General Time-Frequency Distribution Series

In this paper, we present a new and robust method, general time-frequency distribution series (TFDS), for time-frequency analysis. We also introduce the concept of local interference and global interference and show that the local interference is important and the global interference is less important in TF analysis. The TFDS is very general. With only one parameter, order of the TFDS, one can easily balance different and competing requirements for the best result. I. Time-Frequency Distribution Series A signal's time-frequency properties can also be obtained by mapping the time-domain signal x(t ) to a set of elementary function hmn(t) in the TF domain directly by an expansion formula: x(t> = C C Cmn hmn(t) (1) where m and n are indices in the TF domain, and Cmn is the coefficient associated with hpn(t). Eq. (1) will be referred to as the TF expansion series in the rest of this article. A well-known example of the TF expansion series is the Gabor expansion. By applying a Cohen's class [ 1) bilinear transformation to both sides of Eq. (l), and truncate the series to a finite number of terms, we obtain: m n 0-7803-2127-8194 $4.00 01994 IEEE 437 px(t, f;p) 4 C cmncm'n' Ph,h' (t, f;A) k 0 A p 2 0, (2) where Ph,h' ( 6 f;a) = Ph,,,,h,~,, ( t , 81) for brevity, A is a set of indices of m, n, m', n' and A is the cross-term order. Each term in the series is a cross term of a Cohen's distribution on a pair of elementary functions, obtained from the TF expansion series from Eq. (1). The terms are arranged in the cross-term order, A, which is proportional to the distance between the two elementary functions in the TF domain. p is the highest order of the cross terms included in the series. In rest of the paper, Eq. (2) will be referred to as the general TF distribution series, or GTFDS. When p + 00, the series converges to Cohen's class. When p = 0, the series becomes either the Gabor spectrogram [5] or the adaptive TF distributions [3, 41 depending on the choice of the elementary functions. For other orders p E [1, -), a new set of TF distributions can be defined. The key to the GTFDS is to find a set of very well-behaved elementary functions in the TF expansion series Eq. (1) that satisfies two requirements: 1) it is well localized in both the TF domains, and 2) it describes the signal's local behavior. In general, we found the Gabor functions obtained by the orthogonal-like Gabor expansion are a good choice. [9] Well-Behaved Elementary Functions The Gabor functions has a form: