Design of High-Speed Simulation Models for Mobile Fading Channels ...
Design of High-Speed Simulation Models for Mobile Fading Channels by Using Table Look-Up Techniques MATTHIAS PA TZOLD,
SENIOR MEMBER, IEEE,
RAQUEL GARCIA, and FRANK LAUE
Technical University of Hamburg-Harburg Department of Digital Communication Systems Denickestr. 17 D-21071 Hamburg Germany Tel. : (+49)40-42878-3149 Fax : (+49)40-42878-2941 E-Mail : [email protected]
Abstract
This paper describes a procedure for the design of fast simulation models for Rayleigh fading channels. The presented method is based on an ecient implementation of Rice's sum of sinusoids by using table look-up techniques. The proposed channel simulator is composed of a few number of adders, storage elements, and simple modulo operators, whereas time-consuming operations like multiplications and trigonometric operations are not required. Such a multiplier-free simulation model is introduced as high-speed channel simulator. It is shown that the high-speed channel simulator can be interpreted as a nite state machine which generates deterministic output envelope sequences with approximately Rayleigh distribution. The statistical properties of the designed high-speed channel simulator are investigated analytically and compared with the statistics of the underlying Rayleigh reference model. Results of experiments for measuring the speed of the presented and other types of channel simulators are also presented. Index Terms { Deterministic channel modeling, high-speed fading channel simulator, Rayleigh fading, Rice's sum of sinusoids, statistics.
Submitted to IEEE Transactions on Vehicular Technology First Version: June 19, 1998 Revised Version: August 17,1999 Final Version: October 8, 1999
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I. Introduction
Computer simulation experiments are an established method to design and evaluate digital communication systems. The advantages of a block-oriented subroutine library for each part of the communication system have been widely recognized, since one can easily set up one's own transmission system. Especially for the simulation of mobile communication systems, ecient algorithms for di erent types of fading channels are mandatory for user-friendly tool-box libraries. In past, several methods have been developed for the design of simulation models for mobile radio channels. These methods can be classi ed in four categories. The rst category comprises design methods employing a nite number of low-frequency oscillators to generate pseudo-random noise sequences (e.g. [1], [2], [3], [4]). Such procedures go originally back to an early work of Rice [5], [6], who has shown that narrowband Gaussian noise processes can be modeled by an in nite number of properly designed sinusoids. The second class comprises all those methods which are basing on low-pass ltering of zeromean white Gaussian noise processes by using linear time-invariant lters (e.g. [7], [8], [9]). The class of channel design methods which are basing on Markov processes de nes the third category (e.g. [10], [11], [12]). Finally, the fourth category comprises channel models using the stored channel principle (e.g. [13], [14]). The aim of this paper is to show that all simulation models corresponding to the rst above mentioned category can be eciently implemented on a computer or hardware platform by using table look-up techniques. Thereby, we exploit the fact that the harmonic functions generating the fading signals are periodic functions. The data related to one period of each harmonic function of Rice's sum of sinusoids is stored in a table. The resulting table system is a multiplier-free high-speed channel simulator which can easily be realized by using adders, storage elements, and simple modulo operators. Throughout the paper, we make use of the complex equivalent baseband notation, and, to simplify matters, we consider the transmission of a non-modulated carrier over a Rayleigh
at fading channel. The organization of this paper is as follows. Section II reviews the statistics of Rayleigh processes. Section III presents an overview of the concept of deterministic channel modeling. In Section IV, the new high-speed channel simulator is derived, and its statistical properties are analytically investigated in Section V. Section VI is devoted to compare the performance of the high-speed channel simulator with the underlying conventional deterministic simulation model as well as with a lter method based Rayleigh at fading channel simulator. Finally, the conclusions are drawn in Section VII. II. Review of Rayleigh Processes
The statistics of Rayleigh processes is brie y reviewed in this section. The Rayleigh process is a statistical model that gures in Section V as reference model when discussing and evaluating the performance of the proposed high-speed channel simulator. In typical mobile radio environments, the transmitted signal propagates over di erent signal paths from the transmitter to the receiver. At the antenna of the receiver the signals are superposed to produce an environment speci c standing waves eld. When
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the receiver and/or the transmitter moves in the standing waves eld, the received signal experiences random variations in the envelope and in the phase [15]. An often used and reasonable statistical model for describing the random variations in the complex equivalent baseband is a zero-mean complex Gaussian noise process [1]
(t) = 1(t) + j2(t) (1) where 1(t) and 2(t) are narrowband real Gaussian noise processes with zero mean and variance Varfi(t)g = 02 (i = 1; 2). The Gaussian processes 1(t) and 2(t) are often assumed to be statistically independent. If that is the case, the cross-correlation function (CCF) of 1(t) and 2(t) is zero. In general, a real Gaussian noise process is completely described by its mean and autocorrelation function (ACF) [16]. For an omnidirectional receiving antenna in a two-dimensional isotropic scattering environment, it was shown in the pioneering work of Clark [17] that the ACF of i(t) can be represented by ri i ( ) = 02 J0 (2fmax ) ; i = 1; 2 (2) where fmax denotes the maximum Doppler frequency, and J0() is the zeroth order Bessel function of the rst kind. The Fourier transform of (2) gives the Doppler power spectral density (PSD) Si i (f ) =
(
fmax
0;
p1?(f=f 2 0
max )2
; jf j fmax jf j > fmax
(3)
which is often called in literature Jakes PSD [1]. A Rayleigh process, (t), is de ned as the envelope of (1), i.e.,
(t) = j(t)j: (4) The probability density function (PDF) of (t) is known as Rayleigh density [16] p (z) = z2 exp[?z2 =(202)] ; z 0: (5) 0 Furthermore, the corresponding cumulative distribution function (CDF), de ned by P (r) = Probf (t) rg, can be expressed by [18] P (r) = 1 ? exp[?r2=(202)] ; r 0: (6) The phase of (t) de nes a further random process, namely #(t) = arg f(t)g, with uniform distribution over the interval (?; ], that is, the PDF of the phase #(t) becomes (7) p#() = 21 ; 2 (?; ]: Fig. 1 shows the structure of an analytical model for a Rayleigh at fading channel. Thereby, ni(t) denotes a zero-mean white Gaussian noise (WGN) process with normalized variance Varfni (t)g = 1, and the transferpfunction Hi(f ) of each lter is adapted to the Doppler PSD (3) according to Hi(f ) = Si i (f ) (i = 1; 2). Henceforth, we designate channel design methods employing linear lter techniques as lter methods.
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III. Overview of deterministic channel modeling
Aside from the lter method, there exists another fundamental method for modeling narrowband Gaussian noise processes: the Rice method. The principle of Rice's method [5], [6] implies that a Gaussian noise process i(t) can be modeled by a superposition of an in nite number of weighted harmonic functions with equidistant frequencies and random phases according to
i(t) = Nlim !1 i
Ni X n=1
ci;n cos(2fi;nt + i;n)
(8)
where the gains ci;n and discrete Doppler frequencies fi;n are given by
q
ci;n = 2 fi Si i (fi;n) fi;n = n fi
(9a) (9b)
respectively. Thereby, the random phases i;n are uniformly distributed over the interval (?; ], and the quantity fi is chosen such that (9b) covers the whole frequency range of interest, where fi ! 0 if Ni ! 1. Equation (8) describes an analytical model for Gaussian processes that cannot be implemented on a computer due to the in nite number of sinusoids. When Ni is nite, we obtain another stochastic process
^i(t) =
Ni X n=1
ci;n cos(2fi;nt + i;n)
(10)
which is strictly speaking non-Gaussian distributed. But nevertheless the PDF of ^i(t) is close to a Gaussian density if Ni is sucient, say Ni 7. Only in the limit Ni ! 1, the process ^i(t) follows the ideal Gaussian distribution, and we may write ^i(t) = i(t). It should be noted that ^i(t) is even now a stochastic process due to the assumed random phases i;n. Its implementation on a computer can easily be performed and results in a stochastic simulation model. During the simulation, the phases i;n and all other model parameters are kept constant, so that ^i(t) is now neither more a random process but a sample function which is per de nition completely determined for all time t. To distinguish between the stochastic process ^i(t) with random phases and the corresponding deterministic process (sample function), we use for the latter one the notation
~i(t) =
Ni X n=1
~i;n(t) =
Ni X n=1
ci;n cos(2fi;nt + i;n)
(11)
where the phases i;n are now constant quantities, and the gains ci;n and discrete Doppler frequencies fi;n are -at the moment- furthermore given by (9a) and (9b), respectively. Let Ni ! 1, then the deterministic process ~i(t) tends to a sample function of the stochastic Gaussian process i(t). To emphasize this property, the function ~i(t) has been introduced in [19] as deterministic Gaussian process. By analogy with (1),
~(t) = ~1(t) + j ~2(t) is called complex deterministic Gaussian process, and, consequently ~(t) = j~(t)j
(12) (13)
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represents a deterministic Rayleigh process [20]. A direct realization of deterministic Rayleigh processes leads to the time-continuous structure shown in Fig. 2. The problem with (9a) and (9b) is now twofold. First, the period of ~i(t) is relatively small due to the equally spaced discrete Doppler frequencies fi;n. Second, the density of ~i(t) is non-optimal Gaussian distributed. From the central limit theorem it follows that the density of ~i(t) is for a given number of sinusoids Ni < 1 only optimal Gaussian distributed by keeping the power constraint if all gains ci;n are equal [19]. This equal gain requirement is not ful lled when Rice's original method is used as can be seen immediately by substituting (3) in (9a). To overcome the problems caused by (9a) and (9b) several new methods for the computation of the model parameters (ci;n; fi;n; i;n) have been proposed in recent years. For example, the Monte Carlo method [21], [2], the method of equal areas [4], the Lp-norm method [19], and the method of exact Doppler spread [19]. Even the in-phase and quadrature components of the well-known Jakes simulator (see [1], p. 70) can be brought onto the form (11) [22]. For short, we restrict our investigations in the sequel to the method of exact Doppler spread (MEDS) and the Jakes method (JM). 1) Parameters by using the MEDS [19]: Applying the MEDS to the Jakes PSD (3) gives the following closed form solutions for the gains ci;n and discrete Doppler frequencies fi;n:
r2
ci;n = 0 N (14a) i (14b) fi;n = fmax sin 2N n ? 21 i for all n = 1; 2; ; Ni (i = 1; 2). The phases i;n are obtained by taking Ni drawings from a random generator uniformly distributed over the interval (?; ]. Thereby, the number of sinusoids N1 and N2 are in general related by N2 = N1 + 1 what guarantees the design of zero cross-correlated deterministic Gaussian processes ~1(t) and ~2(t). 2) Parameters by using the JM [1]: When the JM is used, the gains ci;n , discrete Doppler frequencies fi;n, and phases i;n can be expressed in closed form as follows [22]:
n 8 2 > p > < 2Ni? sin Nni?1 ; n = 1; 2; ; Ni ? 1; i = 1 pNi? cos Ni?1 ; n = 1; 2; ; Ni ? 1; i = 2 > > : pNi? ; n = Ni ; i = 1; 2 0
ci;n =
0
0
fi;n =
(
1 2 1 2
(15a)
1 2
fmax cos fmax ;
n 2Ni ?1
;
n = 1; 2; ; Ni ? 1; i = 1; 2 n = Ni ; i = 1; 2
(15b)
i;n = 0; n = 1; 2; ; Ni; i = 1; 2 (15c) where N1 = N2 . According to the JM, the discrete Doppler frequencies f1;n and f2;n are identical for all n = 1; 2; ; N1(N2), and, thus, the processes ~1(t) and ~2 (t) are correlated.
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Of further interest are some basic statistical properties of deterministic processes like the ACF, PSD, CCF, PDF, and CDF. The statistics of deterministic processes has been investigated in detail in [19]. Here, we only present some basic results as far as they are of importance for the understanding of the rest of the paper. A. ACF, PSD, and CCF The ACF r~i i ( ) and the corresponding Doppler PSD S~i i (f ) of the deterministic Gaussian process ~i(t) may be represented by [19]
r~ii ( ) = S~i i (f ) =
Ni c2 X i;n n=1 Ni
2
cos(2fi;n ); i = 1; 2
(16a)
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[(f ? fi;n) + (f + fi;n)]; i = 1; 2
(16b)
X c2i;n n=1
respectively. For the MEDS one can show by putting (14a) and (14b) in (16a) that r~i i ( ) tends to ri i ( ) if Ni ! 1. This is in contrast to the JM, where the inequality r~i i ( ) 6= ri i ( ) holds even if Ni ! 1, as can be proved by substituting (15a) and (15b) in (16a). The CCF r~1 2 ( ) of the deterministic Gaussian processes ~1(t) and ~2(t) is given by [19]
8 PN < 1 c1;nc2;m cos(2f1;n ? 1;n 2;m); r~ ( ) = : 2 n=1 0;
if f1;n= f2;m (17) if f1;n6= f2;m for all n = 1; 2; ; N1 and m = 1; 2; ; N2, where N indicates the minimum value of N1 and N2 , i.e., N = min fN1; N2g. For the MEDS it can easily be guaranteed that the designed processes ~1(t) and ~2(t) are uncorrelated, i.e., r~1 2 ( ) = 0, e.g., by de ning N2 = N1 + 1. For the JM, where N1 is per de nition equal to N2 , we obtain r~1 2 ( ) 6= 0. 1 2
B. PDF and CDF General analytical expressions for the density of the envelope ~ = j~(t)j and phase #~(t) = arg f~(t)g of complex deterministic Gaussian processes ~(t) = ~1(t) + j ~2(t) have been derived in [19]. There the following results are presented:
Z
p~ (z) = z p~1 (z cos ) p~2 (z sin ) d; z 0 ?
