Downlink Performance of Distributed Antenna Systems in MIMO Composite Fading Channel

Abstract

In this paper, the capacity and BER performance of downlink distributed antenna systems (DAS) with transmit antenna selection and multiple receive antennas are investigated in MIMO composite channel, where path loss, Rayleigh fading and lognormal shadowing are all considered. Based on the performance analysis, using the probability density function (PDF) of the effective SNR and numerical integrations, tightly-approximate closed-form expressions of ergodic capacity and average BER of DAS are derived, respectively. These expressions have more accuracy than the existing expressions, and can match the simulation well. Besides, the outage capacity of DAS is also analyzed, and a tightly-approximate closed-form expression of outage capacity probability is derived. Moreover, a practical iterative algorithm based on Newton's method for finding the outage capacity is proposed. To avoid iterative calculation, another approximate closed-form outage capacity is also derived by utilizing the Gaussian distribution approximation. With these theoretical expressions, the downlink capacity and BER performance of DAS can be effectively evaluated. Simulation results show that the theoretical analysis is valid, and consistent with the corresponding simulation.

1. Introduction

As a promising technique for future wireless communications, the distributed antenna system (DAS) has received much interests in recent years [1][2][3][4][5][6][7][8][9][10]. In DAS, the distributed antennas are separated geographically and connected to a central control module via dedicated wires, fiber optics, or an exclusive radio frequency link. Compared to the traditional co-located antenna system (CAS), DAS can effectively exploit both the advantages of micro- and macro-diversity, and significantly improve the system capacity and cell coverage, and reduce power consumption. Recent studies on application of DAS have drawn considereable attention in both academic research and standardization activities like LTE-Advanced [8][9], DAS cellular system [6], and distriubted multi-input multi-output (MIMO) [8]. Moreover, as a promising application, DAS may solve the problems arising from frequency selective channel and limited transmit power [10]. Besides, the antenna selection (AS) transmission scheme with maximal-ratio combining (MRC) is an effective diversity scheme to increase the performane while significantly decreasing the hardware complexity and cost [11][12]. Therefore, effective combination of DAS and AS technique will improve the system performance greatly with efficient hardware complexity and cost.

Many academic works have been done to analyze the capacity performance of DAS, and proved that DAS can enhance the system capacity effectively [13][14][15]. The impact of MRC-based macrodiversity on the capacity of code division multiple access (CDMA) DAS uplink is studied in [13]. The downlink capacity of DAS over Rayleigh fading and Nakagami fading in multicell environment are analyzed in [14] and [15], respectively. The closed-form expressions of ergodic capacity are derived in the above two references, but the derived expression in [14] has minor error, while [15] needs iterative calculation. For this, an accurate closed-form expression of ergodic capacity of downlink DAS in Nakagami fading multicell environment is derived in [16]. However, the above literatures do not consider the effect of shadow fading on the performance. Based on this, the ergodic capacity performance of downlink DAS is investigated in [17] by using different cooperation strategies, but the influence of noise is neglected for analysis simplicity. In [18], [19] and [20], by using a log-normal distribution to substitute the gamma-log-normal distribution, approximate expressions of the ergodic capacity are derived for downlink DAS over shadowed fading channel, where Rayleigh fading and Nakagami fading are respectively considered, but the derived theoretical capacity are not accurate enough to reflect the actual values, and the analysis is limited in single receive antenna case. By utilizing the high signal to noise ratio (SNR) analysis, a simple ergodic capacity expression is derived in [21], and can be close to the actual cell average ergodic capacity in high SNR (even in moderate SNR). By assuming double-sided spatial correlation, an analytical lower bound of the ergodic capacity of DAS in composite Rayleigh/lognormal fading channel is provided in [22], but the capacity is accurate in the low and high SNR regimes only. Considering that system outage probability is an important performance measure of communication systems, the outage performance of DAS in fading channel is studied in [23][24][25][26][27][28][29]. For the DAS uplink, the approximate outage probability is analyzed and derived for DAS in composite Rayleigh/ lognormal fading channels [23][24], in composite Nakagami/lognormal fading channels [25], and in composite generalized-gamma /lognormal fading channels [26][27], where the mobile terminal (MT) is equipped with one antenna only. In [28], an approximate outage probability is derived for downlink DAS with single receive antenna in composite Rayleigh and lognormal fading channels [28]. In [29], a closed-form approximation of the outage probability for the DAS downlink and uplink is derived, and this expression will be tight and valid only when the rate or the number of nodes and antennas of the system become large. Besides, [30] and [31] investigate the BER performance of DAS in composite fading channel, and give the approximate BER expressions in single-cell and multi-cell environment, respectively. However, the obtained BER in [30] is not accurate since it adopts the gamma distribution to approximate log-normal distribution for easy derivation, and the BER is not easy to calculate because of employing the complicated function (Meijer function). Whereas the derived BER in [31] is suitable for BPSK modulation only, and does not consider the superiority of AS scheme. If the AS is adopted, the derivation of average BER will become difficult.

