Design of high-speed simulation models for mobile fading channels ...

1178 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 49, NO. 4, JULY 2000 

Design of High-Speed Simulation Models for Mobile 

Fading Channels by Using Table Look-Up Techniques 

Matthias Pätzold, Senior Member, IEEE, Raquel García, and Frank Laue 

Abstract—This paper describes a procedure for the design 

of fast simulation models for Rayleigh fading channels. The 

presented method is based on an efficient 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 

finite 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. 

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, efficient algorithms for different types of fading 

channels are mandatory for user-friendly tool-box libraries. 

In the past, several methods have been developed for the 

design of simulation models for mobile radio channels. These 

methods can be classified in four categories. The first category 

comprises design methods employing a finite number of 

low-frequency oscillators to generate pseudo-random noise 

sequences (e.g., [1]–[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 infinite number 

of properly designed sinusoids. The second class comprises 

all those methods which are basing on low-pass filtering of 

zero-mean white Gaussian noise processes by using linear 

time-invariant filters (e.g., [7]–[9]). The class of channel design 

methods which are basing on Markov processes defines the 

third category (e.g., [10]–[12]). Finally, the fourth category 

Manuscript received June 19, 1998; revised August 17, 1999. 

The authors are with the Department of Digital Communication Systems, 

Technical University of Hamburg-Harburg, D-21071 Hamburg, Germany 

(e-mail: paetzold@tu-harburg.de). 

Publisher Item Identifier S 0018-9545(00)03687-2. 

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 first above mentioned category can be efficiently 

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 nonmodulated carrier over a Rayleigh flat 

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 filter method 

based Rayleigh flat fading channel simulator. Finally, the conclusions 

are drawn in Section VII. 

II. REVIEW OF RAYLEIGH PROCESSES 

The statistics of Rayleigh processes is briefly reviewed in 

Section II. The Rayleigh process is a statistical model that figures 

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 different signal paths from the transmitter to 

the receiver. At the antenna of the receiver the signals are superposed 

to produce an environment specific standing waves field. 

When the receiver and/or the transmitter moves in the standing 

waves field, 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] 

where and are narrowband real Gaussian noise processes 

with zero mean and variance Var ( 

(1) 

0018-9545/00$10.00 © 2000 IEEE

PÄTZOLD et al.: DESIGN OF HIGH-SPEED SIMULATION MODELS 1179 

Fig. 1. 

Analytical model for Rayleigh processes. 

). The Gaussian processes and are often assumed 

to be statistically independent. If that is the case, the 

cross-correlation function (CCF) of and 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 can be represented by 

where denotes the maximum Doppler frequency, and 

is the zeroth order Bessel function of the first kind. The Fourier 

transform of (2) gives the Doppler power spectral density (PSD) 

(2) 

each filter is adapted to the Doppler PSD (3) according to 

. Henceforth, we designate 

channel design methods employing linear filter techniques as 

filter methods. 

III. OVERVIEW OF DETERMINISTIC CHANNEL MODELING 

Aside from the filter 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 can be 

modeled by a superposition of an infinite number of weighted 

harmonic functions with equidistant frequencies and random 

phases according to 

which is often called in literature Jakes PSD [1]. 

A Rayleigh process, , is defined as the envelope of (1), 

i.e., 

The probability density function (PDF) of 

Rayleigh density [16] 

(3) 

(4) 

is known as 

Furthermore, the corresponding cumulative distribution function 

(CDF), defined by Prob , can be expressed 

by [18] 

The phase of defines a further random process, namely 

, with uniform distribution over the interval 

, that is, the PDF of the phase becomes 

Fig. 1 shows the structure of an analytical model for a Rayleigh 

flat fading channel. Thereby, denotes a zero-mean 

white Gaussian noise (WGN) process with normalized variance 

Var , and the transfer function of 

(5) 

(6) 

(7) 

where the gains and discrete Doppler frequencies are 

given by 

(8) 

(9a) 

(9b) 

respectively. Thereby, the random phases are uniformly 

distributed over the interval , and the quantity is 

chosen such that (9b) covers the whole frequency range of interest, 

where if . Equation (8) describes an 

analytical model for Gaussian processes that cannot be implemented 

on a computer due to the infinite number of sinusoids. 

