Limitations of sum-of-sinusoids fading channel ... - Semantic Scholar
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 4, APRIL 2001
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Limitations of Sum-of-Sinusoids Fading Channel Simulators Marius F. Pop and Norman C. Beaulieu, Fellow, IEEE
Abstract—Rayleigh signal fading due to multipath propagation in wireless channels is widely modeled using sum-of-sinusoids simulators. In particular, Jakes’ simulator and derivatives of Jakes’ simulator have gained widespread acceptance. Despite this, few in-depth studies of the simulators’ statistical behaviors have been reported in the literature. Here, the extent to which Jakes’ simulator adequately models the multipath Rayleigh fading propagation environment is examined. The results show that Jakes’ simulator does not reproduce some important properties of the physical fading channel. Some possible improvements to Jakes’ simulator are examined. The significances of the number and the symmetries of the Doppler frequency shifts on the validity of the simulator’s reproduction of the physical fading channel are elucidated. Index Terms—Fading channels, multipath channels, simulation.
I. INTRODUCTION
S
IMULATION of Rayleigh fading channels requires generation of correlated Rayleigh faded samples. Jakes has given a sum-of-sinusoids (SOS)-based fading channel simulator design [1] that has been widely used for almost three decades. Recently, Dent et al. [2] proposed an improvement to Jakes’ simulator, and Pätzold et al. [3] and Pätzold et al. [4] determined the time-average statistics of such simulators. Despite widespread acceptance and application of Jakes’ simulator and conclusions drawn in [4], there are indications that the limitations of the simulator are not well understood. For example, [5] uses 500 sinusoids to match the performance of an SOS simulator to measured results, whereas existing theory suggests a surprisingly smaller number of sinusoids are sufficient to adequately model the fading channel, such estimates ranging from 6 to 20 [1], [3], [4]. In this paper, we examine Jakes’ simulator in detail; in particular, the statistical properties of the simulator are derived and
Paper approved by R. A. Valenzuela, the Editor for Transmission Systems of the IEEE Communications Society. Manuscript received March 29, 1999; revised April 15, 1999 and March 27, 2000. This work was supported in part by an NSERC Strategic Grant STR0181058, NSERC Research Grant OGP0003986, and CITO funding from the High Bandwidth Ultra-Long Wireline Access Technology. This paper was presented in part at the IEEE Global Conference on Communications (GLOBECOM), Rio de Janeiro, Brazil, December 1999, and at the IEEE International Symposium on Wireless Communications (ISWC), Victoria, BC, Canada, June 1999. M. F. Pop was with the Department of Electrical and Computer Engineering, Queen’s University, Kingston, ON K7L 3N6, Canada. He is now with MD Robotics, Branpton, ON L6S 4J3, Canada. N. C. Beaulieu was with the Department of Electrical and Computer Engineering, Queen’s University, Kingston, ON K7L 3N6, Canada. He is now with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2G7, Canada (e-mail: [email protected]). Publisher Item Identifier S 0090-6778(01)03141-5.
some unexpected conclusions are mandated. In Section II, we present Jakes’ simulator design and the basis of this design. We also present formulae for determining the statistics of this simulator’s signal. We find that the signal generated by Jakes’ simulator is not stationary, as previously assumed by some authors. This section concludes by summarizing some of the problems of Jakes’ simulator. In Section III, we determine a relationship between the oscillator gains of Jakes’ simulator and the phase shifts of the physical model. An important design shortcoming in Jakes’ model is found, namely, that rays experiencing the same Doppler frequency shift are correlated. This causes the generated signal to be nonstationary. We seek an improvement to Jakes’ design, i.e., a simple modification to the simulator, such that it generates a wide-sense stationary (WSS) signal. We find that introduction of random phases in the low-frequency oscillators ensures the generation of a WSS signal. Section IV concludes the paper. II. JAKES’ SIMULATOR DESIGN In a multipath fading wireless channel, the transmitted signal is reflected and refracted, such that the received signal is made up of a superposition of waves. These waves may add constructively or destructively giving rise to received signal fading. It is of interest to characterize the channel response, in order that systems may be designed which operate at acceptable performance levels during fades. Since the orientation and material properties of the obstacles between transmitter and receiver are not known in advance, or may be time varying, it is common to characterize the received signal as stochastic. In the case of SOS simulators, the received signal is a sum of randomly-phased sinusoids. The idea that the received signal may be usefully represented as a superposition of a finite number of waves has existed for three decades. Among the first to propose such models of multipath fading channels are Bello [6], Gilbert [7], and Clarke [8]. Jakes designed an SOS simulator a decade after Bello introduced his model [1]. Jakes’ fading channel simulator has been used widely and studied intermittently over the past decades [2]–[4], [9]–[12]. Prior to examining the simulator design, we note some properties of the Rayleigh flat fading narrow-band signal. It is well established that the envelope probability density function (pdf) of a transmitted carrier, at the receiver, in an environment with no direct line-of-sight between transmitter and receiver, is given by the Rayleigh distribution
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while the phase pdf is given by the uniform distribution [1], [8], [12], [13] (2) These pdfs are born out by measurements, as well as theory; see, for example, the work of Young [14]. It is equally well established that the autocorrelation function of the received signal for two-dimensional (2-D) isotropic scattering and an omnidirectional receiving antenna is given by [1], [8], [12], [13] (3) Any model that attempts to model the Rayleigh flat fading narrow-band wireless channel has to exhibit the statistical behaviors given in (1)–(3). A. The Reference Model and Jakes’ Simulator In developing his simulator, Jakes starts with an expression as a superposition of representing the received signal waves, similar to that given by Bello (with the path delays set ) equal to zero, i.e., Fig. 1. Jakes’ fading channel simulator, after [1, p. 70]. Note that the normalization constants of (8a) and (8b) are not included to avoid cluttering the diagram.
(4) is the amplitude of the transmitted cosine wave, where is a random variable (RV) representing the attenuation of the th path, is an RV representing the angle of arrival of the th ray, with respect to the direction of motion of the receiver, is an RV representing the phase shift undergone by the th is the transmitted cosine’s radian frequency, is the ray, , maximum Doppler radian frequency shift, i.e., is the wavelength where is the speed of the receiver and of the transmitted cosine wave, and is the complex constant, . We use uppercase letters to denote RVs, and i.e., thus emphasize the stochastic character of the received fading signal. Furthermore, to aid in comparisons to be made later, we such that it has unit power, and thus normalize the signal (4) becomes
and (6b) Furthermore, Jakes chooses
of the form (6c)
so that the number of distinct Doppler frequency shifts is re. Thus, the fading signal may be generduced from to low-frequency oscillators. ated through the use of only The block diagram of the simulator is given in Fig. 1. From the block diagram, we can readily write the simulator output signal in terms of quadrature components (7) where
(5) is the normalized received signal. We take this as where repour reference model. Note that the stochastic signal resenting the flat fading signal can be characterized through sets of triples . The RVs , , and are assumed independent; this assumption may be verified from the physical nature of the fading phenomenon. Note that to simulate low-frequency oscillators, the signal of (5) one would need . one for each Doppler frequency In the development of his simulator, Jakes makes some assumptions which have the goal of reducing the number of lowfrequency oscillators needed to generate the flat fading signal of (5). Thus, he selects (6a)
(8a) and
(8b) is a The use of the tilde “ ” is intended to emphasize that , with similar reduced realization of the stochastic process and . Furthermore, note that the interpretations for normalization constants of (8a) and (8b) are not used in Jakes’ presentation [1]. In the sequel, we will determine whether the assumptions and simplifications made by Jakes affect the performance and statistics of the SOS simulator.