Although in all these studies, the channel capacity and outage probability are well analyzed, the obtained capacity expressions in composite fading channel are not accurate because of using inaccurate approximation. Moreover, only limited work has been carried out for the BER and outage performance analysis of DAS downlink. Especially, the outage capacity of DAS over composite fading channels is not studied in the existing literatures. Motivated by the reasons above, we will address the performance analysis of downlink DAS with transmit antenna selection and multiple receive antennas in composite channel which takes path loss, lognormal shadowing and Rayleigh fading into account, and focus on the derivation of the ergodic capacity, average BER and outage capacity of downlink DAS in MIMO composite fading channel. In terms of the channel state information and maximum SNR criterion, a selection transmission scheme is presented to maximize the output SNR, i.e., only the ‘best’ distributed antenna is selected for data transmission. Based on this scheme and the performance analysis, using some numerical integrations (Gauss-Hermite integration, Gauss-Laguerre integration, etc.), the probability density function (PDF) and cumulative distribution function (CDF) of the effective output SNR of DAS are respectively obtained. With the obtained CDF and PDF, tightly-approximate closed-form expressions of ergodic capacity and BER of DAS are derived, respectively. These expressions have more accuracy than the existing ones. To analyze the outage performance of the system well, we also derive a tightly-approximate closed-form expression of outage probability for DAS for a given outage capacity. Meanwhile, a practical iterative algorithm based on Newton method for calculating the system outage capacity is proposed for a given outage probability. As a result, accurate outage capacity is achieved. Moreover, by utilizing the Gaussian distribution approximation, an approximate closed-form expression of outage capacity of DAS is also derived. With these theoretical expressions, the capacity and BER performance of DAS downlink can be effectively assessed. Computer simulation shows that the presented theoretical analysis is valid, and has the values very close to the simulated results.

The notations we use throughout this paper are as follows. Bold upper case and lower case letters denote matrices and column vectors, respectively. E{·} denotes the expectation. The superscripts (·)T and (·)H are used to stand for the transpose and Hermitian transpose, respectively.

 

2. System Model

In this paper, we consider a DAS with multiple remote antennas (RAs) and multiple receive antennas in a single cell whose covering area is approximated by a circle of radius R as shown in Fig. 1, the RAs are distributed in the cell and connected with the central base station (BS, also named as RA0) via coaxial cables, fiber optics or radio links, where Nt RAs are considered, and the i-th RA is denoted as RAi. The MT is equipped with Nr antennas, and it is assumed to be independent and uniformly distributed in the cell as in [14] [21]. To improve the system performance, the antenna selection diversity scheme is employed for RAs. If RAi is selected to transmit the signals, the received signals at MT can be expressed as

Fig. 1.Diagram of DAS structure.

where is the received signal from the j-th antenna at MT, Pt is the transmit power. hi(j) is the element of channel vector hi, and it denotes the composite channel fading coefficient between RAi and the j-th antenna of MT. b is the transmitted signal from RAi with unity energy. w is the noise vector, whose elements are independent, identically distributed (i.i.d) complex Gaussian random variables with zero-mean and variance σw2. hi(j) can be modeled as [18][19].

where κi(j) represents the small-scale fading between RAi and the j-th receive antenna of MT. For Rayleigh fading channel, {κi(j)} are modeled as independent complex Gaussian random variables with zero-mean and unit-variance. Li and Si denote the pass loss and shadowing effect between RAi and MT. The path loss term Li is expressed as where βi is the path loss exponent, dr is the reference distance and di represents the distance from RAi to MT. The shadowing effect term Si is log-normally distributed, and the mean of 10lgSi is assumed to be zero (in dB). As a result, the PDF of Si can be expressed as [32]

where s is the variable of shadowing effect term Si, σi (in dB) is the standard deviation of 10lgSi and γ=10/ln10.