When is finite, we obtain another stochastic process 

(10) 

which is strictly speaking non-Gaussian distributed. But nevertheless 

the PDF of is close to a Gaussian density if 

is sufficient, say . Only in the limit , the 

process follows the ideal Gaussian distribution, and we 

may write . It should be noted that is even 

now a stochastic process due to the assumed random phases 

. Its implementation on a computer can easily be performed


Fig. 2. 

Structure of deterministic Rayleigh processes (direct realization form). 

and results in a stochastic simulation model. During the simulation, 

the phases and all other model parameters are kept 

constant, so that is now neither more a random process 

but a sample function which is per definition completely determined 

for all time . To distinguish between the stochastic 

process with random phases and the corresponding deterministic 

process (sample function), we use for the latter one the 

notation 

(11) 

where the phases are now constant quantities, and the 

gains and discrete Doppler frequencies are—at the 

moment—furthermore given by (9a) and (9b), respectively. 

Let , then the deterministic process tends to a 

sample function of the stochastic Gaussian process .To 

emphasize this property, the function has been introduced 

in [19] as deterministic Gaussian process. By analogy with (1) 

(12) 

is called complex deterministic Gaussian process, and, consequently 

(13) 

represents a deterministic Rayleigh process [20]. A direct realization 

of deterministic Rayleigh processes leads to the timecontinuous 

structure shown in Fig. 2. 

The problem with (9a) and (9b) is now twofold. First, the period 

of is relatively small due to the equally spaced discrete 

Doppler frequencies . Second, the density of 

is nonoptimal Gaussian distributed. From the central limit theorem 

it follows that the density of is for a given number 

of sinusoids only optimal Gaussian distributed by 

keeping the power constraint if all gains are equal [19]. 

This equal gain requirement is not fulfilled 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 

have been proposed in recent years. For 

example, the Monte Carlo method [21], [2], the method of equal 

areas [4], the -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 and discrete Doppler frequencies 

(14a) 

(14b)


for all . The phases are 

obtained by taking drawings from a random generator 

uniformly distributed over the interval . Thereby, the 

number of sinusoids and are in general related by 

what guarantees the design of zero cross-correlated 

deterministic Gaussian processes and . 

Parameters by Using the JM [1]: When the JM is used, the 

gains , discrete Doppler frequencies , and phases 

can be expressed in closed form as follows [22]: 

, 

, 

(15a) 

, 

(15b) 

(15c) 

where 

. According to the JM, the discrete 

Doppler frequencies and are identical for all 

, and, thus, the processes and 

are correlated. 

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 and the corresponding Doppler PSD 

of the deterministic Gaussian process may be 

represented by [19] 

(16a) 

respectively. For the MEDS, one can show by putting (14a) and 

(14b) in (16a) that tends to if . 

This is in contrast to the JM, where the inequality 

holds even if , as can be proved by substituting 

(15a) and (15b) in (16a). The CCF of the deterministic 

Gaussian processes and is given by [19] 

if 

if 

(17) 

for all and , where 

indicates the minimum value of and , i.e., 

. For the MEDS it can easily be guaranteed 

that the designed processes and are uncorrelated, 

i.e., , e.g., by defining .For 

the JM, where is per definition equal to , we obtain 

. 

B. PDF and CDF 

General analytical expressions for the density of the envelope 

and phase 

of complex deterministic 

Gaussian processes 

have been derived 

in [19]. There the following results are presented: 

where 

(18a) 

(18b) 

(19) 

denotes the PDF of the deterministic Gaussian process as 

introduced by (11). Note that , and thus and 

are only functions of the gains and the number of sinusoids 

. From the central limit theorem it follows for both methods 

(MEDS and JM) that (19) tends in the limit 

to a 

Gaussian density with zero mean and variance . Hence, when 

then it follows immediately from the expressions 

(18a) and (18b) that approaches the Rayleigh density (5), 

whereas approaches the uniform density (7). However, for 

finite values of , we may write and 

, 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 Prob of 

: 

(16b) 

(20)


Fig. 3. 

Structure of the high-speed channel simulator. 

Thus, the above result enables an analytical investigation of the 

CDF of deterministic Rayleigh processes . It should be clear 

that we may write if . 

