Performance Analysis and Location Optimization for Massive MIMO Systems with Circularly Distributed Antennas
Abstract
In this paper, we analyze the achievable rate of the uplink of a singlecell multiuser distributed massive multipleinputmultipleoutput (MIMO) system. The multiple users are equipped with single antenna and the base station (BS) is equipped with a large number of distributed antennas. We derive an analytical expression for the asymptotic ergodic achievable rate of the system under zeroforcing (ZF) detector. In particular, we consider circular antenna array, where the distributed BS antennas are located evenly on a circle, and derive an analytical expression and closedform tight bounds for the achievable rate of an arbitrarily located user. Subsequently, closedform bounds on the average achievable rate per user are obtained under the assumption that the users are uniformly located in the cell. Based on the bounds, we can understand the behavior of the system rate with respect to different parameters and find the optimal location of the circular BS antenna array that maximizes the average rate. Numerical results are provided to assess our analytical results and examine the impact of the number and the location of the BS antennas, the transmit power, and the pathloss exponent on system performance. It is shown that circularly distributed massive MIMO system largely outperforms centralized massive MIMO system.
Massive MIMO, distributed MIMO, achievable rate analysis, antenna location optimization.
I Introduction
With the demands of the wireless data services nowadays, high spectrum efficiency or data rate is undoubtedly an important feature of future wireless systems [1, 2]. In order to improve the data rate of wireless systems, various innovative ideas have been proposed and investigated. Among the most successful ones in recently years is the multipleinputmultipleoutput (MIMO) concept [3, 4].
Recently, distributed MIMO systems, or distributed antenna systems, was proposed to further improve the data rate, in which multiple transmit or receive antennas are distributively located to reduce the physical transmission distance between the transmitter and the receiver [9, 7, 5, 13, 6, 10, 8, 11, 12, 14]. The distributive antennas are assumed to be connected to the central unit via highbandwidth and lowdelay backhaul such as optical fiber channels. It has been proved that distributed MIMO outperforms centralized MIMO in outage probability and achievable rate. In [9], for a singlecell singleuser distribute MIMO system with arbitrary antenna topology, the authors analyzed the outage performance as well as the diversity and multiplexing gains. For a singlecell distributed MIMO systems with multiple uniformly distributed users, upper bounds on the ergodic capacity of one user and approximate expressions of sum capacity of the cell were derived in [10]. In [11], both singlecell and twocell distributed MIMO systems with multiple uniformly distributed users per cell are considered. The cells are assumed to be circular and the distributed base station (BS) antennas have circular layout. The locations of the distributive antennas were optimized to maximize lower bounds on the expected signal to noise ratio (SNR) and signal to leakage ratio. The resource (including power, subcarrier, and bit) allocation problems in singlecell multiuser distributed antenna systems were investigated in [12]. In [13], different radio resource management schemes were compared for multicell multiuser distributed antenna systems. In [14], for multicell networks with multiple remote antennas and one multiantenna user in each cell, the input covariances for all users were jointly optimized to maximize the achievable ergodic sum rate.
In the above literature of distributed MIMO, single user or multiple users with orthogonal channels are assumed, so there is no interuser interference. However in current and future wireless systems, it is expected to have multiple users sharing the same timefrequency resource. In such systems, one user will suffer from the interferences of other users in the cell, which can largely degrade the system rate. To conquer the intracell user interference problem, the concept of massive MIMO, where the BS is equipped with a very large number of antennas usually of hundreds or higher, was proposed and attracted considerable attention recently [2, 15, 16, 17, 18, 19, 21, 22, 23, 24, 20, 25, 26]. With a large number of antennas at the BS, according to the law of very long vectors, transmission channels for different users are orthogonal to each other. Userinterference diminishes and very high data rate can be achieved with low complexity signal processing. The ergodic achievable rates of the singlecell multiuser massive MIMO uplink with linear detectors, i.e. maximum ratio combining (MRC), zeroforcing (ZF), minimummeansquareerror (MMSE), have been derived in [20]. The achievable rates of both the uplink and downlink of multicell multiuser massive MIMO systems with linear precoders and detectors were analyzed in [21].