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B. Time-Average Analyses In [3], simulators similar to that of Jakes are presented. The approach taken in [3] is to generate independent Gaussian stochastic processes which are then added in quadrature to generate the desired Rayleigh process. The Gaussian stochastic processes are obtained as finite sums of appropriately weighted randomly-phased sinusoids. Four methods for determining the path and angles of arrival are proposed. In [3], attenuations [4], and [9], the word deterministic is used to emphasize that once the simulator parameters, i.e., the RVs , , and , are chosen, they remain fixed for the duration of the simulation run (and usually all uses of the simulator). In these works, quantities similar to statistical properties are derived. The time-average and quadrature autocorrelation functions of the in-phase components are found. Expressions for the envelope and are derived by assuming phase pdfs of the fading signal that time is an RV (in the probability theory sense), having uniform distribution on a finite closed interval. It is observed that a single sinusoid with fixed amplitude and random phase is both ergodic1 and stationary. Hence, for a single sinusoid, stochastic averages may be computed as appropriate time averages. In [4] and [9], it is tacitly assumed this is also true of sums of fixed amplitude, random-phase sinusoids. However, since sums of fixed amplitude, random-phase sinusoids are not ergodic and stationary (this is proved in [16, pp. 24–25]), the time averages may or may not equal the stochastic (statistical) averages in the . case of the flat fading signal Our study here of SOS simulators is different than the studies in [4] and [9]. In these past works, it is assumed that all random, , and are ness has been removed, i.e., the values of fixed and known. Thus, the only variable quantity left is time , which is reinterpreted as an RV in order to apply the techniques of stochastic processes and obtain approximations to the time-average behaviors of the simulators. An important question is whether the simulators adequately model the physical , , and are RVs in the physical channel. Recall that model but are fixed constants in the simulators. It is not clear a priori that fixing these RVs does not vitiate some or all of the important statistical properties of the simulators. It is this question that is addressed in the present paper. From the physical description of the flat fading channel, it seems reasonable that and can be fixed, but the same is not true for the . , , and results in simulators We show here that fixing that do not adequately model the real channel, but that simulator designs that fix only and can adequately model the real channel. A crucial flaw in [4] and [9] is the assumption that the signal produced by Jakes’ simulator is WSS and ergodic, i.e., that first- and second-order time averages may be substituted for stochastic ones. References [4] and [9] provide neither justification for this assumption, nor mathematical verification. Furthermore, [1], [2], and [12] interchange time and statistical averages implying that the simulated fading signal is at least WSS and ergodic. We show below that the signal produced by Jakes’ simulator is not WSS, and thus substitution of time averages for statistical ones is unjustified. 1We distinguish two definitions of ergodicity here. One comes from the field of mathematics and statistics, while the other is common in electrical engineering [15]. The two definitions are not the same; often they may be a source of confusion. Ergodic here is used as defined in the mathematical context.
Fig. 2. Envelope pdf of the signal generated by the reference model for various numbers of low-frequency oscillators.
A second shortcoming in [4] lies in the fact that the results and are developed under the assumption that the in-phase components are independent. An equivalent quadrature condition, derived in [9], is that the in-phase and quadrature components share no common Doppler frequency shifts. However, this is not true of Jakes’ simulator, because the in-phase and quadrature components do share common frequencies, as seen in Fig. 1, or equivalently, in (8a) and (8b). The effects of this invalid assumption on the results obtained in [4] regarding the envelope and phase pdfs are not assessed. C. Statistics of the Reference Model and Jakes’ Fading Channel Simulator Jakes’ simulator may be an efficient way to implement a hardware or software simulation of the Rayleigh fading multipath channel. However, for simulation results to be meaningful, they must reproduce the important statistics of the real world. This is the question that we are addressing in detail here, noting that past studies of Jakes’ simulator based on time averaging do not address this fundamental question. Before we determine the statistics of Jakes’ simulator, we determine the statistical behaviors of the reference model. Then, upon comparisons of the two sets of statistics we may determine whether the assumptions made by Jakes are valid. Applying the results developed in the Apof (5) pendix to the reference model signal, i.e., the signal subject to the constraints of (6a) and (6b), we obtain, upon substitution in (A.16), the envelope pdf (9) and together and plot this in Fig. 2 for with the standard Rayleigh pdf. The envelope pdf in (9) tends to the standard Rayleigh pdf with increasing , as expected. Note that the envelope pdf does not depend on time . The phase pdf is given by (A.15) for as required by the physical model in (2).
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Low-frequency component of the autocorrelation function of R(t), for = 1, and various numbers of low-frequency oscillators.
Fig. 3.
!
The stochastic autocorrelation of , again subject to the constraints of (6a) and (6b), is given by [16, eq. (3.42)]
Fig. 4. Variation with time of the envelope pdf of the signal produced by Jakes’ simulator averaged across the fading ensemble. Here, M = 8 and ! = 2 .
fading process is time varying. That is, Jakes’ simulator model averaged across the ensemble of physical fading channels is not stationary, not even WSS. The nonstationary character of the autocorrelation function of the signal will be shown below. The phase pdf is given by (A.15)
(11)
(15)
From (11), we observe that the autocorrelation function is a function of only the time difference , i.e., is WSS. For even and , the fading signal as usually used in simulator implementations, (11) simplifies to
It is interesting to note that this result is different from the result in [4] based on time averages. Here, we have computed a stochastic average. We have also shown that the first-order pdf is time varying, and thus suspect the random process of is not WSS; we will show this directly later. This point further illustrates that one may not readily substitute time averages for statistical ones. As discussed earlier, such substitutions are meaningful only for random processes which are ergodic and stationary, or equivalently satisfy some ergodic theorem [15]. The stochastic autocorrelation of the signal of Jakes’ simuis [16, eq. (3.70)] lator
(12) The low-frequency term of (12) is shown in Fig. 3 for and . Note that the low-frequency term tends to the Bessel function expression [1, eq. (1.7-10)] as becomes large. Finally, note that if we remove the restriction of (6b), i.e., the are allowed to be uniform independent and angles of arrival , then the autocorrelaidentically distributed (i.i.d.) over becomes [16, eq. (3.56)] tion function of (13) and the low-frequency term is the Bessel function expression, as established above. We now turn our attention to Jakes’ simulator. Applying the formulae derived in the Appendix to the signal generated by Jakes’ simulator, we obtain upon appropriate substitution in (A.16)
(14) We note that the pdf of the envelope of the signal produced by Jakes’ simulator averaged across the fading ensemble (i.e., stochastic average pdf) is a function of the time as well as the envelope level . We plot the result of (14) in Fig. 4. Studying Fig. 4, we note that the statistical variance of the simulator
(16) Plots of the low-frequency term of the autocorrelation function of the signal produced by Jakes’ simulator are given in Fig. 5 for , equivalent to , for three different values of . Upon analysis of Fig. 5, the nonstationary character of the signal produced by Jakes’ simulator is readily obvious. Furthermore, the form of (16) emphasizes the dependence of the autocorrela, when averaged over the tion function of Jakes’ simulator ensemble of physical fading channels, on both the time differand the time sum . From this, we are forced ence to conclude the signal produced by Jakes’ simulator is not stationary when averaged across the fading ensemble. Indeed, it is not even WSS. Observe that the signal will, in general, not
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in a form similar to that of the fading signal
in (5),
i.e.,
(18) ’s and the ’s, To determine the relation between the we equate cosines of the same frequency. Thus, considering the and , we obtain particular cases for (19)
Fig. 5.
! t.
~ (t), for Low-frequency component of the autocorrelation function of R M = 8, corresponding to N = 34, at three different time instants,
= 1 and
be cyclostationary either, as the frequencies in the fading signal are incommensurate. In particular, note that the variance is time varying and converges to of the process
. Note From (19), it follows that Jakes forces that these phase shifts are no longer independent, as assumed in the reference model. We may obtain similar relationships for the remaining variand ables. In particular, it can be shown that are related through one of the four equations [16, eq. (2.21)] (20a) (20b)
(17) , or equivalently , where we have used as [1, eq. (1.7-10)] to determine the limit of the sum. This is seen graphically in Fig. 4.
(20c) (20d) and and these four equations [16, eq. (2.22)]
through one of
III. IMPROVING JAKES’ DESIGN
(21a)
We have seen in the previous section that the signal produced by Jakes’ fading channel simulator is not WSS. In this section, we investigate why. We determine a relationship between the of Jakes’ simulator and the phase shifts oscillator gains of the reference model. Once the cause of the nonstationarity problem is determined, we attempt to improve the performance of the simulator by introducing random phase shifts in the lowfrequency oscillators. Finally, we conclude with some remarks relating to the reduction of the number of Doppler frequency shifts in the SOS model and how this should be implemented in a simulator.
