We consider the form of a physically motivated where simple one path time-varying channel in the time-varying impulse in (7W (T n response, time-frequency characterization and time-scale charach(t) h(t, T) w dT;terization settings. The focus, in general, is to determine which n (t-n) setting allows for the most efficient (i.e., sparse) discrete channel representation. We measure how well the discrete representation time-frequency domain, reconstructs the channel when a limited number of discretecoefficients are available and compare the result among the W channel models. m n where I. OVERVIEW Smn =I/S(O, ')sinc (n -7W) sinc(mn -OT)e j7r(moT)dOd7; Which discrete time-varying channel model is the most JJ efficient? Several channel models exist, but which of them cap-(7) tures most efficiently the action of the channel? We consider and time-scale domain, in this work three models: a time domain characterization (i.e., t2 Lmn (t -nboaom 88 the time-varying impulse response), a time-frequency charac-(tY) S m/2X Y am ) (8) terization, and a time-scale characterization. The time-domain m, 0 characterization represents the channel output as a series of where weighted discrete delayed versions of the input signal, the I b ._ si a -. dadb.time-frequency characterization represents the channel output Im,m jj L(a, b)sinc -in ao) abo J as a series of weighted discrete delayed and frequency shifted (9) versions of the input signal, and the time-scale characterization In the above models, W, T, ao, bo are related to channel and represents the channel output as a series of weighted discrete signal characteristics [1][2][3][4]. Each of these three models and delayed and dilated versions of the input signal. Given that their discrete expansions have been studied, in some cases, for practical receivers must model the channel using a limited nearly 50 years [1-9]. In each case, the channel is captured by number of channel coefficients, we examine how accurately a set of coefficients (hn, Sm,n, and Lm,n). In this paper, we the channel is captured for the three models as a function of quantify how well these coefficients represent the channel for the number of coefficients available. a simple one path time-varying channel in the important case In continuous time, the three models considered here are when only a finite number of coefficients are available. The time-domain, channel we consider is one where a transmitter and receiver move with constant radial velocity relative to one another. For y (t) t / (t, T)X(t -T)dT7,(1) ease of presentation, we consider in this work exclusively that the signal is an audio signal so that we may ignore relativistic time-frequency domain, effects. This simplifies the derivation of the Doppler effect, although the resulting channel for electromagnetic waves has a y(t) = S(O, T)X(t -T)ej27OtdTdO,(2) similar, but not identical, form. The results presented here can be extended to apply to wireless signals in a straightforward and time-scale domain, manner.The paper is organized as follows. In Section II ...