IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 8, AUGUST 2000 2321 Elimination of Interference Terms of the Discrete Wigner Distribution Using Nonlinear Filtering Gonzalo R. Arce, Fellow, IEEE, and Syed Rashid Hasan Abstract—Methods for interference reduction in the Wigner distribution (WD) have traditionally relied on linear filtering. This paper introduces a new nonlinear filtering approach for the removal of cross terms in the discrete WD. Realizing that linear smoothing kernels are unable to completely cancel the cross-terms without compromising time–frequency concentration and resolution of the auto-terms, a nonlinear filtering algorithm is devised where the filter automatically adapts to the rapidly changing nature of the WD plane. Varying the filter behavior from an identity operation at one extreme to a lowpass linear filter at the other, a near-optimal removal of cross terms is achieved. Unlike traditional smoothing and optimal kernel design techniques, this algorithm does not reduce the time–frequency resolution and concentration of the auto-terms and performs equally well for a very large variety of signals. Index Terms—Nonlinear filters, time–frequency representation, Wigner–Ville distribution. I. INTRODUCTION Q UADRATIC time–frequency representations (TFR’s) are powerful tools for the analysis of signals [1]. Among quadratic TFR’s, the Wigner distribution (WD) WD (1) satisfies a number of desirable mathematical properties and fea- tures optimal time–frequency concentration [2]–[6]. By using the WD, more subtle signal features may be detected, especially for signals having short length and high time–frequency varia- tion. However, despite the desirable properties of the Wigner distribution, its use in practical applications has often been lim- ited by the presence of cross terms. The Wigner distribution of the sum of two signals WD WD WD WD (2) Manuscript received December 3, 1998; revised March 9, 2000. This work was supported in part by the National Science Foundation under Grant MIP- 9530923. The associate editor coordinating the review of this paper and ap- proving it for publication was Dr. Shubha Kadambe. G. R. Arce is with the Department of Electrical and Computer En- gineering, University of Delaware, Newark, DE 19716 USA (e-mail: arce@eecis.udel.edu). S. R. Hasan was with the Department of Electrical and Computer Engi- neering, University of Delaware, Newark, DE 19716 USA. He is now with Nokia Mobile Phones, USA. Publisher Item Identifier S 1053-587X(00)05977-8. has a “cross term” WD in addition to the two auto components, where the cross WD is defined as WD (3) The cross or interference terms are often a grave problem in practical applications, especially if a WD outcome is to be visually analyzed by a human analyst. Cross terms have been extensively studied and many of their properties are known [5], [7]. Cross terms lie between two auto components and are oscillatory with their frequencies increasing with the increasing distance in time–frequency between the two auto components. In practice, for real-valued bandpass signals, the WD of the analytic signal is generally used because removal of the negative frequency components also eliminates cross terms between positive and negative frequency components. Although helpful in eliminating cross terms, cross terms between multiple components in the analytic signal still make interpretation difficult. Since cross terms have oscillations of relatively high frequency, they can be attenuated by means of a smoothing operation that corresponds to the convolution of the WD with a 2-D “smoothing kernel.” Quite generally, the smoothing tends to produce the following effects: 1) a (desired) partial attenuation of the interference terms; 2) an (undesired) broadening of signal terms i.e., a loss of time-frequency concentration; 3) a (sometimes undesired) loss of some of the mathematical properties of the WD (for example, the WD preserves the time and frequency marginals of a signal [1]). The design of a “good” smoothing kernel is, hence, an attempt to achieve effect 1) while avoiding, as far as possible, effect 2) and, if mathematical properties are important, effect 3) as well. The 2-D linear lowpass filtering of the WD, leads us to a TFR of the Cohen’s class [8], which can be written as TFR WD (4) where is a 2-D function called the kernel, which is a term coined by Classen and Mecklenbrauker [2]. However, as described above, this smoothing also produces a less accu- rate time–frequency localization of the signal components. It is well known that a fixed kernel cannot achieve a good rep- resentation, as defined by minimal smearing of auto terms and strong suppression of cross component interference, for every type of signal encountered [9]–[12]. Signal-dependent kernel 1053–587X/00$10.00 © 2000 IEEE