(21b) (21c) (21d) and Next, we require relationships between and . Observe that and satisfy one of the four equalities [16, eq. (2.24)]
, and must (22a) (22b)
A. A Note on the PDF of
(22c)
One drawback to Jakes’ design is that there is no obvious reand the physical nalationship between the oscillator gains and ture of the fading channel, i.e., no relationship between . As well, the pdf of is not obvious from the phase shifts Jakes’ development; the only requirement Jakes makes is that selection of the oscillator gains be made such that the in-phase and quadrature components have equal power. This guarantees the phase pdf of the resulting signal is uniform over . is meant to be an efficient realization of . Recall that Thus, to obtain the required relationships, we rewrite the signal
(22d) and and that the four equalities [16, eq. (2.26)]
must satisfy one of (23a) (23b) (23c) (23d)
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The signal produced by this simulator can be written, upon analysis of Fig. 6 and under the constraint that it has unit power, as
(24) are uniform i.i.d. RVs over . where It can be shown that the autocorrelation function of given by [16, eq. (4.24)]
is
Fig. 6. Improving Jakes’ simulator by the introduction of random phases in the low-frequency oscillators.
(25) From (20)–(23), we conclude that the four-tuple of phase for shifts are no longer independent as in the reference model. In [16], it is proved that the dependence of the phase shifts leads to waves experiencing the same Doppler frequency shift being correlated. This in turn leads to stationarity problems, as outlined in Section II-C. depend Returning to (22a)–(22d), we note that . Also, because the are uniform i.i.d. directly on RVs over , we conclude that are also independent and uniformly distributed over some interval of length . However, since the ’s appear only as arguments to eior , we may take the as uniform i.i.d. over ther , without loss of generality.
We observe that the stochastic autocorrelation of the signal produced by this improved simulator depends on only the time dif, i.e., the signal is WSS, and thus, we may write ference (26) As expected, the introduction of the random phases eliminates the correlation observed earlier between the low-frequency oscillators, and thus leads to the generation of a WSS signal. The envelope and phase pdfs of the improved simulator’s signal are derived using formulae developed in the Appendix. They are, from (A.16) and (A.15), respectively
B. Introduction of Random Phases in the Low-Frequency Oscillators A plausible improvement to Jakes’ design is the introduction using of sine terms. That is, generate the in-phase term using sines. In [16, Sec. cosines and the quadrature term 4.2], we found, however, that this improvement does not lead to the generation of a WSS signal; thus, this possible change is reported but not pursued further. A second plausible modification is the introduction of random phases in the low-frequency oscillators. An intuitive justification for using this method is the fact that for small values of time , the values produced by the low-frequency . By oscillators are highly correlated; they are equal at adding the random phases, this correlation is eliminated. The block diagram of this improved simulator is given in Fig. 6.
(27a) and (27b) for and as well as Fig. 7 shows ; as exthe standard Rayleigh pdf, corresponding to converges to the Rayleigh pdf pected, the envelope pdf with increasing . As well, the pdf does not depend on time, is unisatisfying the WSS requirement. The phase pdf as desired for the Rayleigh flat fading signal. form over apFinally, we should determine whether the signal , i.e., we need to determine proximates the fading signal
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Fig. 8. Fig. 7. Envelope pdf of the signal generated by the simulator of Fig. 6, for various numbers of low-frequency oscillators.
how, if at all, the introduction of the random phases has changed as the fading signal. We rewrite the signal
Symmetry of arriving rays in Jakes’ design.
We start with the fading signal and, subject to the constraints imposed by Jakes in (6a)–(6c), we rewrite the in-phase as component
(28) in the fading signal Recall that the phase shifts are uniform i.i.d. over . In the case of the improved simulator of Fig. 6, the phase shifts are represented , and by the random variables , as evidenced from (28). It is , the phase shifts are interesting to note that in the case of no longer independent, as was the case with the phase shifts of . In particular, it can be shown that the phases and are dependent, for [16]; however, they are uncorrelated. Note that although the envelope and phase pdfs, as well as converge to the desired the autocorrelation function, of values, i.e., the Rayleigh and uniform distributions, and the Bessel function expression, respectively, the improved model does not completely represent the fading signal of (5). In other words, despite the fact the first- and second-order statistics of this improved simulator match the desired ones, it is likely that higher-order statistics will not, due to the dependence of the phase shifts. Of course, if only first- and second-order statistics are of interest, use of this improved simulator is warranted. C. Reducing the Complexity of the Physical Model of (5). In general, this equaReturn to the fading signal tion may not be reduced in the manner done by Jakes. However, if some symmetry exists, whether warranted by the physical phenomenon, or construction as in Jakes’ case, care must be taken that all phase shifts are included for each Doppler frequency shift. This point merits further consideration.
(29) and the quadrature component as
(30) has four phase shifts Note that each Doppler frequency shift associated with it, except for the maximum Doppler frequency which has only two phase shifts. This point is further ilshift , or equivalently , in Fig. 8. lustrated for the case Studying Fig. 8, observe that the rays numbered 1, 4, 6, and 9 experience the same Doppler frequency shift, rays 1 and 9, a positive Doppler shift, and rays 4 and 6, a negative Doppler shift. Thus, the oscillator generating this Doppler frequency shift corresponds to four arriving rays with the associated four phase shifts. A similar condition holds for the rays numbered 2, 3, 7,
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To conclude, the savings incurred by forcing the angles of arrival to be symmetric about the - and -axes, as in Jakes’ design, occur through the inclusion of only the smallest number of different Doppler frequency shifts. However, we must include the appropriate phase shifts corresponding to each Doppler frequency magnitude. Failure to do so leads to stationarity problems, as was illustrated in Section II-C for Jakes’ simulator.
IV. CONCLUSION
= 4 + 2 and =
Fig. 9. Simplified simulator obtained for the case n M n=N; n ; ; N . The gains are defined in (31a)–(31h).
2
= 1 ...
and 8. Note that only two rays, 5 and 10, experience the maximum Doppler frequency shift, with ray 10 a positive shift and ray 5 a negative shift. Hence, the oscillator generating the maximum Doppler frequency shift has two phase shifts associated with it. The simulator derived based on the simplified equations for and is shown in Fig. 9. The gains corresponding to each sinusoid, or oscillator, in (29) and (30) are too cumbersome to include in a block diagram. Thus, we make the substitutions
(31a) (31b) (31c) (31d) (31e) (31f) (31g) (31h)
The work of Clarke [8] summarizes the important characteristics of fading channels and gives a useful mathematical model. This mathematical model has led to various simulator designs. One method of simulating the multipath fading encountered on Rayleigh flat fading wireless channels is based upon the SOS model. It is generally desired to have an efficient method for generating fading signals. Viewed in this light, Jakes’ approach presents an interesting point. That is, if we are able to reduce the complexity of the model, i.e., the number of low-frequency oscillators, then the generation of the signal is more efficient. On the other hand, this reduction was shown to lead to the generation of a nonstationary signal. Also shown here is that introduction of random phase shifts in the low-frequency oscillators removes the stationarity problems; that is, the resulting signal is WSS. Finally, it has been shown that the smallest number of low-frequency oscillators required is equal to the number of distinct Doppler frequency shifts, counting positive and negative shifts as one. However, care must be taken in determining the gain of each branch for each low-frequency oscillator, i.e., all phase shifts corresponding to a particular Doppler frequency shift must be included in the simulator design.
