MARGINALS VS. COVARIANCE IN JOINT ... - Semantic Scholar
MARGINALS VS. COVARIANCE IN JOINT DISTRIBUTION THEORY Richard G. Baraniuk * Department of Electrical and Computer Engineering Rice University Houston, T X 77251-1892. USA
ABSTRACT Recently, Cohen has proposed a method for constructing joint distribut,ions of arbitrary physical quantities, in direct generalization of joint. time-frequency representations. In this paper, we investigate the covariance properties of this procedure and caution that in( its present form it cannot generate all possible distributions. Using group theory, we extend Cohen’s construction to a more general form that can be customized to satisfy specific marginal and covariance requirements. 1.
INTRODUCTION
paper aims t o understand the connections between margin - 7 . and covariance in the context of this powerful method for generating joint distribution classes.
2.
ENERGY DENSITIES
An energy density measures the content of a physical quantity (time, frequency, scale, for example) in a signal s.’ 2.1.
D e n s i t y of a S i n g l e Variable
H e r m i t i a n Representations. From quantum mechanics we appropriate the associatian of physical quantities with Hermitian operators on the vector space L2(R) of square integrable signals. We use capital letters t o represent these operators, with A and B corresponding t o the arbitrary variables a and b. Examples include [6-81: Time (Ts)(z) = z ~ ( z ) Fre; q u e n c y ( F s ) ( z )= &i(z); Time Scale (Log Time) ( D s ) ( z )= (log T s ) ( z )= log(z) s(z), z > 0 ; Frequency Scale (Log Frequency) (D’s)(z) = (logFs)(z) = F-llog(f)S(f), f > 0; Mellin ( H s ) ( z )= i ( T F F T ) ; Fourier-MeZZin H i = F-’HF; and Power Frequency (Chirp) F ” , including Inverse Frequency R = F - ’ . (We use F t o denote the usual Fourier transform S(f)= ( F s ) ( f ) . )
Joint distributions of arbitrary variables extend the notion of time-frequency analysis t o quantities such as scale, Mellin, chirp rate, and inverse frequency. Two complementary approaches t o constructing distributions have been developed. Covariance-based methods [l-41 concentrate on certain canonical signal transformations that leave the form of the distribution unchanged, while marginal-based methods [5,6] aim for the properity that integrating out one variable leaves the valid density alf the other. Both approaches have their merits, but also their limitations. Covariance approaches rely on a group structure that may not be present in general, while the generality of marginal methods makes characterizing the Averages of a physical quantity a can be computed resulting distributions difficult. In particular, the difficulties using the operator representation A through (s1Als) = introduced in dealing with the noncommuting operators that (5, A s ) = s s * ( z ) ( A s ) ( z )d ~ For . ~ example, the average represent most physical quantities has lead to a simplified (mean) frequency of a time signal s is given by ( s l F l s ) = marginal methLod, the kernel method of Cohen and Scullys s * ( z ) i ( z ) dz/27rj = IS(f)12df. This generalizes in the Cohen [5,6], that if not used carefully can limit the range of obvious way t o the average value of a function g ( a ) . possible joint clistributions. T h e case of time ls(d)l’ and inverse frequency 1r-l S ( T - ~ ) ~ ’ Densities. T h e square of the expansion onto the eigenof the Hermitian representation measures distributions illustrates the essence of our discussion. T h e functions the quantity of the associated concept in the signal. Define kernel method yields a class of distributions [6, p. 2391 (FAs)(u)= (5, for the variable a. T h e transforms measuring the quantities introduced above are [6,7]: Time ( F ~ s ) ( t=) s ( t ) ; Frequency (Fps)(f) = S(f); Time Scale (Fos)(d) = parameterized by a kernel function that is assumed t o e d / ’ s ( e d ) ; Frequency Scale (Fps)(d) = F - ’ e q / ’ S ( e q ) ; MelIin ) Ss(z) e--jZrhlogr dl Fourier-Mellin FHI= FHF. contain a!l distributions having time and inverse frequency ( F ~ s ) ( h= marginals. (Here A , is the narrow-band ambiguity func- and Power Frequency (FFnS)(r) = rl/nS(rl’n). tion of 3.) However, this class does not contain all of them, one notable outcast being [l], [6, p. 2401 ( P s ) ( t , r )= U n i t a r y Representations of physical quantities are obS S * ( X ( u ) / r S(X(-u)/r) ) eJ2aut/rjL2(u) du, with X(u) = tained by exponentiating Hermitian representations, via Note and p’(u) = X(u)X(-U). In contrast t o the time and the formal Taylor series e J Z f f a Afrequency shift covariance forced on distributions of the form that lF~s1’is invariant t o the unitary operator eJ21raA Z s , Ps is covatriant to time shifts and scale changes. This IFAelaraA SI’ = lF~s1’- and thus e l z x a Acannot correspond
-+
sf
et(.)
et)
x;
5
w.
*This work was supported by the National Science Foundation, grant no. MIP-9457438, and by the State of Texas, grant no. TXATP 003604-002. Email address: richberice. edu
See [S-S] for detailed discussions on this topic. ’ We adopt the physicists’ notation of inner product linear in second element.
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0-7803-2431 4 / 9 5 $4.00 0 1995 IEEE
directly t o the physical quantity a. (In fact, it corresponds to some "orthogonal" concept [7].) To find the unitary operator representing the variable a , we solve the operator equation FAA,, = T : I F ~ ,with (~:s)(z) = s(z - k) the additive translation operator [8]. T h e density FA is covariant to A, and thus a , A , and A are equivalent representations of the same physical concept. Covariances other than additive can be handled by group methods [SI; covariance via multiplicative translation is defined using (T:s)(z) = 3 ( z / k ) / & in the above. We will use script letters to denote unitary operators. Continuing the examples from above, we have for unitary representations: Time (7ts)(z) = s(z - t ) = (e--@"'Fs)(z); Frequency (.Tfs)(z) = e J Z r f ss(z) =, ( e 3 2 " f T s ) ( x ) ; T i m e ; Scale ( Z ) d S ) ( Z ) = e - d / 2 s ( z e - d ) = ( e 3 2 " d H s ) ( x ) Frequency Scale Z); = V - d ; MeZlin (~Fth3)(z)= eJ2ah10gzs(x) = ( e J Z n h D s ) ( zFourier-Mellin ); 7-l; = F - ' X h F ; and Power Frequency ( R : s ) ( z ) = F - ' S [ ( f " - r)''"] (f" - r)'ln f'/".
some sense still measure joint a-b energy content. Example. Time and Frequency: Employing the operator pair T , F in (1) yields the classical Cohen's class of time-frequency representations having time and frequency marginals [ 5 , 6 ] . Included among t h e generalized distributions t h a t do not marginalize are such workhorse representations as the spectrogram and the smoothed pseudo Wigner distribution.