As the combination of the two promising concepts, massive MIMO and distributed MIMO, distributed massive MIMO systems are of great potential in fulfilling the increasing demands of next generation communication systems [27, 28, 29, 30, 31]. The authors in [27] focused on an uplink massive MIMO system consisting of multiple users and one BS with several large scale distributed antenna sets. In this system, the deterministic equivalence of the ergodic sum rate was derived and an iterative waterfilling algorithm was proposed for finding the capacityachieving input covariance matrices. [28] analyzed the sum rates of distributed MIMO systems, in which multipleantenna radio ports form a virtual transmitter and jointly transmit data to a centralized multipleantenna BS. The capacity and spatial degrees of freedom of a large distributed MIMO system were investigated in [29], where wireless users with single transmit and receive antenna cooperate in clusters to form distributed transmit and receive antenna arrays. In [30], a simplistic matched filtering scheme and a subspace projection filtering scheme were investigated in a fixed size singlecell network, where the BS antennas are assumed uniformly and randomly located in the cell serving singleantenna users. In [31], for networks where one largescale distributed BS with a grid antenna layout serving singleantenna users, the energy efficiency maximization problem was formulated under perantenna transmit power and peruser rate constraints. low complexity channelgainbased antenna selection method and interferencebased user clustering method were proposed to improve the system energy efficiency.
In this paper, we consider the uplink of a singlecell multiuser distributed massive MIMO system in which the BS equipped with a large number of distributed antennas receives information from multiple singleantenna users. The asymptotic system achievable rate under linear ZF detector is analyzed for arbitrary but known antenna locations and for circular antenna layout. Based on the analysis, the location of the distributed antennas for circular antenna layout is optimized. Compared to [27, 28], the topology of the distributed BS antennas used in our work is different, as the BS in [27] is the combination of several centralized BS and the BS in [28] is centralized. Compared to [29], the transmitters in this work are noncooperative singleantenna users, while the users in [29] cooperate with each other and form a virtual transmitter with distributed antenna arrays. Thus our system model and derivations of achievable rate are different from [27, 28, 29]. Compared to [30, 31], we focus on the derivations of achievable rate and the location optimization of the distributive BS antennas, while [30] faces the problem of interference control through the use of secondorder channel statistics and [31] works on the energy efficiency maximization problem. The major contributions of this paper are summarized as follows.

We provide new results for independent but nonidentically distributed (i.n.i.d) random vectors with very high dimension (see Lemma 1). Based on the results, an analytical expression for the asymptotic achievable rate of multiuser distributed massive MIMO systems is derived for arbitrary but known user location and antenna deployment (see Proposition 1).

Furthermore, for the circular antenna layout, tight closedform bounds on the average rate of the cell with uniform user location are obtained (see Theorem 3). These results can be used to predict the system performance and understand its behavior with respect to the number of antennas, the location of antenna, the cell size, and the transmit power.

Based on the acquired tight bounds, we derive the optimal radius of the distributed antennas for the maximum average achievable rate (see Lemma 2), which guides the fundamental and practical problem of antenna placement for distributed MIMO systems.

Finally, numerical results are provided to assess our analysis. It is shown that multiuser distributed massive MIMO is largely superior to centralized massive MIMO in achievable uplink rate.
The remaining of the paper is organized as follows. The system model and asymptotic achievable rate analysis of a general distributed massive MIMO system are presented in Section II. Asymptotic achievable rate analysis of the circularly distributed massive MIMO is present in Section III. Location optimization of the circularly distributed antennas is provided in Section IV. Simulation results are presented in Section V. Conclusions are drawn in Section VI. Involved proofs are included in the appendices.
Notation: Boldface lowercase letters denote vectors, while boldface uppercase letters denote matrices. We use , and to denote the transpose, conjugate and conjugate transpose of a matrix or a vector, respectively. For a matrix , is its trace. The symbol denotes the identity matrix, while denotes the matrix whose entries are zeros. The symbol denotes the statistical expectation operation. The symbol denotes Frobenius norm of a matrix or a vector. The function is the base2 logarithm and is the natural logarithm.