APPENDIX COMPUTATION OF THE ENVELOPE AND PHASE PDFS We are interested here in computing the envelope and phase and . Previous methods pdfs of SOS signals, such as entailed computing the pdfs of the underlying approximating Gaussian processes, and then combining them in quadrature to obtain the desired envelope pdf using probability transformation theory. This approach requires making some unjustified assumptions, such as the independence of the in-phase and quadrature components. Furthermore, the results obtained for the envelope and phase pdfs are computed by evaluating double integrals [9, eq. (42)]. The method proposed in [16] is based upon the computation of the pdf of the sum of independent 2-D RVs. This method avoids making any independence requirements on the quadrature components. Furthermore, in some cases, a double integral evaluated numerically in [9, eq. (42)] can often be reduced to a single integral [16, eq. (3.22)]. A similar result is provided without proof by Goldman [17]. Here, we give a simple proof. The problem we are attempting to solve is that of computing the joint pdf (jpdf) of the sum of independent 2-D RVs. Ultimately, however, we want to determine the pdf of the magnitude of the sum of the random vectors, and the pdf of the angle
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the final sum vector makes with the positive -axis. i.e., the pdfs of the RVs2
707
The independence of and , and we have
allows us to separate the jpdf
(A.1) where
and
are defined by (A.2a)
and (A.2b)
(A.7)
and are independent. When applying the We assume that results obtained herein, we must make sure that this assumption is justified; this justification is provided by the physical nature of the problem. It is not obvious from this assumption that and are also independent; we will see that they are indeed indeand are correlated. pendent below. Note that, in general, From [18, Sec. 7-2], we note that either the jpdf or the joint characteristic function is sufficient to describe a random vector and the two are Fourier transform pairs. Thus, we seek to deterand then, via the mine the joint characteristic function of . Next, the inverse Fourier transform, obtain the jpdf of is obtained by converting to polar coordinates. jpdf of and , respecFrom this, envelope and phase pdfs tively, are obtained as appropriate marginal pdfs. Because the random vectors being summed are independent, is the product the joint characteristic function of the sum of joint characteristic functions of the random vectors being added [18, eq. (7–25)]. With the notation of (A.2a) and (A.2b), we have
. To evaluate the integral inside the where square brackets, we use [19, eq. (8.411.7)] to obtain the equality
(A.3)
(A.11) We choose to perform the double integration of (A.11) by changing to polar coordinates. Letting
is the characteristic function of the final sum where , and is the characteristic function of vector . The joint characteristic the th vector in the sum is [18, eq. (7–23)] function of a random vector
(A.4) As assumed above, when expressed in polar coordinates, the and , are independent. Thus, we perform RVs involved, the double integration in (A.4) by converting to polar coordinates. Using [18, eq. (6–72)] (A.5) in (A.4) we obtain
(A.8) Performing the substitution (A.8), we obtain3
in (A.7) and using (A.9)
The joint characteristic function of the sum by substituting (A.9) in (A.3)
is obtained
(A.10) , we take the inverse Fourier transTo obtain the jpdf of according to [18, eq. (7-24)] form of
in (A.11), yields
where . The negative sign in the exponential may be absorbed by the cosine via a shift of . We now use (A.8) to evaluate the integral in the right set of square brackets. We thus have
(A.12) (A.6) 2The
arctan (x; y ) function returns the proper angle on the interval [0; 2 ).
3Integrals of the form of (A.9), i.e., in which we integrate against the Bessel function J (1), are referred to in the literature as Hankel transforms.
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Converting the jpdf of (A.12) to polar coordinates, we obtain . Letting the jpdf of the random vector in (A.12) and substituting in (A.5) gives
(A.13) Note that we can separate the jpdf of
and
as (A.14)
and are the appropriate marginal pdfs, where proving that, indeed, and are independent RVs. Furtherin (A.13) does not depend on . From this, we more, conclude that is uniformly distributed over (A.15) regardless of the number of random vectors added. To obtain the pdf of the magnitude of the sum of random , we integrate the jpdf over the variable vectors to obtain
[12] G. L. Stüber, Principles of Mobile Communication. Norwell, MA: Kluwer, 1996. [13] M. J. Gans, “A power-spectral theory of propagation in the mobile radio environment,” IEEE Trans. Veh. Technol., vol. VT-21, pp. 27–38, Feb. 1972. [14] W. R. Young Jr, “Comparison of mobile radio transmission at 150, 450, 900, and 3700 Mc.,” Bell Syst. Tech. J., vol. 31, no. 6, pp. 1068–1085, Apr. 1952. [15] R. M. Gray and L. D. Davisson, Random Processes: A Mathematical Approach for Engineers. Englewood Cliffs, NJ: Prentice-Hall, 1986. [16] M. F. Pop, “Statistical Analysis of Sum-of-Sinusoids Fading Channel Simulators,” M. Sc. thesis, Queen’s Univ., Kingston, ON, Canada, 1999. [17] J. Goldman, “Statistical properties of a sum of sinusoids and Gaussian noise and its generalization to higher dimensions,” Bell Syst. Tech. J., vol. 53, no. 4, pp. 557–580, Apr. 1974. [18] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3 ed. Toronto, ON, Canada: McGraw-Hill, 1991. [19] I. S. Gradshteyn, I. M. Ryzhik, and A. Jeffrey, Eds., Table of integrals, series, and products, 5 ed. New York: Academic Press, 1994.
Marius F. Pop received the B.Sc. (honors) degree in mathematics and engineering and the M. Eng. degree in electrical engineering from Queen’s University, Kingston, ON, Canada, in 1996 and 1999, respectively. Since April 1999, he has been a Member of the Technical Staff at MacDonald Dettwiler’s Space and Advanced Robotics Division. His current research interests include fading channels and characterization of electronic circuits and components in harsh environments.
(A.16) , to We use (A.16) in this paper, with appropriate pdfs determine the envelope pdfs in (9) and (14), as well as (27a). REFERENCES [1] W. C. Jakes, Microwave Mobile Communications. Piscataway, NJ: IEEE Press, 1994. [2] P. Dent, G. E. Bottomley, and T. Croft, “Jakes’ fading model revisited,” Electron. Lett., vol. 29, no. 3, pp. 1162–1163, June 1993. [3] 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. [4] M. Pätzold and F. Laue, “Statistical properties of Jakes’ fading channel simulator,” in Vehicular Technology Conf. Rec. (VTC’98), vol. II, Ottawa, ON, Canada, May 1998, pp. 712–718. [5] P. Hoeher, “A statistical discrete-time model for the WSSUS multipath channel,” IEEE Trans. Veh. Technol., vol. 41, pp. 461–468, Nov. 1992. [6] P. A. Bello, “Characterization of randomly time-variant linear channels,” IEEE Trans. Commun., vol. COM-11, pp. 360–393, Dec. 1963. [7] E. N. Gilbert, “Energy reception for mobile radio,” Bell Syst. Tech. J., pp. 1779–1803, Oct. 1965. [8] R. H. Clarke, “A statistical theory of mobile-radio reception,” Bell Syst. Tech. J., pp. 957–1000, July–Aug. 1968. [9] 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. [10] E. F. Casas and C. Leung, “A simple digital fading simulator for mobile radio,” in Proc. IEEE Vehicular Technology Conf., Sept. 1988, pp. 212–217. [11] T. Eyceoz, A. Duel-Hallen, and H. Hallen, “Deterministic channel modeling and long range prediction of fast fading mobile radio channels,” IEEE Commun. Lett., vol. 2, pp. 254–256, Sept. 1998.
Norman C. Beaulieu (S’82–M’86–SM’89–F’99) received the B.A.Sc. (honors), M.A.Sc., and Ph.D. degrees in electrical engineering from the University of British Columbia, Vancouver, BC, Canada, in 1980, 1983, and 1986, respectively. In 1980, he was awarded the University of British Columbia Special University Prize in Applied Science. He held an appoinment as Queen’s National Scholar Assistant Professor in the Department of Electrical Engineering at Queen’s University, Kingston, ON, Canada, from September 1986 to June 1988. From July 1988 to June 1993, he held the position of Associate Professor, and from July 1993 to August 2000, the position of Professor, both at Queen’s University. In September 2000, he became the iCORE Research Chair in Broadband Wireless Communications at the University of Alberta, Edmonton, AB, Canada. His current research interests include digital communications over fading channels, interference effects in digital modulations, channel modeling and simulation, decision-feedback equalization, and digital synchronization in sampled receivers. Dr. Beaulieu is a member of the IEEE Communication Theory Committee. He served as the Committee’s representative to the Technical Program Committee of the 1991 International Conference on Communications (ICC’91) and as the Committee’s co-representative to the Technical Program Committee of ICC’93 and ICC’96. He was General Chair of the Sixth Communication Theory Mini-Conference, held in association with GLOBECOM’97, and Co-Chair of the 1999 Canadian Workshop on Information Theory. Since January 1992, he has served as Editor for Wireless Communication Theory of the IEEE TRANSACTIONS ON COMMUNICATIONS. Since November 1996, he has served as Associate Editor for Wireless Communication Theory of the IEEE COMMUNICATIONS LETTERS. Effective January 2000, he was appointed Editor-in-Chief of the IEEE TRANSACTIONS ON COMMUNICATIONS. He was a recipient of the Natural Science and Engineering Research Council (NSERC) E.W.R. Steacie Memorial Fellowship in 1999.
