Characterization of Randomly Time-Variant Line;ar Channels Summary-This paper is concerned with various aspects of the characterization of randomly time-variant linear channels. At the outset it is demonstrated that time-varying linear channels (or filters) may be characterized in an interesting symmetrical manner in time and frequency variables by arranging system functions in (timefrequency) dual pairs. Following this a statistical characterization of randomly &.e-variant linear channels is carried out in terms of correlation functions for the various system functions. These results are specialized by considering three classes of practically interesting channels. These are the wide-sense stationary (WSS) channel, the uncorrelated scattering (US) channel, and the wide-sense stationary uncorrelated scattering (WSSUS) channel. The WSS and US channels are shown to be (time-frequency) duals. Previous discussions of channel correlation functions and their relationships have dealt exclusively with the WSSUS channel. The point of view presented here of dealing with the dually related system functions and starting with the unrestricted linear channels is considerably more general and places in proper perspective previous results on the WSSUS channel. Some attention is given to the problem of characterizing radio channels. A model called the Quasi-WSSUS channel is presented to model the behavior of such channels. All real-lie channels and signals have an essentially finite number of degrees of freedom due to restrictions on time duration and bandwidth. This fact may be used to derive useful canonical channel models with the aid of sampling theorems and power series expansions. Several new canonical channel models are derived in this paper, some of which are dual to those of Kailath. I. INTRODUCTION D URING RECENT YEARS there has been an increasing amount of attention given to the study of randomly time-variant linear channels. This attention has been motivated to a large extent by the advent of troposcatter, ionoscatter, chaff and moon communication links and radar astronomy systems. The determination of optimum modulation and demodulation techniqu.es and the. analytical determination of the efficacy of optimum and suboptimum communication (or radar) techniques for such channels depends heavily upon a satisfactory characterization of the transmission channel. Thus, the characterization of randomly timevariant linear channels is of some interest. The chara,cterization of time-variant linear filters (whether random or not) in terms of system functions received its first general analytical treatment by Zadeh,' who introduced the Time-Variant Transfer Function and the Bi-Frequency Function as frequency domain methods of characterizing time-variant linear filters to performed, in part, under Subcontract 480.117D with ITT Com
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