p~#() =
Z1
z p~1 (z cos ) p~2 (z sin ) dz; jj
(18a) (18b)
0
where
p~i (x) = 2
Z1 "Y Ni 0
n=1
#
J0 (2ci;n ) cos(2x) d
(19)
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denotes the PDF of the deterministic Gaussian process ~i(t) as introduced by (11). Note that p~i (x), and thus p~ (z) and p~#() are only functions of the gains ci;n and the number of sinusoids Ni. From the central limit theorem it follows for both methods (MEDS and JM) that (19) tends in the limit Ni ! 1 to a Gaussian density with zero mean and variance 02. Hence, when Ni ! 1 then it follows immediately from the expressions (18a) and (18b) that p~ (z) approaches the Rayleigh density (5), whereas p~#() approaches the uniform density (7). However, for nite values of Ni, we may write p~ (z) p (z) and p~#() p#(), thereby it is worth mentioning that the approximations are better for the MEDS than for the JM. After performing some algebraic manipulations, we can obtain from (18a) the following CDF P~ (r) = Probf~(t) rg of ~(t):
Z1
P~ (r) = 4r J1(2rz) 0
"Y N 2
n=1
Z=2"Y N 1
0
n=1
J0(2c1;nz cos )
#
#
J0(2c2;nz sin ) d dz; r 0:
(20)
Thus, the above result enables an analytical investigation of the CDF of deterministic Rayleigh processes ~(t). It should be clear that we may write P~ (r) = P (r) if Ni ! 1. IV. Design of High-Speed Channel Simulators
An ecient implementation of the conventional deterministic model described in the previous section can be achieved by applying table look-up techniques. Each sinusoidal function ~i;n(t) in (11) is substituted by a table which stores the data related to one period of ~i;n(t). For the resulting discrete-time simulation model, an address generator is required to have access to the registers of the tables. The outputs of each block of Ni tables are added in order to obtain an equivalent discrete-time realization of the deterministic Gaussian processes ~i(t) (i = 1; 2). Fig. 3 shows the discrete-time structure of the resulting tables system. The equivalent discrete-time realization of the deterministic Gaussian processes ~i(t) is called discrete deterministic Gaussian process i[k] which can be expressed by
i[k] =
Ni X n=1
i;n[k] =
Ni X n=1
ci;n cos(2fi;nkTs + i;n) ; i = 1; 2
(21)
where Ts denotes the sampling interval. Obviously, the discrete deterministic Gaussian process i[k] is obtained by sampling the continuous-time deterministic Gaussian process ~i(t) at t = kTs (k = 0; 1; ) , whereby additionally the discrete Doppler frequencies fi;n and phases i;n are replaced by fi;n and i;n, respectively. It will be shown below that the new parameters fi;n and i;n are slightly modi ed versions of the original parameters fi;n and i;n, in the order mentioned. In general, a Rayleigh process is obtained by taking the absolute value of a complex Gaussian noise process. According to our method, we can approximately model the statistics of a sample function of a Rayleigh process by combining two real table blocks to a so-called discrete complex deterministic Gaussian process
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[k] = 1 [k] + j 2[k]
(22)
and then taking at each instant k the absolute value according to
[k] = j[k]j = j1[k] + j 2 [k]j
(23)
where [k] is named discrete deterministic Rayleigh process. The table that corresponds to the sinusoidal function i;n[k] is referred in Fig. 3 as Tabi;n. The number of entries of the table Tabi;n corresponds to the table length which will be denoted as Li;n. To nd an expression for the table length Li;n, recall that it is sucient to store only one period of the underlying sinusoidal function ~i;n(t). Thereby, we have to take into account the in uence of the sampling interval Ts. In order to obtain for any given sampling interval Ts an integer number for the table length Li;n, the discrete Doppler frequency fi;n of the original sinusoid has to be slightly modi ed according to fns o fi;n = (24) round ffi;ns where fs is the sampling frequency, and the operation roundfxg denotes the nearest integer to x. The period of the resulting discrete-time sinusoid i;n[k] corresponds to the table length Li;n which is now given by f f s (25) Li;n = = round f s : fi;n i;n By selecting a suciently large value for the sampling frequency fs = 1=Ts, the modi ed discrete Doppler frequency fi;n is very close to the original quantity fi;n. The quality of the approximation fi;n fi;n and, consequently, of the statistics of the resulting discrete-time simulation model, depends on the sampling frequency fs. A measure for the approximation error is the relative error of fi;n, i.e., (26) "fi;n = fi;nf? fi;n : i;n The behavior of "fi;n is shown in Fig. 4 over the sampling interval Ts for a discrete Doppler frequency fi;n of 91 Hz. The presented result shows us that the relative error "fi;n is less than 5 % if the sampling interval Ts is less than (10fi;n)?1. Note that it follows from (24) that we may write fi;n = fi;n if Ts ! 0 (fs ! 1). The complexity of the tables system is closely related with the tables lengths Li;n. From (25), we realize that Li;n is proportional to the sampling frequency fs. Hence, a compromise between the precision and the complexity of the tables system must be found by choosing an adequate value for fs. Experimental results have revealed that an adequate value for the sampling frequency fs is given if fs is within the range of 20fmax fs 30fmax. Since the number of samples of the discrete sinusoid i;n[k] stored in the table Tabi;n is Li;n, then the phase i;n can only take values from a discrete set of Li;n equidistant phases within the interval [0; 2), i.e.,
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i;n 2 i;n = f0; 2fi;nTs; ; 2fi;n(Li;n ? 1)Tsg
(27)
whereby each of these values corresponds to a speci c initial position in the table Tabi;n. The phases i;n are chosen from the above set i;n in such a way that i;n is nearest to i;n. Hence, it follows for Ts ! 0 that we may write i;n = i;n. From the fact that fi;n and i;n are tending to fi;n and i;n, respectively, if Ts approaches to zero, it follows: i[k] ! ~i(t) if Ts ! 0. Moreover, by taking the results of Section III into account, we can say that i[k] tends to a sample function of the Gaussian process i(t) if Ts ! 0 and Ni ! 1. A. Tables System The tables system is composed of N1 + N2 tables, whereby each one is equivalent to a sinusoidal function in the conventional deterministic model. The data stored in the tables contain the full information to compute the corresponding discrete deterministic Gaussian process i;n[k] at any instant k = 0; 1; .
The entry of the table Tabi;n at position l 2 f0; 1; ; Li;n ? 1g corresponds to the value of the discrete-time sinusoidal function i;n[k] at instant k = l, i.e.,
i;n[l] = ci;n cos(2fi;nlTs + i;n)
(28)
where n = 1; 2; ; Ni and i = 1; 2. If the entries of the table Tabi;n are sequentially read from the beginning, then the table output sequence is i;n[0], i;n[1], , i;n[Li;n ? 1], i;n[Li;n] = i;n[0], , and, therefore, the discrete-time sinusoid i;n[k] can completely be reconstructed for all k = 0; 1; . The address generator (AG) shown in Fig. 3 is required to enable the access to the data stored in the tables. The number of addresses that the AG generates is N1 + N2 , i.e., one address for each table. Let us denote the address of the table Tabi;n at instant k by ai;n[k], then the way of operation of the AG can easily be described as follows. At instant k, the AG points to a certain register in the table at position l = ai;n[k]. At the next instant k + 1, the generated address corresponds to the position of the next lower register in the table shown in Fig. 3. When the last position of the table is reached, then the AG points at the next instant to the rst register again. Let us start from the initial addresses ai;n[0] which are determined by the phases i;n, then we can compute recursively the addresses ai;n[k] at each instant k > 0 by using
ai;n[k] = (ai;n[k ? 1] + 1) mod Li;n
(29)
for all n = 1; 2; ; Ni (i = 1; 2), where the notation mod denotes the modulo operation. It should be mentioned that the modulo operation has been introduced here only for mathematical convenience. Its implementation on a computer can easily be achieved by using an adder and an if &else operation.
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B. Matrix System The new implementation form of the conventional deterministic model can also be interpreted as a matrix system. Therefore, we de ne a matrix i for the block of Ni tables which determine the discrete deterministic Gaussian process i[k] (i = 1; 2). The number of rows of the matrix i is equal to the number of tables Ni. Thereby, the nth row of i contains the data of the table Tabi;n . Hence, the length of the largest table de nes i . The rst L elements the number of columns Li;max of i, i.e., Li;max = maxfLi;ngNn=1 i;n of the nth row of i are determined by the contents of the registers of table Tabi;n, whereas the rest of the row is padded with zeros. Thus, the matrix i 2 R Ni Li;max can be represented by
M
M
M
M
M
M
0 i;1[0] i;1[1] M i = BB@ i;2..[0] i;2..[1]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i;1[Li;1 ? 1] . . . . . . . . . . . . . . . . . . i;2[Li;2 ? 1] 0 0 ... . .. . . i;Ni [0] i;Ni [1] i;Ni [Li;Ni ? 1] 0 0
1 CC A
(30) where we have assumed { without loss in generality { that the number of columns Li;max is given by the rst table, i.e., Li;max = Li;1. This is in fact the case by using the MEDS, since fi;1 is the smallest discrete Doppler frequency obtained from (14b). Next, we introduce a further matrix i which is called selection matrix. This matrix is closely related with the addresses generated by the AG. The elements of i can take the values 0 or 1 and they are changing their positions at each instant k, just like the addresses of the AG. The elements of the selection matrix i = (sl;n) 2 f0; 1gLi;maxNi at instant k are given by
S
S
S
sl;n = sl;n[k] =
1
if l = ai;n[k] 0 if l = 6 ai;n[k]
(31)
for all l = 0; 1; ; Li;max ? 1 and n = 1; 2; ; Ni (i = 1; 2). The discrete deterministic Gaussian process i[k] can now be obtained from the product of the matrices i and i as follows:
M
S
M i Si) ;
i[k] = tr (
i = 1; 2
(32)
where tr() denotes trace. Consequently, the discrete complex deterministic Gaussian process (22) can be expressed alternatively by
M S ) + j tr (M S ) :
[k] = tr (
(33) Finally, an equivalent representation of the discrete deterministic Rayleigh process [k] introduced by (23) is given by 1
1
2
2
M S ) + j tr (M S ) j:
[k] = j[k]j = jtr (
1
1
2
2
(34)
Although the tables system and the matrix system are equivalent from the statistical point of view, the realization of the later one is less ecient. Therefore, we restrict our further investigations to the tables system.
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V. Performance of the high-speed Channel Simulator
A. Period The sequence i[k] (i = 1; 2) generated by the tables system is designed to approximate the statistics of a stochastic process, in our case, a narrowband Gaussian noise process which is of course a non-periodic stochastic process. But due to the limited number of tables, whereby each having a nite number of elements, the generated output sequence of the tables system is a periodic sequence. Nevertheless, we will see in this section that the period of the output sequence is extremely large, i.e., the tables system generates quasi non-periodic sequences which are sucient for most practical applications.
Recall that the period of a single sinusoid i;n[k] was introduced as Li;n. The period of the sum of sinusoids i[k] is denoted henceforth as Li . In the following, we show that the i , period Li of i[k] is the least common multiple (LCM) of the elements of the set fLi;ngNn=1 i.e., i ; i = 1; 2: Li = lcmfLi;ngNn=1
(35)
Therefore, we have to prove the validity of
i[k] = i[k + Li ] 8 k 2 Z:
(36)
i , L is a multiple of each L , and we can write Since Li is the LCM of the set fLi;ngNn=1 i i;n Li;n = dLi (37) i;n
where di;n is a natural number which can be di erent for each Li;n. It is clear that if Li;n is the period of i;n[k], then the product di;nLi;n satis es also
i;n[k] = i;n[k + di;nLi;n] 8 k 2 Z:
(38)
Consequently, from (21), (36), and (38), we obtain
i[k + Li ] = = =
Ni P i;n[k + Li ] n=1 Ni P [k + d L n=1 Ni
i;n
P [k] i;n n=1
i;n i;n ]
(39)
= i[k]:
i , it is assured that this is the minimum value Since Li is de ned as the LCM of fLi;ngNn=1 which satis es (36), and therefore, Li is the period of i[k].