Ii System model and Asymptotic Achievable Rate Analysis
Iia MultiUser Distributed Massive MIMO System Model
We consider a singlecell multiuser distributed massive MIMO system. As shown in the left side of Figure 1, in this system, there is one BS equipped with antennas which are spatially distributed [30, 31]. The number of antennas is assumed to be large, e.g., a few hundreds. This is different to the multiuser centralized MIMO system [15, 16, 17, 18, 19, 21, 22, 23, 24, 20, 25, 26], where the BS antennas are centralized and spatially colocated (shown in the right side of Figure 1). Compared with centralized MIMO systems, distributed MIMO systems provide macrodiversity and have enhanced network coverage and capacity, due to their open and flexible infrastructure [6, 7, 27]. We assume that the distributed BS antennas are connected with high capacity backhaul and have ideal cooperation with each other. There are users, each equipped with single antenna. We assume that .
Denote the channel coefficient between the th antenna of the BS and the th user as . We consider both the pathloss and smallscale fading as follows [31]
(1) 
where is the smallscale fading coefficient, which is model as a random variables with zeromean and unitvariance. Without loss of generality, Rayleigh fading is adopted in the simulation, where follows circularly symmetric complex Gaussian (CSCG) distribution, i.e., . ’s are assumed to be mutually independent. models the pathloss. We assume that
(2) 
where is the distance between th antenna of the BS and the th user and is the pathloss exponent with typical values ranging from to , i.e., . Let
which is the channel vector between th user and all BS antennas. Let
which is the channel matrix between all users and all BS antennas. Perfect channel state information (CSI) is assumed at the BS, that is, the BS knows precisely.
The uplink communication is studied, where the users transmit their data in the same timefrequency resource to the BS. Let be the signal vector containing the user data, where its th entry is the information symbol of the th user. is normalized as . Let be the average transmit power of each user. This implies that in this work, we assume that all users have the same transmit power. But the derived results can be directly employed to nonequal power case. The vector of the received signals at the BS is
(3) 
where is the noise vector, whose entries are assumed to be independent and identically distributed (i.i.d.) CSCG random variables with zeromean and unitvariance, that is, .
ZF linear detector is used at the receiver, which has lowcomplexity and achieves comparable sumrate performance to other more complicated designs such as the minimummeansquareerror (MMSE) detector in massive MIMO systems [20, 21]. ZF separates data streams from different users by multiplying the received signal vector with . From (3), we have
(4) 
Focusing on the th element of , we have, from (4),
(5) 
where is the th column of the matrix . Since , the equivalent noise is a CSCG random variable with zero mean and variance , i.e., . There is no interference term in (5) due to the ZF matrix structure. It is noteworthy that ZF detector is chosen here for the simplicity of the presentation and our work can be straightforwardly extended to MMSE linear detector.
IiB Asymptotic Achievable Rate Analysis
In this subsection, we analyze the asymptotic achievable rate of the multiuser distributed massive MIMO uplink when , for a general BS antenna deployment and user location.
To assist the analysis, we first prove the following results for very long random vectors.
Lemma 1.
Let and be independent vectors whose elements are i.n.i.d. zeromean random variables. Assume that , and , for . We have, when ,
(6) 
and
(7) 
where denotes the almost sure convergence.
Proof.
See Appendix A. ∎
The results in Lemma 1 can be seen as generalizations of the results in Eqs. (4) and (5) in [20]. More specifically, the results in [20] are for very long random vectors with i.i.d. elements, while the results in Lemma 1 can be applied to very long random vectors with i.n.i.d. elements.
With the results in Lemma 1, we can obtain the following expression for the asymptotic achievable rate of distributed massive MIMO systems.
Proposition 1.
When , the average achievable rate of the th user in the multiuser distributed MIMO system has the following asymptotic behavior:
(8) 
Proof.
The result in Eq. (8) of Proposition 1 can be applied to massive MIMO systems with arbitrary antenna deployment and user location. Proposition 1 is also applicable to MMSE detector, where . With the MMSE detector, we can change Eqs. (4), (5), and (9) accordingly to include the interference. But due to the large number of antennas, the interference term will diminish and the same achievable rate result can be obtained.