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Limitations of Sum-of-Sinusoids Fading Channel Simulators Marius F. Pop and Norman C. Beaulieu, Fellow, IEEE
Abstract—Rayleigh signal fading due to multipath propagation in wireless channels is widely modeled using sum-of-sinusoids simulators. In particular, Jakes’ simulator and derivatives of Jakes’ simulator have gained widespread acceptance. Despite this, few in-depth studies of the simulators’ statistical behaviors have been reported in the literature. Here, the extent to which Jakes’ simulator adequately models the multipath Rayleigh fading propagation environment is examined. The results show that Jakes’ simulator does not reproduce some important properties of the physical fading channel. Some possible improvements to Jakes’ simulator are examined. The significances of the number and the symmetries of the Doppler frequency shifts on the validity of the simulator’s reproduction of the physical fading channel are elucidated. Index Terms—Fading channels, multipath channels, simulation.
I. INTRODUCTION
S
IMULATION of Rayleigh fading channels requires generation of correlated Rayleigh faded samples. Jakes has given a sum-of-sinusoids (SOS)-based fading channel simulator design [1] that has been widely used for almost three decades. Recently, Dent et al. [2] proposed an improvement to Jakes’ simulator, and Pätzold et al. [3] and Pätzold et al. [4] determined the time-average statistics of such simulators. Despite widespread acceptance and application of Jakes’ simulator and conclusions drawn in [4], there are indications that the limitations of the simulator are not well understood. For example, [5] uses 500 sinusoids to match the performance of an SOS simulator to measured results, whereas existing theory suggests a surprisingly smaller number of sinusoids are sufficient to adequately model the fading channel, such estimates ranging from 6 to 20 [1], [3], [4]. In this paper, we examine Jakes’ simulator in detail; in particular, the statistical properties of the simulator are derived and
Paper approved by R. A. Valenzuela, the Editor for Transmission Systems of the IEEE Communications Society. Manuscript received March 29, 1999; revised April 15, 1999 and March 27, 2000. This work was supported in part by an NSERC Strategic Grant STR0181058, NSERC Research Grant OGP0003986, and CITO funding from the High Bandwidth Ultra-Long Wireline Access Technology. This paper was presented in part at the IEEE Global Conference on Communications (GLOBECOM), Rio de Janeiro, Brazil, December 1999, and at the IEEE International Symposium on Wireless Communications (ISWC), Victoria, BC, Canada, June 1999. M. F. Pop was with the Department of Electrical and Computer Engineering, Queen’s University, Kingston, ON K7L 3N6, Canada. He is now with MD Robotics, Branpton, ON L6S 4J3, Canada. N. C. Beaulieu was with the Department of Electrical and Computer Engineering, Queen’s University, Kingston, ON K7L 3N6, Canada. He is now with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2G7, Canada (e-mail: [email protected]). Publisher Item Identifier S 0090-6778(01)03141-5.
some unexpected conclusions are mandated. In Section II, we present Jakes’ simulator design and the basis of this design. We also present formulae for determining the statistics of this simulator’s signal. We find that the signal generated by Jakes’ simulator is not stationary, as previously assumed by some authors. This section concludes by summarizing some of the problems of Jakes’ simulator. In Section III, we determine a relationship between the oscillator gains of Jakes’ simulator and the phase shifts of the physical model. An important design shortcoming in Jakes’ model is found, namely, that rays experiencing the same Doppler frequency shift are correlated. This causes the generated signal to be nonstationary. We seek an improvement to Jakes’ design, i.e., a simple modification to the simulator, such that it generates a wide-sense stationary (WSS) signal. We find that introduction of random phases in the low-frequency oscillators ensures the generation of a WSS signal. Section IV concludes the paper. II. JAKES’ SIMULATOR DESIGN In a multipath fading wireless channel, the transmitted signal is reflected and refracted, such that the received signal is made up of a superposition of waves. These waves may add constructively or destructively giving rise to received signal fading. It is of interest to characterize the channel response, in order that systems may be designed which operate at acceptable performance levels during fades. Since the orientation and material properties of the obstacles between transmitter and receiver are not known in advance, or may be time varying, it is common to characterize the received signal as stochastic. In the case of SOS simulators, the received signal is a sum of randomly-phased sinusoids. The idea that the received signal may be usefully represented as a superposition of a finite number of waves has existed for three decades. Among the first to propose such models of multipath fading channels are Bello [6], Gilbert [7], and Clarke [8]. Jakes designed an SOS simulator a decade after Bello introduced his model [1]. Jakes’ fading channel simulator has been used widely and studied intermittently over the past decades [2]–[4], [9]–[12]. Prior to examining the simulator design, we note some properties of the Rayleigh flat fading narrow-band signal. It is well established that the envelope probability density function (pdf) of a transmitted carrier, at the receiver, in an environment with no direct line-of-sight between transmitter and receiver, is given by the Rayleigh distribution
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while the phase pdf is given by the uniform distribution [1], [8], [12], [13] (2) These pdfs are born out by measurements, as well as theory; see, for example, the work of Young [14]. It is equally well established that the autocorrelation function of the received signal for two-dimensional (2-D) isotropic scattering and an omnidirectional receiving antenna is given by [1], [8], [12], [13] (3) Any model that attempts to model the Rayleigh flat fading narrow-band wireless channel has to exhibit the statistical behaviors given in (1)–(3). A. The Reference Model and Jakes’ Simulator In developing his simulator, Jakes starts with an expression as a superposition of representing the received signal waves, similar to that given by Bello (with the path delays set ) equal to zero, i.e., Fig. 1. Jakes’ fading channel simulator, after [1, p. 70]. Note that the normalization constants of (8a) and (8b) are not included to avoid cluttering the diagram.
(4) is the amplitude of the transmitted cosine wave, where is a random variable (RV) representing the attenuation of the th path, is an RV representing the angle of arrival of the th ray, with respect to the direction of motion of the receiver, is an RV representing the phase shift undergone by the th is the transmitted cosine’s radian frequency, is the ray, , maximum Doppler radian frequency shift, i.e., is the wavelength where is the speed of the receiver and of the transmitted cosine wave, and is the complex constant, . We use uppercase letters to denote RVs, and i.e., thus emphasize the stochastic character of the received fading signal. Furthermore, to aid in comparisons to be made later, we such that it has unit power, and thus normalize the signal (4) becomes
and (6b) Furthermore, Jakes chooses
of the form (6c)
so that the number of distinct Doppler frequency shifts is re. Thus, the fading signal may be generduced from to low-frequency oscillators. ated through the use of only The block diagram of the simulator is given in Fig. 1. From the block diagram, we can readily write the simulator output signal in terms of quadrature components (7) where
(5) is the normalized received signal. We take this as where repour reference model. Note that the stochastic signal resenting the flat fading signal can be characterized through sets of triples . The RVs , , and are assumed independent; this assumption may be verified from the physical nature of the fading phenomenon. Note that to simulate low-frequency oscillators, the signal of (5) one would need . one for each Doppler frequency In the development of his simulator, Jakes makes some assumptions which have the goal of reducing the number of lowfrequency oscillators needed to generate the flat fading signal of (5). Thus, he selects (6a)
(8a) and
(8b) is a The use of the tilde “ ” is intended to emphasize that , with similar reduced realization of the stochastic process and . Furthermore, note that the interpretations for normalization constants of (8a) and (8b) are not used in Jakes’ presentation [1]. In the sequel, we will determine whether the assumptions and simplifications made by Jakes affect the performance and statistics of the SOS simulator.