Note that the upper limit of the period Li is de ned by the product L^ i = Li;1 Li;2 Li;Ni which also ful lls (36), i.e.,
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L^ i =
Ni Y n=1
i : Li;n Li = lcm fLi;ngNn=1
(40)
The period L of the discrete deterministic Rayleigh process [k] = j1[k] + j 2[k]j can directly be derived from the LCM of L1 and L2 , i.e., L = lcmfL1 ; L2g. The period Li of the output sequences i[k] of the tables system is dependent on the sampling frequency fs, because the period Li;n of each single discrete sinusoid i;n[k] is a function of fs [see (25)]. Consequently, fs has also an in uence on the period L of the discrete deterministic Rayleigh process [k]. In order to achieve long periods Li and L, the value selected for fs should be suciently large. Observe that in the limit fs ! 1, the periods Li and L are approaching in nity, provided that Ni is suciently large. The evaluation of the period Li and its upper limit L^ i as function of the normalized sampling frequency fs=fmax is shown in Fig. 5 for a small and a large number of tables Ni. From the results of Fig. 5 it can be observed that in many cases the period Li is close to its upper limit L^ i . Moreover, it can also be realized that the period Li is extremely large even for relatively small sampling frequencies fs. This fact gives us an account of saying that i[k] can be considered as a quasi non-periodic deterministic sequence. B. ACF, PSD, and CCF Due to the deterministic behavior of i[k], we have to compute the ACF ri i [] of i[k] by using time averages instead of statistical averages. An analytical expression for the ACF ri i [] is obtained by using the de nition [23]
1 rii [] = Klim !1 2K + 1
K X
k=?K
i[k]i[k + ]; i = 1; 2:
(41)
Substituting (21) in (41) and taking (16a) into account, the following ACF ri i [] can be derived Ni c2 X i;n
cos(2fi;nTs) 2 n=1 = r~ii (Ts)jfi;n =fi;n ; i = 1; 2:
rii [] =
(42)
Obviously, the ACF ri i [] is a sampled version of the ACF r~ii ( ) as introduced by (16a), whereby additionally the discrete Doppler frequencies fi;n have to be substituted by their modi ed quantities fi;n. Taking the discrete-time Fourier transform of (42) and using (16b) allows us to present the Doppler PSD Si i (f ) of i[k] as function of S~i i (f ) as follows Ni c2 1 X X i;n [(f ? fi;n ? fs) + (f + fi;n ? fs)] Si i (f ) = T1 s =?1 n=1 4
13
1 X 1 ~ = T Si i (f ? fs) f =f ; i = 1; 2: i;n i;n s =?1
(43)
A consequence of the property that the modi ed discrete Doppler frequencies fi;n are depending on the sampling interval Ts is that the performance of the ACF rii [] also depends on Ts. Figs. 6(a) and 6(b) show the results for the ACF ri i [] for a small and large value of Ts, respectively. For the sake of comparison, the ACF r~i i ( ) of the conventional deterministic model and the ACF ri i ( ) of the reference model { both evaluated at = Ts { are also depicted in these gures. The simulation model parameters have been designed according to the MEDS, whereby for the maximum Doppler frequency fmax the value fmax = 91 Hz has been selected, the variance 02 was equal to one, and the number of sinusoids (number of tables) Ni was in both cases equal to eight. It can be seen in Fig. 6(a) that the ACF ri i [] of the tables system coincides very closely with the ACF r~i i ( ) of the deterministic model if the sampling frequency fs is suciently large. But severe degradations with respect to the underlying ACF r~i i ( ) occur if fs is relatively small, say fs 10fmax, as can be observed in Fig. 6(b). The Gaussian processes 1(t) and 2(t) used to describe the Rayleigh process (t) are assumed to be uncorrelated. The same property can easily be imposed on the discrete deterministic Gaussian processes 1[k] and 2[k]. Therefore, we consider the CCF r1 2 [] between 1[k] and 2[k] which can be computed by using time averages instead of statistical averages yielding
8 PN < 1 c1;nc2;m cos(2f1;nTs ? 1;n 2;m) ; r [] = : 2 n=1 0 ; 1 2
if f1;n = f2;m if f1;n 6= f2;m
(44)
for all n = 1; 2; ; N1 and m = 1; 2; ; N2, where N = minfN1; N2 g. Hence, the discrete 1 deterministic Gaussian processes 1 [k] and 2[k] are uncorrelated if the sets ff1;ngNn=1 2 and ff2;ngNn=1 are disjunct. By using the MEDS, the set of discrete Doppler frequencies N 1 ff1;ngn=1 is di erent from the set ff2;ngNn=12 if N1 6= N2 , e.g., by choosing N2 = N1 + 1. Provided that the sampling interval Ts is suciently small, we obtain for the modi ed discrete Doppler frequencies f1;n 6= f2;m for all n, m. Thus, the CCF r1 2 [] of the processes 1[k] and 2[k] becomes zero. C. PDF and CDF The statistical properties of the discrete deterministic Rayleigh process [k] are investigated in this section. Especially, we are deriving analytical expressions for the PDF and CDF of [k]. The obtained results are compared with those of the reference model as well as the with the underlying continuous-time deterministic model.
Let us start with the derivation of the PDF pi;n (x) of a single discrete sinusoidal sequence i;n[k] = ci;n cos(2fi;nkTs + i;n) with known and constant quantities ci;n, fi;n, and i;n. To derive the density of the deterministic sequence i;n[k], we assume that the instant k itself is a discrete random variable which is uniformly distributed over the period Li;n. Hence, i;n[k] is no more a deterministic sequence but a random variable with possible outcomes (realizations) i;n[0], i;n[1], , i;n[Li;n ? 1], whereby each outcome occurs with probability 1=Li;n. The PDF pi;n (x) of i;n[k] is thus a discrete density which can be expressed by
14
LX i;n ?1 1 pi;n (x) = L (x ? i;n[l]) i;n l=0
(45)
for all n = 1; 2; ; Ni (i = 1; 2). Similar arguments allow us to derive an analytical expression for the PDF pi (x) of the discrete deterministic Gaussian process i[k]. The result for pi (x) is given by
pi (x) = L1
X
Li ?1
i l=0
(x ? i[l]);
i = 1; 2:
(46)
Obviously, the PDF pi (x) is a weighted sum of delta functions. The weighting factor is the reciprocal of the period Li and the locations of the delta functions are determined by all possible combinations of the sum of table outputs i[l] allowed by the AG. On the analogy of the above, the envelope PDF p (z) and the phase PDF p#() of the complex discrete deterministic Gaussian process [k] = 1[k]+ j 2 [k] can be derived. The PDFs p (z) and p#() are given by
and
L?1 X 1 p (z) = L (z ? [l]); z 0 l=0
(47)
L?1 X p# () = L1 ( ? #[l]); 2 (?; ]
(48)
l=0
respectively, where [l] = j[l]j is the envelope and #[l] = arg f[l]g is the phase of the complex discrete deterministic Gaussian process [k] = 1[k] + j 2[k] at instant k = l 2 f0; 1; ; L ? 1g. In contrast to the behavior of the continuous PDFs p~ (z) and p (z) of the deterministic and reference model, respectively, the PDF p (z) of the envelope [k] of the tables system is a discrete density. The reason is that the sequence [k] can only take values from a set with an extremely large but limited number of elements given by the period L. From Section V-A, we know that L depends on L1 and L2 , and, thus, on the sampling interval Ts. If Ts is suciently small, then L will be large enough to consider p (z) as an almost continuous density, and we can write p (z) p~ (z). Clearly, in the limit Ts ! 0, we obtain p (z) = p~ (z). Moreover, for Ts ! 0 and Ni ! 1, the PDF p (z) becomes the Rayleigh density (5), i.e. p (z) = p (z). It is also important to note that in general all model parameters Ni, ci;n, fi;n, and even i;n have an in uence on the PDFs (45)-(48). This is in contrast to the continuous-time deterministic model, where the corresponding PDFs [cf. (18a), (18b), (19)] are only functions of Ni and ci. Nevertheless, the in uence of fi;n and i;n on the statistics of the tables system's output sequence can be neglected if Ts is small enough.
15
Next, we carry out a comparison with the conventional deterministic model and the reference model on the basis of the CDF of the envelope. From (47) it follows for the CDF P (r) = Probf(t) rg of the tables system the expression L?1 Z X 1 P (r) = L (z ? [l])dz; r 0: l=0 r
(49)
0
The evaluation of the above expression results for a given signal level r in a rational number for the CDF P (r) that is simply the quotient of the number of values [k] which are less than or equal to r and the length of the period L. Figs. 7(a) and 7(b) present the CDF P (r) of the tables system for small values (Ts = 0:5 ms) and large values (Ts = 5 ms) of the sampling interval Ts, respectively. The results are obtained by simulating K L samples of the output sequence [k]. For the tables system with Ts = 0:5 ms, the number of evaluated samples K was K = 5 105 which is less than the period L 9:8 1015. For the case Ts = 5 ms, we have simulated a full period of the output sequence [k], i.e., the number of samples K was K = L = 9240. For the sake of comparison, we have also plotted the CDF P (r) of the reference model and the CDF P~ (r) of the deterministic model. The results presented in Fig. 7(a) show that the CDFs P (r) and P~ (r) are nearly identical if the sampling interval Ts is small (Ts = 0:5 ms). In this case, the in uence of the phases i;n can be neglected, i.e., di erent realizations of [k] with almost identical statistical properties can be generated by using di erent sets i . On the other hand, if the sampling interval T increases, then the of phases fi;ngNn=1 s N i in uence of di erent realizations for the sets fi;ngn=1 becomes more and more evident. Note that large degradations can be obtained [see Fig. 7(b)] if Ts is non-sucient even though the sampling theorem is ful lled. D. Realization Expenditure and Speed In this section, we will compare the eciency of the conventional deterministic simulation model with the therefrom derived tables system. The simulation of the fading behavior of mobile radio channels by using these types of simulators can basically be partitioned into two steps.
The rst step is called the simulation set-up phase. For the conventional deterministic model, the simulation model parameters (ci;n, fi;n, and i;n) are calculated in this step for a given number of sinusoids Ni. The set-up phase for the tables system requires additional computations. Besides deriving the modi ed parameters fi;n and i;n from those of the underlying conventional deterministic model, the tables entries must be computed and stored. Due to this fact, the memory resources required for the tables system are greater than for the deterministic model. In order to achieve moderate tables lengths, a reasonable value for the sampling interval Ts has to be selected. The simulation run phase is the next step after the simulation set-up phase, where the output sequence of the channel simulator is generated for k = 0; 1; ; K . From Fig. 2, we see that a direct implementation of the conventional deterministic model needs the operations listed in Table I to generate the complex channel output sequence ~[k] := ~(kTs) = ~1(kTs) + j ~2(kTs) at each instant k. Thereby, it is worth mentioning that we have used normalized discrete Doppler frequencies Fi;n = 2Tsfi;n in order to
16
avoid unnecessary multiplications in the argument of the sinusoidal functions (11). By considering the tables system in Fig. 3, we realize that only additions and modulo operations are required. The additions are necessary to generate the addresses and to add the table outputs, whereas the modulo operations are used within the AG. The number of operations required for the tables system are also listed in Table I. TABLE I
number of operations required to compute ~[k] (conventional deterministic simulation model) and [k] (tables system)
Number of Op. Conv. Det. Sim. Model Tables System # Mult. 2(N1 + N2) 0 # Add. 2(N1 + N2 ) ? 2 2(N1 + N2 ) ? 2 # Trig. Op. N 1 + N2 0 # Modulo Op. 0 N 1 + N2 The results shown in Table I can be interpreted as follows. All multiplications can be avoided, the number of additions remain unchanged, and the trigonometric operations can be replaced by modulo operations by using table look-up techniques. Remember that a modulo operation can eciently be implemented on a computer by using a simple if &else command. It should therefore not be surprising that the algorithm of the tables system is considerably faster than that of a direct implementation of the conventional deterministic simulation model. In the following, we present here some results about the speed of both channel simulators. An appropriate measure for the speed is the iteration time (time per output sample) de ned by sim Tsim = TK (50) where Tsim is the simulation time required to generate K samples of the complex channel output sequence. Fig. 8 presents for both simulation models Tsim as function of the number of sinusoids (tables) N1 . For the computation of the parameters of the underlying deterministic simulation model, we have used the MEDS with N2 = N1 + 1 and the JM with N2 = N1 . The number of samples K was always K = 104. The algorithms have been implemented by using MATLAB and the simulation results for Tsim are obtained by running the programs on a workstation (HP 730). It can be observed by considering Fig. 8 that the algorithm of the tables system is for all relevant values of N1 , i.e., N1 7, faster than the algorithm of the conventional deterministic simulation model. By using the MEDS, the improvement of the speed of the simulator is approximately a factor of 3:8 which is nearly independent of N1 . By using the JM, we can exploit the fact that the discrete Doppler frequencies f1;n and f2;n are identical and the corresponding phases (1;n and 2;n) are zero for all n = 1; 2; ; N1 (N1 = N2). This enables a drastic reduction of the complexity of both simulation models. The consequence for the tables system is that the number of tables and the tables lengths coincide for the inphase and quadrature components of the complex discrete Gaussian process [k] = 1[k] + j 2 [k]. Hence, the AG only needs to generate half of the addresses.
