IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 8, AUGUST 2001
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Optimization of Power Control Parameters for DS-CDMA Cellular Systems Andrea Abrardo, Member, IEEE, Giovanni Giambene, Member, IEEE, and David Sennati, Member, IEEE
Abstract—This paper envisages a cellular system based on code-division multiple access and investigates the performance of a strength-based closed-loop power control (CLPC) scheme on the basis of different parameters, such as the number of bits of the power command, the quantization step size, and the user speed. On the basis of a log-linear CLPC model, an analytical approach has been developed that has allowed to determine the optimum quantization step size to be used for each value of the number of power command bits. Simulation results have permitted to support the analytical framework developed in this paper. Index Terms—Code-division multiple access, mobile radio systems, power control, third-generation mobile systems.
I. INTRODUCTION
D
IRECT-SEQUENCE code-division multiple access (DS-CDMA) is the dominant air interface solution in the global IMT-2000 (International Mobile Telecommunications—Year 2000) standard [1]. CDMA is an attracting technique, since it allows high capacity and supports variable bit-rate user transmissions. New aspects need to be carefully taken into consideration when a CDMA air interface is designed. In particular, equal power levels must be received at the base station (BS) from different mobile terminals (MTs) in a cell (uplink). Therefore, open-loop power control (OLPC) schemes have been implemented, where MTs adjust their power levels on the basis of the received power of a pilot signal broadcast by the BS (downlink direction). These techniques are able to counteract path loss variations and shadowing effects, but they are not effective against multipath fading, since it entails fast uplink power fluctuations, that are uncorrelated with respect to downlink [2], [3]. A strength-based closed-loop power control (CLPC) scheme has been considered in this paper1 [4], where the BS measures the received power from the desired MT and sends back a command to equalize the received level. In our study, we refer to a CDMA cellular system, where OLPC fully removes slow power level fluctuations due to shadowing and path loss. Hence, we focus on the CLPC performance that depends on the following parameters. Paper approved by S. L. Ariyavisitakul, the Editor for Wireless Techniques and Fading of the IEEE Communications Society. Manuscript received April 14, 2000; revised September 19, 2000. The authors are with the Department of Information Engineering, University of Siena, 53100 Siena, Italy (e-mail:
[email protected]). Publisher Item Identifier S 0090-6778(01)06932-X. 1As stated in [2], if the short-term variation of interference plus noise is negligible compared to that of the desired signal, strength-based CLPC schemes and signal-to-interference ratio (SIR)-based [5] ones attain similar performance. Therefore, this paper focuses on the strength-based case, since it allows an easier analytical approach.
• The frequency according to which the BS sends power commands to the MT. • The number of bits of the power command that specifies the quantized power level to be used by the MT. • The power quantization step size , i.e., the difference between two adjacent power levels. • The loop delay which accounts for the following contri; 2) the delay butions: 1) the uplink propagation delay ; 3) due to the power measurement process at the BS ; and 4) the downthe downlink propagation delay link command transmission time . • The characteristics of the radio mobile channel, i.e., the Doppler spectrum (with maximum Doppler frequency ) and the multipath intensity profile ( being the shift number of resolvable paths by the RAKE receiver at the BS). A given downlink capacity has to be reserved for sending power control bits to an active MT every . The BS estimates the received power level from the MT by means of an integrator (thus reducing the that averages the received signal over power estimation error). This procedure introduces a delay equal to [6]. We will assume that the MT receives a kHz, i.e., power command with a rate ms. We have assumed a QPSK modulation with a symbol, rate of 32 ksymbol/s (i.e., 64 kbit/s). In a typical cellular environment the propagation delay is on the order of few tens of microseconds, thus it can be ne. The downlink transmission time glected with respect to can be reasonably neglected with respect to , bits (i.e., , if ), so that is if . The power control command sent to the MT about equal to is affected by errors that are due to the following aspects: 1) the loop delay that causes the MT to receive an out-of-date command (i.e., the command is not related to the present situation, but to an old one); 2) measurement errors at the BS for the received power from the MT; and 3) the quantization of the power command due to the use of a finite number of bits . Since the s, there is a power control command is transmitted every further error that is caused by channel variations in this time interval. Therefore, there is a difference (i.e., an error) between the controlled power level and the reference one, both expressed in logarithmic scale. This error will be modeled through a random variable that will be characterized by its standard deviation . Since parameter has a significant impact on the CDMA system capacity [7], the choice of and has to be optimized . in order to reduce , , ) on The impact of the CLPC parameters (i.e., the performance of CDMA cellular systems has been mainly
0090–6778/01$10.00 © 2001 IEEE
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Fig. 1.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 8, AUGUST 2001
Block diagram of the log-linear CLPC scheme.