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B. Time-Average Analyses In [3], simulators similar to that of Jakes are presented. The approach taken in [3] is to generate independent Gaussian stochastic processes which are then added in quadrature to generate the desired Rayleigh process. The Gaussian stochastic processes are obtained as finite sums of appropriately weighted randomly-phased sinusoids. Four methods for determining the path and angles of arrival are proposed. In [3], attenuations [4], and [9], the word deterministic is used to emphasize that once the simulator parameters, i.e., the RVs , , and , are chosen, they remain fixed for the duration of the simulation run (and usually all uses of the simulator). In these works, quantities similar to statistical properties are derived. The time-average and quadrature autocorrelation functions of the in-phase components are found. Expressions for the envelope and are derived by assuming phase pdfs of the fading signal that time is an RV (in the probability theory sense), having uniform distribution on a finite closed interval. It is observed that a single sinusoid with fixed amplitude and random phase is both ergodic1 and stationary. Hence, for a single sinusoid, stochastic averages may be computed as appropriate time averages. In [4] and [9], it is tacitly assumed this is also true of sums of fixed amplitude, random-phase sinusoids. However, since sums of fixed amplitude, random-phase sinusoids are not ergodic and stationary (this is proved in [16, pp. 24–25]), the time averages may or may not equal the stochastic (statistical) averages in the . case of the flat fading signal Our study here of SOS simulators is different than the studies in [4] and [9]. In these past works, it is assumed that all random, , and are ness has been removed, i.e., the values of fixed and known. Thus, the only variable quantity left is time , which is reinterpreted as an RV in order to apply the techniques of stochastic processes and obtain approximations to the time-average behaviors of the simulators. An important question is whether the simulators adequately model the physical , , and are RVs in the physical channel. Recall that model but are fixed constants in the simulators. It is not clear a priori that fixing these RVs does not vitiate some or all of the important statistical properties of the simulators. It is this question that is addressed in the present paper. From the physical description of the flat fading channel, it seems reasonable that and can be fixed, but the same is not true for the . , , and results in simulators We show here that fixing that do not adequately model the real channel, but that simulator designs that fix only and can adequately model the real channel. A crucial flaw in [4] and [9] is the assumption that the signal produced by Jakes’ simulator is WSS and ergodic, i.e., that first- and second-order time averages may be substituted for stochastic ones. References [4] and [9] provide neither justification for this assumption, nor mathematical verification. Furthermore, [1], [2], and [12] interchange time and statistical averages implying that the simulated fading signal is at least WSS and ergodic. We show below that the signal produced by Jakes’ simulator is not WSS, and thus substitution of time averages for statistical ones is unjustified. 1We distinguish two definitions of ergodicity here. One comes from the field of mathematics and statistics, while the other is common in electrical engineering [15]. The two definitions are not the same; often they may be a source of confusion. Ergodic here is used as defined in the mathematical context.
Fig. 2. Envelope pdf of the signal generated by the reference model for various numbers of low-frequency oscillators.
A second shortcoming in [4] lies in the fact that the results and are developed under the assumption that the in-phase components are independent. An equivalent quadrature condition, derived in [9], is that the in-phase and quadrature components share no common Doppler frequency shifts. However, this is not true of Jakes’ simulator, because the in-phase and quadrature components do share common frequencies, as seen in Fig. 1, or equivalently, in (8a) and (8b). The effects of this invalid assumption on the results obtained in [4] regarding the envelope and phase pdfs are not assessed. C. Statistics of the Reference Model and Jakes’ Fading Channel Simulator Jakes’ simulator may be an efficient way to implement a hardware or software simulation of the Rayleigh fading multipath channel. However, for simulation results to be meaningful, they must reproduce the important statistics of the real world. This is the question that we are addressing in detail here, noting that past studies of Jakes’ simulator based on time averaging do not address this fundamental question. Before we determine the statistics of Jakes’ simulator, we determine the statistical behaviors of the reference model. Then, upon comparisons of the two sets of statistics we may determine whether the assumptions made by Jakes are valid. Applying the results developed in the Apof (5) pendix to the reference model signal, i.e., the signal subject to the constraints of (6a) and (6b), we obtain, upon substitution in (A.16), the envelope pdf (9) and together and plot this in Fig. 2 for with the standard Rayleigh pdf. The envelope pdf in (9) tends to the standard Rayleigh pdf with increasing , as expected. Note that the envelope pdf does not depend on time . The phase pdf is given by (A.15) for as required by the physical model in (2).
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Low-frequency component of the autocorrelation function of R(t), for = 1, and various numbers of low-frequency oscillators.
Fig. 3.
!
The stochastic autocorrelation of , again subject to the constraints of (6a) and (6b), is given by [16, eq. (3.42)]
Fig. 4. Variation with time of the envelope pdf of the signal produced by Jakes’ simulator averaged across the fading ensemble. Here, M = 8 and ! = 2 .
fading process is time varying. That is, Jakes’ simulator model averaged across the ensemble of physical fading channels is not stationary, not even WSS. The nonstationary character of the autocorrelation function of the signal will be shown below. The phase pdf is given by (A.15)
(11)
(15)
From (11), we observe that the autocorrelation function is a function of only the time difference , i.e., is WSS. For even and , the fading signal as usually used in simulator implementations, (11) simplifies to
It is interesting to note that this result is different from the result in [4] based on time averages. Here, we have computed a stochastic average. We have also shown that the first-order pdf is time varying, and thus suspect the random process of is not WSS; we will show this directly later. This point further illustrates that one may not readily substitute time averages for statistical ones. As discussed earlier, such substitutions are meaningful only for random processes which are ergodic and stationary, or equivalently satisfy some ergodic theorem [15]. The stochastic autocorrelation of the signal of Jakes’ simuis [16, eq. (3.70)] lator
(12) The low-frequency term of (12) is shown in Fig. 3 for and . Note that the low-frequency term tends to the Bessel function expression [1, eq. (1.7-10)] as becomes large. Finally, note that if we remove the restriction of (6b), i.e., the are allowed to be uniform independent and angles of arrival , then the autocorrelaidentically distributed (i.i.d.) over becomes [16, eq. (3.56)] tion function of (13) and the low-frequency term is the Bessel function expression, as established above. We now turn our attention to Jakes’ simulator. Applying the formulae derived in the Appendix to the signal generated by Jakes’ simulator, we obtain upon appropriate substitution in (A.16)
(14) We note that the pdf of the envelope of the signal produced by Jakes’ simulator averaged across the fading ensemble (i.e., stochastic average pdf) is a function of the time as well as the envelope level . We plot the result of (14) in Fig. 4. Studying Fig. 4, we note that the statistical variance of the simulator
(16) Plots of the low-frequency term of the autocorrelation function of the signal produced by Jakes’ simulator are given in Fig. 5 for , equivalent to , for three different values of . Upon analysis of Fig. 5, the nonstationary character of the signal produced by Jakes’ simulator is readily obvious. Furthermore, the form of (16) emphasizes the dependence of the autocorrela, when averaged over the tion function of Jakes’ simulator ensemble of physical fading channels, on both the time differand the time sum . From this, we are forced ence to conclude the signal produced by Jakes’ simulator is not stationary when averaged across the fading ensemble. Indeed, it is not even WSS. Observe that the signal will, in general, not
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in a form similar to that of the fading signal
in (5),
i.e.,
(18) ’s and the ’s, To determine the relation between the we equate cosines of the same frequency. Thus, considering the and , we obtain particular cases for (19)
Fig. 5.
! t.
~ (t), for Low-frequency component of the autocorrelation function of R M = 8, corresponding to N = 34, at three different time instants,
= 1 and
be cyclostationary either, as the frequencies in the fading signal are incommensurate. In particular, note that the variance is time varying and converges to of the process
. Note From (19), it follows that Jakes forces that these phase shifts are no longer independent, as assumed in the reference model. We may obtain similar relationships for the remaining variand ables. In particular, it can be shown that are related through one of the four equations [16, eq. (2.21)] (20a) (20b)
(17) , or equivalently , where we have used as [1, eq. (1.7-10)] to determine the limit of the sum. This is seen graphically in Fig. 4.
(20c) (20d) and and these four equations [16, eq. (2.22)]
through one of
III. IMPROVING JAKES’ DESIGN
(21a)
We have seen in the previous section that the signal produced by Jakes’ fading channel simulator is not WSS. In this section, we investigate why. We determine a relationship between the of Jakes’ simulator and the phase shifts oscillator gains of the reference model. Once the cause of the nonstationarity problem is determined, we attempt to improve the performance of the simulator by introducing random phase shifts in the lowfrequency oscillators. Finally, we conclude with some remarks relating to the reduction of the number of Doppler frequency shifts in the SOS model and how this should be implemented in a simulator.
