SCATTERING FUNCTION AND TIME-FREQUENCY SIGNAL PROCESSING Linh-Trung Nguyen, Bouchra Senadji & Boualem Boashash Signal Processing Research Centre Queensland University of Technology GPO Box 2434, Brisbane, Qld 4001, Australia trung.nguyen@qut.edu.au ABSTRACT The estimation of the scattering function in time-frequency selective fading mobile environment is considered. The scat- tering function explicitly reveals the time-frequency selec- tive behavior of the fading channel under the well-known WSSUS assumption. We propose two classes of estimators based on a time-frequency framework that generalize the existing estimators while giving an extra freedom according to different criteria wanted to be achieved in the estimation of the scattering function. Instead of using Woodward am- biguity function or symmetric ambiguity function, we use the generalized ambiguity function which comes from the general class of quadratic time-frequency distributions. 1. INTRODUCTION One of the major effects to wideband transmission in mobile radio communications due to multipath propagation is the time and frequency dispersion as the results of time-delays over the multipaths and Doppler-shifts from random motion of scatterers. This effect is known as time-frequency selec- tive fading [1, 2]. In practice, this type of channels is often modeled as a random linear time-varying filters. Its second order statis- tics is completely characterized by its scattering function under the wide-sense stationary Gaussian process with un- correlated scattering (WSSUS) assumption [3]. The scat- tering function explicitly reveals the time-frequency selec- tive behavior of the fading channel. The importance of the scattering function is emphasized by the extensive literature [4, 5, 6, 7, 8, 9, 10, 11] (and references therein). The estimation of the scattering function of the random linear time-varying is considered. A common approach is to relate the scattering function with the symmetric ambi- guity function [4] or Woodward ambiguity ambiguity func- tion [8, 9] of the input signal. However, a classical problem faced in this approach is the division of zero. In order to solve it, thresholding method and its derivatives have been approached (review of this can be found in [8]). We propose two classes of estimators based on a time- frequency framework that generalize the existing estimators while giving an extra freedom according to different crite- ria wanted to be achieved in the estimation of the scattering function. Instead of using Woodward ambiguity function or symmetric ambiguity function, we use the generalized ambiguity function which comes from a general class of quadratic time-frequency distributions [12]. 2. WSSUS CHANNEL A complex baseband received signal, , through a wire- less mobile communication channel can be modeled 1 as fol- lows [3] with (1) (2) where is the channel impulse response representing the linear time-varying behavior; is the complex base- band transmitted signal; is the symbol duration; is the additive white Gaussian noise with zero mean and vari- ance ; and denote the time-delay and Doppler shift variables; and , the Fourier transform of from to , is called Delay-Doppler Spread function of the LTV channel. By applying the Fourier transform among the variables , , and , we can define several system func- tions [2, 3] with their relationship shown on Fig. 1. The Delay-Doppler Spread function is often modeled as a wide-sense stationary Gaussian process with uncorrelated scattering (WSSUS) [2, 3] whose second-order statistics can be represented by 2 (3) 1 In practice, the double integral is bounded by the ranges of delays and Doppler-shifts, however, without loss of generality, we the full range and drop them for short notation. 2 denotes the expected value operator.