studied through simulations due to the analytical complexity in modeling these aspects [2], [7]–[9]. Intuitively, if the value is too small or too high, the CLPC system is not able to track values are high) [7]. The choice of fading variations (i.e., the value also depends on the MT mobility conditions. Moreover, has to be increased if the power command is sent less . However, a frequently. Of course, the greater , the lower too high value entails a waste of bandwidth to transport the CLPC signaling information. The effort made in this paper is to propose an analytical framework for the CLPC performance evaluation. In particular, on the basis of a suitable power control model, we have analytically obtained the distribution of . Different values of , , and symbol energy-to-interference power spectral density ratio have been considered. Correspondingly, we have identified value that realizes the minimum value the optimum by assuming a large range of MT speeds. Simulation results have permitted to validate the optimum values predicted by analysis. The study carried out in this paper is of significant interest for the implementation phase of CDMA air interfaces, since it may help the designer to define an efficient CLPC scheme. This paper is organized as follows. The system model is described in Section II. The analytical approach for the derivation of the optimum quantization step size is given in Section III. Finally, Section IV presents simulation results and validates the proposed analytical approach. II. SYSTEM MODEL The log-linear CLPC model we refer to is shown in Figs. 1 and 2. In particular, we consider a power control scheme based on the strength received at the BS. This is a discrete-time model (i.e., the with sample period equal to the symbol duration ; we system is sampled at the symbol rate). Let consider as a fixed integer. The BS compares the estimated reand performs a ceived signal power with the target level2 -bit quantization of the difference (in log-scale). This process is updated for every symbol. For the sake of simplicity, we normalize all power levels to . Thus, the power terms will be con2In actual systems, the target power level P is estimated through an outerloop. Since the outer-loop update time is tipically much greater than T , we will consider, for our purposes, P as a constant.
Fig. 2. Transmission, channel and measurement model.
Fig. 3.
B -bit uniform midrise quantizer with step size q.
sidered as dimensionless quantities. If normalized power levels dB. Accordingly, the target are considered, we have will be neglected in the following analysis. power level In this paper, we refer to the -bit uniform midrise quantizer shown in Fig. 3 [10], which is characterized by quantization with intervals for for for
(1)
and quantized levels in decibels (2) being the quantization step size.
ABRARDO et al.: OPTIMIZATION OF POWER CONTROL PARAMETERS FOR DS-CDMA CELLULAR SYSTEMS
The midrise quantizer described above is optimum for uniformly distributed signals with even number of quantization levels. The measured power at the BS is far to be uniformly distributed. However, we will show in Section IV that low values are sufficient to achieve a good CLPC performance. Thus, the adoption of a nonuniform quantization would not significantly improve the CLPC performance [10]. Hence, the midrise quantizer can be reasonably considered as a near-optimal solution. The zero order hold block in Fig. 1 performs a decimation is sent to the MT every samso that a power command ples. The MT sets the transmission power by subtracting a to the power transmitted samples power correction term before. The transmission/channel/measurement block in Fig. 1 accounts for symbol transmission, multipath radio channel, and power measurement procedure at the BS. This block is sketched with more details in Fig. 2(a). As for the transmission part, we assume a baseband model with rectangular pulse shape. Ac. The information cordingly, the pulse amplitude is set to are denoted by . We assume pure symbols at instants . The multipath propphase-modulated signals, i.e., agation environment has been modeled by replicating times the MT signal and by multiplying each th replica by a complex fading term that is assumed to be Rayleigh distributed. Each fading term can be expressed as (3) and are independent Gaussian processes with zero where . mean and variance Let denote the envelope of each multipath contribution, i.e., . We consider that variables , , are statistically independent (uncorrelated scattering). We have made the assumption of equal-strength paths in this paper, so . In such environment, . that Thus, if we consider, without loss of generality, a multipath . In a widely fading signal with unitary mean, we have and accepted channel model [11], the Gaussian processes have a band-limited nonrational power spectrum given by
for
(4)
