2796 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 10, OCTOBER 1998 [4] P. Craven and G. Wahba, “Smoothing noisy data with spline functions,” Numerische Mathematik, vol. 31, pp. 377–403, 1979. [5] M. Stone, “Cross-validatory choice and assessment of statistical predic- tions,” J. R. Stat. Soc., vol. B36, pp. 111–147, 1974. [6] L. R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975. [7] T. W. Parks and J. H. McClellan, “A program for the design of linear phase finite impulse response digital filters,” IEEE Trans. Audio Electroacoust., vol. AU-20, pp. 195–199, 1972. [8] W. F. Trench, “Inversion of Toeplitz band matrices,” Math. Comput., vol. 28, pp. 1089–1095, 1974. High Spectral Resolution Time–Frequency Distribution Kernels Moeness G. Amin and William J. Williams Abstract—A new class of time–frequency distribution (TFD) kernels is introduced. Members in this class satisfy the desirable TFD properties and simultaneously provide local autocorrelation functions (LAF) that are amenable to high-frequency resolution modeling techniques. It is shown that members of the proposed class are product kernels, fast implementa- tion multiplication-free kernels, recursive kernels, and optimum kernels with respect to autoterm localization. I. INTRODUCTION Time–frequency distributions (TFD’s) along with their temporal and spectral resolutions are uniquely defined by the employed – kernels. Potential kernels seek to map, at every time sample, the time-varying signals in the data into approximately fixed frequency sinusoids in the local autocorrelation function (LAF). Applying the FT to the LAF, therefore, provides a peaky spectrum where the location of the peaks are indicative to the signals’ instantaneous power concentrations. The sinusoidal components in the LAF, however, gen- erally appear with some type of amplitude modulations (AM), which are highly dependent on the kernel composition. Such modulation presents a limitation on spectral resolution in the – plane, as it is likely to spread both the auto and crossterms to localizations over a wide a range of frequencies. Because of the kernel modulation effects on the various terms, closely spaced frequencies may not be resolved. Further, since TFD’s are Fourier-based, then in addition to the amplitude modu- lation imposed by the kernels, the spectral resolution is limited by and highly dependent on the extent of LAF, i.e., the lag window employed. However, increasing the length of the LAF will not always yield improved resolution. We maintain that events occurring over short periods of time do not require large kernels, which Manuscript received March 20, 1997; revised April 13, 1998. This work was supported in part by Rome Laboratory under Contract F30602-96-C-0077 and by ONR under Grants N00014-90-J-1654 and N000014-97-1-0072. The associate editor coordinating the review of this paper and approving it for publication was Dr. Frans M. Coetzee. M. G. Amin is with the Department of Electrical and Computer En- gineering, Villanova University, Villanova, PA 19085 USA (e-mail: moe- ness@ece.vill.edu). W. J. Williams is with the Department of Electrical Engineering and Computer Science, The University of Michigan, Ann Arbor, MI 48019 USA (e-mail: wjw@gabor.eecs.umich.edu). Publisher Item Identifier S 1053-587X(98)07078-0. Fig. 1. Binomial kernel. (a) (b) Fig. 2. High-resolution exponential kernel. (a) . (b) . may only lead to increased crossterm contributions from distant events and obscure the local autoterms. Limited availability of data samples may also provide another reason for using small- extent kernels. In these cases, improving spectral resolution of a TFD can be achieved by parameterizing its local autocorrelation 1053–587X/98$10.00  1998 IEEE