The multiuser centralized MIMO system considered in [15, 21, 22, 20] can be seen as a special case of our multiuser distributed MIMO system with all the distributed antennas located in the same place, i.e., for the th user. Thus, from (8), we can obtain its achievable rate result as
(13) 
which is the same as Eq. (13) in [20].
Iii Asymptotic Analysis for the Achievable Rate of Circularly Distributed Antennas
Theoretically speaking, antennas in a distributed massive MIMO system can take arbitrary locations and topology. The optimization of the antenna locations can be highly challenging, if not intractable, due to the large number of antennas and design parameters. On the other hand, arbitrary antenna locations or optimal topology may have prohibitive backhaul cost and installation cost. In real applications, it is more practical to consider manageable antenna topology. In this work, we consider circularly located BS antennas, where all antennas are on a circle centered at the cell enter. Circular topology has ideal symmetry and low dimension (radius and angle). Compared with the line topology, it is expected to have superior performance due to better symmetry. Compared with the grid topology, it is expected to have lower implementation cost and more tractability in analysis. Circular antenna layout has been considered in the literature [11, 32, 33] and shown to have good performance.
In this section, for distributed massive MIMO systems with circular antenna layout, we first specify the system model, then analyze the asymptotic achievable rate for an arbitrary user, and finally derive the average sumrate per user assuming uniform user location.
The singlecell multiuser distributed massive MIMO system is shown in Figure 2. We consider a circular cell with radius . The center of the cell is denoted as . Circular cell is widely used [11, 32, 34] and has been shown to have similar performance to hexagonal cell. But it enjoys more tractable analysis. The distributed BS antennas are located evenly on a smaller circle with radius , whose center is the same as the center of the cell. We denote the location of the th BS antenna as . Thus, the length of the segment is . Notice that since the BS antennas are evenly located on the circle and the antenna number is assumed to be large, the angle dimension of the antenna location (for example, the angle of the segment and the horizontal axis) has little effect on the system performance and only the radius of the antenna circle matters. We denote the location of an arbitrarily user as . Let be the distance between the user and the cell center, i.e., the length of the segment .
Iiia Asymptotic Achievable Rate of an Arbitrary User for Circularly Distributed Massive MIMO
The following results on the asymptotic achievable rate of an arbitrary user at distance for the circularly distributed massive MIMO system are obtained.
Theorem 1.
For the circularly distributed massive MIMO system with antenna radius , when , the ergodic achievable rate of the user at distance from the cell center has the following asymptotic behavior:
(14) 
where
(15) 
with the Legendre function [37].
For , closedform expressions for the achievable rate can obtained as follows:
(16) 
Proof.
See Appendix B. ∎
In this theorem, to facilitate the presentation, we introduce a new notation, for the asymptotic ergodic achievable rate of a user when . Note that , where is the Gauss hypergeometric function [37]. Many software for scientific computations such as Matlab have this function. Thus the result in (14) and (15) of Theorem 1 can be easily calculated numerically. However, due to the special function, for , the achievable rate is not in closedform and little insight can be obtained on the performance behavior of the circularly distributed massive MIMO system with respect to the cell size and antenna location. Thus, in what follows, we derive bounds on the achievable rate in closedform.
Theorem 2.
Define
(17)  
(18) 
For the circularly distributed massive MIMO system with antenna radius , the asymptotic achievable rate of a user at distance from the cell center, denoted as , can be bounded as follows
(19) 
Proof.
See Appendix C. ∎
Theorem 2 provides both lower and upper bounds on the achievable rate. When , is a lower bound and is an upper bound; when , is an upper bound and is a lower bound. When , and are the same and equal the achievable rate of the user. Moreover, it is evident that the bounds in Theorem 2 are in closedform. We note that the coefficient in (18) only depends on , the pathloss exponent and can be easily calculated offline.
