DISPLACEMENT-COVARIANT TIME-FREQUENCY ENERGY DISTRIBUTIONS - Acoustics, Speech, and Signal Processing, 1995. ICASSP-95., 1995 International Conference on

-’We present a theory of quadratic time-frequency (TF) energy distributions that satisfy a covariance property and generalized marginal properties. The theory coincides with the characteristic function method of Cohen and Earaniuk in the special case of ‘‘conjugate operators.” 1 I N T R O D U C T I O N A N D O U T L I N E Important c1.asses of quadratic time-frequency representations (QTFRs), such as Cohen’s class’ and the affine, hyperbolic, and power classes [1]-[8], are special cases within a general theory of displacement-covariant Q T F R s [9]. This theory (briefly reviewed in Section 2) is based on the concept of t imefrequlency displacement operators (DOS). In Section 3, we shall consider the important separable case where a DO can be decomposed into two “partial DOS” (PDOs). Section 4 defines marginal properties associated to the PDOs and derives constraints on the QTFR kernels. Section 5 shows that, for “conjugate” PDOs, our theory coincides with ithe characteristic function method of [lo, 111. 2 D I S P L A C E M E N T C O V A R I A N T Q T F R s Time-Frequency Displacement Operators . A DO is a family of unitary, linear operators De defined on a linear space X c &(R) of finite-energy signals z ( t ) , and indexed by the 2D “displacement parameter” 6‘ = ( c u , ~ ) E 2) with D c R2. By definition, De obeys a composition law (1) De$@, = ej‘(el,eZ~ D eloez where o is a binary operation such that V and o form a group’ with identity element 60 and inverse element B-’. The TF displacements produced by a DO are described by its displacement func t ion (DF) d(z, 6 ) : if a signal z ( t ) is localized about a TF point z = (ti f), then (De z ) ( t ) is localized about some other T F point z’ = (t’ , f’) given by z/ = d(z, e ) , which is short for t‘ = d l ( t , f ; c u , p ) , f’ = &(t , f ;a ,P) . The DF’s construmction is discussed in [9]. The DF is assumed to be an invertible, area-preserving mapping of 2 onto 2 (where 2 IIt2 denotes the set of TF points z = ( t , f)), and to obey the csomposition law (cf. (1)) d(d(2, 61), e,) = d(z, 6 1 0 B z ) . (2) The parameter func t ion p(z’,z) of De yields the displacement parameter B that maps z into z’,