According to (1) and (2), when single RAi is selected, the output SNR after MRC can be obtained as

where ρi(j) denotes the instantaneous SNR at the j-th receive antenna, and From (4), it is shown that αi is also log-normally distributed. Thus, by a transformation of random variables, its PDF can be given by

where is the mean of 10lgαi . Considering that is chi-squared distribution with 2Nr degrees of freedom, the PDF of ρi can be obtained as

Due to the difficulty of integration, (6) is not easy to calculate and obtain the closed form. For this reason, [18] and [19] resort to the log-normal distribution, and use it to approximate the Gamma-log-normal distribution in (6) and derive the closed-form PDF of ρi. Unfortunately, using this approximate PDF to derive the capacity and BER will result in the inaccuracy of the obtained theoretical formulae in some cases. For this, we will recalculate Eq.(6) using the numerical integration and the transformation of variable.

Let , then (6) can be rewritten as

where the Gauss-Hermite integration [33] is utilized , and . tn and Hn are the base point and weight factor of the Np-order Hermite polynomial, respectively. Eq.(7) is a tightly approximate PDF, and has more accuracy than the provided approximate PDFs in [18][19].

From (7), we can obtain the CDF of ρi as follows:

where Γ(·,·) is the incomplete Gamma function [34].

 

3. Capacity performance analysis

3.1 Ergodic capacity

In this subsection, we will give the downlink capacity analysis of the DAS in a composite Rayleigh and lognormal shadowing fading channel. Ergodic capacity is an appropriate capacity metric for channels that vary quickly, or where the channel is ergodic over the time period of interest [35], it is equal to Shannon capacity in an Additive White Gaussian Noise (AWGN) channel with log2(1+ρ), averaged over the distribution of ρ. Thus, the ergodic capacity can be expressed as

where C=log2(1+ρ) is the channel capacity, and f(ρ) is PDF of output SNR ρ.

According to the analyzed in section 2, the selective diversity scheme is applied to the transmitter, i.e., only one ‘best’ RA is selected for transmission to maximize the output SNR. Thus, we have: Considering large space among remote antennas, it is reasonable to assume that {ρi} are independent. Hence, with (8), the CDF of ρ, F(ρ), can be given by

Using (10) and (7) as well as (8), we can obtain the PDF, f(ρ), as follows:

Substituting (11) into (9) yields

Utilizing the transformation of variable z=ρcin, and Gaussian-Laguerre integration in [33], (12) can be changed to

where Fρk(ρ) is the CDF of ρk , as shown in (8) , {Au} are the weights associated with the zeros {zu} of the one dimensional Nq-th order Laguerre polynomial [33]. Eq.(13) is a tightly approximate closed-form expression of ergodic capacity of downlink DAS in composite fading channel, it has more accuracy than the existing expressions in [18][19] because the latter employs the approximate PDF for capacity derivation. It is shown that the derived (13) is in good agreement with the simulation.

3.2 Outage capacity

The outage capacity and probability are important performance measure of communication systems in fading channels, and are also important to network scheduling and antenna layout [23][24][25][26][27]. However, the related research is relatively less, especially the outage capacity analysis is much less. For this reason, we will give the outage performance analysis of the DAS downlink in a composite channel, and derive the closed-form outage capacity and probability. Since the channel capacity is a random variable, it is meaningful to consider its statistical distribution. A useful measure of statistical characteristic is the outage capacity [36]. For the given outage capacity Co, P(Co) is defined as the outage probability that the channel capacity C=log2(1+ρ) falls below Co. Thus, the outage probability can be expressed as

Substituting (10) into (14) gives

Eq.(15) is a tightly-approximate closed-form expression of outage probability of downlink DAS in composite fading channel for the given outage capacity Co, which is shown to accord with the simulation.

For a given outage probability ε=P(Co), we propose a practical iterative algorithm based on Newton’s method for finding the outage capacity Co. In the following, we express solving the outage capacity from (15) as finding the root of G(Co)=0 with

where ε is the given outage probability. We now show that G(Co) is a monotonic function with G'(Co)＞0. Differentiating G(Co) with respect to Co gives

Considering that ln2 and the PDF are all positive, the derivative G'(Co) in (17) is also positive. Thus F(Co) is a strictly monotonically increasing function of Co.