17
From Fig. 8 it can also be seen that the speed of the algorithm of the tables system can be improved by a constant factor of approximately 1:5 by using the JM instead of the MEDS. VI. Comparison with the Filter Method
Apart from Rice's sum of sinusoids, the lter method is also widely in use to simulate the typical fading behavior of mobile radio channels [8], [9]. In this section, we will compare the performance and the eciency of a lter method based Rayleigh fading channel simulator with those of the conventional deterministic simulation model and the tables system. A Rayleigh at fading channel simulator by using the lter method is obtained from Fig. 1 by substituting the ideal non-realizable WGN processes ni(t) by non-ideal but therefore realizable pseudo-random sequences n~ i(kTs) that approximate the required stochastic characteristics. Moreover, we have to replace the ideal lter transfer functions Hi(f ) by rational lter transfer functions H~ i(ej2f=fs ). As already mentioned in Section II, the Jakes PSD (3) is often used as proper Doppler PSD for the narrowband Gaussian noise processes 1(t) and 2(t). In order to approximate the square-root Jakes PSD with a rational lter transfer function, eighth-order IIR digital lters are usually used [8], [24]. The system function H~ (z) = H~ 1 (z) = H~ 2(z) of the proposed digital lter is given by
Y4
H~ (z) = Ko
i=1 4
Y i=1
[z ? 0i exp(j'0i)] [z ? 0i exp(?j'0i )]
[z ? 1i exp(j'1i)] [z ? 1i exp(?j'1i )]
(51)
where the constant Ko is introduced to normalize the mean power of the output sequence of the digital lter to 02. Table II presents the parameters describing the system function H~ (z) for a corner frequency fc of fc = 1=(110:5Ts) [24]. TABLE II
parameters of the eight-order IIR filter [24]
i 1 2 3 4
0i 1:0 1:0 1:0 1:0
'0i 5:730778 10?2 7:151706 10?2 0:105841 0:264175
1i 0:991177 0:980664 0:998042 0:999887
'1i 4:542547 10?2 1:912862 10?2 5:507401 10?2 5:670618 10?2
The resulting magnitude-squared frequency response jH~ (ej2f=fs )j2 of the designed digital lter is depicted in Fig. 9 together with the desired shape of the Jakes PSD according to (3). In order to avoid non-linear frequency distortions due to lowpass-lowpass transformations, we have selected a sampling interval of Ts = 0:1 ms which results in a corner frequency of fc = 90:5 Hz. By using the lter coecients listed in Table II, the magnitude-squared frequency response of the digital lter approximates very well the Jakes PSD. The corresponding ACF
18
r~HH (Ts), de ned as the inverse Fourier transform of jH~ (ej2f=fs )j2, also coincides closely with the ideal ACF according to (2) as can be seen by considering Fig. 10. Detailed investigations have shown that good approximations to the desired channel statistics can be achieved by all of the presented simulators. One can say that the main statistical properties (density, level-crossing rate, average duration of fades) of the generated channel output sequence by using: (i) the conventional deterministic simulation model, (ii) the tables system, and (iii) the simulator based on the lter method are completely equivalent provided that the model parameters are accurately designed. Next, we investigate the speed of the channel simulator based on the lter method. In order to enable a fair comparison with the results presented in Section V-D, we have implemented the eight-order IIR lter on the same workstation (HP 730) by using the MATLAB software tool. The simulator has been designed by using a maximum Doppler frequency fmax of fmax = 91 Hz and a variance 02 of 02 = 1. The sampling interval Ts was equal to 0:1 ms so that the corner frequency fc of the lter coincides very closely with the selected maximum Doppler frequency fmax what allows a direct implementation of the digital lter without lowpass-lowpass transformations. In order to compare the speed of the simulators, we use again the iteration time Tsim (FM ) as de ned in the previous section. For the lter method, the simulation time Tsim is composed of the time to generate the lter input noise sequences n~i(kTs) (i = 1; 2) and the time to lter them. The comparison is made with a tables system which has been designed by employing the MEDS with N1 = 7 and N2 = 8 tables. Our simulations have shown that the ratio between the iteration time by using the lter method Tsim (FM ) and that of the tables system Tsim(TS) is (FM ) Tsim 3: (TS ) Tsim
(52)
From the above, we may conclude that the tables system is a faster channel simulator than that based on the lter method. In particular, for typical values of the number of tables, i.e., N1 = 7 and N2 = 8, the speed improvement is approximately a factor of three. We have also observed that the algorithm to generate the lter input noise sequences n~i (kTs) (FM ) (i = 1; 2) requires approximately seventy per cent of the whole simulation time Tsim , i.e., a reduction of the lter order does not automatically lead to a signi cant speed improvement. As we have already shown for the tables system, the iteration time Tsim (TS) increases linearly with the number of tables Ni . Thus, the following interesting question arises. For which values N1 and N2 is the ratio (52) equal to one? Our investigations have shown that this is the case if the number of tables are N1 = 27 and N2 = 28. VII. Conclusions
The concept of deterministic channel modeling is based on Rice's sum of sinusoids. We have shown that the class of deterministic simulation models can eciently be implemented on a computer or hardware platform by using table look-up techniques. The basic idea of the proposed procedure is to store the data of one period of each harmonic function in a
19
table. During simulation, a simple AG points at each instant k in a deterministic manner to the table entries which are summed up to produce a complex fading envelope. The method is very simple and easy to implement due to closed formulas for the parameters of the tables system. It was shown that the tables system can be expressed mathematically as a matrix system which is statistically equivalent but computationally less ecient. Both systems, namely the table and the matrix system, are describing a nite state machine that generates a discrete deterministic process which can be interpreted as deterministic Markov process. The analysis of the statistics of such types of simulation models was also a topic of the paper. Analytical expressions have been derived especially for the ACF, PSD, and even for the PDF of the amplitude and phase. Our results have shown that the sampling frequency fs has a dominant in uence on the performance of the tables system. As good compromise between the model's precision and complexity, we propose to select fs within the range 20fmax fs 30fmax. Our investigations concerning the speed have shown that the conventional deterministic simulation model with (N1; N2) = (7; 8) and simulation models using eight-order IIR lters are roughly speaking equivalent with respect to speed, whereas the tables system is approximately three times faster, what legitimates us to designate the tables system as high-speed channel simulator. References
[1] W.C. Jakes, Ed., Microwave Mobile Communications. New Jersey: IEEE Press, 1993. [2] P. Hoher, \A statistical discrete-time model for the WSSUS multipath channel," IEEE Trans. Veh. Technol., vol. VT-41, no. 4, pp. 461{468, Nov. 1992. [3] P.M. Crespo and J. Jimenez, \Computer simulation of radio channels using a harmonic decomposition technique," IEEE Trans. Veh. Technol., vol. VT-44, no. 3, pp. 414{419, Aug. 1995. [4] M. Patzold, U. Killat, and F. Laue, \A deterministic digital simulation model for Suzuki processes with application to a shadowed Rayleigh land mobile radio channel," IEEE Trans. Veh. Technol., vol. VT-45, no. 2, pp. 318{331, May 1996. [5] S.O. Rice, \Mathematical analysis of random noise," Bell Syst. Tech. J., vol. 23, pp. 282{332, Jul. 1944. [6] S.O. Rice, \Mathematical analysis of random noise," Bell Syst. Tech. J., vol. 24, pp. 46{156, Jan. 1945. [7] E. Lutz and E. Plochinger, \Generating Rice processes with given spectral properties," IEEE Trans. Veh. Technol., vol. VT-34, no. 4, pp. 178{181, Nov. 1985. [8] H. Brehm, W. Stammler, and M. Werner, \Design of a highly exible digital simulator for narrowband fading channels," in Signal Processing III: Theories and Applications, Amsterdam: Elsevier Science Publishers (North-Holland), EURASIP, Sept. 1986, pp. 1113{1116. [9] S.A. Fechtel, \A novel approach to modeling and ecient simulation of frequency-selective fading radio channels," IEEE J. Select. Areas Commun., vol. SAC-11, no. 3, pp. 422{431, Apr. 1993. [10] F. Swarts and H.C. Ferreira, \Markov characterization of channels with soft decision outputs," IEEE Trans. Commun., vol. 41, pp. 678{682, May 1993. [11] H.S. Wang and N. Moayeri, \Finite-state Markov channel - A useful model for radio communication channels," IEEE Trans. Veh. Technol., vol. VT-44, pp. 163{171, Feb. 1995. [12] W. Turin and R. Nobelen, \Hidden Markov modeling of fading channels," in Proc. IEEE 48th Veh. Technol. Conf., VTC'98, Ottawa, Ontario, Canada, May 1998, pp. 1234{1238. [13] J. Hagenauer and W. Papke, \The stored channel - A simulation method for fading channels (in German)," FREQUENZ, vol. 36, pp. 122{129, Apr. /May 1982. [14] J. Hagenauer and W. Papke, \Data transmission for maritime and land mobiles using stored channel simulation," in Proc. IEEE 32nd Veh. Technol. Conf., VTC'82, San Diego, CA, USA, 1982, pp. 379{383. [15] Y. Akaiwa, Introduction to Digital Mobile Communication. New York: John Wiley & Sons, 1997. [16] A. Papoulis, Probability, Random Variables, and Stochastic Processes. New York: McGraw-Hill, 3rd edition, 1991. [17] R.H. Clarke, \A statistical theory of mobile-radio reception," Bell Syst. Tech. J., vol. 47, pp. 957{1000, Jul. /Aug. 1968. [18] J. Proakis, Digital Communications. New York: McGraw-Hill, 3rd edition, 1995. [19] M. Patzold, U. Killat, F. Laue, and Y. Li, \On the statistical properties of deterministic simulation models for mobile fading channels," IEEE Trans. Veh. Technol., vol. VT-47, no. 1, pp. 254{269, Feb. 1998. [20] M. Patzold, Mobile Radio Channels - Modelling, Analysis, and Simulation (in German). Wiesbaden: Vieweg, 1999.
20
[21] H. Schulze, \Stochastic models and digital simulation of mobile channels (in German)," in U.R.S.I/ITG Conf. in Kleinheubach 1988, Germany (FR), Proc. Kleinheubacher Reports of the German PTT, Darmstadt, Germany, 1989, vol. 32, pp. 473{483. [22] M. Patzold and F. Laue, \Statistical properties of Jakes' fading channel simulator," in Proc. IEEE 48th Veh. Technol. Conf., VTC'98, Ottawa, Ontario, Canada, May 1998, pp. 712{718. [23] A.V. Oppenheim and R.W. Schafer, Digital Signal Processing. Englewood Cli s, New Jersey: Prentice-Hall, 1975. [24] R. Hab, Coherent reception by data transmission over frequency-nonselective fading channels (in German). Ph.D. thesis, Rheinisch-Westfalische Technische Hochschule Aachen, Aachen, Germany, Apr. 1988. [25] I.S. Gradstein and I.M. Ryshik, Tables of Series, Products, and Integrals, vol. I and II. Frankfurt: Harri Deutsch, 5th edition, 1981.
21
H1 (f)
WGN
µ 1 (t)
n (t) ~ N(0,1) 1
. H2 (f)
WGN
ζ (t)
µ 2 (t)
n (t) ~ N(0,1) 2
Fig. 1. Analytical model for Rayleigh processes.
c 1,1 cos(2π f1,1 t + θ1,1 ) c 1,2
~ µ (t)
cos(2π f1,2 t + θ1,2 ) ...
...
1
c 1,N1 cos(2π f1,N1 t + θ1,N1 ) . c 2,1
~ µ (t)
cos(2π f2,1 t + θ2,1 )
~ ζ (t)
c 2,2 ~ µ 2 (t) ...
...
cos(2π f2,2 t + θ2,2 )
c 2,N
2
cos(2π f2,N t + θ2,N ) 2
2
Fig. 2. Structure of deterministic Rayleigh processes (direct realization form).
22 Tab1,1 0 1
µ 1,1[k]
. . . L 1,1-1
ADDRESS
a 1,1[k] . . . a 1,N [k] 1
. . . Tab1,N
µ 1 [k] 1
0 1 . . .
µ 1,N1[k]
L 1,N1-1 µ [k]
a GENERATOR
2,1
Tab2,1
. . . a
.
[k] 0 1
[k]
2,N2
. . . L 2,1-1
µ 2,1[k]
. . . Tab2,N2
0 1 . . .
µ 2 [k]
µ 2,N [k] 2
L 2,N -1 2
Fig. 3. Structure of the high-speed channel simulator.
ζ [k]
23
Relative Error, ε f¯
i,n
0.05
0
-0.05 0
0.2
0.4 0.6 0.8 Sampling Interval, T s (ms)
1
Fig. 4. Evaluation of the relative error "fi;n according to (26) for fi;n = 91 Hz as function of the sampling interval Ts.
ˆL i Li ˆL i Li
100
Periods, L i , ˆL i
10
for N i =21 for N i =21 for N i =7 for N i =7
50
10
0
10
-5
10
-4
-3
-2
10 10 10 Normalized Sampling Interval, T S ⋅ f max
-1
10
Fig. 5. Evaluation of the period Li of i[k] and its upper limit L^ i as function of the normalized sampling interval Tsfmax (MEDS, fmax = 91 Hz, 02 = 1).
24
[κ]
1 0.8
r¯µ
iµi
0.6
r˜µ
iµi
rµ
iµi
[κ] [see (16a)] [κ] [see (2)]
ACF, r¯µ
iµi
0.4
[κ] [see (42)]
0.2 0 -0.2 -0.4 -0.6 0
200
400 600 Time Difference, κ
800
1000
(a)
r¯µ µ [κ] [see (42)] i i r˜µ µ (κT s ) [see (16a)]
0.8 0.6
i i
rµ
ACF, r¯µ
iµi
[κ]
0.4
iµi
(κT s ) [see (2)]
0.2 0 -0.2 -0.4 -0.6 -0.8 0
20
40 60 Time Difference, κ
80
100
(b) Fig. 6. ACF ri i [] of the discrete deterministic Gaussian process i[k] for (a) Ts = 0:1 ms and (b) Ts = 1 ms (MEDS, Ni = 8, fmax = 91 Hz, 02 = 1).
25
1
CDF, P¯ ζ (r)
0.8
0.6
0.4 P¯ ζ (r) for T= 0.5 ms(Sim.) P˜ (r) [see (20)] ζ
0.2
0 0
P ζ (r) [see (6)] 1
2 Level, r
3
4
(a)
1 1. Realization 2. Realization
0.8
CDF, P¯ ζ (r)
3. Realization 0.6
0.4 P¯ ζ (r) for T= 5 ms (Sim.) P˜ ζ (r) [see (20)]
0.2
P ζ (r) [see (6)] 0 0
1
2 Level, r
3
4
(b) Fig. 7. CDF P (r) of the discrete deterministic Rayleigh process [k] for (a) Ts = 0:5 ms and (b) Ts = 5 ms (MEDS, N1 = 7, N2 = 8, fmax = 91 Hz, 02 = 1).
26
0.12 Convent. Sim. Model (MEDS) Convent. Sim. Model (JM) Tables System (MEDS) Tables System (JM)
Iteration Time, ∆T sim (ms)
0.1 0.08 0.06 0.04 0.02 0 0
5
10 15 20 Number of Sinusoids, N 1
25
30
Fig. 8. Iteration time Tsim as function of the number of sinusoids (tables) N1 (02 = 1, fmax = 91 Hz, Ts = 0:1 ms).
250
˜ j2 π f/f s )| 2 |H(e
200
150
100
50
0 -0.015
-0.01
-0.005
0 f/f s
0.005
0.01
0.015
Fig. 9. Magnitude-squared frequency response jH~ (ej2f=fs )j2 of the 8th-order IIR lter [8], [24].
27
1 Ideal ACF r µ
iµi
(κ T s )
Real ACF r˜HH (κ T s ) ACF, r˜HH (κ T s )
0.5
0
-0.5 0
200
400 600 Time Difference, κ
800
1000
Fig. 10. ACF r~HH (Ts) of the 8th-order IIR lter presented in [8], [24].