(21b) (21c) (21d) and Next, we require relationships between and . Observe that and satisfy one of the four equalities [16, eq. (2.24)]
, and must (22a) (22b)
A. A Note on the PDF of
(22c)
One drawback to Jakes’ design is that there is no obvious reand the physical nalationship between the oscillator gains and ture of the fading channel, i.e., no relationship between . As well, the pdf of is not obvious from the phase shifts Jakes’ development; the only requirement Jakes makes is that selection of the oscillator gains be made such that the in-phase and quadrature components have equal power. This guarantees the phase pdf of the resulting signal is uniform over . is meant to be an efficient realization of . Recall that Thus, to obtain the required relationships, we rewrite the signal
(22d) and and that the four equalities [16, eq. (2.26)]
must satisfy one of (23a) (23b) (23c) (23d)
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The signal produced by this simulator can be written, upon analysis of Fig. 6 and under the constraint that it has unit power, as
(24) are uniform i.i.d. RVs over . where It can be shown that the autocorrelation function of given by [16, eq. (4.24)]
is
Fig. 6. Improving Jakes’ simulator by the introduction of random phases in the low-frequency oscillators.
(25) From (20)–(23), we conclude that the four-tuple of phase for shifts are no longer independent as in the reference model. In [16], it is proved that the dependence of the phase shifts leads to waves experiencing the same Doppler frequency shift being correlated. This in turn leads to stationarity problems, as outlined in Section II-C. depend Returning to (22a)–(22d), we note that . Also, because the are uniform i.i.d. directly on RVs over , we conclude that are also independent and uniformly distributed over some interval of length . However, since the ’s appear only as arguments to eior , we may take the as uniform i.i.d. over ther , without loss of generality.
We observe that the stochastic autocorrelation of the signal produced by this improved simulator depends on only the time dif, i.e., the signal is WSS, and thus, we may write ference (26) As expected, the introduction of the random phases eliminates the correlation observed earlier between the low-frequency oscillators, and thus leads to the generation of a WSS signal. The envelope and phase pdfs of the improved simulator’s signal are derived using formulae developed in the Appendix. They are, from (A.16) and (A.15), respectively
B. Introduction of Random Phases in the Low-Frequency Oscillators A plausible improvement to Jakes’ design is the introduction using of sine terms. That is, generate the in-phase term using sines. In [16, Sec. cosines and the quadrature term 4.2], we found, however, that this improvement does not lead to the generation of a WSS signal; thus, this possible change is reported but not pursued further. A second plausible modification is the introduction of random phases in the low-frequency oscillators. An intuitive justification for using this method is the fact that for small values of time , the values produced by the low-frequency . By oscillators are highly correlated; they are equal at adding the random phases, this correlation is eliminated. The block diagram of this improved simulator is given in Fig. 6.
(27a) and (27b) for and as well as Fig. 7 shows ; as exthe standard Rayleigh pdf, corresponding to converges to the Rayleigh pdf pected, the envelope pdf with increasing . As well, the pdf does not depend on time, is unisatisfying the WSS requirement. The phase pdf as desired for the Rayleigh flat fading signal. form over apFinally, we should determine whether the signal , i.e., we need to determine proximates the fading signal
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Fig. 8. Fig. 7. Envelope pdf of the signal generated by the simulator of Fig. 6, for various numbers of low-frequency oscillators.
how, if at all, the introduction of the random phases has changed as the fading signal. We rewrite the signal
Symmetry of arriving rays in Jakes’ design.
We start with the fading signal and, subject to the constraints imposed by Jakes in (6a)–(6c), we rewrite the in-phase as component
(28) in the fading signal Recall that the phase shifts are uniform i.i.d. over . In the case of the improved simulator of Fig. 6, the phase shifts are represented , and by the random variables , as evidenced from (28). It is , the phase shifts are interesting to note that in the case of no longer independent, as was the case with the phase shifts of . In particular, it can be shown that the phases and are dependent, for [16]; however, they are uncorrelated. Note that although the envelope and phase pdfs, as well as converge to the desired the autocorrelation function, of values, i.e., the Rayleigh and uniform distributions, and the Bessel function expression, respectively, the improved model does not completely represent the fading signal of (5). In other words, despite the fact the first- and second-order statistics of this improved simulator match the desired ones, it is likely that higher-order statistics will not, due to the dependence of the phase shifts. Of course, if only first- and second-order statistics are of interest, use of this improved simulator is warranted. C. Reducing the Complexity of the Physical Model of (5). In general, this equaReturn to the fading signal tion may not be reduced in the manner done by Jakes. However, if some symmetry exists, whether warranted by the physical phenomenon, or construction as in Jakes’ case, care must be taken that all phase shifts are included for each Doppler frequency shift. This point merits further consideration.
(29) and the quadrature component as
(30) has four phase shifts Note that each Doppler frequency shift associated with it, except for the maximum Doppler frequency which has only two phase shifts. This point is further ilshift , or equivalently , in Fig. 8. lustrated for the case Studying Fig. 8, observe that the rays numbered 1, 4, 6, and 9 experience the same Doppler frequency shift, rays 1 and 9, a positive Doppler shift, and rays 4 and 6, a negative Doppler shift. Thus, the oscillator generating this Doppler frequency shift corresponds to four arriving rays with the associated four phase shifts. A similar condition holds for the rays numbered 2, 3, 7,
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To conclude, the savings incurred by forcing the angles of arrival to be symmetric about the - and -axes, as in Jakes’ design, occur through the inclusion of only the smallest number of different Doppler frequency shifts. However, we must include the appropriate phase shifts corresponding to each Doppler frequency magnitude. Failure to do so leads to stationarity problems, as was illustrated in Section II-C for Jakes’ simulator.
IV. CONCLUSION
= 4 + 2 and =
Fig. 9. Simplified simulator obtained for the case n M n=N; n ; ; N . The gains are defined in (31a)–(31h).
2
= 1 ...
and 8. Note that only two rays, 5 and 10, experience the maximum Doppler frequency shift, with ray 10 a positive shift and ray 5 a negative shift. Hence, the oscillator generating the maximum Doppler frequency shift has two phase shifts associated with it. The simulator derived based on the simplified equations for and is shown in Fig. 9. The gains corresponding to each sinusoid, or oscillator, in (29) and (30) are too cumbersome to include in a block diagram. Thus, we make the substitutions
(31a) (31b) (31c) (31d) (31e) (31f) (31g) (31h)
The work of Clarke [8] summarizes the important characteristics of fading channels and gives a useful mathematical model. This mathematical model has led to various simulator designs. One method of simulating the multipath fading encountered on Rayleigh flat fading wireless channels is based upon the SOS model. It is generally desired to have an efficient method for generating fading signals. Viewed in this light, Jakes’ approach presents an interesting point. That is, if we are able to reduce the complexity of the model, i.e., the number of low-frequency oscillators, then the generation of the signal is more efficient. On the other hand, this reduction was shown to lead to the generation of a nonstationary signal. Also shown here is that introduction of random phase shifts in the low-frequency oscillators removes the stationarity problems; that is, the resulting signal is WSS. Finally, it has been shown that the smallest number of low-frequency oscillators required is equal to the number of distinct Doppler frequency shifts, counting positive and negative shifts as one. However, care must be taken in determining the gain of each branch for each low-frequency oscillator, i.e., all phase shifts corresponding to a particular Doppler frequency shift must be included in the simulator design.