is the maximum Doppler frequency shift, where is the MT speed, is the speed of light, and is the carrier frequency. According to previous considerations, each fading term has been generated by filtering a complex Gaussian noise with zero mean and unitary variance through a digital Doppler filter with transfer function obtained so that its squared amplitude is equal to (4). In order to take into account both the system noise and the interference caused by other transmitting MTs, a Gaussian complex noise has been added to each multipath term. The interference will be considered as a Gaussian noise with a constant variance. Let us evaluate the variance of , re. Since we have considered a system sampled ferred to as at the symbol rate, spreading and despreading processes do not appear in Fig. 2(a). Thus, represents the noise contribution
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at the output of the matched filter (despreading). As shown in Fig. 2(a), terms are also the noise contributions at the input of the maximal ratio combiner. Accordingly, the decision variat the input of the detector (evaluated at the th time able instant) is given by (5) The average of the squared envelope of the signal contribution in (5) divided by the variance of the interference term gives the symbol energy-to-interference power spectral density ratio at , i.e., instant ,
(6) By averaging (6) over and considering that (7) we obtain the symbol energy-to-interference power spectral as density ratio (8) . Hence, directly from (8) we have As for the power measurement process at the BS, let us refer to the dotted box in Fig. 2(a). Each multipath component is (i.e., the complex conjugate of the transmitted multiplied by symbol) obtained from the demodulator, which is assumed error-free. This feedback has been introduced to make the received signal independent of the transmitted symbols and suitable for a power measure. Each resulting multipath component is averaged over samples to smooth both interference and noise fluctuations (Avg blocks). The squared envelopes of the outputs of the Avg blocks are then combined to obtain a measure of the total multipath power received at the BS. III. OPTIMUM QUANTIZATION STEP SIZE FOR POWER CONTROL COMMANDS In order to evaluate analytically the optimum quantization step size , we consider a simplified model, where we assume . As exthat the averaging filter introduces a delay ; since , plained in Section I, . For the sake of simplicity, we consider we have that that is an integer number. Besides, assuming that the power measurement error is small compared to the received power, the scheme in Fig. 2(a) leads to the log-linear model with additive noise shown in Fig. 2(b) (see Appendix I). Hence, referring to the simplified model in Fig. 2(b), the power measured at the ( being an integer dimensionless number) is BS at instants [dB] (9)
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where and denote the received and the transmitted , respectively. denotes the global mulpowers at instants , expressed in decibels tipath term at instants
(10) represents the power measurement error at the BS. Acand cording to the considerations made in Appendix I, this term will be modeled by a Gaussian random variable with mean and vari. ance that depend on , , and The power control command is given by the quantization of denote the difference the measured power (see Fig. 1). Let between the power control command and the actual received . This term is the global power, i.e., estimation error, which depends on both the power measurement error and the quantization error. The power control command can be written as
Note that, since dB, coincides with the power control error (PCE), i.e, . Assuming that is a stationary process and that the global is independent3 of , the estimation error pdf of the process given and can be easily derived from (17) as
(18) where represents the pdf of the global estimation error . . Let Let the quantizer output be denote the pdf of the power measurement error given , , and the received power . This pdf can be derived according to the considerations shown in Appendix I. In particular, by setting (19)
[dB] (11) , actually sent to the MT, The power control command at instants , with coincides with , (see the zero order hold in Fig. 1); hence, we have
and
(20) (12) we have is the floor operator. where , is obtained by The transmission power at instants the MT on the basis of both the power control command and the previously transmitted power (see Fig. 1) [dB]
(13)
Directly from (13) and (9), we have
(21) The cumulative distribution function (cdf) of considering the following equality: from
can be derived
(22)
[dB] (14) It results
at instants Let us refer to the received power , i.e., . Since , we have from (12) , thus yielding
v qLB [dB] (15)
s
(23)
By substituting (11) in (15), we obtain is the following region in : . The pdf of can be obtained by differentiating (23) with respect to
where [dB] (16) . Hence,
denote the process Let (16) can be rewritten as follows: [dB]
(17)
Appendix II shows an analytical approach to derive an approximate expression for the probability density function (pdf) of conditioned on both the MT speed and the number of paths , . denote the random process obtained by Let . sampling the received power in decibels at instants
(24) and can be derived by solving the system of The pdfs (18) and (24) by means of the following iterative approach. At , we set as in step 3Such approximation allows the analytical tractability of the problem. The goodness of this assumption has been validated through the agreement between simulation and analytical results (see Section IV).