IV. DESIGN OF HIGH-SPEED CHANNEL SIMULATORS 

An efficient implementation of the conventional deterministic 

model described in Section III can be achieved by applying table 

look-up techniques. Each sinusoidal function in (11) is 

substituted by a table which stores the data related to one period 

of . 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 tables are added 

in order to obtain an equivalent discrete-time realization of the 

deterministic Gaussian processes ( ). Fig. 3 shows 

the discrete-time structure of the resulting tables system. 

The equivalent discrete-time realization of the deterministic 

Gaussian processes is called discrete deterministic 

Gaussian process which can be expressed by 

(21) 

where denotes the sampling interval. Obviously, the discrete 

deterministic Gaussian process is obtained by sampling 

the continuous-time deterministic Gaussian process at 

, whereby additionally the discrete Doppler 

frequencies and phases are replaced by and , 

respectively. It will be shown below that the new parameters 

and are slightly modified versions of the original parameters 

and , 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 

(22) 

and then taking at each instant the absolute value according to 

(23) 

where is named discrete deterministic Rayleigh process. 

The table that corresponds to the sinusoidal function 

is referred in Fig. 3 as Tab . The number of entries of the table 

Tab corresponds to the table length which will be denoted as 

. To find an expression for the table length , recall that 

it is sufficient to store only one period of the underlying sinusoidal 

function . Thereby, we have to take into account


Since the number of samples of the discrete sinusoid 

stored in the table Tab is , then the phase can only 

take values from a discrete set of equidistant phases within 

the interval , i.e., 

Fig. 4. Evaluation of the relative error " according to (26) for f = 91 

Hz as function of the sampling interval T . 

the influence of the sampling interval . In order to obtain for 

any given sampling interval an integer number for the table 

length , the discrete Doppler frequency of the original 

sinusoid has to be slightly modified according to 

round 

(24) 

where is the sampling frequency, and the operation round 

denotes the nearest integer to . The period of the resulting discrete-time 

sinusoid corresponds to the table length 

which is now given by 

round (25) 

By selecting a sufficiently large value for the sampling frequency 

, the modified discrete Doppler frequency 

is very close to the original quantity . The quality of 

the approximation 

and, consequently, of the statistics 

of the resulting discrete-time simulation model, depends on 

the sampling frequency . A measure for the approximation 

error is the relative error of , i.e., 

(26) 

The behavior of is shown in Fig. 4 over the sampling interval 

for a discrete Doppler frequency of 91 Hz. The 

presented result shows us that the relative error is less than 

5% if the sampling interval is less than . Note that 

it follows from (24) that we may write 

if 

. 

The complexity of the tables system is closely related with 

the tables lengths . From (25), we realize that is proportional 

to the sampling frequency . Hence, a compromise 

between the precision and the complexity of the tables system 

must be found by choosing an adequate value for . Experimental 

results have revealed that an adequate value for the sampling 

frequency is given if is within the range of 

. 

(27) 

whereby each of these values corresponds to a specific initial 

position in the table Tab . The phases are chosen from 

the above set in such a way that is nearest to . 

Hence, it follows for that we may write . 

From the fact that and are tending to and , 

respectively, if approaches to zero, it follows: 

if . Moreover, by taking the results of Section III into 

account, we can say that tends to a sample function of the 

Gaussian process if and . 

A. Tables System 

The tables system is composed of 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 at any instant 

. 

The entry of the table Tab at position 

corresponds to the value of the 

discrete-time sinusoidal function at instant , i.e., 

(28) 

where and . If the entries of the 

table Tab are sequentially read from the beginning, then the 

table output sequence is , , , , 

, , and, therefore, the discrete-time sinusoid 

can completely be reconstructed for all 

. 

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 , i.e., one address 

for each table. Let us denote the address of the table Tab at 

instant by , then the way of operation of the AG can 

easily be described as follows. At instant , the AG points to a 

certain register in the table at position . At the next 

instant , 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 first register again. 

Let us start from the initial addresses which are determined 

by the phases , then we can compute recursively the 

addresses at each instant by using 

(29) 

for all 

, where the notation 

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 operation.