APPENDIX COMPUTATION OF THE ENVELOPE AND PHASE PDFS We are interested here in computing the envelope and phase and . Previous methods pdfs of SOS signals, such as entailed computing the pdfs of the underlying approximating Gaussian processes, and then combining them in quadrature to obtain the desired envelope pdf using probability transformation theory. This approach requires making some unjustified assumptions, such as the independence of the in-phase and quadrature components. Furthermore, the results obtained for the envelope and phase pdfs are computed by evaluating double integrals [9, eq. (42)]. The method proposed in [16] is based upon the computation of the pdf of the sum of independent 2-D RVs. This method avoids making any independence requirements on the quadrature components. Furthermore, in some cases, a double integral evaluated numerically in [9, eq. (42)] can often be reduced to a single integral [16, eq. (3.22)]. A similar result is provided without proof by Goldman [17]. Here, we give a simple proof. The problem we are attempting to solve is that of computing the joint pdf (jpdf) of the sum of independent 2-D RVs. Ultimately, however, we want to determine the pdf of the magnitude of the sum of the random vectors, and the pdf of the angle
POP AND BEAULIEU: LIMITATIONS OF SOS FADING CHANNEL SIMULATORS
the final sum vector makes with the positive -axis. i.e., the pdfs of the RVs2
707
The independence of and , and we have
allows us to separate the jpdf
(A.1) where
and
are defined by (A.2a)
and (A.2b)
(A.7)
and are independent. When applying the We assume that results obtained herein, we must make sure that this assumption is justified; this justification is provided by the physical nature of the problem. It is not obvious from this assumption that and are also independent; we will see that they are indeed indeand are correlated. pendent below. Note that, in general, From [18, Sec. 7-2], we note that either the jpdf or the joint characteristic function is sufficient to describe a random vector and the two are Fourier transform pairs. Thus, we seek to deterand then, via the mine the joint characteristic function of . Next, the inverse Fourier transform, obtain the jpdf of is obtained by converting to polar coordinates. jpdf of and , respecFrom this, envelope and phase pdfs tively, are obtained as appropriate marginal pdfs. Because the random vectors being summed are independent, is the product the joint characteristic function of the sum of joint characteristic functions of the random vectors being added [18, eq. (7–25)]. With the notation of (A.2a) and (A.2b), we have
. To evaluate the integral inside the where square brackets, we use [19, eq. (8.411.7)] to obtain the equality
(A.3)
(A.11) We choose to perform the double integration of (A.11) by changing to polar coordinates. Letting
is the characteristic function of the final sum where , and is the characteristic function of vector . The joint characteristic the th vector in the sum is [18, eq. (7–23)] function of a random vector
(A.4) As assumed above, when expressed in polar coordinates, the and , are independent. Thus, we perform RVs involved, the double integration in (A.4) by converting to polar coordinates. Using [18, eq. (6–72)] (A.5) in (A.4) we obtain
(A.8) Performing the substitution (A.8), we obtain3
in (A.7) and using (A.9)
The joint characteristic function of the sum by substituting (A.9) in (A.3)
is obtained
(A.10) , we take the inverse Fourier transTo obtain the jpdf of according to [18, eq. (7-24)] form of
in (A.11), yields
where . The negative sign in the exponential may be absorbed by the cosine via a shift of . We now use (A.8) to evaluate the integral in the right set of square brackets. We thus have
(A.12) (A.6) 2The
arctan (x; y ) function returns the proper angle on the interval [0; 2 ).
3Integrals of the form of (A.9), i.e., in which we integrate against the Bessel function J (1), are referred to in the literature as Hankel transforms.
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 4, APRIL 2001
Converting the jpdf of (A.12) to polar coordinates, we obtain . Letting the jpdf of the random vector in (A.12) and substituting in (A.5) gives
(A.13) Note that we can separate the jpdf of
and
as (A.14)
and are the appropriate marginal pdfs, where proving that, indeed, and are independent RVs. Furtherin (A.13) does not depend on . From this, we more, conclude that is uniformly distributed over (A.15) regardless of the number of random vectors added. To obtain the pdf of the magnitude of the sum of random , we integrate the jpdf over the variable vectors to obtain
[12] G. L. Stüber, Principles of Mobile Communication. Norwell, MA: Kluwer, 1996. [13] M. J. Gans, “A power-spectral theory of propagation in the mobile radio environment,” IEEE Trans. Veh. Technol., vol. VT-21, pp. 27–38, Feb. 1972. [14] W. R. Young Jr, “Comparison of mobile radio transmission at 150, 450, 900, and 3700 Mc.,” Bell Syst. Tech. J., vol. 31, no. 6, pp. 1068–1085, Apr. 1952. [15] R. M. Gray and L. D. Davisson, Random Processes: A Mathematical Approach for Engineers. Englewood Cliffs, NJ: Prentice-Hall, 1986. [16] M. F. Pop, “Statistical Analysis of Sum-of-Sinusoids Fading Channel Simulators,” M. Sc. thesis, Queen’s Univ., Kingston, ON, Canada, 1999. [17] J. Goldman, “Statistical properties of a sum of sinusoids and Gaussian noise and its generalization to higher dimensions,” Bell Syst. Tech. J., vol. 53, no. 4, pp. 557–580, Apr. 1974. [18] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3 ed. Toronto, ON, Canada: McGraw-Hill, 1991. [19] I. S. Gradshteyn, I. M. Ryzhik, and A. Jeffrey, Eds., Table of integrals, series, and products, 5 ed. New York: Academic Press, 1994.
Marius F. Pop received the B.Sc. (honors) degree in mathematics and engineering and the M. Eng. degree in electrical engineering from Queen’s University, Kingston, ON, Canada, in 1996 and 1999, respectively. Since April 1999, he has been a Member of the Technical Staff at MacDonald Dettwiler’s Space and Advanced Robotics Division. His current research interests include fading channels and characterization of electronic circuits and components in harsh environments.
(A.16) , to We use (A.16) in this paper, with appropriate pdfs determine the envelope pdfs in (9) and (14), as well as (27a). REFERENCES [1] W. C. Jakes, Microwave Mobile Communications. Piscataway, NJ: IEEE Press, 1994. [2] P. Dent, G. E. Bottomley, and T. Croft, “Jakes’ fading model revisited,” Electron. Lett., vol. 29, no. 3, pp. 1162–1163, June 1993. [3] 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. [4] M. Pätzold and F. Laue, “Statistical properties of Jakes’ fading channel simulator,” in Vehicular Technology Conf. Rec. (VTC’98), vol. II, Ottawa, ON, Canada, May 1998, pp. 712–718. [5] P. Hoeher, “A statistical discrete-time model for the WSSUS multipath channel,” IEEE Trans. Veh. Technol., vol. 41, pp. 461–468, Nov. 1992. [6] P. A. Bello, “Characterization of randomly time-variant linear channels,” IEEE Trans. Commun., vol. COM-11, pp. 360–393, Dec. 1963. [7] E. N. Gilbert, “Energy reception for mobile radio,” Bell Syst. Tech. J., pp. 1779–1803, Oct. 1965. [8] R. H. Clarke, “A statistical theory of mobile-radio reception,” Bell Syst. Tech. J., pp. 957–1000, July–Aug. 1968. [9] 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. [10] E. F. Casas and C. Leung, “A simple digital fading simulator for mobile radio,” in Proc. IEEE Vehicular Technology Conf., Sept. 1988, pp. 212–217. [11] T. Eyceoz, A. Duel-Hallen, and H. Hallen, “Deterministic channel modeling and long range prediction of fast fading mobile radio channels,” IEEE Commun. Lett., vol. 2, pp. 254–256, Sept. 1998.
Norman C. Beaulieu (S’82–M’86–SM’89–F’99) received the B.A.Sc. (honors), M.A.Sc., and Ph.D. degrees in electrical engineering from the University of British Columbia, Vancouver, BC, Canada, in 1980, 1983, and 1986, respectively. In 1980, he was awarded the University of British Columbia Special University Prize in Applied Science. He held an appoinment as Queen’s National Scholar Assistant Professor in the Department of Electrical Engineering at Queen’s University, Kingston, ON, Canada, from September 1986 to June 1988. From July 1988 to June 1993, he held the position of Associate Professor, and from July 1993 to August 2000, the position of Professor, both at Queen’s University. In September 2000, he became the iCORE Research Chair in Broadband Wireless Communications at the University of Alberta, Edmonton, AB, Canada. His current research interests include digital communications over fading channels, interference effects in digital modulations, channel modeling and simulation, decision-feedback equalization, and digital synchronization in sampled receivers. Dr. Beaulieu is a member of the IEEE Communication Theory Committee. He served as the Committee’s representative to the Technical Program Committee of the 1991 International Conference on Communications (ICC’91) and as the Committee’s co-representative to the Technical Program Committee of ICC’93 and ICC’96. He was General Chair of the Sixth Communication Theory Mini-Conference, held in association with GLOBECOM’97, and Co-Chair of the 1999 Canadian Workshop on Information Theory. Since January 1992, he has served as Editor for Wireless Communication Theory of the IEEE TRANSACTIONS ON COMMUNICATIONS. Since November 1996, he has served as Associate Editor for Wireless Communication Theory of the IEEE COMMUNICATIONS LETTERS. Effective January 2000, he was appointed Editor-in-Chief of the IEEE TRANSACTIONS ON COMMUNICATIONS. He was a recipient of the Natural Science and Engineering Research Council (NSERC) E.W.R. Steacie Memorial Fellowship in 1999.