Substituting (11) into (17) yields

Eq.(18) is the closed-form expression of the derivative G'(Co). From (16), we can obtain

and

This is because outage probability ε ∈(0, 1). Based on the analysis above, the equation G(Co)=0 is shown to have a unique solution for Co>0.

There are many methods such as bisection for finding the root of a strictly monotonic function. We propose to use Newton’s method to find the root iteratively because it has the quadratic convergence rate. Newton’s method is described as follows.

where Co,l is the outage capacity value at the l-th iteration. G(Co,l) and G'(Co,l) are computed by (16) and (18), respectively. With (21), the accurate outage capacity can be achieved, and is shown to match the simulation well. However, this capacity needs iterative calculation. For this, we will give closed-form capacity calculation in the following.

With (11), we can calculate the second moment of the channel capacity C=log2(1+ρ) as follows:

From (13) and (22), the variance of C can be evaluated as

With the mean value (13) and the variance (23), using a Gaussian distribution approximation, the CDF of C can be approximated as

where erfc{ ﹒} denotes the complementary error function. By setting the above F(c) equal to the given outage probability ε, the outage capacity can be obtained as

where erfc-1{ ﹒} denotes the inverse complementary error function which can be evaluated by table look-up. Equation (25) is a closed-form expression of outage capacity, and has the value close to the simulation. From (25) and (15), it is found that Co will increase as ε increases, and decreases as ε decreases, which accords with the existing knowledge. Thus, the derived theoretical formulae provide an effective method to evaluate the outage performance of DAS downlink.Moreover, (25) is relatively simpler since it may provide closed-form calculation and does not need any iterative calculation.

 

4. BER performance analysis

Considering that BER is an another important performance measure of communication systems in fading channel, we will give the BER performance analysis of the DAS over composite fading channel in this section, and derive the average BER expressions of the system with MQAM and MPSK, respectively. For the given SNR ρ , utilizing [37, Eq.(16)] and [38, Eq.(8)], the average BER of DAS with MQAM and Gray codes can be given by

where ξl and τl are constants which depend on M (i.e., the constellation size), and the constant sets {ξl, τl} for MQAM can be found in [37][32]. f(ρ) and F(ρ) are the PDF and CDF of ρ, respectively.

Using (10) and the Gauss-Laguerre integration [33], (26) can be changed to

where . Eq.(27) is a tightly-approximate closed-form expression of average BER of DAS with MQAM over composite fading channel.

In the following, we give the derivation of average BER of DAS with MPSK. According to [39, Eq.(12)] and [38, Eq.(1)], the average BER using MPSK and Gray codes can be expressed as

where ζl=sin2((2l-1)π/M) , and M is the constellation size. Using (10) and the Gauss-Laguerre integration [33], (28) can be rewritten as

Eq.(29) is a tightly-approximate closed-form expression of average BER of DAS with MPSK over composite fading channel. Eqs.(27) and (29) provide closed-form calculations of the system BER in theory, and have more accuracy than the existing expression in [30]. It is because the latter employs the gamma distribution to approach the log-normal shadowing in (5) for the derivation of PDF and BER, which will result in the inaccuracy of the derived BER in some cases. Besides, unlike the derived BER in [31], the expression is limited in BPSK modulation. It is shown that (27) and (29) will be very close to the corresponding simulated values. Hence, using them, the error performance of DAS with MQAM and MPSK will be effectively evaluated.

 

5. Simulation Results and theoretical evaluation

In this section, we use the derived theoretical performance formulae and computer simulation to evaluate the capacity and BER performance of the DAS over composite channels, where path loss, Rayleigh fading and lognormal shadowing are all considered. In simulation, for analytical convenience, the cell shape is assumed to be circle with radius R. The BS (RA0) is in the center of the cell, and the Nt RAs are uniformly distributed over a circle with radius d. Correspondingly, the polar coordinate of the ith RA is (d, 2π(i-1)/Nt), i=1,2,…,Nt. The main parameters are listed as: Nt=6; d=2R/3; the reference distance dr=50m; radius of the cell R=103m. The simulation results are illustrated in Figs.2-8.