B. Matrix System 

The new implementation form of the conventional deterministic 

model can also be interpreted as a matrix system. Therefore, 

we define a matrix for the block of tables which determine 

the discrete deterministic Gaussian process 

. The number of rows of the matrix is equal to the 

number of tables . Thereby, the th row of contains the 

data of the table Tab . Hence, the length of the largest table 

defines the number of columns of , i.e., 

. The first elements of the th row of 

are determined by the contents of the registers of table Tab , 

whereas the rest of the row is padded with zeros. Thus, the matrix 

can be represented by (30), shown at 

the bottom of the page, where we have assumed—without loss 

in generality—that the number of columns is given by 

the first table, i.e., 

. This is in fact the case by 

using the MEDS, since is the smallest discrete Doppler frequency 

obtained from (14b). Next, we introduce a further matrix 

which is called selection matrix. This matrix is closely 

related with the addresses generated by the AG. The elements 

of can take the values 0 or 1 and they are changing their positions 

at each instant , just like the addresses of the AG. The elements 

of the selection matrix 

at instant are given by 

if 

if 

(31) 

for all 

and 

. 

The discrete deterministic Gaussian process can now be 

obtained from the product of the matrices and as follows: 

V. PERFORMANCE OF THE HIGH-SPEED CHANNEL SIMULATOR 

A. Period 

The sequence ( ) 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 nonperiodic stochastic process. But due to the limited 

number of tables, whereby each having a finite 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-nonperiodic sequences which are 

sufficient for most practical applications. 

Recall that the period of a single sinusoid was introduced 

as . The period of the sum of sinusoids is denoted 

henceforth as . In the following, we show that the period 

of is the least common multiple (LCM) of the elements 

of the set , i.e., 

lcm (35) 

Therefore, we have to prove the validity of 

(36) 

Since is the LCM of the set , is a multiple of 

each , and we can write 

(37) 

where is a natural number which can be different for each 

. It is clear that if is the period of , then the 

product satisfies also 

(38) 

Consequently, from (21), (36), and (38), we obtain 

tr (32) 

where tr denotes trace. Consequently, the discrete complex 

deterministic Gaussian process (22) can be expressed alternatively 

by 

tr tr (33) 

Finally, an equivalent representation of the discrete deterministic 

Rayleigh process introduced by (23) is given by 

tr tr (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 efficient. Therefore, we restrict our further investigations 

to the tables system. 

(39) 

Since is defined as the LCM of , it is assured that 

this is the minimum value which satisfies (36), and therefore, 

is the period of . 

Note that the upper limit of the period is defined by the 

product 

which also fulfills (36), i.e., 

lcm (40) 

. 

. 

. 

. 

. .. 

. 

. 

(30)


Obviously, the ACF is a sampled version of the ACF 

as introduced by (16a), whereby additionally the discrete 

Doppler frequencies have to be substituted by their 

modified quantities . 

Taking the discrete-time Fourier transform of (42) and using 

(16b) allows us to present the Doppler PSD of 

as function of as follows: 

Fig. 5. Evaluation of the period L of [k] and its upper limit ^L as function 

of the normalized sampling interval T f (MEDS, f =91Hz, =1). 

The period of the discrete deterministic Rayleigh process 

can directly be derived from the LCM 

of and , i.e., lcm . 

The period of the output sequences of the tables 

system is dependent on the sampling frequency , because the 

period of each single discrete sinusoid is a function 

of [see (25)]. Consequently, has also an influence on the 

period of the discrete deterministic Rayleigh process .In 

order to achieve long periods and , the value selected for 

should be sufficiently large. Observe that in the limit , 

the periods and are approaching infinity, provided that 

is sufficiently large. 

The evaluation of the period and its upper limit as function 

of the normalized sampling frequency is shown in 

Fig. 5 for a small and a large number of tables . From the 

results of Fig. 5 it can be observed that in many cases the period 

is close to its upper limit . Moreover, it can also be 

realized that the period is extremely large even for relatively 

small sampling frequencies . This fact gives us an account of 

saying that can be considered as a quasi-nonperiodic deterministic 

sequence. 