To justify the tightness of the two closedform bounds, we analyze their difference as follows:
where (a) is obtained by software calculations for . This shows that the two bounds are close to each other with less than 0.6 bits/s/Hz difference. The difference is negligible for massive MIMO systems as both bounds and increase in . Thus, either and can function as a tight closedform approximation of the achievable rate when the number of BS antennas is large. Our simulation results in Section V also justify the tightness of the bounds.
Since both bounds and increase in where is the transmit power and where is the number of BS antennas, the user achievable rate is proved to increase in and .
IiiB Asymptotic Average Achievable Rate of the Cell for Circularly Distributed Massive MIMO
In the previous subsection, we have analyzed the asymptotic rate of an arbitrarily located user in the cell. In this subsection, we derive the asymptotic average rate of a user in the cell, which indicates the average experience of user service. The users are assumed to be randomly and uniformly located [11, 36]. Thus, the probability distribution function of a user’s distance to the cell center, denoted as , is
(20) 
The angle of the user’s location (to the horizonal axis) is uniform distributed on . The following theorem on the asymptotic average rate of a user is proved.
Theorem 3.
Define
(21)  
(22) 
For the circularly distributed massive MIMO system with uniformly distributed users in the cell, the asymptotic average achievable rate per user of the cell, denoted as , can be bounded as follows:
(23) 
Proof.
With uniformly distributed user location and the probability density function of the user distance to the cell center in (20), the asymptotic average achievable rate per user of the cell can be calculated as:
(24) 
where is the asymptotic achievable rate for a user at distance . From (19) in Theorem 2, we have (23), where
From (17) in Theorem 2, we have
(25)  
(26)  
from which we can obtain (3) via simple rewriting.
It is evident that our derived bounds on the asymptotic average achievable rate in Theorem 3 are in closedform. Also, calculating the difference between the two bounds, we have
(28) 
Thus, either bound can be used as a tight approximation of with the error being less than 0.6 bits/s/Hz. The error is negligible when since increases in logarithm in . The tightness of the bounds will also be justified by our simulation results in Section V.
Iv Location Optimization of the Circular Antenna Array
In the previous section, the ergodic achievable rate of an arbitrarily located user and the average achievable rate per user of the cell for uniformly located users are derived when . We can see from the results that other than the transmit power and the number of the distributive antennas , the radius of the distributed antennas largely affects the average achievable rate. In this section, we turn to derive the optimal radius of the circularly distributed antenna array to maximize the average achievable rate of the cell, which is one of the most important measures of wireless system performance.
In Theorem 3, both the upper and lower bounds, and , are derived for the average achievable rate per user. The bounds are in closedform and shown to be close to each other. Thus, in the radius optimization, we aim at maximizing . The same result can be obtained if is used since the difference only depends on and is independent of , the radius of the circular antenna array.
Lemma 2.
The radius of the circular antenna array for the distributed massive MIMO system that maximizes is:
(29) 
where is the solution of the following equation:
(30) 
Proof.
The equation in (30) only depends on the pathloss exponent and can be easily solved offline by many softwares such as Matlab. Using the result in Lemma 2, the radius of the distributed circular BS antenna array can be designed for the maximum average achievable rate for the distributed massive MIMO system.
From (29) we have . Thus, the ratio of the antenna radius and the cell radius depends on the pathloss exponent only and is independent of the transmit power and the BS antennas size . This is very appealing in wireless network designs and implementation. Based on this fact, we further note that the improvement of the hardware in distributed MIMO, such as increasing the number of the distributed antennas, will not affect the optimal location of the distributive antennas.
V Numerical results
In this section, we present numerical results to show the performance of the distributed massive MIMO system with circular BS antenna array and justify the accuracy of our theoretical results. The impacts of different parameters, such as the number and the location of the distributive antennas, the transmit power, and the pathloss exponent, on the achievable rate are also investigated. We consider the uplink of a circular cell whose cell radius is set as meters. There is a massive BS with circularly distributed antennas located on a circle of radius . There are users. The users have the same transmit power, which is set to be where meters. So, if a user is located 500 meters away from a BS antenna, the average received SNR of the antenna from the user is . The normalization with in the transmit power does not affect the behavior of the simulation curves but only affects the position of the curves on the axis. The smallscale channel fading is generated as circularly symmetric complex Gaussian with zeromean and unitvariance, thus Rayleigh fading.