2.3. Extended Covariance T h e a-indicating transform FA can be covariant t o multiple operators in addition t o A. For instance, we might have FAA:, = TZFA with A' # A and T G a unitary group translation different from addition. Thus, in addition t o A content, FA also measures A' content, in a subtlely different way. In certain applications, this extended covariance of the transform FA can be more important than A covariance. Time and frequency provide an excellent example of this behavior, since both the signal 3 ( t ) and its Fourier transform are covariant t o scale changes z ) d by multiplicative transC h a r a c t e r i s t i c Functions provide a direct route t o densities s(f) circumventing eigenanalysis [ 5 , 6 ] . Given a density [(FAs)(a)I2, lation. Thus, s ( t ) and S ( f ) can each be interpreted as in some its inverse Fourier transform defines a characteristic (ambigu- sense measuring the "scale content" of the signal in the time eJ2nQada. Since the and frequency domains [9].4 T h e scale covariance of the power ity) function ( M s ) ( a ) = I(FAS)(U)I~ right side of this ex ression corresponds t o an average of the and inverse frequency transform F p proves far more useful with respect t o the density of a, it than its complicated additive covariance t o X:. quantity g ( a ) = e3 can be computed directly from the signal via (see "Averages" 3. COVARIANCE OF DISTRIBUTIONS above) ( M s ) ( a ) = (s e J 2 x " A s). Combining these two In many applications, the covariance of a distribution rivals equations yields I(FAs)(a.)12 = s ( s e J Z r a A 3 ) da. the marginals in importance. A joint distribution Cs i s covariant to a unitary operator X , if ( C X s s ) ( a ,b ) = ( C s ) ( a ' ,b'), 2.2. J o i n t D e n s i t i e s V i a C h a r a c t e r i s t i c F u n c t i o n s Joint densities attempt t o indicate the simultaneous content where a', b' are functions of a, b, and z,preferably a group law. of two (or more) physical quantities in a signal. Scuily and With joint distributions, it seems reasonable t o expect covariCohen, guided by the one-dimensional chwacteristic function ance to two operators X and y , one relating t o each variable. it can be method, derived a formula for the joint density of a and b [6] Since Cs simply "shifts" in response t o s H Xzyys, interpreted as somehow measuring the X-y content of signals in addition t o the a-b content. Options for the pair X , y in(Cs)(a,b ) = (s e ' 2 x ( o A + P B ) s e 3 2 n ( a a + P b ) da d p . (1) A, 0 , and any d u d e various combinations of extended covariance operators for FA and FB. Cohen's class Like a true density, this functional marginalizes to the indi- of time-frequency distributions provides a fine example of covidual densities of a and 6 ; t h a t is, integration over b yields f) = ( C s ) ( t - z,f - y ) , changes in variance; since (CTZFys)(t, the time-frequency origin do not affect the properties of the / ( C s ) ( a , b ) d b = (s e J a x a A13) e-J2xoada = I ( F ~ s ) ( a ) l ~ ,distributions. Covariance is deemphasized in the characteristic function with a similar result for integration over a. method of Scully and Cohen (Section 2.2.). While (1) can Joint distributions are not unique. Since in general A and B construct all distributions with correct marginals irrespective do not commute ( A B # B A ) , the e x p o n e h a 1 eJ2n(aA+BB)in of the choice of variables, the covariance properties of the re(1) can be evaluated in many ways, giving different distribu- sults are neither predicted nor ensured. Furthermore, recall tions satisfying the same marginals. T h e t~hreesimplest corre- from the Introduction that there exist variables a, b (time and spondence rules are [5,6]: the symmetric Weyl correspondence inverse frequency, for example) for which a simple weighting of one fixed correspondence rule with a kernel @ ( a , cannot e 3 2 r ( a A t B B ) , where the sum A+B is exponentiatcd ensemble, mimic all possible rules. In order t o explain this apparent the distributed correspondence e 3 2 " 3 Ae J Z s p Bej21rqA, where inconsistency, we now undertake a study of the relationship A a n d E are e x p o n e n t i a t e d s e p a r a t c l y a n d t h e n c o m p o s e d , between the marginal and covariance properties of the charand the similar simpler correspondence e J Z n a Ae32"pB. De- acteristic function method. spite the ordering differences, every correspondence rule yields T h e Hermitian operators A , B that determine the marginal a distribution marginalizing t o IFAS/'and l F ~ s 1 ~ . properties of the distributions constructed by the characSince keeping track of all possible correspondence rules is an arduous task, Cohen fixes a single correspondence and then inScale: Covariance and Confusion. Since the Fourier transform serts a fixed kernel function3 @(alp)in (1) t o take care of the FF is invariant to time shifts It in a signal, we know that it does not measure time content. Similarly, since the Mellin transform FH other possibilities [ 5 , 6 ] . Constraining @ ( a0) , = @(a,p ) = 1 is invariant to scale changes z ) d , it does not and c a n n o t measure V C Y , p generates distributions with correct marginals. Leav- the scale content of signals, contrary to popular belief. The density ing @ unconstrained generalizes the concept of distribution t o that is covariant to Z)d - and thus measures scale content - was representations that may not have correct marginals yet in derived in [9] as precisely the transform ~ ( F D s ) . ( z=) e5 Is(e5)I2 indicating the amount of signal stretching required to move the point 3: to the reference point z o = 1. We consider only signal-independent kernels in this paper.