B. ACF, PSD, and CCF 

Due to the deterministic behavior of , we have to compute 

the ACF of by using time averages instead 

of statistical averages. An analytical expression for the ACF 

is obtained by using the definition [23] 

(41) 

Substituting (21) in (41) and taking (16a) into account, the following 

ACF can be derived 

(42) 

(43) 

A consequence of the property that the modified discrete 

Doppler frequencies are depending on the sampling 

interval is that the performance of the ACF also 

depends on . Fig. 6(a) and (b) show the results for the ACF 

for a small and large value of , respectively. For 

the sake of comparison, the ACF of the conventional 

deterministic model and the ACF of the reference 

model—both evaluated at —are also depicted in these 

figures. The simulation model parameters have been designed 

according to the MEDS, whereby for the maximum Doppler 

frequency the value Hz has been selected, 

the variance was equal to one, and the number of sinusoids 

(number of tables) was in both cases equal to eight. It 

can be seen in Fig. 6(a) that the ACF of the tables 

system coincides very closely with the ACF of the 

deterministic model if the sampling frequency is sufficiently 

large. But severe degradations with respect to the underlying 

ACF occur if is relatively small, say , 

as can be observed in Fig. 6(b). 

The Gaussian processes and used to describe the 

Rayleigh process are assumed to be uncorrelated. The same 

property can easily be imposed on the discrete deterministic 

Gaussian processes and . Therefore, we consider the 

CCF between and which can be computed 

by using time averages instead of statistical averages yielding 

if 

if 

(44) 

for all and , where 

. Hence, the discrete deterministic Gaussian processes 

and are uncorrelated if the sets 

and 

are disjunct. By using the MEDS, the set of 

discrete Doppler frequencies 

is different from the 

set if , e.g., by choosing . 

Provided that the sampling interval is sufficiently small, we


. The PDF of is thus a discrete density 

which can be expressed by 

(45) 

for all . 

Similar arguments allow us to derive an analytical expression 

for the PDF of the discrete deterministic Gaussian 

process . The result for is given by 

(46) 

Obviously, the PDF is a weighted sum of delta functions. 

The weighting factor is the reciprocal of the period and the 

locations of the delta functions are determined by all possible 

combinations of the sum of table outputs allowed by the 

AG. 

On the analogy of the above, the envelope PDF and 

the phase PDF of the complex discrete deterministic 


can be derived. The 

PDFs and are given by 

and 

(47) 

(48) 

Fig. 6. ACF r [] of the discrete deterministic Gaussian process [k] for: 

(a) T =0:1 ms and (b) T =1ms (MEDS, N =8, f =91Hz, =1). 

obtain for the modified discrete Doppler frequencies 

for all , . Thus, the CCF of the processes 

and becomes zero. 

C. PDF and CDF 

The statistical properties of the discrete deterministic 

Rayleigh process are investigated in Section V-C. Especially, 

we are deriving analytical expressions for the PDF and 

CDF of . 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 of a single 

discrete sinusoidal sequence 

with known and constant quantities , , and . 

To derive the density of the deterministic sequence ,we 

assume that the instant itself is a discrete random variable 

which is uniformly distributed over the period . Hence, 

is no more a deterministic sequence but a random variable 

with possible outcomes (realizations) , , , 

, whereby each outcome occurs with probability 

respectively, where 

is the envelope and 

is the phase of the complex discrete deterministic 


at instant 

. 

In contrast to the behavior of the continuous PDFs and 

of the deterministic and reference model, respectively, the 

PDF of the envelope of the tables system is a discrete 

density. The reason is that the sequence can only take 

values from a set with an extremely large but limited number of 

elements given by the period . From Section V-A, we know 

that depends on and , and, thus, on the sampling interval 

.If is sufficiently small, then will be large enough 

to consider as an almost continuous density, and we can 

write Clearly, in the limit , we obtain 

. Moreover, for and , the PDF 

becomes the Rayleigh density (5), i.e., . 

It is also important to note that in general all model parameters 

, , , and even have an influence on the PDFs 

(45)–(48). This is in contrast to the continuous-time deterministic 

model, where the corresponding PDFs [cf. (18a), (18b), and 

(19)] are only functions of and . Nevertheless, the influence 

of and on the statistics of the tables system’s output 

sequence can be neglected if is small enough. 