Fig. 2.Ergodic capacity of DAS with different receive antennas and path loss exponents (Nr=1,2, β=2,3, σ=8dB)

Fig. 3.Average BER of DAS with different receive antennas and modulation modes (β=3, σ=8dB, Nr=1,3).

Fig. 4.Average BER of DAS with two receive antennas and different path-loss exponents (Nr=2, σ=10dB, β=2,3)

Fig. 5.Average BER of DAS with four receive antennas and modulations (Nr=4, β=2, σ=10dB).

Fig. 6.Outage capacity of DAS with different receive antennas and outage probability (β=3, σ=6dB, Nr=2, 4, P0=0.15, 0.3).

Fig. 7.Outage capacity of DAS with different path loss exponents and outage probability(Nr=3, σ=6dB, β=2, 4, P0=0.15, 0.3).

Fig. 8.Outage probability of DAS with different receive antennas and path loss exponents (Nr=2, 3, σ=6dB, C0=2bit/s/Hz, β=2, 3).

In Fig. 2, we plot the theoretical ergodic capacity and corresponding simulation of the DAS system with different receive antennas and path loss exponents, where standard deviation σ=σi=8dB. The derived (13) is used for theoretical calculation (referred as ‘theory 1’), and the existing theoretical formula [18][19] (referred as ‘theory 2’) is used for comparison. As shown in Fig.2, the theoretical values from ‘theory 1’ are all in good agreement with the simulated ones for different receive antennas and path loss cases. Whereas the theoretical values from ‘theory 2’ have some difference with the corresponding simulations, and the difference will be more obvious when the antenna number is smaller. This is because the latter uses the log-normal distribution to replace the Gamma-log-normal distribution for deriving the PDF, which brings about the inaccuracy of the derived ergodic capacity in some cases. Thus, our derived ‘theory 1’ has more accuracy. Besides, it is observed that the systems with Nr=2 outperform those with Nr=1 because the former has greater diversity than the latter, and the ergodic capacity of DAS with β=3 is lower than that with β=2 due to larger path loss, as expected. The above results show that the derived capacity expression of DAS is effective and reasonable.

In Fig. 3, we plot the theoretical average BER and corresponding simulation of DAS with different receive antennas and modulation modes, where standard deviation σ=8dB, path loss exponent β=βi=3, BPSK and 4QAM are used for the system modulation. The theoretical BER are evaluated by (27) and (29) for 4QAM and BPSK modulation respectively, which is referred as ‘theory 3’. For comparison, the existing theoretical formula [30] (referred as ‘theory 4’) is also employed for BER calculation. It is found that the derived ‘theory 3’ can match the corresponding simulation very well, whereas the existing ‘theory 4’ is close to the simulated value for single receive antenna case only. When the number of the receive antenna is large, the ‘theory 4’ will be different with the simulation. The reason for this is that ‘theory 4’ uses the gamma distribution instead of the log-normal shadowing to derive the PDF and the corresponding BER. As a result, the obtained BER is not accurate. Besides, the average BER of DAS with three receive antennas is lower than that with single receive antenna due to the achievable higher diversity gain. Moreover, the BER performance of DAS with BPSK is superior to that of DAS with 4QAM. It is because the latter employs higher modulation, while the constellation points in high modulation are densely packed and prone to errors in fading channel. The above results indicate that the derived BER expression of DAS is also valid for error performance evaluation.

Fig. 4 shows the BER performance of DAS with two receive antennas and different path loss exponents, where σ=10dB, 4QAM and BPSK are used. The ‘theory 3’ and ‘theory 4’ are employed for theoretical calculation of average BER. As shown in Fig.4, the derived ‘theory 3’ still has the values very close to the corresponding simulated ones, but the theoretical evaluation from ‘theory 4’ is different with the simulation result. Especially for small path loss exponent case, the difference is much more obvious. Besides, it is found that DAS with β=3 is worse than that with β=2 because larger path loss happens, which accords with the existing knowledge. The above results further testify that the derived BER formulae are effective and reasonable.