F
1
11 I J
I
I
I
1)-
1
a)
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teristic function method also control the covariance properties, through the unitary operator G(a,P ) = e J 2 r ( u A + P B ) that anchors the characteristic function in (1) [3,8]. Covariance depends almost entirely on whether this operator is a representation1 of some 2-d group [lo]; that is, on whether G(ai,PI) B(a2,P 2 ) = G[(ai,Bi)*c(a2, P z ) ] with group law * g . Equivalently, covariance demands that the sum A + B generate a 2-d group ~ e p r e s e n t a t i o n . ~ E x a m p l e s . T h e CBH expansions (see footnote 5 below) of the pairs T+ F , log T + H , and log F + H ‘ all terminate after three distinct terms, with the third a simple phase factor. T h e group action, ( Q ~ , P I ) * ( Q Z , P = Z ) ( a 1 + a z 1 P 1 + P 2 ) neglecting the phase, is that of the Weyl-Heisenberg group. T h e pairs T + H and F S H ‘ each generate the “az+b” affine group, with action (a,, e o ” ) 0 ( 0 2 , e p z ) = (a1 e P 1 a z ,eP1 e a 2 ) . T h e CBH expansion of the pair T + F 2 terminates after three distinct terms in T , l”, and F . Thus, while these three operators generate a 3-d group, T + F 2 does not generate a 2-d group on its own. Similarly, the expansion of T+R comprises terms i i i T , R , F-2, and F-3. T h e expansion of l o g T + F becomes a complicated mess after three terms.
and eJ2rrBB,either employ the group symmetrization procedure of Kirillov [ 3 ] ,solve an eigenequation [6], or use the C R r I Theorem. Because of the strong parallel to the time-frqiipr case, we will call this special distribution the a-b Wrgner drstrzbutzon ( W s ) ( a ,b ) . Step 2. To generate a class of distributions covariant to 9. “smooth” the Wigner distribution. T h e coadjoint group law supports two types of smoothed distributions. T h e first, p ;
(c:s)(a, b) =
4.
COVARIANCE VS. M A R G I N A L S
+
4.1. Best Case: A B Generates a 2-d Group In this, the best of all worlds, the characteristic function method coincides with the group-based approach of Shenoy and Parks [3] and succeeds both on the marginal and covariance fronts. All distributions constructed from the various correspondences boast covariance t o the operator Q appearing in the characteristic function. T h e covariance is of the form ( c G (~~) ,3 ) ( ab ), = ( ~ 3 [)( a , b ) 0; (U, v ) ] , with 0 ;the coadjoint group action [3]. It is interesting t o note t h a t the Cs are covariant to the operators e J Z r f f A and e J Z n P Bt o which the marginal distributions lFAs12 and ~ F B u ~are ’ tnvartant, rather than covariant t o A and B . In fact, A,& is not even a valid group representation in general,6 which has some interesting ramifications for operator pairs not unitarily equivalent t o T and F [7]. T h e group structure of B elicits an extension of the characteristic function method that will generate not only all distributions with correct marginals but also all distributions covariant to the signal transformation s H Bs:
Step 1.
T h e symmetrical
Weyl correspondence rule
e J 2 * ( a A t P B ) yields a unitary distribution [3]. (Note that some
correspondenc’es yield nonunitary distributions, so this choice is nonarbitrarjr.) To evaluate eJ2*(uA+pe)in terms of eJ2ruA
5A useful test of whether A+B generates a group entails expanding G ( 01, P;I using the Campbell-Baker-Hausdorff (CBH) Theorem [lo], which states that ,A+B = ,A e B ,2&[A,BI,&[A,[A311 &[BJA,BIl e&[BJAJA9B111 , , . If this expansion terminates after N distinct terms, then A + B generates a subset of an N-parameter group. ‘For example, T+H generates a representation e J 2 = f Te J 2 r r d H= FfDd of the affine group, but the operators It and ‘HJ, to which the FT and FH marginals are covariant do not represent any group.
[(a,b)
U)] 41(% 8)a
u , U),
with 41 E L1(Rz)and d ( u , U ) the invariant measure, remainc in the a-b domain, while the second,
+
We are now ini a position t o classify the covariance properties of the characlteristic function method. Our discussion partitions naturally into three cases, depending on what type of group structuie the operator representations A and B can support. In addition t o providing new insight into Scully and Cohen’s construction, our analysis indicates its limitations and suggests modifications and extensions.
//(Ws)
(Cis)(u, U ) = / / ( W s ) [ ( a , b ) ~ , z ( uU)],
b) 4 a ,b),
lives in the domain of the arameters a,P of the covariance operators eJ21raA and e J z n P g .With Cis, the marginal distributions are obtained by integrating along curves in the ( U ,U ) plane. Since W is a unitary map, these constructions reach all distributions having G covariance, including all those with FA and FB marginals. A l t e r n a t i v e Step 2. In many cases, the group convolutions of Step 2 can be rewritten using the 2-d Fourier transform in terms of a kernel @(a,,&a , b ) weighting the characteristic function in (l).’ In this case, the functional form of @ ( ap; , a, b ) can be determined directly by imposing the requisite covariance constraints on this generalized version of (1). Analogous t o the spectrogram in the time-frequency case, this class of distributions contains the squared magnitude of a linear “group transform” ( L a ) ( u ,w) = (s I G ( u , U ) I g) that corresponds t o (Cis)(u,U ) from above with 4 2 = W g . Examples. Time and Frequency: T + F generates the WeylHeisenberg group, the usual Wigner distribution, and the classical Cohen’s class of time and frequency covariant distributions. In this case, the two smoothing methods coincide, with 7 and F serving simultaneously as invariant and covariant transformations for the marginals. This follows because time and frequency are in a sense orthogonal concepts [7]. Kernels of the form @(a,p ) can reach all distributions in this class. Frequency Scale and Fourier-Mellin: log F + H ’ generates the Weyl-Heisenberg group, the Altes Q distribution, and the prehyperbolic class [4, 71 of scale and hyperbolic time-shift (NI)covariant distributions. Kernels of the form @ ( ap, ) can reach all distributions in this class. Frequency and Fourier-Mellin: F+H’ generates the affine group, the Bertrand distribution [I], and two classes of time and scale covariant distributions. Distributions C: s have frequency and Fourier-Mellin coordinates. Distributions C; s have time and scale coordinates, and hence inhabit the “affine class” defined in [l,21. Both parameterizations require kernels more general than @(a,p);for example, iP2 must be of the p e d ) , with d the scale variable. form
+
Worst Case: A B Does Not Generate a Group Unfortunately, A and B must be very well matched in order t o generate a 2-d group, as only two 2-parameter groups a p 4.2.
71n general the kernel will not take the form @(a,/3).While Cohen anticipated the value of a-b-dependent kernels in even his original paper [ 5 ] , such kernels have not been identified as critical for some choices of A , B until now.