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 

Prob of the tables system the expression 

(49)


TABLE I 

NUMBER OF OPERATIONS REQUIRED TO 

COMPUTE ~[k] (CONVENTIONAL DETERMINISTIC SIMULATION MODEL) AND 

[k] (TABLES SYSTEM) 

glected, i.e., different realizations of with almost identical 

statistical properties can be generated by using different sets of 

phases . On the other hand, if the sampling interval 

increases, then the influence of different realizations for the 

sets becomes more and more evident. Note that large 

degradations can be obtained [see Fig. 7(b)] if is nonsufficient 

even though the sampling theorem is fulfilled. 

Fig. 7. CDF P (r) of the discrete deterministic Rayleigh process [k] for: (a) 

T =0:5 ms and (b) T =5ms (MEDS, N =7, N =8, f =91Hz, 

=1). 

The evaluation of the above expression results for a given signal 

level in a rational number for the CDF that is simply 

the quotient of the number of values which are less than or 

equal to and the length of the period . 

Fig. 7(a) and (b) presents the CDF of the tables system 

for small values ( ms) and large values ( ms) of 

the sampling interval , respectively. The results are obtained 

by simulating samples of the output sequence . 

For the tables system with ms, the number of evaluated 

samples was which is less than the period 

. For the case ms, we have simulated a 

full period of the output sequence , i.e., the number of samples 

was . For the sake of comparison, we 

have also plotted the CDF of the reference model and 

the CDFs of the deterministic model. The results presented 

in Fig. 7(a) show that the CDFs and are 

nearly identical if the sampling interval is small ( 

ms). In this case, the influence of the phases can be ne- 

D. Realization Expenditure and Speed 

In Section V-D, we will compare the efficiency 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 first step is called the simulation set-up phase. For the 

conventional deterministic model, the simulation model parameters 

( , , and ) are calculated in this step for a given 

number of sinusoids . The set-up phase for the tables system 

requires additional computations. Besides deriving the modified 

parameters and 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 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 

. 

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 

at each instant . 

Thereby, it is worth mentioning that we have used normalized 

discrete Doppler frequencies 

in order to 

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. 

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 efficiently be implemented 

on a computer by using a simple command.


Fig. 8. Iteration time 1T as function of the number of sinusoids (tables) 

N ( =1, f =91Hz, T =0:1 ms). 

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) defined by 

(50) 

where is the simulation time required to generate samples 

of the complex channel output sequence. Fig. 8 presents 

for both simulation models as function of the number 

of sinusoids (tables) . For the computation of the parameters 

of the underlying deterministic simulation model, we have used 

the MEDS with and the JM with . The 

number of samples was always . The algorithms 

have been implemented by using MATLAB and the simulation 

results for 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 , i.e., , 

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 . By using the JM, we can exploit the fact 

that the discrete Doppler frequencies and are identical 

and the corresponding phases ( and ) are zero for all 

( ). 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 . 

Hence, the AG only needs to generate half of the addresses. 

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. 

Fig. 9. Magnitude-squared frequency response j ~ H(e )j of the 

8th-order IIR filter [8], [24]. 

VI. COMPARISON WITH THE FILTER METHOD 

Apart from Rice’s sum of sinusoids, the filter method is also 

widely in use to simulate the typical fading behavior of mobile 

radio channels [8], [9]. In Section VI, we will compare 

the performance and the efficiency of a filter method based 

Rayleigh fading channel simulator with those of the conventional 

deterministic simulation model and the tables system. 

A Rayleigh flat fading channel simulator by using the filter 

method is obtained from Fig. 1 by substituting the ideal nonrealizable 

WGN processes by nonideal but therefore realizable 

pseudo-random sequences that approximate 

the required stochastic characteristics. Moreover, we have to replace 

the ideal filter transfer functions by rational filter 

transfer functions . 

As already mentioned in Section II, the Jakes PSD (3) is 

often used as proper Doppler PSD for the narrowband Gaussian 

noise processes and . In order to approximate the 

square-root Jakes PSD with a rational filter transfer function, 

eighth-order IIR digital filters are usually used [8], [24]. 

The system function 

of the proposed 

digital filter is given by 

(51) 

where the constant is introduced to normalize the mean 

power of the output sequence of the digital filter to . Table II 

presents the parameters describing the system function for 

a corner frequency of [24]. 