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the case with no quantization error and no power measurement error. Then, at the generic th step, we evaluate
(25) The iterative procedure described in (25) is repeated until convergence is achieved; recursions are stopped at the step where is lower than a prethe integral of [12]. Numerical quadrature has defined threshold been used to solve the integrals in (25). The validity of the pdfs obtained by this algorithm has been verified through the good agreement with simulation results (see Fig. 7 in Section IV). We assume that the MT speed is uniformly distributed beand and we denote by the corresponding tween [see pdf of the speed. Hence, the dependence on in the pdf (25)] can be removed as follows:
Fig. 4. Simulation results and analytical ones for the standard deviation of PCE, , as a function of the quantization step size, q , for different B values,
= 7 dB, and L = 2.
(26) of can be derived from Finally, the standard deviation as given in (27), shown (26) as a function of , , , and at the bottom of the page. The optimum quantization step size, , is the value that minimizes (27), once the number of bits , the number of resolvable paths and the symbol energy-to-interference power spectral density ratio are given. IV. RESULTS In order to verify the goodness of the analysis proposed in the previous section, we have simulated the CLPC scheme described in Figs. 1 and 2(a). The following parameter values have been considered to obtain the results shown in this section: ms, s (for a transmission rate of 32 kilosymbols/s; hence, a power command is sent every 20 GHz, km/h, and km/h symbols), (urban scenario). For these MT speed values, the product ranges from 0.005 to 0.1 with GHz.4 In the assimulations, we have derived the pdf to with a step of 5 km/h; suming a fixed speed from then, we have removed the dependence on . The obtained pdf . has been used to evaluate the variance behaviors obtained from both simulaFigs. 4–6 show the tions and analysis as a function of for different values in the dB. Figs. 4–6 refer to , , , case 4As shown in [2], we must have f T < 0:1 in order to obtain an effective CLPC scheme. Accordingly, v = 90 km/h may be considered as the maximum MT speed value for which the CLPC scheme is effective, once T is set to 0.625 ms.
Fig. 5. Simulation results and analytical ones for the standard deviation of PCE, , as a function of the quantization step size, q , for different B values,
= 7 dB, and L = 3.
respectively. The vertical dash-dot lines shown in these figures , which highlight the optimum theoretical values, minimize (27) for the different and values. It is worth noting approaches very well the value of that minthat estimated from simulations, thus validating the analimizes ysisproposedinSectionIII. The values obtainedfromanalysis are slightly higher than those from simulations (upper bound) for bit and close to simulations for bit. This is due to the first-order Taylor approximation used to derive the pdf of in Appendix II that lends to overestimate channel fluctuations for high MT speed values [13]. This effect is more evident for high
(27)
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TABLE I OPTIMUM q VALUES FROM ANALYSIS AND COMPARISON BETWEEN THEORY AND SIMULATIONS IN TERMS OF THE MINIMUM
Fig. 6. Simulation results and analytical ones for the standard deviation of PCE, , as a function of the quantization step size, q , for different B values,
= 7 dB, and L = 4.