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pear in nature: IR’ (essentially the Weyl-Heisenberg group) and the affine group. Thus, group theoretic approaches break down in the general case for arbitrary variables, meaning t h a t the characteristic function has no guaranteed covariance p r o p erties. In most cases, the best we can d o is either keep track of a l l correspondence rules or introduce a generalized kernel a,b) in (1) and hope for the best. function @(a,P;
Examples. Since the pair T
+ F 2 generates a nongroup s u b
set of a three-parameter group requiring F , we cannot expect covariance from time and chirp distributions without also including frequency shift as a (third) variable. T h e pair log T+F likewise does not appear t o support any useful covariances.
4.3. Intermediate Case: A + B Salvageable In some cases, we can rescue variables t h a t do not generate groups by dividing their associated distributions into s u b classes that obey certain alternative covariances. Although the characteristic function operator 4 is not a group representation in general, it is potentially compatible with another unitary operator pair X,Y, (there might exist several) that simultaneously: (1) represents a 2-d group, and (2) is physically reasonable in t h a t it acts as an invariance, covariance or extended covariance operator for the marginal distributions. In this case, given a unitary distribution having both the marginals and this alternative covariance, smoothing via the group convolution induced by X,Y, will generate a useful class of distributions. Each such class will contain all distributions covariant t o X,Y, as well as some of the distributions having a,b marginals. Different operator pairs will generate different, potentially nonoverlapping, distribution classes. A procedure for generating a covariance-based subclass of distriL ~ I ~ ~ runs u I L as ~ follows: Step 1. Construct an a-b Wigner distribution using the Weyl or symmetrical correspondence. T h e eigenanalysis method of Cohen [6] can be applied t o reduce the characteristic function operator B into manageable components. T h e resulting distribution is unitary and has correct marginals. Step 2. Find a 2-d unitary group representation X,Y, such that X and Y transform the marginals in meaningful ways (this is where the art comes in). Verify t h a t the Wigner distribution shares these covariances. (If not, recompute using another correspondence, being sure t o verify unitarity.) Smooth W s over the corresponding group as in Section 4.1. In certain cases, the smoothing operation can be implemented in the characteristic function domain by inserting a kernel @ ( c r , / ? ; a , b )in (1). T h e resulting class contains distributions covariant t o X Y , of which will have the correct FA and FB marginals. Examples. Tzme a n d Inverse Frequency illustrate the tradeoff of marginals for covariance. Since the operator pair T+R does not generate a 2-d group, the covariance of t-r distributions rests on shaky ground. However, in addition to representing the affine group, the time-scale operator ?;Dd performs a natural transformation in ( t ,.) coordinates, with FT covariant t o T and 2, and FR invariant t o 7 and covariant to D. Smoothing over the affine group induces the affine class [ l ,21 centered about the distribution P s from the I n t r e duction. This class also contains the popular scalogram [2], the squared magnitude of the continuous wavelet transform (L,s)(t,.) = I J s ( z ) h * ( G ) some
&I2.
T h e time-frequency operator Z3f performs similar duties for the operator pair T+R. Smoothing over the
Weyl-Heisenberg group induces the reparameterized Cohen’s class 2s from the Introduction. Included in = this class is the remapped spectrogram (L’s)(t,.) I ~ s ( zg* ) (z- t ) e-J2*fo/‘ d z
1.’
5.
CONCLUSIONS
By ranging through every possible correspondence rule representing G(a,p ) = eJ’ff(aAtpB)in (l),the characteristic function method of Scully and Cohen can generate every possible distribution having a and b marginals. While certainly general enough, this approach lacks practicality, however, because studying distribution classes through their correspondence rules would be exasperating. Furthermore, this approach can construct only distributions satisfying the marginals, leaving external to the theory most of the useful distributions (spectrogram, scalogram, pseudo Wigner distribution, etc.) employed in practice. Cohen’s elegant kernel method takes care of many of these practical issues, but since it introduces limitations of its own, it must be approached with caution. T h e three-way classification of the operators A and B representing the variables has significant ramifications for joint distribution design. When A+B generates a 2-d group, then a generalized version of the kernel method will succeed and generate a single unique class containing marginalizing and covariant distributions. Since there are only two 2-d groups in nature, most of the work for this case has already been completed: Cohen’s class represents the Weyl-Heisenberg group [5] and the affine class represents the affine group [l-31. Simple coordinate changes (unitary equivalence) can generate all other distribution classes resembling these two [7]. When A+B fails t o generate a group, the kernel method does not appear foolproof. T h e approach of imposing order on the chaos of correspondences through alternative “pseudocovariances” appears promising, because it has generated the affine class, an important representation class that has remained outside the marginal-based theory until this time.’
REFERENCES [l] J. Bertrand, P. Bertrand, ‘‘A class of affine Wigner functions,” J. Math. Phys., vol. 33, pp. 2515-2527, July 1992. I21 0. Rioul, P. Flandrin, “Time-scale energy distributions,” IEEE Trans. Signal PTOC.,vol. 40, pp. 1746-1757, July 1992. [3] R. G. Shenoy, T. W. Parks, “Wide-bandambiguity and affine Wigner distributions,” Signal PTOC.,vol. 41, no. 3, 1995. [4] A. Papandreou, et al, “The hyperbolic class of quadratic timefrequencyrepresentations,”IEEE Trans. Signal Proc., vol. 41, pp. 3425-3444, Dec. 1993. [SI L. Cohen, “Generalized phase space distribution functions,” J. Math. Phys., vol. 7, pp. 781-786, 1966. [SI L. Cohen, Time-Fregzlency Analysis, Prentice-Hall, 1995. [7] R. Baraniuk, D. Jones, “Unitary equivalence: A new twist on signal processing,” IEEE Trans. Signal Proc., 1993. Preprint. [8] R. Baraniuk, “Beyond time-frequency analysis: Energy densities in one and many dimensions,” in PTOC.ICASSP ’94, vol. 111, pp. 357-360, 1994. [9] R. Baraniuk, “A signal transform covariant to scale changes,” Electronics Lefters, vol. 29, pp. 1675-1676, Sept. 17, 1993. [lo] L. Corwin, F. Greenleaf, Representations of Nilpotent Lie GTOUPS and Their Applications, Part I, 1990. ‘Thanks to 3. Bertrand, L. Cohen, P. GonsalvL.s, A. Sayeed, R. Shenoy, and R. Wells for insighlful discussions regarding this material.