The resulting magnitude-squared frequency response 

of the designed digital filter is depicted in 

Fig. 9 together with the desired shape of the Jakes PSD according 

to (3). In order to avoid nonlinear frequency distortions 

due to lowpass–lowpass transformations, we have selected a


TABLE II 

PARAMETERS OF THE EIGHT-ORDER IIR FILTER [24] 

method, the simulation time is composed of the time to 

generate the filter input noise sequences ( ) and 

the time to filter them. 

The comparison is made with a tables system which has been 

designed by employing the MEDS with and 

tables. Our simulations have shown that the ratio between the 

iteration time by using the filter method and that of the 

tables system is 

(52) 

Fig. 10. ACF ~r (T ) of the 8th-order IIR filter presented in [8] and [24]. 

sampling interval of ms which results in a corner 

frequency of Hz. 

By using the filter coefficients listed in Table II, the 

magnitude-squared frequency response of the digital filter 

approximates very well the Jakes PSD. The corresponding 

ACF , defined as the inverse Fourier transform of 

, 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 filter method are completely 

equivalent provided that the model parameters are accurately 

designed. 

Next, we investigate the speed of the channel simulator based 

on the filter method. In order to enable a fair comparison with 

the results presented in Section V-D, we have implemented the 

eight-order IIR filter on the same workstation (HP 730) by using 

the MATLAB software tool. The simulator has been designed 

by using a maximum Doppler frequency of Hz 

and a variance of . The sampling interval was 

equal to 0.1 ms so that the corner frequency of the filter coincides 

very closely with the selected maximum Doppler frequency 

what allows a direct implementation of the digital 

filter without lowpass–lowpass transformations. 

In order to compare the speed of the simulators, we use again 

the iteration time as defined in Section V. For the filter 

From the above, we may conclude that the tables system is a 

faster channel simulator than that based on the filter method. 

In particular, for typical values of the number of tables, i.e., 

and , the speed improvement is approximately 

a factor of three. We have also observed that the algorithm to 

generate the filter input noise sequences ( ) requires 

approximately seventy per cent of the whole simulation 

time , i.e., a reduction of the filter order does not automatically 

lead to a significant speed improvement. As we have 

already shown for the tables system, the iteration time 

increases linearly with the number of tables . Thus, the following 

interesting question arises. For which values and 

is the ratio (52) equal to one? Our investigations have shown 

that this is the case if the number of tables are and 

. 

VII. CONCLUSION 

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 efficiently 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 table. During 

simulation, a simple AG points at each instant 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 efficient. Both systems, 

namely the table and the matrix system, are describing a finite 

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 has a dominant influence on 

the performance of the tables system. As good compromise 

between the model’s precision and complexity, we propose to 

select within the range . 

Our investigations concerning the speed have shown that the 

conventional deterministic simulation model with 

and simulation models using eight-order IIR filters 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. Piscataway, 

NJ: IEEE Press, 1993. 

[2] P. Höher, “A statistical discrete-time model for the WSSUS multipath 

channel,” IEEE Trans. Veh. Technol., vol. 41, pp. 461–468, Nov. 1992. 

[3] P. M. Crespo and J. Jiménez, “Computer simulation of radio channels 

using a harmonic decomposition technique,” IEEE Trans. Veh. Technol., 

vol. 44, pp. 414–419, Aug. 1995. 

[4] M. Pätzold, 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. 45, 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] , “Mathematical analysis of random noise,” Bell Syst. Tech. J., vol. 

24, pp. 46–156, Jan. 1945. 

[7] E. Lutz and E. Plöchinger, “Generating Rice processes with given spectral 

properties,” IEEE Trans. Veh. Technol., vol. VT-34, pp. 178–181, 

Nov. 1985. 

[8] H. Brehm, W. Stammler, and M. Werner, “Design of a highly flexible 

digital simulator for narrowband fading channels,” in Signal Processing 

III: Theories and Applications. Amsterdam, The Netherlands: Elsevier 

Science Publishers (North-Holland), EURASIP, Sep. 1986, pp. 

1113–1116. 

[9] S. A. Fechtel, “A novel approach to modeling and efficient simulation 

of frequency-selective fading radio channels,” IEEE J. Select. Areas 

Commun., vol. 11, 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. 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., Ottawa, ON, 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, 

April/May 1982. 

