A New Procedure for the Design of Fast Simulation Models for Rayleigh Fading Channels

Models for Rayleigh Fading Channels MATTHIAS P ATZOLD and RAQUEL GARCIA1 Abstract | A new method for the design of fast simulation models for mobile fading channels by using table look-up techniques is presented. For short, we assume frequency nonselective Rayleigh fading, but the proposed design procedure can easily be extended to modeling frequency selective mobile radio channels. The resulting Rayleigh fading channel simulator is only based on some few adders, storage elements, and logical operations, whereas multipliers and trigonometric operations are not required. SUMMARY In general, Rayleigh fading is obtained from a complex Gaussian noise process (t) = 1(t) + j 2(t) by taking its absolute value, i.e., (t) = j (t)j = j 1(t)+j 2(t)j, where 1(t) and 2(t) are uncorrelated colored Gaussian noise processes each having zero mean and variance V arf i(t)g = 2 0 (i = 1; 2). Our procedure for the design of highly e cient (Rayleigh fading) channel simulators can be partitioned into two steps. In the rst step, we apply the concept of deterministic channel modeling that has been introduced in [1]. According to that principle, we have to replace the stochastic Gaussian processes 1(t) and 2(t) by so-called deterministic Gaussian processes ~ i(t) = Ni Xn=1 ~ i;n(t) = Ni Xn=1 ci;n cos(2 fi;nt+ i;n) (1) for i = 1; 2, where Ni denotes the number of harmonic functions that determines the realization expenditure of the resulting channel simulator. The gains ci;n, discrete Doppler frequencies fi;n, and phases i;n are constant quantities which are computed, e.g., by applying the method of exact Doppler spread (MEDS) [2]. In [2], closed formulas can be found for the model parameters ci;n, fi;n, and i;n. From the substitution i(t) ! ~ i(t) it follows (t) ! ~ (t) = j~ 1(t) + j~ 2(t)j, where ~ (t) is called deterministic Rayleigh process. By using the MEDS, it can be shown that ~ (t) tends in the limit Ni ! 1 to a sample function of (t). Nevertheless, the statistics of ~ (t) is extremely close to that of (t) if Ni is su cient, say Ni 7. In the second step, we sample the deterministic Gaussian processes ~ 1(t) and ~ 2(t) at t = kT , where k = 0; 1; 2; : : : and T is the sampling interval. Next, we apply table look-up techniques with the object to store only a single period of each of the Ni weighted harmonic functions ~ i;n(kT ) introduced in (1). This results in the following discrete deterministic Gaussian process: i(kT ) = Ni Xn=1 i;n(kT ) = Ni Xn=1 ci;n cos(2 fi;nkT + i;n) (2) 1The authors are with the Department of Digital Communication Systems, Technical University of HamburgHarburg, Denickestr. 17, D-21071 Hamburg, Germany (e-mail: paetzold@tu-harburg.de). for i = 1; 2, where fi;n is closely related to the sampling interval T and the table length Li;n = roundf1=(fi;nT )g 2 according to fi;n = 1=(TLi;n). Note that the table lenght Li;n is identical with the period of i;n(kT ). For the phases i;n, we obtain the relation i;n = i;n + i;n, where i;n is a correction quantity to approximate i;n to the nearest value in the set f i;ng = f0; 2 fi;nT; : : : ; 2 fi;n(Li;n 1)Tg. The structure for the resulting Rayleigh fading channel simulator by using table look-up techniques is presented in Fig. 1. The design of the address generator (see Fig. 1) is quite simple and requires only some few logical operations and additions. Obviously, the resulting channel simulator is multiplier-free. The computation of the table entries has to be done once during the simulation set-up phase. Afterwards, during the simulation run phase, only 2(N1 +N2 1) additions and N1 +N2 logical operations have to be carried out in order to obtain (kT ) = 1(kT ) + j 2(kT ) at a given value for k = 1; 2; : : :. Our investigations concerning the speed of the two channel simulators have revealed that the model in Fig. 1 is more than three times faster than the original model.