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ABSTRACT Recently, Cohen has proposed a method for constructing joint distribut,ions of arbitrary physical quantities, in direct generalization of joint. time-frequency representations. In this paper, we investigate the covariance properties of this procedure and caution that in( its present form it cannot generate all possible distributions. Using group theory, we extend Cohen’s construction to a more general form that can be customized to satisfy specific marginal and covariance requirements. 1.
INTRODUCTION
paper aims t o understand the connections between margin - 7 . and covariance in the context of this powerful method for generating joint distribution classes.
2.
ENERGY DENSITIES
An energy density measures the content of a physical quantity (time, frequency, scale, for example) in a signal s.’ 2.1.
D e n s i t y of a S i n g l e Variable
H e r m i t i a n Representations. From quantum mechanics we appropriate the associatian of physical quantities with Hermitian operators on the vector space L2(R) of square integrable signals. We use capital letters t o represent these operators, with A and B corresponding t o the arbitrary variables a and b. Examples include [6-81: Time (Ts)(z) = z ~ ( z ) Fre; q u e n c y ( F s ) ( z )= &i(z); Time Scale (Log Time) ( D s ) ( z )= (log T s ) ( z )= log(z) s(z), z > 0 ; Frequency Scale (Log Frequency) (D’s)(z) = (logFs)(z) = F-llog(f)S(f), f > 0; Mellin ( H s ) ( z )= i ( T F F T ) ; Fourier-MeZZin H i = F-’HF; and Power Frequency (Chirp) F ” , including Inverse Frequency R = F - ’ . (We use F t o denote the usual Fourier transform S(f)= ( F s ) ( f ) . )
Joint distributions of arbitrary variables extend the notion of time-frequency analysis t o quantities such as scale, Mellin, chirp rate, and inverse frequency. Two complementary approaches t o constructing distributions have been developed. Covariance-based methods [l-41 concentrate on certain canonical signal transformations that leave the form of the distribution unchanged, while marginal-based methods [5,6] aim for the properity that integrating out one variable leaves the valid density alf the other. Both approaches have their merits, but also their limitations. Covariance approaches rely on a group structure that may not be present in general, while the generality of marginal methods makes characterizing the Averages of a physical quantity a can be computed resulting distributions difficult. In particular, the difficulties using the operator representation A through (s1Als) = introduced in dealing with the noncommuting operators that (5, A s ) = s s * ( z ) ( A s ) ( z )d ~ For . ~ example, the average represent most physical quantities has lead to a simplified (mean) frequency of a time signal s is given by ( s l F l s ) = marginal methLod, the kernel method of Cohen and Scullys s * ( z ) i ( z ) dz/27rj = IS(f)12df. This generalizes in the Cohen [5,6], that if not used carefully can limit the range of obvious way t o the average value of a function g ( a ) . possible joint clistributions. T h e case of time ls(d)l’ and inverse frequency 1r-l S ( T - ~ ) ~ ’ Densities. T h e square of the expansion onto the eigenof the Hermitian representation measures distributions illustrates the essence of our discussion. T h e functions the quantity of the associated concept in the signal. Define kernel method yields a class of distributions [6, p. 2391 (FAs)(u)= (5, for the variable a. T h e transforms measuring the quantities introduced above are [6,7]: Time ( F ~ s ) ( t=) s ( t ) ; Frequency (Fps)(f) = S(f); Time Scale (Fos)(d) = parameterized by a kernel function that is assumed t o e d / ’ s ( e d ) ; Frequency Scale (Fps)(d) = F - ’ e q / ’ S ( e q ) ; MelIin ) Ss(z) e--jZrhlogr dl Fourier-Mellin FHI= FHF. contain a!l distributions having time and inverse frequency ( F ~ s ) ( h= marginals. (Here A , is the narrow-band ambiguity func- and Power Frequency (FFnS)(r) = rl/nS(rl’n). tion of 3.) However, this class does not contain all of them, one notable outcast being [l], [6, p. 2401 ( P s ) ( t , r )= U n i t a r y Representations of physical quantities are obS S * ( X ( u ) / r S(X(-u)/r) ) eJ2aut/rjL2(u) du, with X(u) = tained by exponentiating Hermitian representations, via Note and p’(u) = X(u)X(-U). In contrast t o the time and the formal Taylor series e J Z f f a Afrequency shift covariance forced on distributions of the form that lF~s1’is invariant t o the unitary operator eJ21raA Z s , Ps is covatriant to time shifts and scale changes. This IFAelaraA SI’ = lF~s1’- and thus e l z x a Acannot correspond
-+
sf
et(.)
et)
x;
5
w.
*This work was supported by the National Science Foundation, grant no. MIP-9457438, and by the State of Texas, grant no. TXATP 003604-002. Email address: richberice. edu
See [S-S] for detailed discussions on this topic. ’ We adopt the physicists’ notation of inner product linear in second element.
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0-7803-2431 4 / 9 5 $4.00 0 1995 IEEE
directly t o the physical quantity a. (In fact, it corresponds to some "orthogonal" concept [7].) To find the unitary operator representing the variable a , we solve the operator equation FAA,, = T : I F ~ ,with (~:s)(z) = s(z - k) the additive translation operator [8]. T h e density FA is covariant to A, and thus a , A , and A are equivalent representations of the same physical concept. Covariances other than additive can be handled by group methods [SI; covariance via multiplicative translation is defined using (T:s)(z) = 3 ( z / k ) / & in the above. We will use script letters to denote unitary operators. Continuing the examples from above, we have for unitary representations: Time (7ts)(z) = s(z - t ) = (e--@"'Fs)(z); Frequency (.Tfs)(z) = e J Z r f ss(z) =, ( e 3 2 " f T s ) ( x ) ; T i m e ; Scale ( Z ) d S ) ( Z ) = e - d / 2 s ( z e - d ) = ( e 3 2 " d H s ) ( x ) Frequency Scale Z); = V - d ; MeZlin (~Fth3)(z)= eJ2ah10gzs(x) = ( e J Z n h D s ) ( zFourier-Mellin ); 7-l; = F - ' X h F ; and Power Frequency ( R : s ) ( z ) = F - ' S [ ( f " - r)''"] (f" - r)'ln f'/".
some sense still measure joint a-b energy content. Example. Time and Frequency: Employing the operator pair T , F in (1) yields the classical Cohen's class of time-frequency representations having time and frequency marginals [ 5 , 6 ] . Included among t h e generalized distributions t h a t do not marginalize are such workhorse representations as the spectrogram and the smoothed pseudo Wigner distribution.