SENIOR MEMBER, IEEE,
RAQUEL GARCIA, and FRANK LAUE
Technical University of Hamburg-Harburg Department of Digital Communication Systems Denickestr. 17 D-21071 Hamburg Germany Tel. : (+49)40-42878-3149 Fax : (+49)40-42878-2941 E-Mail : [email protected]
Abstract
This paper describes a procedure for the design of fast simulation models for Rayleigh fading channels. The presented method is based on an ecient implementation of Rice's sum of sinusoids by using table look-up techniques. The proposed channel simulator is composed of a few number of adders, storage elements, and simple modulo operators, whereas time-consuming operations like multiplications and trigonometric operations are not required. Such a multiplier-free simulation model is introduced as high-speed channel simulator. It is shown that the high-speed channel simulator can be interpreted as a nite state machine which generates deterministic output envelope sequences with approximately Rayleigh distribution. The statistical properties of the designed high-speed channel simulator are investigated analytically and compared with the statistics of the underlying Rayleigh reference model. Results of experiments for measuring the speed of the presented and other types of channel simulators are also presented. Index Terms { Deterministic channel modeling, high-speed fading channel simulator, Rayleigh fading, Rice's sum of sinusoids, statistics.
Submitted to IEEE Transactions on Vehicular Technology First Version: June 19, 1998 Revised Version: August 17,1999 Final Version: October 8, 1999
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I. Introduction
Computer simulation experiments are an established method to design and evaluate digital communication systems. The advantages of a block-oriented subroutine library for each part of the communication system have been widely recognized, since one can easily set up one's own transmission system. Especially for the simulation of mobile communication systems, ecient algorithms for di erent types of fading channels are mandatory for user-friendly tool-box libraries. In past, several methods have been developed for the design of simulation models for mobile radio channels. These methods can be classi ed in four categories. The rst category comprises design methods employing a nite number of low-frequency oscillators to generate pseudo-random noise sequences (e.g. [1], [2], [3], [4]). Such procedures go originally back to an early work of Rice [5], [6], who has shown that narrowband Gaussian noise processes can be modeled by an in nite number of properly designed sinusoids. The second class comprises all those methods which are basing on low-pass ltering of zeromean white Gaussian noise processes by using linear time-invariant lters (e.g. [7], [8], [9]). The class of channel design methods which are basing on Markov processes de nes the third category (e.g. [10], [11], [12]). Finally, the fourth category comprises channel models using the stored channel principle (e.g. [13], [14]). The aim of this paper is to show that all simulation models corresponding to the rst above mentioned category can be eciently implemented on a computer or hardware platform by using table look-up techniques. Thereby, we exploit the fact that the harmonic functions generating the fading signals are periodic functions. The data related to one period of each harmonic function of Rice's sum of sinusoids is stored in a table. The resulting table system is a multiplier-free high-speed channel simulator which can easily be realized by using adders, storage elements, and simple modulo operators. Throughout the paper, we make use of the complex equivalent baseband notation, and, to simplify matters, we consider the transmission of a non-modulated carrier over a Rayleigh
at fading channel. The organization of this paper is as follows. Section II reviews the statistics of Rayleigh processes. Section III presents an overview of the concept of deterministic channel modeling. In Section IV, the new high-speed channel simulator is derived, and its statistical properties are analytically investigated in Section V. Section VI is devoted to compare the performance of the high-speed channel simulator with the underlying conventional deterministic simulation model as well as with a lter method based Rayleigh at fading channel simulator. Finally, the conclusions are drawn in Section VII. II. Review of Rayleigh Processes
The statistics of Rayleigh processes is brie y reviewed in this section. The Rayleigh process is a statistical model that gures in Section V as reference model when discussing and evaluating the performance of the proposed high-speed channel simulator. In typical mobile radio environments, the transmitted signal propagates over di erent signal paths from the transmitter to the receiver. At the antenna of the receiver the signals are superposed to produce an environment speci c standing waves eld. When
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the receiver and/or the transmitter moves in the standing waves eld, the received signal experiences random variations in the envelope and in the phase [15]. An often used and reasonable statistical model for describing the random variations in the complex equivalent baseband is a zero-mean complex Gaussian noise process [1]
(t) = 1(t) + j2(t) (1) where 1(t) and 2(t) are narrowband real Gaussian noise processes with zero mean and variance Varfi(t)g = 02 (i = 1; 2). The Gaussian processes 1(t) and 2(t) are often assumed to be statistically independent. If that is the case, the cross-correlation function (CCF) of 1(t) and 2(t) is zero. In general, a real Gaussian noise process is completely described by its mean and autocorrelation function (ACF) [16]. For an omnidirectional receiving antenna in a two-dimensional isotropic scattering environment, it was shown in the pioneering work of Clark [17] that the ACF of i(t) can be represented by ri i ( ) = 02 J0 (2fmax ) ; i = 1; 2 (2) where fmax denotes the maximum Doppler frequency, and J0() is the zeroth order Bessel function of the rst kind. The Fourier transform of (2) gives the Doppler power spectral density (PSD) Si i (f ) =
(
fmax
0;
p1?(f=f 2 0
max )2
; jf j fmax jf j > fmax
(3)
which is often called in literature Jakes PSD [1]. A Rayleigh process, (t), is de ned as the envelope of (1), i.e.,
(t) = j(t)j: (4) The probability density function (PDF) of (t) is known as Rayleigh density [16] p (z) = z2 exp[?z2 =(202)] ; z 0: (5) 0 Furthermore, the corresponding cumulative distribution function (CDF), de ned by P (r) = Probf (t) rg, can be expressed by [18] P (r) = 1 ? exp[?r2=(202)] ; r 0: (6) The phase of (t) de nes a further random process, namely #(t) = arg f(t)g, with uniform distribution over the interval (?; ], that is, the PDF of the phase #(t) becomes (7) p#() = 21 ; 2 (?; ]: Fig. 1 shows the structure of an analytical model for a Rayleigh at fading channel. Thereby, ni(t) denotes a zero-mean white Gaussian noise (WGN) process with normalized variance Varfni (t)g = 1, and the transferpfunction Hi(f ) of each lter is adapted to the Doppler PSD (3) according to Hi(f ) = Si i (f ) (i = 1; 2). Henceforth, we designate channel design methods employing linear lter techniques as lter methods.
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III. Overview of deterministic channel modeling
Aside from the lter method, there exists another fundamental method for modeling narrowband Gaussian noise processes: the Rice method. The principle of Rice's method [5], [6] implies that a Gaussian noise process i(t) can be modeled by a superposition of an in nite number of weighted harmonic functions with equidistant frequencies and random phases according to
i(t) = Nlim !1 i
Ni X n=1
ci;n cos(2fi;nt + i;n)
(8)
where the gains ci;n and discrete Doppler frequencies fi;n are given by
q
ci;n = 2 fi Si i (fi;n) fi;n = n fi
(9a) (9b)
respectively. Thereby, the random phases i;n are uniformly distributed over the interval (?; ], and the quantity fi is chosen such that (9b) covers the whole frequency range of interest, where fi ! 0 if Ni ! 1. Equation (8) describes an analytical model for Gaussian processes that cannot be implemented on a computer due to the in nite number of sinusoids. When Ni is nite, we obtain another stochastic process
^i(t) =
Ni X n=1
ci;n cos(2fi;nt + i;n)
(10)
which is strictly speaking non-Gaussian distributed. But nevertheless the PDF of ^i(t) is close to a Gaussian density if Ni is sucient, say Ni 7. Only in the limit Ni ! 1, the process ^i(t) follows the ideal Gaussian distribution, and we may write ^i(t) = i(t). It should be noted that ^i(t) is even now a stochastic process due to the assumed random phases i;n. Its implementation on a computer can easily be performed and results in a stochastic simulation model. During the simulation, the phases i;n and all other model parameters are kept constant, so that ^i(t) is now neither more a random process but a sample function which is per de nition completely determined for all time t. To distinguish between the stochastic process ^i(t) with random phases and the corresponding deterministic process (sample function), we use for the latter one the notation
~i(t) =
Ni X n=1
~i;n(t) =
Ni X n=1
ci;n cos(2fi;nt + i;n)
(11)
where the phases i;n are now constant quantities, and the gains ci;n and discrete Doppler frequencies fi;n are -at the moment- furthermore given by (9a) and (9b), respectively. Let Ni ! 1, then the deterministic process ~i(t) tends to a sample function of the stochastic Gaussian process i(t). To emphasize this property, the function ~i(t) has been introduced in [19] as deterministic Gaussian process. By analogy with (1),
~(t) = ~1(t) + j ~2(t) is called complex deterministic Gaussian process, and, consequently ~(t) = j~(t)j
(12) (13)
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represents a deterministic Rayleigh process [20]. A direct realization of deterministic Rayleigh processes leads to the time-continuous structure shown in Fig. 2. The problem with (9a) and (9b) is now twofold. First, the period of ~i(t) is relatively small due to the equally spaced discrete Doppler frequencies fi;n. Second, the density of ~i(t) is non-optimal Gaussian distributed. From the central limit theorem it follows that the density of ~i(t) is for a given number of sinusoids Ni < 1 only optimal Gaussian distributed by keeping the power constraint if all gains ci;n are equal [19]. This equal gain requirement is not ful lled when Rice's original method is used as can be seen immediately by substituting (3) in (9a). To overcome the problems caused by (9a) and (9b) several new methods for the computation of the model parameters (ci;n; fi;n; i;n) have been proposed in recent years. For example, the Monte Carlo method [21], [2], the method of equal areas [4], the Lp-norm method [19], and the method of exact Doppler spread [19]. Even the in-phase and quadrature components of the well-known Jakes simulator (see [1], p. 70) can be brought onto the form (11) [22]. For short, we restrict our investigations in the sequel to the method of exact Doppler spread (MEDS) and the Jakes method (JM). 1) Parameters by using the MEDS [19]: Applying the MEDS to the Jakes PSD (3) gives the following closed form solutions for the gains ci;n and discrete Doppler frequencies fi;n:
r2
ci;n = 0 N (14a) i (14b) fi;n = fmax sin 2N n ? 21 i for all n = 1; 2; ; Ni (i = 1; 2). The phases i;n are obtained by taking Ni drawings from a random generator uniformly distributed over the interval (?; ]. Thereby, the number of sinusoids N1 and N2 are in general related by N2 = N1 + 1 what guarantees the design of zero cross-correlated deterministic Gaussian processes ~1(t) and ~2(t). 2) Parameters by using the JM [1]: When the JM is used, the gains ci;n , discrete Doppler frequencies fi;n, and phases i;n can be expressed in closed form as follows [22]:
n 8 2 > p > < 2Ni? sin Nni?1 ; n = 1; 2; ; Ni ? 1; i = 1 pNi? cos Ni?1 ; n = 1; 2; ; Ni ? 1; i = 2 > > : pNi? ; n = Ni ; i = 1; 2 0
ci;n =
0
0
fi;n =
(
1 2 1 2
(15a)
1 2
fmax cos fmax ;
n 2Ni ?1
;
n = 1; 2; ; Ni ? 1; i = 1; 2 n = Ni ; i = 1; 2
(15b)
i;n = 0; n = 1; 2; ; Ni; i = 1; 2 (15c) where N1 = N2 . According to the JM, the discrete Doppler frequencies f1;n and f2;n are identical for all n = 1; 2; ; N1(N2), and, thus, the processes ~1(t) and ~2 (t) are correlated.
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Of further interest are some basic statistical properties of deterministic processes like the ACF, PSD, CCF, PDF, and CDF. The statistics of deterministic processes has been investigated in detail in [19]. Here, we only present some basic results as far as they are of importance for the understanding of the rest of the paper. A. ACF, PSD, and CCF The ACF r~i i ( ) and the corresponding Doppler PSD S~i i (f ) of the deterministic Gaussian process ~i(t) may be represented by [19]
r~ii ( ) = S~i i (f ) =
Ni c2 X i;n n=1 Ni
2
cos(2fi;n ); i = 1; 2
(16a)
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[(f ? fi;n) + (f + fi;n)]; i = 1; 2
(16b)
X c2i;n n=1
respectively. For the MEDS one can show by putting (14a) and (14b) in (16a) that r~i i ( ) tends to ri i ( ) if Ni ! 1. This is in contrast to the JM, where the inequality r~i i ( ) 6= ri i ( ) holds even if Ni ! 1, as can be proved by substituting (15a) and (15b) in (16a). The CCF r~1 2 ( ) of the deterministic Gaussian processes ~1(t) and ~2(t) is given by [19]
8 PN < 1 c1;nc2;m cos(2f1;n ? 1;n 2;m); r~ ( ) = : 2 n=1 0;
if f1;n= f2;m (17) if f1;n6= f2;m for all n = 1; 2; ; N1 and m = 1; 2; ; N2, where N indicates the minimum value of N1 and N2 , i.e., N = min fN1; N2g. For the MEDS it can easily be guaranteed that the designed processes ~1(t) and ~2(t) are uncorrelated, i.e., r~1 2 ( ) = 0, e.g., by de ning N2 = N1 + 1. For the JM, where N1 is per de nition equal to N2 , we obtain r~1 2 ( ) 6= 0. 1 2
B. PDF and CDF General analytical expressions for the density of the envelope ~ = j~(t)j and phase #~(t) = arg f~(t)g of complex deterministic Gaussian processes ~(t) = ~1(t) + j ~2(t) have been derived in [19]. There the following results are presented:
Z
p~ (z) = z p~1 (z cos ) p~2 (z sin ) d; z 0 ?