[14] , “Data transmission for maritime and land mobiles using stored 

channel simulation,” in Proc. IEEE 32nd Veh. Technol. Conf., San Diego, 

CA, 1982, pp. 379–383. 

[15] Y. Akaiwa, Introduction to Digital Mobile Communication. New York: 

Wiley, 1997. 

[16] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 

3rd ed. New York: McGraw-Hill, 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, 3rd ed. New York: McGraw-Hill, 

1995. 

[19] M. Pätzold, U. Killat, F. Laue, and Y. Li, “On the statistical properties 

of deterministic simulation models for mobile fading channels,” IEEE 

Trans. Veh. Technol., vol. 47, pp. 254–269, Feb. 1998. 

[20] M. Pätzold, Mobile Fading Channels—Modeling, Analysis, and Simulation 

(in German). Wiesbaden, Germany: Vieweg, 1999. 

[21] H. Schulze, “Stochastic models and digital simulation of mobile channels 

(in German),” in U.R.S.I/ITG Conf. in Kleinheubach 1988, vol. 32, 

Darmstadt, Germany (FR), 1989, Proc. Kleinheubacher Reports of the 

German PTT, pp. 473–483. 

[22] M. Pätzold and F. Laue, “Statistical properties of Jakes’ fading channel 

simulator,” in Proc. IEEE 48th Veh. Technol. Conf., Ottawa, ON, 

Canada, May 1998, pp. 712–718. 

[23] A. V. Oppenheim and R. W. Schafer, Digital Signal Processing. 

Englewood Cliffs, NJ: Prentice-Hall, 1975. 

[24] R. Häb, “Coherent reception by data transmission over frequency-nonselective 

fading channels,” Ph.D. dissertation (in German), Rheinisch- 

Westfälische Technische Hochschule Aachen, Aachen, Germany, Apr. 

1988. 

[25] I. S. Gradstein and I. M. Ryshik, Tables of Series, Products, and Integrals, 

5th ed. Frankfurt, Germany: Harri Deutsch, 1981, vol. I/II. 

Matthias Pätzold (M’94–SM’98) was born in 

Engelsbach, Germany, in 1958. He received the 

Dipl.-Ing. and Dr.-Ing degrees in electrical engineering 

from Ruhr-University Bochum, Bochum, 

Germany, in 1985 and 1989, respectively. In 

1998, he received the Dr.-Ing. habil. degree in 

communications from the Technical University of 

Hamburg-Harburg. 

From 1990 to 1992, he was with ANT Nachrichtentechnik 

GmbH, Backnang, where he was engaged 

in digital satellite communications. Since 1992, 

he has been with the Digital Communication Systems Department at the 

Technical University Hamburg-Harburg, Hamburg, Germany, where he is a 

Private Educator. He is author of the book Mobile Radio Channels—Modeling, 

Analysis, and Simulation (in German) (Wiesbaden: Vieweg, 1999). His current 

research interests include mobile radio communications, especially multipath 

fading channel modeling, channel parameter estimation, and coded-modulation 

techniques for fading channels. 

Dr. Pätzold received 1998 Neal Shepherd Memorial Best Propagation Paper 

Award from the IEEE Vehicular Technology Society. 

Raquel García was born in Madrid, Spain, in 

1973. She received the engineering degree from the 

Escuela Técnica Superior de Ingenieros de Telecomunicación 

(ET-SIT), Universidad Politécnica de 

Madrid (UPM), Madrid in 1998. 

From 1997 to 1998, she was with the Digital Communication 

Systems Department at the Technical 

University of Hamburg-Harburg, Germany, working 

in the field of mobile radio channel modeling. 

In 1998, she joined Telefónica Investigación y 

Desarrollo in Madrid, where she is working in the 

Radiocommunication Research Group. Her research interests include mobile 

communications, especially channel modeling and radio propagation modeling. 

Frank Laue was born in Hamburg, Germany, in 1961. He received his 

Dipl.-Ing. (FH) degree in electrical engineering from the Fachhochschule 

Hamburg, Hamburg, Germany, in 1992. 

Since 1993, he has been a CAE Engineer at the Digital Communications Systems 

Department at the Technical University of Hamburg-Harburg, Hamburg, 

where he is involved in mobile radio communications.

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