values, i.e., when the quantization error becomes negligible with respect to channel fluctuations.Conversely, the quantization error behavior for bit. becomes the predominant effect in the A similar CLPC system has been studied in [3] through simulations. Considering the case referred to as “1NZ” in [3], a fair com. In particular, parison can be made with our approach for and we have (i.e., the power step size for is equal to 1.25). This value is consistent with the results shown in [3, Fig. 3] for the “1NZ” case. values obtained in correspondence of The in Figs. 4–6, referred to as , have been shown in Table I values obtained from that also gives the standard deviation simulations of the received power . Note that is obtained by . Since is fixed, the sampling at intervals of length power control scheme can only modify ; anyway, is the actual parameter upon which depends the communication quality. is quite close to , so conThe results in Table I show that firming the effectiveness of the CLPC scheme with the selected value. According to Table I, as one could expect, the value decreases as the diversity degree increases, of due to the better channel conditions that require smaller power
VALUE FOR = 7 dB
adjustments. Moreover, the value decreases as inbits. This is creases, but it tends to a limiting value when due to the fact that the intrinsic error (due to channel variations and interference) between two subsequent power commands is bits. much greater than the quantization error when bits seems to be the best tradeoff Hence, the choice of between the good CLPC behavior and the downlink capacity values of the midrise quanrequired. Correspondingly, the dB are 2.25, 1.75, and 1.25 dB, for , tizer for respectively (see Table I). bit and on the basis of (2), the MT adIn the case justs its transmitted power by an optimum power step size dB, that is 1.875, 1.375, 1.25 dB, respectively, for and dB. If in these cases would be set to 1 dB, as proposed in [7] and [14], the CLPC performance would be worse increases). In [7], qualitative considerations lead to iden(i.e., dB, by considering simtify an optimum power step size values from 0.001 up to 0.2. However, ulations for fixed values greater than 0.1, the CLPC scheme becomes for ineffective (the power update can not adequately track channel variations [2]) and, hence, the optimum power step size should tend to 0 (i.e., no power control case).5 Therefore, by considering range in which the CLPC scheme is effective, only the more accurate results can be found, as shown in Table I. As a further proof of the analytical approach developed in Section III, we compare in Fig. 7 theoretical and simulation behavfor , dB with , and iors of (extreme cases), in correspondence of the values shown in Table I. From these results it is evident that the iterative approach converges to a pdf that is quite close to that obtained from simulations (differences are appreciable only in the queues, i.e., for probability values lower than 10 ). , for , Fig. 8 shows the cdf of , , dB (from Table I, ) obtained from both simulations and the analytical approach proposed in Section III. Since in the literature a log-normal assumption is made for the distribution of PCE [15], [16], Fig. 8 contains (in log(at a parity of arithmic scale) the Gaussian fitting for obtained from simulations). From this figure, we can note that 5For fixed MT speeds from low-to-medium values (i.e., below 90 km/h, for the assumed T value), q increases with the MT speed; however if the MT speed becomes too high (i.e., if f T > 0:1), the q value decreases.
ABRARDO et al.: OPTIMIZATION OF POWER CONTROL PARAMETERS FOR DS-CDMA CELLULAR SYSTEMS
Fig. 7. Comparison between theoretical and simulation results for the pdf of the measured power, Y with L = 3, = 7 dB, and different B values.
Fig. 8. CDF of Y obtained from simulations and theory and fitting with Gaussian and generalized Gaussian distributions for L = 3, = 7 dB, and B = 1.
the Gaussian fitting is not very good. Moreover, we have also from simulations proposed in Fig. 8 the fitting between and a generalized Gaussian distribution, obtained according to the maximum-likelihood (ML) criterion shown in [18]; see Appendix III for more details. This generalized distribution allows a better fit than the Gaussian one, but the cdf of derived according to the method proposed in Section III yields the best fit for with simulations and also permits a good estimation of low values (i.e., the critical cases where the received power is much lower than the required value). on the MT speed, In order to highlight the dependence of , we show in Fig. 9 the comparison between the behavior obtained from both simulations (meshed surface) and analysis (continuous surface) as a function of and in the case of , dB and . It can be noted that the analysis is in good agreement with simulation results. In particular, for low values the meshed surface is above the continuous one, i.e., the
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Fig. 9. Simulation results and analytical ones for the standard deviation of PCE, , as a function of both the quantization step size, q , and the MT speed v for B = 2, L = 3, = 7 dB.