p~#() =
Z1
z p~1 (z cos ) p~2 (z sin ) dz; jj
(18a) (18b)
0
where
p~i (x) = 2
Z1 "Y Ni 0
n=1
#
J0 (2ci;n ) cos(2x) d
(19)
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denotes the PDF of the deterministic Gaussian process ~i(t) as introduced by (11). Note that p~i (x), and thus p~ (z) and p~#() are only functions of the gains ci;n and the number of sinusoids Ni. From the central limit theorem it follows for both methods (MEDS and JM) that (19) tends in the limit Ni ! 1 to a Gaussian density with zero mean and variance 02. Hence, when Ni ! 1 then it follows immediately from the expressions (18a) and (18b) that p~ (z) approaches the Rayleigh density (5), whereas p~#() approaches the uniform density (7). However, for nite values of Ni, we may write p~ (z) p (z) and p~#() p#(), thereby it is worth mentioning that the approximations are better for the MEDS than for the JM. After performing some algebraic manipulations, we can obtain from (18a) the following CDF P~ (r) = Probf~(t) rg of ~(t):
Z1
P~ (r) = 4r J1(2rz) 0
"Y N 2
n=1
Z=2"Y N 1
0
n=1
J0(2c1;nz cos )
#
#
J0(2c2;nz sin ) d dz; r 0:
(20)
Thus, the above result enables an analytical investigation of the CDF of deterministic Rayleigh processes ~(t). It should be clear that we may write P~ (r) = P (r) if Ni ! 1. IV. Design of High-Speed Channel Simulators
An ecient implementation of the conventional deterministic model described in the previous section can be achieved by applying table look-up techniques. Each sinusoidal function ~i;n(t) in (11) is substituted by a table which stores the data related to one period of ~i;n(t). For the resulting discrete-time simulation model, an address generator is required to have access to the registers of the tables. The outputs of each block of Ni tables are added in order to obtain an equivalent discrete-time realization of the deterministic Gaussian processes ~i(t) (i = 1; 2). Fig. 3 shows the discrete-time structure of the resulting tables system. The equivalent discrete-time realization of the deterministic Gaussian processes ~i(t) is called discrete deterministic Gaussian process i[k] which can be expressed by
i[k] =
Ni X n=1
i;n[k] =
Ni X n=1
ci;n cos(2fi;nkTs + i;n) ; i = 1; 2
(21)
where Ts denotes the sampling interval. Obviously, the discrete deterministic Gaussian process i[k] is obtained by sampling the continuous-time deterministic Gaussian process ~i(t) at t = kTs (k = 0; 1; ) , whereby additionally the discrete Doppler frequencies fi;n and phases i;n are replaced by fi;n and i;n, respectively. It will be shown below that the new parameters fi;n and i;n are slightly modi ed versions of the original parameters fi;n and i;n, in the order mentioned. In general, a Rayleigh process is obtained by taking the absolute value of a complex Gaussian noise process. According to our method, we can approximately model the statistics of a sample function of a Rayleigh process by combining two real table blocks to a so-called discrete complex deterministic Gaussian process
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[k] = 1 [k] + j 2[k]
(22)
and then taking at each instant k the absolute value according to
[k] = j[k]j = j1[k] + j 2 [k]j
(23)
where [k] is named discrete deterministic Rayleigh process. The table that corresponds to the sinusoidal function i;n[k] is referred in Fig. 3 as Tabi;n. The number of entries of the table Tabi;n corresponds to the table length which will be denoted as Li;n. To nd an expression for the table length Li;n, recall that it is sucient to store only one period of the underlying sinusoidal function ~i;n(t). Thereby, we have to take into account the in uence of the sampling interval Ts. In order to obtain for any given sampling interval Ts an integer number for the table length Li;n, the discrete Doppler frequency fi;n of the original sinusoid has to be slightly modi ed according to fns o fi;n = (24) round ffi;ns where fs is the sampling frequency, and the operation roundfxg denotes the nearest integer to x. The period of the resulting discrete-time sinusoid i;n[k] corresponds to the table length Li;n which is now given by f f s (25) Li;n = = round f s : fi;n i;n By selecting a suciently large value for the sampling frequency fs = 1=Ts, the modi ed discrete Doppler frequency fi;n is very close to the original quantity fi;n. The quality of the approximation fi;n fi;n and, consequently, of the statistics of the resulting discrete-time simulation model, depends on the sampling frequency fs. A measure for the approximation error is the relative error of fi;n, i.e., (26) "fi;n = fi;nf? fi;n : i;n The behavior of "fi;n is shown in Fig. 4 over the sampling interval Ts for a discrete Doppler frequency fi;n of 91 Hz. The presented result shows us that the relative error "fi;n is less than 5 % if the sampling interval Ts is less than (10fi;n)?1. Note that it follows from (24) that we may write fi;n = fi;n if Ts ! 0 (fs ! 1). The complexity of the tables system is closely related with the tables lengths Li;n. From (25), we realize that Li;n is proportional to the sampling frequency fs. Hence, a compromise between the precision and the complexity of the tables system must be found by choosing an adequate value for fs. Experimental results have revealed that an adequate value for the sampling frequency fs is given if fs is within the range of 20fmax fs 30fmax. Since the number of samples of the discrete sinusoid i;n[k] stored in the table Tabi;n is Li;n, then the phase i;n can only take values from a discrete set of Li;n equidistant phases within the interval [0; 2), i.e.,
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i;n 2 i;n = f0; 2fi;nTs; ; 2fi;n(Li;n ? 1)Tsg
(27)
whereby each of these values corresponds to a speci c initial position in the table Tabi;n. The phases i;n are chosen from the above set i;n in such a way that i;n is nearest to i;n. Hence, it follows for Ts ! 0 that we may write i;n = i;n. From the fact that fi;n and i;n are tending to fi;n and i;n, respectively, if Ts approaches to zero, it follows: i[k] ! ~i(t) if Ts ! 0. Moreover, by taking the results of Section III into account, we can say that i[k] tends to a sample function of the Gaussian process i(t) if Ts ! 0 and Ni ! 1. A. Tables System The tables system is composed of N1 + N2 tables, whereby each one is equivalent to a sinusoidal function in the conventional deterministic model. The data stored in the tables contain the full information to compute the corresponding discrete deterministic Gaussian process i;n[k] at any instant k = 0; 1; .
The entry of the table Tabi;n at position l 2 f0; 1; ; Li;n ? 1g corresponds to the value of the discrete-time sinusoidal function i;n[k] at instant k = l, i.e.,
i;n[l] = ci;n cos(2fi;nlTs + i;n)
(28)
where n = 1; 2; ; Ni and i = 1; 2. If the entries of the table Tabi;n are sequentially read from the beginning, then the table output sequence is i;n[0], i;n[1], , i;n[Li;n ? 1], i;n[Li;n] = i;n[0], , and, therefore, the discrete-time sinusoid i;n[k] can completely be reconstructed for all k = 0; 1; . The address generator (AG) shown in Fig. 3 is required to enable the access to the data stored in the tables. The number of addresses that the AG generates is N1 + N2 , i.e., one address for each table. Let us denote the address of the table Tabi;n at instant k by ai;n[k], then the way of operation of the AG can easily be described as follows. At instant k, the AG points to a certain register in the table at position l = ai;n[k]. At the next instant k + 1, the generated address corresponds to the position of the next lower register in the table shown in Fig. 3. When the last position of the table is reached, then the AG points at the next instant to the rst register again. Let us start from the initial addresses ai;n[0] which are determined by the phases i;n, then we can compute recursively the addresses ai;n[k] at each instant k > 0 by using
ai;n[k] = (ai;n[k ? 1] + 1) mod Li;n
(29)
for all n = 1; 2; ; Ni (i = 1; 2), where the notation mod denotes the modulo operation. It should be mentioned that the modulo operation has been introduced here only for mathematical convenience. Its implementation on a computer can easily be achieved by using an adder and an if &else operation.
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B. Matrix System The new implementation form of the conventional deterministic model can also be interpreted as a matrix system. Therefore, we de ne a matrix i for the block of Ni tables which determine the discrete deterministic Gaussian process i[k] (i = 1; 2). The number of rows of the matrix i is equal to the number of tables Ni. Thereby, the nth row of i contains the data of the table Tabi;n . Hence, the length of the largest table de nes i . The rst L elements the number of columns Li;max of i, i.e., Li;max = maxfLi;ngNn=1 i;n of the nth row of i are determined by the contents of the registers of table Tabi;n, whereas the rest of the row is padded with zeros. Thus, the matrix i 2 R Ni Li;max can be represented by
M
M
M
M
M
M
0 i;1[0] i;1[1] M i = BB@ i;2..[0] i;2..[1]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i;1[Li;1 ? 1] . . . . . . . . . . . . . . . . . . i;2[Li;2 ? 1] 0 0 ... . .. . . i;Ni [0] i;Ni [1] i;Ni [Li;Ni ? 1] 0 0
1 CC A
(30) where we have assumed { without loss in generality { that the number of columns Li;max is given by the rst table, i.e., Li;max = Li;1. This is in fact the case by using the MEDS, since fi;1 is the smallest discrete Doppler frequency obtained from (14b). Next, we introduce a further matrix i which is called selection matrix. This matrix is closely related with the addresses generated by the AG. The elements of i can take the values 0 or 1 and they are changing their positions at each instant k, just like the addresses of the AG. The elements of the selection matrix i = (sl;n) 2 f0; 1gLi;maxNi at instant k are given by
S
S
S
sl;n = sl;n[k] =
1
if l = ai;n[k] 0 if l = 6 ai;n[k]
(31)
for all l = 0; 1; ; Li;max ? 1 and n = 1; 2; ; Ni (i = 1; 2). The discrete deterministic Gaussian process i[k] can now be obtained from the product of the matrices i and i as follows:
M
S
M i Si) ;
i[k] = tr (
i = 1; 2
(32)
where tr() denotes trace. Consequently, the discrete complex deterministic Gaussian process (22) can be expressed alternatively by
M S ) + j tr (M S ) :
[k] = tr (
(33) Finally, an equivalent representation of the discrete deterministic Rayleigh process [k] introduced by (23) is given by 1
1
2
2
M S ) + j tr (M S ) j:
[k] = j[k]j = jtr (
1
1
2
2
(34)
Although the tables system and the matrix system are equivalent from the statistical point of view, the realization of the later one is less ecient. Therefore, we restrict our further investigations to the tables system.
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V. Performance of the high-speed Channel Simulator
A. Period The sequence i[k] (i = 1; 2) generated by the tables system is designed to approximate the statistics of a stochastic process, in our case, a narrowband Gaussian noise process which is of course a non-periodic stochastic process. But due to the limited number of tables, whereby each having a nite number of elements, the generated output sequence of the tables system is a periodic sequence. Nevertheless, we will see in this section that the period of the output sequence is extremely large, i.e., the tables system generates quasi non-periodic sequences which are sucient for most practical applications.
Recall that the period of a single sinusoid i;n[k] was introduced as Li;n. The period of the sum of sinusoids i[k] is denoted henceforth as Li . In the following, we show that the i , period Li of i[k] is the least common multiple (LCM) of the elements of the set fLi;ngNn=1 i.e., i ; i = 1; 2: Li = lcmfLi;ngNn=1
(35)
Therefore, we have to prove the validity of
i[k] = i[k + Li ] 8 k 2 Z:
(36)
i , L is a multiple of each L , and we can write Since Li is the LCM of the set fLi;ngNn=1 i i;n Li;n = dLi (37) i;n
where di;n is a natural number which can be di erent for each Li;n. It is clear that if Li;n is the period of i;n[k], then the product di;nLi;n satis es also
i;n[k] = i;n[k + di;nLi;n] 8 k 2 Z:
(38)
Consequently, from (21), (36), and (38), we obtain
i[k + Li ] = = =
Ni P i;n[k + Li ] n=1 Ni P [k + d L n=1 Ni
i;n
P [k] i;n n=1
i;n i;n ]
(39)
= i[k]:
i , it is assured that this is the minimum value Since Li is de ned as the LCM of fLi;ngNn=1 which satis es (36), and therefore, Li is the period of i[k].
