 0 ) e j( 0f 0t) e j( 0 h 1 A g ( 0 ; 0 ) e j() (;; 0 ; 0 ) i ) d d d 0 d 0 ; where () (; ; 0 ; 0 ) = 0 0 ( ) and the domains of integration are (; ) [ ; ] [ ; ] and ( 0 ; 0 ) [ ; ] [ ; ] since is assumed to be strictly underspread. The magnitude of 0 (t; f) can now be bounded as j 0 (t; f)j js (; )j S ( + 0 ; + 0 ) 1 Ag ( 0 ; 0 ) e j() 0 0 tr n 1= o where we have used js (; )j trn 1= 6 4 o (;; 0 ; 0 ) 1 A g ( 0 ; 0 ) e j() (;; 0 ; 0 ) d d d d d 0 d d 0 d 0 ; [3]. Since the domain of integration is very concentrated around the origin, we use the approimation A g (; ) 1 (a more rigorous but lengthier derivation could be given by using a higher-order Taylor epansion of A g (; ) around (0; 0)). Furthermore using j1 e j() (;; 0 ; 0 ) j jsin( () (; ; 0 ; 0 ))j j () (; ; 0 ; 0 )j [(1 + jj)(j 0 j + j 0 j) + jjj 0 0 j] (1 + jj) (on the integration domain) leads, after integrating, to the nal bound j 0 (t; f)j with 8 (1 + n o jj) 3 tr 1=. This bound approaches zero with decreasing spread of the innovations system,, which proves the approimation (). Appendi B: Proof of the Bounds (3) The dierence between two GES based on the same innovations system can be epressed as (t; f) = GES ( 1) (t; f) GES ( ) (t; f) = L ( 1) (t; f) 1) L( (t; f) L( ) (t; f) ) L( (t; f) 5 The basic technique of proof used here and also in Appendi B has been developed in [3]. We note that other bounds and approimations mentioned in this paper can be derived in a similar manner.

23 = 0 0 h i S (1) (; ) 1) S( ( 0 ; 0 ) 1 e j( 0 0 ) e j[( 0 )t( 0 )f ] d d d 0 d 0 ; where we have used (4) and the relation S ( ) (; ) = S( 1) (; ) ej with = 1. The magnitude of (t; f) can then be bounded as j(t; f)j = js (; )j S ( 0 ; 0 ) 1 e j( 0 0 ) d d d 0 d js (; )j S ( 0 ; 0 ) sin ( 0 0 ) d d d 0 d 0 : We now assume that is strictly underspread, which means that the (eective) support of S () (; ) is enclosed by the rectangle [ ; ] [ ; ], which limits the domain of integration accordingly. Using j sin j jj and the fact that j 0 0 j within the domain of integration, we obtain further Using js (; )j trn 1= trn 1= j(t; f)j jj 4 o [3] it follows that js (; )j d d js (; )j d d tr S ( 0 ; 0 ) d 0 d 0 : n o 1= o n 4. This nally yields the rst bound in (3), j(t; f)j jj (4 ) 3 tr 1= We net consider the L -norm of (t; f). With (4) and Schwarz' inequality, we obtain kk = = t f j(t; f)j dt df = 6 44 t f L ( 1) (t; f) L( 1) (t; f) L( ) (t; f) ) L( (t; f) dt df o. d d = h i S ( 1) ( + 0 ; + 0 ) S ( 1) ( 0 ; 0 ) 1 e j( ) d 0 d 0 d d 3 S ( 0 ; 0 ) sin ( ) d 0 d S ( + 0 ; + 0 ) d 0 d d d; where the relation S ( ) (; ) = S( 1) (; ) ej has been used. With sin and the fact that j j 8 within the domain of integration, we obtain further where we have used kk 16 (4 ) = 64 (4 ) 3 tr f g ; S ( 0 ; 0 ) d 0 d d d js (; )j d d = kk = trf g. This proves the second bound in (3). 3