2.3. Extended Covariance T h e a-indicating transform FA can be covariant t o multiple operators in addition t o A. For instance, we might have FAA:, = TZFA with A' # A and T G a unitary group translation different from addition. Thus, in addition t o A content, FA also measures A' content, in a subtlely different way. In certain applications, this extended covariance of the transform FA can be more important than A covariance. Time and frequency provide an excellent example of this behavior, since both the signal 3 ( t ) and its Fourier transform are covariant t o scale changes z ) d by multiplicative transC h a r a c t e r i s t i c Functions provide a direct route t o densities s(f) circumventing eigenanalysis [ 5 , 6 ] . Given a density [(FAs)(a)I2, lation. Thus, s ( t ) and S ( f ) can each be interpreted as in some its inverse Fourier transform defines a characteristic (ambigu- sense measuring the "scale content" of the signal in the time eJ2nQada. Since the and frequency domains [9].4 T h e scale covariance of the power ity) function ( M s ) ( a ) = I(FAS)(U)I~ right side of this ex ression corresponds t o an average of the and inverse frequency transform F p proves far more useful with respect t o the density of a, it than its complicated additive covariance t o X:. quantity g ( a ) = e3 can be computed directly from the signal via (see "Averages" 3. COVARIANCE OF DISTRIBUTIONS above) ( M s ) ( a ) = (s e J 2 x " A s). Combining these two In many applications, the covariance of a distribution rivals equations yields I(FAs)(a.)12 = s ( s e J Z r a A 3 ) da. the marginals in importance. A joint distribution Cs i s covariant to a unitary operator X , if ( C X s s ) ( a ,b ) = ( C s ) ( a ' ,b'), 2.2. J o i n t D e n s i t i e s V i a C h a r a c t e r i s t i c F u n c t i o n s Joint densities attempt t o indicate the simultaneous content where a', b' are functions of a, b, and z,preferably a group law. of two (or more) physical quantities in a signal. Scuily and With joint distributions, it seems reasonable t o expect covariCohen, guided by the one-dimensional chwacteristic function ance to two operators X and y , one relating t o each variable. it can be method, derived a formula for the joint density of a and b [6] Since Cs simply "shifts" in response t o s H Xzyys, interpreted as somehow measuring the X-y content of signals in addition t o the a-b content. Options for the pair X , y in(Cs)(a,b ) = (s e ' 2 x ( o A + P B ) s e 3 2 n ( a a + P b ) da d p . (1) A, 0 , and any d u d e various combinations of extended covariance operators for FA and FB. Cohen's class Like a true density, this functional marginalizes to the indi- of time-frequency distributions provides a fine example of covidual densities of a and 6 ; t h a t is, integration over b yields f) = ( C s ) ( t - z,f - y ) , changes in variance; since (CTZFys)(t, the time-frequency origin do not affect the properties of the / ( C s ) ( a , b ) d b = (s e J a x a A13) e-J2xoada = I ( F ~ s ) ( a ) l ~ ,distributions. Covariance is deemphasized in the characteristic function with a similar result for integration over a. method of Scully and Cohen (Section 2.2.). While (1) can Joint distributions are not unique. Since in general A and B construct all distributions with correct marginals irrespective do not commute ( A B # B A ) , the e x p o n e h a 1 eJ2n(aA+BB)in of the choice of variables, the covariance properties of the re(1) can be evaluated in many ways, giving different distribu- sults are neither predicted nor ensured. Furthermore, recall tions satisfying the same marginals. T h e t~hreesimplest corre- from the Introduction that there exist variables a, b (time and spondence rules are [5,6]: the symmetric Weyl correspondence inverse frequency, for example) for which a simple weighting of one fixed correspondence rule with a kernel @ ( a , cannot e 3 2 r ( a A t B B ) , where the sum A+B is exponentiatcd ensemble, mimic all possible rules. In order t o explain this apparent the distributed correspondence e 3 2 " 3 Ae J Z s p Bej21rqA, where inconsistency, we now undertake a study of the relationship A a n d E are e x p o n e n t i a t e d s e p a r a t c l y a n d t h e n c o m p o s e d , between the marginal and covariance properties of the charand the similar simpler correspondence e J Z n a Ae32"pB. De- acteristic function method. spite the ordering differences, every correspondence rule yields T h e Hermitian operators A , B that determine the marginal a distribution marginalizing t o IFAS/'and l F ~ s 1 ~ . properties of the distributions constructed by the characSince keeping track of all possible correspondence rules is an arduous task, Cohen fixes a single correspondence and then inScale: Covariance and Confusion. Since the Fourier transform serts a fixed kernel function3 @(alp)in (1) t o take care of the FF is invariant to time shifts It in a signal, we know that it does not measure time content. Similarly, since the Mellin transform FH other possibilities [ 5 , 6 ] . Constraining @ ( a0) , = @(a,p ) = 1 is invariant to scale changes z ) d , it does not and c a n n o t measure V C Y , p generates distributions with correct marginals. Leav- the scale content of signals, contrary to popular belief. The density ing @ unconstrained generalizes the concept of distribution t o that is covariant to Z)d - and thus measures scale content - was representations that may not have correct marginals yet in derived in [9] as precisely the transform ~ ( F D s ) . ( z=) e5 Is(e5)I2 indicating the amount of signal stretching required to move the point 3: to the reference point z o = 1. We consider only signal-independent kernels in this paper.