Note that the upper limit of the period Li is de ned by the product L^ i = Li;1 Li;2 Li;Ni which also ful lls (36), i.e.,
12
L^ i =
Ni Y n=1
i : Li;n Li = lcm fLi;ngNn=1
(40)
The period L of the discrete deterministic Rayleigh process [k] = j1[k] + j 2[k]j can directly be derived from the LCM of L1 and L2 , i.e., L = lcmfL1 ; L2g. The period Li of the output sequences i[k] of the tables system is dependent on the sampling frequency fs, because the period Li;n of each single discrete sinusoid i;n[k] is a function of fs [see (25)]. Consequently, fs has also an in uence on the period L of the discrete deterministic Rayleigh process [k]. In order to achieve long periods Li and L, the value selected for fs should be suciently large. Observe that in the limit fs ! 1, the periods Li and L are approaching in nity, provided that Ni is suciently large. The evaluation of the period Li and its upper limit L^ i as function of the normalized sampling frequency fs=fmax is shown in Fig. 5 for a small and a large number of tables Ni. From the results of Fig. 5 it can be observed that in many cases the period Li is close to its upper limit L^ i . Moreover, it can also be realized that the period Li is extremely large even for relatively small sampling frequencies fs. This fact gives us an account of saying that i[k] can be considered as a quasi non-periodic deterministic sequence. B. ACF, PSD, and CCF Due to the deterministic behavior of i[k], we have to compute the ACF ri i [] of i[k] by using time averages instead of statistical averages. An analytical expression for the ACF ri i [] is obtained by using the de nition [23]
1 rii [] = Klim !1 2K + 1
K X
k=?K
i[k]i[k + ]; i = 1; 2:
(41)
Substituting (21) in (41) and taking (16a) into account, the following ACF ri i [] can be derived Ni c2 X i;n
cos(2fi;nTs) 2 n=1 = r~ii (Ts)jfi;n =fi;n ; i = 1; 2:
rii [] =
(42)
Obviously, the ACF ri i [] is a sampled version of the ACF r~ii ( ) as introduced by (16a), whereby additionally the discrete Doppler frequencies fi;n have to be substituted by their modi ed quantities fi;n. Taking the discrete-time Fourier transform of (42) and using (16b) allows us to present the Doppler PSD Si i (f ) of i[k] as function of S~i i (f ) as follows Ni c2 1 X X i;n [(f ? fi;n ? fs) + (f + fi;n ? fs)] Si i (f ) = T1 s =?1 n=1 4
13
1 X 1 ~ = T Si i (f ? fs) f =f ; i = 1; 2: i;n i;n s =?1
(43)
A consequence of the property that the modi ed discrete Doppler frequencies fi;n are depending on the sampling interval Ts is that the performance of the ACF rii [] also depends on Ts. Figs. 6(a) and 6(b) show the results for the ACF ri i [] for a small and large value of Ts, respectively. For the sake of comparison, the ACF r~i i ( ) of the conventional deterministic model and the ACF ri i ( ) of the reference model { both evaluated at = Ts { are also depicted in these gures. The simulation model parameters have been designed according to the MEDS, whereby for the maximum Doppler frequency fmax the value fmax = 91 Hz has been selected, the variance 02 was equal to one, and the number of sinusoids (number of tables) Ni was in both cases equal to eight. It can be seen in Fig. 6(a) that the ACF ri i [] of the tables system coincides very closely with the ACF r~i i ( ) of the deterministic model if the sampling frequency fs is suciently large. But severe degradations with respect to the underlying ACF r~i i ( ) occur if fs is relatively small, say fs 10fmax, as can be observed in Fig. 6(b). The Gaussian processes 1(t) and 2(t) used to describe the Rayleigh process (t) are assumed to be uncorrelated. The same property can easily be imposed on the discrete deterministic Gaussian processes 1[k] and 2[k]. Therefore, we consider the CCF r1 2 [] between 1[k] and 2[k] which can be computed by using time averages instead of statistical averages yielding
8 PN < 1 c1;nc2;m cos(2f1;nTs ? 1;n 2;m) ; r [] = : 2 n=1 0 ; 1 2
if f1;n = f2;m if f1;n 6= f2;m
(44)
for all n = 1; 2; ; N1 and m = 1; 2; ; N2, where N = minfN1; N2 g. Hence, the discrete 1 deterministic Gaussian processes 1 [k] and 2[k] are uncorrelated if the sets ff1;ngNn=1 2 and ff2;ngNn=1 are disjunct. By using the MEDS, the set of discrete Doppler frequencies N 1 ff1;ngn=1 is di erent from the set ff2;ngNn=12 if N1 6= N2 , e.g., by choosing N2 = N1 + 1. Provided that the sampling interval Ts is suciently small, we obtain for the modi ed discrete Doppler frequencies f1;n 6= f2;m for all n, m. Thus, the CCF r1 2 [] of the processes 1[k] and 2[k] becomes zero. C. PDF and CDF The statistical properties of the discrete deterministic Rayleigh process [k] are investigated in this section. Especially, we are deriving analytical expressions for the PDF and CDF of [k]. The obtained results are compared with those of the reference model as well as the with the underlying continuous-time deterministic model.
Let us start with the derivation of the PDF pi;n (x) of a single discrete sinusoidal sequence i;n[k] = ci;n cos(2fi;nkTs + i;n) with known and constant quantities ci;n, fi;n, and i;n. To derive the density of the deterministic sequence i;n[k], we assume that the instant k itself is a discrete random variable which is uniformly distributed over the period Li;n. Hence, i;n[k] is no more a deterministic sequence but a random variable with possible outcomes (realizations) i;n[0], i;n[1], , i;n[Li;n ? 1], whereby each outcome occurs with probability 1=Li;n. The PDF pi;n (x) of i;n[k] is thus a discrete density which can be expressed by
14
LX i;n ?1 1 pi;n (x) = L (x ? i;n[l]) i;n l=0
(45)
for all n = 1; 2; ; Ni (i = 1; 2). Similar arguments allow us to derive an analytical expression for the PDF pi (x) of the discrete deterministic Gaussian process i[k]. The result for pi (x) is given by
pi (x) = L1
X
Li ?1
i l=0
(x ? i[l]);
i = 1; 2:
(46)
Obviously, the PDF pi (x) is a weighted sum of delta functions. The weighting factor is the reciprocal of the period Li and the locations of the delta functions are determined by all possible combinations of the sum of table outputs i[l] allowed by the AG. On the analogy of the above, the envelope PDF p (z) and the phase PDF p#() of the complex discrete deterministic Gaussian process [k] = 1[k]+ j 2 [k] can be derived. The PDFs p (z) and p#() are given by
and
L?1 X 1 p (z) = L (z ? [l]); z 0 l=0
(47)
L?1 X p# () = L1 ( ? #[l]); 2 (?; ]
(48)
l=0
respectively, where [l] = j[l]j is the envelope and #[l] = arg f[l]g is the phase of the complex discrete deterministic Gaussian process [k] = 1[k] + j 2[k] at instant k = l 2 f0; 1; ; L ? 1g. In contrast to the behavior of the continuous PDFs p~ (z) and p (z) of the deterministic and reference model, respectively, the PDF p (z) of the envelope [k] of the tables system is a discrete density. The reason is that the sequence [k] can only take values from a set with an extremely large but limited number of elements given by the period L. From Section V-A, we know that L depends on L1 and L2 , and, thus, on the sampling interval Ts. If Ts is suciently small, then L will be large enough to consider p (z) as an almost continuous density, and we can write p (z) p~ (z). Clearly, in the limit Ts ! 0, we obtain p (z) = p~ (z). Moreover, for Ts ! 0 and Ni ! 1, the PDF p (z) becomes the Rayleigh density (5), i.e. p (z) = p (z). It is also important to note that in general all model parameters Ni, ci;n, fi;n, and even i;n have an in uence on the PDFs (45)-(48). This is in contrast to the continuous-time deterministic model, where the corresponding PDFs [cf. (18a), (18b), (19)] are only functions of Ni and ci. Nevertheless, the in uence of fi;n and i;n on the statistics of the tables system's output sequence can be neglected if Ts is small enough.
15
Next, we carry out a comparison with the conventional deterministic model and the reference model on the basis of the CDF of the envelope. From (47) it follows for the CDF P (r) = Probf(t) rg of the tables system the expression L?1 Z X 1 P (r) = L (z ? [l])dz; r 0: l=0 r
(49)
0
The evaluation of the above expression results for a given signal level r in a rational number for the CDF P (r) that is simply the quotient of the number of values [k] which are less than or equal to r and the length of the period L. Figs. 7(a) and 7(b) present the CDF P (r) of the tables system for small values (Ts = 0:5 ms) and large values (Ts = 5 ms) of the sampling interval Ts, respectively. The results are obtained by simulating K L samples of the output sequence [k]. For the tables system with Ts = 0:5 ms, the number of evaluated samples K was K = 5 105 which is less than the period L 9:8 1015. For the case Ts = 5 ms, we have simulated a full period of the output sequence [k], i.e., the number of samples K was K = L = 9240. For the sake of comparison, we have also plotted the CDF P (r) of the reference model and the CDF P~ (r) of the deterministic model. The results presented in Fig. 7(a) show that the CDFs P (r) and P~ (r) are nearly identical if the sampling interval Ts is small (Ts = 0:5 ms). In this case, the in uence of the phases i;n can be neglected, i.e., di erent realizations of [k] with almost identical statistical properties can be generated by using di erent sets i . On the other hand, if the sampling interval T increases, then the of phases fi;ngNn=1 s N i in uence of di erent realizations for the sets fi;ngn=1 becomes more and more evident. Note that large degradations can be obtained [see Fig. 7(b)] if Ts is non-sucient even though the sampling theorem is ful lled. D. Realization Expenditure and Speed In this section, we will compare the eciency of the conventional deterministic simulation model with the therefrom derived tables system. The simulation of the fading behavior of mobile radio channels by using these types of simulators can basically be partitioned into two steps.
The rst step is called the simulation set-up phase. For the conventional deterministic model, the simulation model parameters (ci;n, fi;n, and i;n) are calculated in this step for a given number of sinusoids Ni. The set-up phase for the tables system requires additional computations. Besides deriving the modi ed parameters fi;n and i;n from those of the underlying conventional deterministic model, the tables entries must be computed and stored. Due to this fact, the memory resources required for the tables system are greater than for the deterministic model. In order to achieve moderate tables lengths, a reasonable value for the sampling interval Ts has to be selected. The simulation run phase is the next step after the simulation set-up phase, where the output sequence of the channel simulator is generated for k = 0; 1; ; K . From Fig. 2, we see that a direct implementation of the conventional deterministic model needs the operations listed in Table I to generate the complex channel output sequence ~[k] := ~(kTs) = ~1(kTs) + j ~2(kTs) at each instant k. Thereby, it is worth mentioning that we have used normalized discrete Doppler frequencies Fi;n = 2Tsfi;n in order to
16
avoid unnecessary multiplications in the argument of the sinusoidal functions (11). By considering the tables system in Fig. 3, we realize that only additions and modulo operations are required. The additions are necessary to generate the addresses and to add the table outputs, whereas the modulo operations are used within the AG. The number of operations required for the tables system are also listed in Table I. TABLE I
number of operations required to compute ~[k] (conventional deterministic simulation model) and [k] (tables system)
Number of Op. Conv. Det. Sim. Model Tables System # Mult. 2(N1 + N2) 0 # Add. 2(N1 + N2 ) ? 2 2(N1 + N2 ) ? 2 # Trig. Op. N 1 + N2 0 # Modulo Op. 0 N 1 + N2 The results shown in Table I can be interpreted as follows. All multiplications can be avoided, the number of additions remain unchanged, and the trigonometric operations can be replaced by modulo operations by using table look-up techniques. Remember that a modulo operation can eciently be implemented on a computer by using a simple if &else command. It should therefore not be surprising that the algorithm of the tables system is considerably faster than that of a direct implementation of the conventional deterministic simulation model. In the following, we present here some results about the speed of both channel simulators. An appropriate measure for the speed is the iteration time (time per output sample) de ned by sim Tsim = TK (50) where Tsim is the simulation time required to generate K samples of the complex channel output sequence. Fig. 8 presents for both simulation models Tsim as function of the number of sinusoids (tables) N1 . For the computation of the parameters of the underlying deterministic simulation model, we have used the MEDS with N2 = N1 + 1 and the JM with N2 = N1 . The number of samples K was always K = 104. The algorithms have been implemented by using MATLAB and the simulation results for Tsim are obtained by running the programs on a workstation (HP 730). It can be observed by considering Fig. 8 that the algorithm of the tables system is for all relevant values of N1 , i.e., N1 7, faster than the algorithm of the conventional deterministic simulation model. By using the MEDS, the improvement of the speed of the simulator is approximately a factor of 3:8 which is nearly independent of N1 . By using the JM, we can exploit the fact that the discrete Doppler frequencies f1;n and f2;n are identical and the corresponding phases (1;n and 2;n) are zero for all n = 1; 2; ; N1 (N1 = N2). This enables a drastic reduction of the complexity of both simulation models. The consequence for the tables system is that the number of tables and the tables lengths coincide for the inphase and quadrature components of the complex discrete Gaussian process [k] = 1[k] + j 2 [k]. Hence, the AG only needs to generate half of the addresses.
