Applications and Interpretations of Reduced Interference Time-Frequency Distributions

Many new time-frequency (1-f) distributions with desirable properties may be designed with relative ease by approaching the problem in t,errns of the ambiguity plane ueprescntatioii of the kernel. Careful atteiition to kernel design principles yicltls kcrnels which result in high resolution tbf dislributions with a considerable reduction of the sometimes troublesome interference ferms ohserved when using other distributions such as the well known Wigncr Distribnlion (WD), which otherwise has many desirable properties. The resiilting high rcsoliit,ion distribut,ion with reduced interference is called t,he Reduced Interference Distribution or RID. The RID rctains marly of the desirable properties of tlie W D a,nd is particularly useful for the analysis of certain multi-component signals. One iiiust take care in the application and interpreta.tion of these distributions and time frcquency distributions in gencral. The spectrogram represents an attempt to apply the Fourier t,ransfwm for a short-time analysis window, within which it is hopcd t,hat the signal behaves reasonably according to the requirements of stationarity. By moving the analysis window aloiig the signal, onc hopes to track and capture the variations of the signal spectrum as a function of time. The s p d r o g r a m has many useful properties including a well deve!oped gcneral theory [l]. The spectrogram often presents serious difficulties when uscd to analyze rapidly varying sign&, howcver [ IG] . If the analysis window is made short cnoL1gh to ca.pture rapid changes ir , the signal it becomes iinposhslble l o resolve frequency components of the signal which are c!ose in freqiiency during the analysis window dura ti on. The Wigrier distribution (WD) has bccn employed as an alkrnativc to overcome this shortcoming. The WD has many important arid iritcresting propcrties i4, 5, 61, It providcs a high resolution reprcsentation in time and in frequency for a norist.ationary signai such as a chirp. In addition, !,lie \VD has the important property of satisfying the time and frequency inarginals in k r m s of the instantaneous power in time and energy spectrum in frcquency. However, its energy distribution is not nonnegative and it of ten posscsses severe cross terms, or interference terms, be1,weeii compontnts in different t-f regions, potentially leading to confusioii a n d misintcrpret,ation. An excellent discussion on tlie geometry of interferences has been provided by Hlawatscli and Flaridrin [IO, 11, 121. 'This rr-search was supportrd in part by grants from the Raclcliam School of Graduatr Srudies arid tho Office of Naval Research, ONR contracts no. N00014-89-J-I723 and N00014-90J-1654 Both the spectrogram and the WD are members of Cohen's class of distributions 171. Coherl has provided a consistent set of definitions for a desirable set of t-f distribution which has been of great value in guiding and clarifying efforts in this area of research. Cohen's Class of Distributions is defined to be: Cf(t,L+I$) = J J J e j ( -o t ru+oto do, 7 ) 2T f ( ~ + 7 / 2 ) f * ( ~ 7/2)dudTdO (1.1) where d(0, T ) is termed the kernel of the distribution.(?'he range of integrals is from -03 to 03 throughout this paper.) A recent comprehensive review by Cohen[8] providcs an excellent overview of time-frequency distributions and recent results using them. This paper addresses a specific subset of t-f distributions belonging to Cohen's class. These are the time shift and frequency shift invariant t-f distributions. The spectrogram, the WD and the RID all have this property. Different distributions can be obtained by selecting different kernel functions in Cohen's class. Boashash lias compared the performances of several time-frequency distributions in terms of resolution 121. Desirable properties of a distribution and associated kernel requirements have been extensively irrvestigated by Ciaasen and Mccklenbraukei [GI. More recently, Choi and Williams int,roduced a new distribution having an exponential-type kernel [ 3 ] . wliicli thcy called the Exponential Distribution or ED. This new distribution overcomes several drawbacks of the Spectrogram and WD while providing high resolution with suppresset! interferences [3> 17, IS]. This distribution has been called the Choi-Williams distribution by other investigators [ L ) , SI. e Reduced Interference