F
1
11 I J
I
I
I
1)-
1
a)
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teristic function method also control the covariance properties, through the unitary operator G(a,P ) = e J 2 r ( u A + P B ) that anchors the characteristic function in (1) [3,8]. Covariance depends almost entirely on whether this operator is a representation1 of some 2-d group [lo]; that is, on whether G(ai,PI) B(a2,P 2 ) = G[(ai,Bi)*c(a2, P z ) ] with group law * g . Equivalently, covariance demands that the sum A + B generate a 2-d group ~ e p r e s e n t a t i o n . ~ E x a m p l e s . T h e CBH expansions (see footnote 5 below) of the pairs T+ F , log T + H , and log F + H ‘ all terminate after three distinct terms, with the third a simple phase factor. T h e group action, ( Q ~ , P I ) * ( Q Z , P = Z ) ( a 1 + a z 1 P 1 + P 2 ) neglecting the phase, is that of the Weyl-Heisenberg group. T h e pairs T + H and F S H ‘ each generate the “az+b” affine group, with action (a,, e o ” ) 0 ( 0 2 , e p z ) = (a1 e P 1 a z ,eP1 e a 2 ) . T h e CBH expansion of the pair T + F 2 terminates after three distinct terms in T , l”, and F . Thus, while these three operators generate a 3-d group, T + F 2 does not generate a 2-d group on its own. Similarly, the expansion of T+R comprises terms i i i T , R , F-2, and F-3. T h e expansion of l o g T + F becomes a complicated mess after three terms.
and eJ2rrBB,either employ the group symmetrization procedure of Kirillov [ 3 ] ,solve an eigenequation [6], or use the C R r I Theorem. Because of the strong parallel to the time-frqiipr case, we will call this special distribution the a-b Wrgner drstrzbutzon ( W s ) ( a ,b ) . Step 2. To generate a class of distributions covariant to 9. “smooth” the Wigner distribution. T h e coadjoint group law supports two types of smoothed distributions. T h e first, p ;
(c:s)(a, b) =
4.
COVARIANCE VS. M A R G I N A L S
+
4.1. Best Case: A B Generates a 2-d Group In this, the best of all worlds, the characteristic function method coincides with the group-based approach of Shenoy and Parks [3] and succeeds both on the marginal and covariance fronts. All distributions constructed from the various correspondences boast covariance t o the operator Q appearing in the characteristic function. T h e covariance is of the form ( c G (~~) ,3 ) ( ab ), = ( ~ 3 [)( a , b ) 0; (U, v ) ] , with 0 ;the coadjoint group action [3]. It is interesting t o note t h a t the Cs are covariant to the operators e J Z r f f A and e J Z n P Bt o which the marginal distributions lFAs12 and ~ F B u ~are ’ tnvartant, rather than covariant t o A and B . In fact, A,& is not even a valid group representation in general,6 which has some interesting ramifications for operator pairs not unitarily equivalent t o T and F [7]. T h e group structure of B elicits an extension of the characteristic function method that will generate not only all distributions with correct marginals but also all distributions covariant to the signal transformation s H Bs:
Step 1.
T h e symmetrical
Weyl correspondence rule
e J 2 * ( a A t P B ) yields a unitary distribution [3]. (Note that some
correspondenc’es yield nonunitary distributions, so this choice is nonarbitrarjr.) To evaluate eJ2*(uA+pe)in terms of eJ2ruA
5A useful test of whether A+B generates a group entails expanding G ( 01, P;I using the Campbell-Baker-Hausdorff (CBH) Theorem [lo], which states that ,A+B = ,A e B ,2&[A,BI,&[A,[A311 &[BJA,BIl e&[BJAJA9B111 , , . If this expansion terminates after N distinct terms, then A + B generates a subset of an N-parameter group. ‘For example, T+H generates a representation e J 2 = f Te J 2 r r d H= FfDd of the affine group, but the operators It and ‘HJ, to which the FT and FH marginals are covariant do not represent any group.
[(a,b)
U)] 41(% 8)a
u , U),
with 41 E L1(Rz)and d ( u , U ) the invariant measure, remainc in the a-b domain, while the second,
+
We are now ini a position t o classify the covariance properties of the characlteristic function method. Our discussion partitions naturally into three cases, depending on what type of group structuie the operator representations A and B can support. In addition t o providing new insight into Scully and Cohen’s construction, our analysis indicates its limitations and suggests modifications and extensions.
//(Ws)
(Cis)(u, U ) = / / ( W s ) [ ( a , b ) ~ , z ( uU)],
b) 4 a ,b),
lives in the domain of the arameters a,P of the covariance operators eJ21raA and e J z n P g .With Cis, the marginal distributions are obtained by integrating along curves in the ( U ,U ) plane. Since W is a unitary map, these constructions reach all distributions having G covariance, including all those with FA and FB marginals. A l t e r n a t i v e Step 2. In many cases, the group convolutions of Step 2 can be rewritten using the 2-d Fourier transform in terms of a kernel @(a,,&a , b ) weighting the characteristic function in (l).’ In this case, the functional form of @ ( ap; , a, b ) can be determined directly by imposing the requisite covariance constraints on this generalized version of (1). Analogous t o the spectrogram in the time-frequency case, this class of distributions contains the squared magnitude of a linear “group transform” ( L a ) ( u ,w) = (s I G ( u , U ) I g) that corresponds t o (Cis)(u,U ) from above with 4 2 = W g . Examples. Time and Frequency: T + F generates the WeylHeisenberg group, the usual Wigner distribution, and the classical Cohen’s class of time and frequency covariant distributions. In this case, the two smoothing methods coincide, with 7 and F serving simultaneously as invariant and covariant transformations for the marginals. This follows because time and frequency are in a sense orthogonal concepts [7]. Kernels of the form @(a,p ) can reach all distributions in this class. Frequency Scale and Fourier-Mellin: log F + H ’ generates the Weyl-Heisenberg group, the Altes Q distribution, and the prehyperbolic class [4, 71 of scale and hyperbolic time-shift (NI)covariant distributions. Kernels of the form @ ( ap, ) can reach all distributions in this class. Frequency and Fourier-Mellin: F+H’ generates the affine group, the Bertrand distribution [I], and two classes of time and scale covariant distributions. Distributions C: s have frequency and Fourier-Mellin coordinates. Distributions C; s have time and scale coordinates, and hence inhabit the “affine class” defined in [l,21. Both parameterizations require kernels more general than @(a,p);for example, iP2 must be of the p e d ) , with d the scale variable. form
+
Worst Case: A B Does Not Generate a Group Unfortunately, A and B must be very well matched in order t o generate a 2-d group, as only two 2-parameter groups a p 4.2.
71n general the kernel will not take the form @(a,/3).While Cohen anticipated the value of a-b-dependent kernels in even his original paper [ 5 ] , such kernels have not been identified as critical for some choices of A , B until now.