17
From Fig. 8 it can also be seen that the speed of the algorithm of the tables system can be improved by a constant factor of approximately 1:5 by using the JM instead of the MEDS. VI. Comparison with the Filter Method
Apart from Rice's sum of sinusoids, the lter method is also widely in use to simulate the typical fading behavior of mobile radio channels [8], [9]. In this section, we will compare the performance and the eciency of a lter method based Rayleigh fading channel simulator with those of the conventional deterministic simulation model and the tables system. A Rayleigh at fading channel simulator by using the lter method is obtained from Fig. 1 by substituting the ideal non-realizable WGN processes ni(t) by non-ideal but therefore realizable pseudo-random sequences n~ i(kTs) that approximate the required stochastic characteristics. Moreover, we have to replace the ideal lter transfer functions Hi(f ) by rational lter transfer functions H~ i(ej2f=fs ). As already mentioned in Section II, the Jakes PSD (3) is often used as proper Doppler PSD for the narrowband Gaussian noise processes 1(t) and 2(t). In order to approximate the square-root Jakes PSD with a rational lter transfer function, eighth-order IIR digital lters are usually used [8], [24]. The system function H~ (z) = H~ 1 (z) = H~ 2(z) of the proposed digital lter is given by
Y4
H~ (z) = Ko
i=1 4
Y i=1
[z ? 0i exp(j'0i)] [z ? 0i exp(?j'0i )]
[z ? 1i exp(j'1i)] [z ? 1i exp(?j'1i )]
(51)
where the constant Ko is introduced to normalize the mean power of the output sequence of the digital lter to 02. Table II presents the parameters describing the system function H~ (z) for a corner frequency fc of fc = 1=(110:5Ts) [24]. TABLE II
parameters of the eight-order IIR filter [24]
i 1 2 3 4
0i 1:0 1:0 1:0 1:0
'0i 5:730778 10?2 7:151706 10?2 0:105841 0:264175
1i 0:991177 0:980664 0:998042 0:999887
'1i 4:542547 10?2 1:912862 10?2 5:507401 10?2 5:670618 10?2
The resulting magnitude-squared frequency response jH~ (ej2f=fs )j2 of the designed digital lter is depicted in Fig. 9 together with the desired shape of the Jakes PSD according to (3). In order to avoid non-linear frequency distortions due to lowpass-lowpass transformations, we have selected a sampling interval of Ts = 0:1 ms which results in a corner frequency of fc = 90:5 Hz. By using the lter coecients listed in Table II, the magnitude-squared frequency response of the digital lter approximates very well the Jakes PSD. The corresponding ACF
18
r~HH (Ts), de ned as the inverse Fourier transform of jH~ (ej2f=fs )j2, also coincides closely with the ideal ACF according to (2) as can be seen by considering Fig. 10. Detailed investigations have shown that good approximations to the desired channel statistics can be achieved by all of the presented simulators. One can say that the main statistical properties (density, level-crossing rate, average duration of fades) of the generated channel output sequence by using: (i) the conventional deterministic simulation model, (ii) the tables system, and (iii) the simulator based on the lter method are completely equivalent provided that the model parameters are accurately designed. Next, we investigate the speed of the channel simulator based on the lter method. In order to enable a fair comparison with the results presented in Section V-D, we have implemented the eight-order IIR lter on the same workstation (HP 730) by using the MATLAB software tool. The simulator has been designed by using a maximum Doppler frequency fmax of fmax = 91 Hz and a variance 02 of 02 = 1. The sampling interval Ts was equal to 0:1 ms so that the corner frequency fc of the lter coincides very closely with the selected maximum Doppler frequency fmax what allows a direct implementation of the digital lter without lowpass-lowpass transformations. In order to compare the speed of the simulators, we use again the iteration time Tsim (FM ) as de ned in the previous section. For the lter method, the simulation time Tsim is composed of the time to generate the lter input noise sequences n~i(kTs) (i = 1; 2) and the time to lter them. The comparison is made with a tables system which has been designed by employing the MEDS with N1 = 7 and N2 = 8 tables. Our simulations have shown that the ratio between the iteration time by using the lter method Tsim (FM ) and that of the tables system Tsim(TS) is (FM ) Tsim 3: (TS ) Tsim
(52)
From the above, we may conclude that the tables system is a faster channel simulator than that based on the lter method. In particular, for typical values of the number of tables, i.e., N1 = 7 and N2 = 8, the speed improvement is approximately a factor of three. We have also observed that the algorithm to generate the lter input noise sequences n~i (kTs) (FM ) (i = 1; 2) requires approximately seventy per cent of the whole simulation time Tsim , i.e., a reduction of the lter order does not automatically lead to a signi cant speed improvement. As we have already shown for the tables system, the iteration time Tsim (TS) increases linearly with the number of tables Ni . Thus, the following interesting question arises. For which values N1 and N2 is the ratio (52) equal to one? Our investigations have shown that this is the case if the number of tables are N1 = 27 and N2 = 28. VII. Conclusions
The concept of deterministic channel modeling is based on Rice's sum of sinusoids. We have shown that the class of deterministic simulation models can eciently be implemented on a computer or hardware platform by using table look-up techniques. The basic idea of the proposed procedure is to store the data of one period of each harmonic function in a
19
table. During simulation, a simple AG points at each instant k in a deterministic manner to the table entries which are summed up to produce a complex fading envelope. The method is very simple and easy to implement due to closed formulas for the parameters of the tables system. It was shown that the tables system can be expressed mathematically as a matrix system which is statistically equivalent but computationally less ecient. Both systems, namely the table and the matrix system, are describing a nite state machine that generates a discrete deterministic process which can be interpreted as deterministic Markov process. The analysis of the statistics of such types of simulation models was also a topic of the paper. Analytical expressions have been derived especially for the ACF, PSD, and even for the PDF of the amplitude and phase. Our results have shown that the sampling frequency fs has a dominant in uence on the performance of the tables system. As good compromise between the model's precision and complexity, we propose to select fs within the range 20fmax fs 30fmax. Our investigations concerning the speed have shown that the conventional deterministic simulation model with (N1; N2) = (7; 8) and simulation models using eight-order IIR lters are roughly speaking equivalent with respect to speed, whereas the tables system is approximately three times faster, what legitimates us to designate the tables system as high-speed channel simulator. References
[1] W.C. Jakes, Ed., Microwave Mobile Communications. New Jersey: IEEE Press, 1993. [2] P. Hoher, \A statistical discrete-time model for the WSSUS multipath channel," IEEE Trans. Veh. Technol., vol. VT-41, no. 4, pp. 461{468, Nov. 1992. [3] P.M. Crespo and J. Jimenez, \Computer simulation of radio channels using a harmonic decomposition technique," IEEE Trans. Veh. Technol., vol. VT-44, no. 3, pp. 414{419, Aug. 1995. [4] M. Patzold, U. Killat, and F. Laue, \A deterministic digital simulation model for Suzuki processes with application to a shadowed Rayleigh land mobile radio channel," IEEE Trans. Veh. Technol., vol. VT-45, no. 2, pp. 318{331, May 1996. [5] S.O. Rice, \Mathematical analysis of random noise," Bell Syst. Tech. J., vol. 23, pp. 282{332, Jul. 1944. [6] S.O. Rice, \Mathematical analysis of random noise," Bell Syst. Tech. J., vol. 24, pp. 46{156, Jan. 1945. [7] E. Lutz and E. Plochinger, \Generating Rice processes with given spectral properties," IEEE Trans. Veh. Technol., vol. VT-34, no. 4, pp. 178{181, Nov. 1985. [8] H. Brehm, W. Stammler, and M. Werner, \Design of a highly exible digital simulator for narrowband fading channels," in Signal Processing III: Theories and Applications, Amsterdam: Elsevier Science Publishers (North-Holland), EURASIP, Sept. 1986, pp. 1113{1116. [9] S.A. Fechtel, \A novel approach to modeling and ecient simulation of frequency-selective fading radio channels," IEEE J. Select. Areas Commun., vol. SAC-11, no. 3, pp. 422{431, Apr. 1993. [10] F. Swarts and H.C. Ferreira, \Markov characterization of channels with soft decision outputs," IEEE Trans. Commun., vol. 41, pp. 678{682, May 1993. [11] H.S. Wang and N. Moayeri, \Finite-state Markov channel - A useful model for radio communication channels," IEEE Trans. Veh. Technol., vol. VT-44, pp. 163{171, Feb. 1995. [12] W. Turin and R. Nobelen, \Hidden Markov modeling of fading channels," in Proc. IEEE 48th Veh. Technol. Conf., VTC'98, Ottawa, Ontario, Canada, May 1998, pp. 1234{1238. [13] J. Hagenauer and W. Papke, \The stored channel - A simulation method for fading channels (in German)," FREQUENZ, vol. 36, pp. 122{129, Apr. /May 1982. [14] J. Hagenauer and W. Papke, \Data transmission for maritime and land mobiles using stored channel simulation," in Proc. IEEE 32nd Veh. Technol. Conf., VTC'82, San Diego, CA, USA, 1982, pp. 379{383. [15] Y. Akaiwa, Introduction to Digital Mobile Communication. New York: John Wiley & Sons, 1997. [16] A. Papoulis, Probability, Random Variables, and Stochastic Processes. New York: McGraw-Hill, 3rd edition, 1991. [17] R.H. Clarke, \A statistical theory of mobile-radio reception," Bell Syst. Tech. J., vol. 47, pp. 957{1000, Jul. /Aug. 1968. [18] J. Proakis, Digital Communications. New York: McGraw-Hill, 3rd edition, 1995. [19] M. Patzold, U. Killat, F. Laue, and Y. Li, \On the statistical properties of deterministic simulation models for mobile fading channels," IEEE Trans. Veh. Technol., vol. VT-47, no. 1, pp. 254{269, Feb. 1998. [20] M. Patzold, Mobile Radio Channels - Modelling, Analysis, and Simulation (in German). Wiesbaden: Vieweg, 1999.
20
[21] H. Schulze, \Stochastic models and digital simulation of mobile channels (in German)," in U.R.S.I/ITG Conf. in Kleinheubach 1988, Germany (FR), Proc. Kleinheubacher Reports of the German PTT, Darmstadt, Germany, 1989, vol. 32, pp. 473{483. [22] M. Patzold and F. Laue, \Statistical properties of Jakes' fading channel simulator," in Proc. IEEE 48th Veh. Technol. Conf., VTC'98, Ottawa, Ontario, Canada, May 1998, pp. 712{718. [23] A.V. Oppenheim and R.W. Schafer, Digital Signal Processing. Englewood Cli s, New Jersey: Prentice-Hall, 1975. [24] R. Hab, Coherent reception by data transmission over frequency-nonselective fading channels (in German). Ph.D. thesis, Rheinisch-Westfalische Technische Hochschule Aachen, Aachen, Germany, Apr. 1988. [25] I.S. Gradstein and I.M. Ryshik, Tables of Series, Products, and Integrals, vol. I and II. Frankfurt: Harri Deutsch, 5th edition, 1981.
21
H1 (f)
WGN
µ 1 (t)
n (t) ~ N(0,1) 1
. H2 (f)
WGN
ζ (t)
µ 2 (t)
n (t) ~ N(0,1) 2
Fig. 1. Analytical model for Rayleigh processes.
c 1,1 cos(2π f1,1 t + θ1,1 ) c 1,2
~ µ (t)
cos(2π f1,2 t + θ1,2 ) ...
...
1
c 1,N1 cos(2π f1,N1 t + θ1,N1 ) . c 2,1
~ µ (t)
cos(2π f2,1 t + θ2,1 )
~ ζ (t)
c 2,2 ~ µ 2 (t) ...
...
cos(2π f2,2 t + θ2,2 )
c 2,N
2
cos(2π f2,N t + θ2,N ) 2
2
Fig. 2. Structure of deterministic Rayleigh processes (direct realization form).
22 Tab1,1 0 1
µ 1,1[k]
. . . L 1,1-1
ADDRESS
a 1,1[k] . . . a 1,N [k] 1
. . . Tab1,N
µ 1 [k] 1
0 1 . . .
µ 1,N1[k]
L 1,N1-1 µ [k]
a GENERATOR
2,1
Tab2,1
. . . a
.
[k] 0 1
[k]
2,N2
. . . L 2,1-1
µ 2,1[k]
. . . Tab2,N2
0 1 . . .
µ 2 [k]
µ 2,N [k] 2
L 2,N -1 2
Fig. 3. Structure of the high-speed channel simulator.
ζ [k]
23
Relative Error, ε f¯
i,n
0.05
0
-0.05 0
0.2
0.4 0.6 0.8 Sampling Interval, T s (ms)
1
Fig. 4. Evaluation of the relative error "fi;n according to (26) for fi;n = 91 Hz as function of the sampling interval Ts.
ˆL i Li ˆL i Li
100
Periods, L i , ˆL i
10
for N i =21 for N i =21 for N i =7 for N i =7
50
10
0
10
-5
10
-4
-3
-2
10 10 10 Normalized Sampling Interval, T S ⋅ f max
-1
10
Fig. 5. Evaluation of the period Li of i[k] and its upper limit L^ i as function of the normalized sampling interval Tsfmax (MEDS, fmax = 91 Hz, 02 = 1).
24
[κ]
1 0.8
r¯µ
iµi
0.6
r˜µ
iµi
rµ
iµi
[κ] [see (16a)] [κ] [see (2)]
ACF, r¯µ
iµi
0.4
[κ] [see (42)]
0.2 0 -0.2 -0.4 -0.6 0
200
400 600 Time Difference, κ
800
1000
(a)
r¯µ µ [κ] [see (42)] i i r˜µ µ (κT s ) [see (16a)]
0.8 0.6
i i
rµ
ACF, r¯µ
iµi
[κ]
0.4
iµi
(κT s ) [see (2)]
0.2 0 -0.2 -0.4 -0.6 -0.8 0
20
40 60 Time Difference, κ
80
100
(b) Fig. 6. ACF ri i [] of the discrete deterministic Gaussian process i[k] for (a) Ts = 0:1 ms and (b) Ts = 1 ms (MEDS, Ni = 8, fmax = 91 Hz, 02 = 1).
25
1
CDF, P¯ ζ (r)
0.8
0.6
0.4 P¯ ζ (r) for T= 0.5 ms(Sim.) P˜ (r) [see (20)] ζ
0.2
0 0
P ζ (r) [see (6)] 1
2 Level, r
3
4
(a)
1 1. Realization 2. Realization
0.8
CDF, P¯ ζ (r)
3. Realization 0.6
0.4 P¯ ζ (r) for T= 5 ms (Sim.) P˜ ζ (r) [see (20)]
0.2
P ζ (r) [see (6)] 0 0
1
2 Level, r
3
4
(b) Fig. 7. CDF P (r) of the discrete deterministic Rayleigh process [k] for (a) Ts = 0:5 ms and (b) Ts = 5 ms (MEDS, N1 = 7, N2 = 8, fmax = 91 Hz, 02 = 1).
26
0.12 Convent. Sim. Model (MEDS) Convent. Sim. Model (JM) Tables System (MEDS) Tables System (JM)
Iteration Time, ∆T sim (ms)
0.1 0.08 0.06 0.04 0.02 0 0
5
10 15 20 Number of Sinusoids, N 1
25
30
Fig. 8. Iteration time Tsim as function of the number of sinusoids (tables) N1 (02 = 1, fmax = 91 Hz, Ts = 0:1 ms).
250
˜ j2 π f/f s )| 2 |H(e
200
150
100
50
0 -0.015
-0.01
-0.005
0 f/f s
0.005
0.01
0.015
Fig. 9. Magnitude-squared frequency response jH~ (ej2f=fs )j2 of the 8th-order IIR lter [8], [24].
27
1 Ideal ACF r µ
iµi
(κ T s )
Real ACF r˜HH (κ T s ) ACF, r˜HH (κ T s )
0.5
0
-0.5 0
200
400 600 Time Difference, κ
800
1000
Fig. 10. ACF r~HH (Ts) of the 8th-order IIR lter presented in [8], [24].