Fig. 10. Standard deviation of Y , , obtained in correspondence of the q value shown in Table I, as a function of for L = 3 and different B values.
analysis slightly tends to underestimate , while the opposite behavior is observed for high values. This is due to the fact that for low values, the Taylor expansion for deriving the pdf of (see Appendix II) is almost exact, and the difference between the two surfaces is mainly caused by the assumption of independence between and , made in Section III. Conversely, for high values the Taylor expansion becomes the main source of approximation, so that the theoretical surface is over the simulation one. However, as previously shown in Fig. 5, by averaging . The on , the proposed analysis gives an upper bound for (cortick curve shown in Fig. 10 represents the minimum ) for different values obtained from analresponding to ysis. Note that this is quite close to the corresponding minima of the meshed surface at different MT speeds. In fact, the two surfaces have the same saddle shape. Eventually, it is worth noting steadily increases until the MT speed reaches about that 60 km/h. For higher speeds the growth is less evident, since the channel decorrelates and the CLPC becomes less effective.
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Finally, we have verified by both simulations and analysis that the optimum values shown in Table I are still valid for from 5 to 11 dB, i.e., the range of practical interest. Accordingly, in behaviors obtained in correFig. 10, we have presented the values shown in Table I, as functions of spondence of the for and different values. We can note that the obtained curves are quite flat, i.e., the power measurement errors (and then PCE) at least for the do not dramatically affect values. considered medium-to-high
the statistical properties of only depend on its mean and variance, respectively, given by (31) (32) where
V. CONCLUSIONS
(33)
This paper proposes a log-linear model to study the CLPC performance in CDMA cellular systems. A multipath fading channel has been considered, so that the RAKE receiver at the BS is able to resolve a given number of paths. Assuming a uniform midrise quantizer, we have achieved a theoretical characterization of the PCE distribution in the presence of power estimation errors at the BS. Hence, it has been possible to evaluate analytically the optimum quantization step size depending on both the number of resolvable paths, the number of bits of the power command and the symbol energy-to-interference power spectral density ratio. The analytical results have been favorably compared with simulation results. Therefore, this study has allowed an optimization of the CLPC scheme, a particularly interesting task for implementing CDMA cellular systems.
in (33) represents the received power at the The process output of the Avg block expressed in linear form, i.e., , being the received power expressed in deci. In (31) and (32), we have exploited the wellbels at instants known properties of the third and fourth moments of zero-mean Gaussian random variables, i.e., if is a zero-mean Gaussian and random variable with variance , we have . in (29) Let us make the assumption that the noisy term . Note that, if this assumption was is small with respect to not true, the CLPC performance would dramatically degrade due to high power measurement errors. Under this assumption, expression (29) may be approximated as
APPENDIX I
(34)
According to the scheme shown in Fig. 2(a) and assuming that the Avg blocks do not alter the received signal (i.e., it merely in), the measured power at instants troduces a delay is
and . where Accordingly, the power measurement error conditioned on can be modeled as a Gaussian random the received power variable, with mean and variance, respectively, given by
(28)
(35)
and represents the where6 , i.e., it is a output of the averaging filter when the input is complex Gaussian random variable with zero mean and variance . Equation (28) may be written as
(36)
APPENDIX II of
Let us consider the process given in (10):
defined as follows in terms
(29) (37) where (30) depends on the sum of We note that the random term real independent random variables. Assuming that the central limit theorem can be applied to the sum of these variables, can be considered as a Gaussian random variable. In this case, 6For the considered pure phase-modulated signal jA j = 1, 8 l . Hence, the term A does not appear in the expression of Z (l).
, the process is highly Under the hypothesis . Thus, the numerator in correlated during the time interval (37) can be suitably approximated by a first-order Taylor expan[13] sion, centered in
(38) is the derivative of with respect to time. where and are evaluated at the same instants in (38), Since we replace the time-varying quantities with the corresponding
ABRARDO et al.: OPTIMIZATION OF POWER CONTROL PARAMETERS FOR DS-CDMA CELLULAR SYSTEMS
random variables in (38) and we characterize variable lows:
as fol-
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Removing the conditioning on
in (46), the cdf of
(47)
(39) In this study, we consider (10), we obtain
, , ,
as fixed parameters. From
is the pdf of the multipath fading term in (10). where The cdf of the variable defined in Section III is (48)
(40) The joint cdf of
and
can be computed as shown in [17]
where is expressed in decibels. is obtained by differentiating (48) Finally, the pdf with respect to .