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pear in nature: IR’ (essentially the Weyl-Heisenberg group) and the affine group. Thus, group theoretic approaches break down in the general case for arbitrary variables, meaning t h a t the characteristic function has no guaranteed covariance p r o p erties. In most cases, the best we can d o is either keep track of a l l correspondence rules or introduce a generalized kernel a,b) in (1) and hope for the best. function @(a,P;
Examples. Since the pair T
+ F 2 generates a nongroup s u b
set of a three-parameter group requiring F , we cannot expect covariance from time and chirp distributions without also including frequency shift as a (third) variable. T h e pair log T+F likewise does not appear t o support any useful covariances.
4.3. Intermediate Case: A + B Salvageable In some cases, we can rescue variables t h a t do not generate groups by dividing their associated distributions into s u b classes that obey certain alternative covariances. Although the characteristic function operator 4 is not a group representation in general, it is potentially compatible with another unitary operator pair X,Y, (there might exist several) that simultaneously: (1) represents a 2-d group, and (2) is physically reasonable in t h a t it acts as an invariance, covariance or extended covariance operator for the marginal distributions. In this case, given a unitary distribution having both the marginals and this alternative covariance, smoothing via the group convolution induced by X,Y, will generate a useful class of distributions. Each such class will contain all distributions covariant t o X,Y, as well as some of the distributions having a,b marginals. Different operator pairs will generate different, potentially nonoverlapping, distribution classes. A procedure for generating a covariance-based subclass of distriL ~ I ~ ~ runs u I L as ~ follows: Step 1. Construct an a-b Wigner distribution using the Weyl or symmetrical correspondence. T h e eigenanalysis method of Cohen [6] can be applied t o reduce the characteristic function operator B into manageable components. T h e resulting distribution is unitary and has correct marginals. Step 2. Find a 2-d unitary group representation X,Y, such that X and Y transform the marginals in meaningful ways (this is where the art comes in). Verify t h a t the Wigner distribution shares these covariances. (If not, recompute using another correspondence, being sure t o verify unitarity.) Smooth W s over the corresponding group as in Section 4.1. In certain cases, the smoothing operation can be implemented in the characteristic function domain by inserting a kernel @ ( c r , / ? ; a , b )in (1). T h e resulting class contains distributions covariant t o X Y , of which will have the correct FA and FB marginals. Examples. Tzme a n d Inverse Frequency illustrate the tradeoff of marginals for covariance. Since the operator pair T+R does not generate a 2-d group, the covariance of t-r distributions rests on shaky ground. However, in addition to representing the affine group, the time-scale operator ?;Dd performs a natural transformation in ( t ,.) coordinates, with FT covariant t o T and 2, and FR invariant t o 7 and covariant to D. Smoothing over the affine group induces the affine class [ l ,21 centered about the distribution P s from the I n t r e duction. This class also contains the popular scalogram [2], the squared magnitude of the continuous wavelet transform (L,s)(t,.) = I J s ( z ) h * ( G ) some
&I2.
T h e time-frequency operator Z3f performs similar duties for the operator pair T+R. Smoothing over the
Weyl-Heisenberg group induces the reparameterized Cohen’s class 2s from the Introduction. Included in = this class is the remapped spectrogram (L’s)(t,.) I ~ s ( zg* ) (z- t ) e-J2*fo/‘ d z
1.’
5.
CONCLUSIONS
By ranging through every possible correspondence rule representing G(a,p ) = eJ’ff(aAtpB)in (l),the characteristic function method of Scully and Cohen can generate every possible distribution having a and b marginals. While certainly general enough, this approach lacks practicality, however, because studying distribution classes through their correspondence rules would be exasperating. Furthermore, this approach can construct only distributions satisfying the marginals, leaving external to the theory most of the useful distributions (spectrogram, scalogram, pseudo Wigner distribution, etc.) employed in practice. Cohen’s elegant kernel method takes care of many of these practical issues, but since it introduces limitations of its own, it must be approached with caution. T h e three-way classification of the operators A and B representing the variables has significant ramifications for joint distribution design. When A+B generates a 2-d group, then a generalized version of the kernel method will succeed and generate a single unique class containing marginalizing and covariant distributions. Since there are only two 2-d groups in nature, most of the work for this case has already been completed: Cohen’s class represents the Weyl-Heisenberg group [5] and the affine class represents the affine group [l-31. Simple coordinate changes (unitary equivalence) can generate all other distribution classes resembling these two [7]. When A+B fails t o generate a group, the kernel method does not appear foolproof. T h e approach of imposing order on the chaos of correspondences through alternative “pseudocovariances” appears promising, because it has generated the affine class, an important representation class that has remained outside the marginal-based theory until this time.’
REFERENCES [l] J. Bertrand, P. Bertrand, ‘‘A class of affine Wigner functions,” J. Math. Phys., vol. 33, pp. 2515-2527, July 1992. I21 0. Rioul, P. Flandrin, “Time-scale energy distributions,” IEEE Trans. Signal PTOC.,vol. 40, pp. 1746-1757, July 1992. [3] R. G. Shenoy, T. W. Parks, “Wide-bandambiguity and affine Wigner distributions,” Signal PTOC.,vol. 41, no. 3, 1995. [4] A. Papandreou, et al, “The hyperbolic class of quadratic timefrequencyrepresentations,”IEEE Trans. Signal Proc., vol. 41, pp. 3425-3444, Dec. 1993. [SI L. Cohen, “Generalized phase space distribution functions,” J. Math. Phys., vol. 7, pp. 781-786, 1966. [SI L. Cohen, Time-Fregzlency Analysis, Prentice-Hall, 1995. [7] R. Baraniuk, D. Jones, “Unitary equivalence: A new twist on signal processing,” IEEE Trans. Signal Proc., 1993. Preprint. [8] R. Baraniuk, “Beyond time-frequency analysis: Energy densities in one and many dimensions,” in PTOC.ICASSP ’94, vol. 111, pp. 357-360, 1994. [9] R. Baraniuk, “A signal transform covariant to scale changes,” Electronics Lefters, vol. 29, pp. 1675-1676, Sept. 17, 1993. [lo] L. Corwin, F. Greenleaf, Representations of Nilpotent Lie GTOUPS and Their Applications, Part I, 1990. ‘Thanks to 3. Bertrand, L. Cohen, P. GonsalvL.s, A. Sayeed, R. Shenoy, and R. Wells for insighlful discussions regarding this material.
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