In Fig. 5, we plot the theoretical BER and corresponding simulation of DAS with different high modulations and four receive antennas, where σ=10dB, β=2, 8PSK and 16QAM are employed for modualtion. The ‘theory 3’ and ‘theory 4’ are used for theoretical evaluation. From Fig.5, we can see that our ‘theory 3’can still match the simulation well, but the ‘theory 4’ has obvious difference with the simulation. Besides, the BER performance of DAS with 16QAM is worse than that of DAS with 8PSK. This is because the former employs higher modualtion, and will be easy to occur error under fading case. The above results show that the derived theoretical BER is also valid for higher order modulation.

In Fig. 6, we plot the theoretical outage capacity and corresponding simulation of DAS with different receive antennas and outage probability, where the outage probability Po is set as 0.15 and 0.3, respectively. β=3, σ=6dB, and Nr=2, 4. The accurate and approximate outage capacities are respectively calculated by (21) and (25), and they are referred as ‘theory 5’ and ‘theory 6’, respectively. Computer simulation results verify the effectiveness of the derived theoretical formula, that is, the theoretical values are all close to the corresponding simulated ones. Especially, the ‘theory 5’ from the Newton’s method has more accuracy than the ‘Theory 6’ from Gaussian distribution approximation, but the latter does not need iterative calculation. It is observed that the outage capacity of DAS with four receive antennas is higher than that with two receive antennas under the same Po because multiple antennas are employed. Besides, under the same receive antenna number, the outage capacity of DAS with Po=0.3 is higher than that with Po =0.15 since larger outage probability is permitted to happen, which accords with the theoretical analysis in section 3.

Fig. 7 shows the outage capacity performance of DAS with three receive antennas under different path loss and outage probability, where σ=6dB, β=2, 4, and Po=0.15, 0.3. ‘Theory 5’ and ‘theory 6’ are employed for the theoretical calculation of outage capacity. From this figure, we can find the results similar to Fig.6. Namely, the derived theoretical outage capacity can match the corresponding simulation well, only some little differences are observed for ‘theory 6’, and the outage capacity of DAS with large Po is higher than that with small Po. Besides, the outage capacity of DAS with β=4 is lower than that with β=2 since the former experiences larger path loss, which agrees with the existing knowledge as well. The above results show that the derived theoretical formulae are valid and reasonable.

In Fig. 8, we plot the theoretical outage probability and corresponding simulation of DAS with different receive antennas and path loss exponents, where the outage capacity Co is set as to 2bit/s/Hz, σ=6dB, and β=2,3. The theoretical outage probability is calculated by (15). As shown in Fig.8, the theoretical outage probability and the corresponding simulation result are very close to each other. It is found that the outage probability of DAS with three receive antennas is lower than that with two receive antennas since the former has greater diversity than the latter. Besides, due to larger path loss, the outage probability of DAS with β=3 is larger than that with β=2 as expected. The above results indicate that the derived outage probability is also valid for the outage performance evaluation.

 

6. Conclusion

We have studied the downlink performance of DAS over MIMO composite fading channel, and the capacity and BER as well as outage performance are all analyzed. Based on the performance analysis, using some numerical integrations, the PDF and CDF of the effective output SNR are respectively derived. With these results, the tightly-approximate closed-form expressions of ergodic capacity and average BER of DAS are achieved, respectively. These expressions have more accuracy than the existing ones, and obtain the values very close to the actual simulations. According to the outage performance analysis, a tightly-approximate closed-form expression of outage probability is also derived for a given outage capacity. Meanwhile, Newton’s method is proposed to find the system outage capacity for a given outage probability. Moreover, to simplify the calculation of the outage capacity, an approximate closed-form expression of outage capacity of DAS is derived by using the Gaussian distribution approximation, which does not need any iterative calculation. Simulation result show that the presented theoretical analysis can match the corresponding simulations very well, and the system performance can be effectively improved as the number of receive antennas increases and/or path loss exponent decreases. Thus, these theoretical expressions provide good performance evaluation for downlink DAS in composite channel, and avoid the conventional requirement for Monte Carlo simulation.

Besides, we also notice that this paper only addresses the performance study of DAS in single-cell, whereas in practice, the DAS will operate in multi-cell scenario. Under multi-cell case, the inter-cell interference (ICI) will affect the system performance inevitably, and bring about the performance degradation. Considering the practicability, in the future work, we will further study the performance of the DAS in multi-cell scenario where ICI is considered. It is expected that some practical results can be achieved.