(41)
represents the second derivative (computed in where corresponding to zero) of the autocorrelation function the power spectrum given in (4)
results
APPENDIX III Let us consider that variable has a generalized Gaussian distribution. Hence, according to [18] the pdf of depends on parameters and as follows: (49)
(42) where being the modified Bessel function of the first kind and zeroth order. and are independent, thus From (41) we have that yielding (43)
, which represents the variance of , is equal The term on the basis of (42). From (40) and under the asto are independent sumption of uncorrelated scattering (i.e., and identically distributed), we have that the random variable is Gaussian with zero mean and variance , re, where ferred to as
(50) denotes the Gamma function. and where Parameter is equal to the standard deviation of the random variable , while controls the pdf shape; and are both real and positive. Note that the generalized Gaussian pdf becomes and , a Laplacian pdf and a Gaussian pdf, for respectively. Parameters and can be fitted to a given set , by means of an ML criterion [18]. In of realizations particular, the ML estimation of the shape parameter can be obtained as the root of the following equation:
(44) (51) Therefore, the random variable conditioned on , , and can be approximated by the truncated version (i.e., only positive [13]. values) of the random variable denote the cdf of a Gaussian random variable Let with mean and standard deviation (45) being the Marcum expressed as
function. Hence, the cdf of
can be
with (52) and where is where is the cardinality of set the Euler constant. Finally, the ML estimation of the standard deviation is given by (53)
ACKNOWLEDGMENT (46)
The authors wish to thank the anonymous reviewers for providing useful suggestions to improve this paper.
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Andrea Abrardo (M’99) was born in Florence, Italy, in 1966. He received the Dr.Ing. degree in electronic engineering from the University of Florence, Florence, Italy, in April 1993, and the Ph.D. degree from the Image Processing and Communications Laboratory of the Department of Electronic Engineering, University of Florence, in June 1998. Since August 1998, he has been with the Department of Information Engineering of the University of Siena, Siena, Italy, where he is an Assistant Professor. His main research interests include the field of computer and communication networks with an emphasis on code-division multiple access for third-generation wireless communications.
Giovanni Giambene (S’94–M’97) received the Dr. Ing. degree in electronics and the Ph.D. degree in telecommunications and informatics from the University of Florence, Florence, Italy, in 1993 and 1997, respectively. From 1994 to 1997, he was with the Department of Electronic Engineering, University of Florence, Florence, Italy. He was the Technical External Secretary of the European Community Project COST 227 Integrated Space/Terrestrial Mobile Networks. He also contributed to the Resource Management activity of the Working Group 3000 within the RACE Project called Satellite Integration in the Future Mobile Network (SAINT, RACE 2117). From 1997 to 1998, he was with OTE of the Marconi Group, Florence, Italy, where he was involved in a GSM development program. In the same period he also contributed to the COST 252 Project (Evolution of Satellite Personal Communications from Second to Future Generation Systems) research activities by studying the performance of PRMA protocols in low earth orbit mobile satellite systems. Presently, he is with the Information Engineering Department, University of Siena, Italy, where he is involved in the activities of the Personalised Access to Local Information and services for tOurists (PALIO) IST Project within the fifth Research Framework of the European Commission. His research interests include third-generation mobile communication systems, medium access control protocols, traffic scheduling algorithms, and queuing theory.
David Sennati (S’98–M’01) was born in Rapolano Terme, Siena, Italy, in 1971. He received the Laurea degree in telecommunication engineering and the Ph.D. in telematics from the University of Siena, Siena, Italy, in 1997 and 2001, respectively. He is currently a Research Associate with the Department of Information Engineering, University of Siena, Siena, Italy. His research interests include power control, medium access protocols, and mobile communication networks.