Va Achievable Rate of an Arbitrarily Located User
In Figure 3 and Figure 4, we show the simulated achievable rate of an arbitrarily user and compare with the derived asymptotic analytical results in (1415), as well as the closedform bounds in (17) and in (18). We set meters. Figure 3 shows the achievable rate for different user location (the distance of the user to the cell center) and pathloss exponent , while and dB. Figure 4 shows the achievable rate for different antenna number and user transmit power , while meters and [38].
We can see from the figures that the curves numerical calculated by (1415) accurately predict the simulated ones. The derived closedform bounds in (17) and (18) are very close to the simulated and numerically calculated achievable rates. The figures show that is a lower bound and is an upper bound when , while is an upper bound and is a lower bound when , which confirms the results of (19).
Figure 3 also shows that the achievable rate is higher for larger pathloss exponent and smaller distance between the user distance and the radius of the circular antenna array . For either or , the achievable rate is a concave function of . We can see from Figure 4 that the achievable rate increases with . For example, increasing from to brings an achievable rate increase of about at dB. For the dB case, increasing from to brings about an achievable rate increase of . The achievable rate also increases with , the user transmit power. For example, increasing from dB to dB results in the achievable rate increase of about at .
VB Average Achievable Rate of the Cell
In Figure 5 and Figure 6, we show the average achievable rate per user of the cell, and compare with the derived bounds in (3) and in (22). We set meters and assume a practical urban scenario with the pathloss exponent . The user location are randomly generated to be uniformly distributed in the cell.
It can be seen from both figures that the simulated achievable rate and the derived closedform bounds match well for all adopted values of and . The average achievable rate of the cell increases with , which indicates that increasing the number of the BS antennas improves the system throughput. For example, the scenario achieves about higher average rate than the scenario at dB. The average achievable rate also increases with . For example, increasing from dB to dB brings an achievable rate advantage of about at .
VC Impact of the Location of the Circularly Antenna Array
Next, we show the impart of the radius of circular antenna array, , on the average achievable rate of the cell. Notice that the case corresponds to centralized massive MIMO system, where the BS antennas are located at the center of the cell.
Figure 7 plots the simulated average rate of the cell and the derived bounds as functions of for three cases: 1) dB, 2) dB, and 3) dB. We set . The figure shows that the radius of the distributed antenna array has significant influence on the average rate of the cell and proper antenna location results in significant improvement in average rate to the centralized case. For example, increasing from to meters boosts up the average rate by about when dB. The figure also indicates that the optimal for different and remains the same, which is about meters. This conforms with our result in Lemma 2 that the optimal is irrelevant to the values of and but only depends on .
To further understand the optimal radius of the circular antenna array, employing (29) in Lemma 2, we plot , the ratio of the optimal antenna array radius to the cell radius, for different pathloss exponent, in Figure 8. It can be seen that is bigger for larger . For example, when , is , while it is when . Thus, as the pathloss is larger, antennas should be installed further away from the cell center for the maximum average rate. For a given value, the radius of the circular antennas array should increases linearly in the cell radius . We can also see that for any , . This shows that the optimal value is far away from the centralized massive MIMO case, where . On the other hand, for different values within the practical range (), has small change. Actually, for any , setting the radius of the circular antenna array as will induce less than loss in the average rate compared to the optimal radius. This result is useful in further simplifying the practical system design of circularly distributed massive MIMO systems.
Vi Conclusions
In this paper, we have considered the uplink of a singlecell multiuser distributed massive MIMO system, where the BS equipped with a large number of distributed antennas receiving information from multiple users equipped with single antenna. Zeroforcing detection is used at the BS. In order to analyze the achievable rate of the system, we provided new results for very long random vectors with independent but nonidentically distributed entries. Based on the results, for circularly distributed base station antennas, we derived analytical expressions of the achievable rate of an arbitrarily located user and two closedform expressions that bound the rate from both sides. The tightness of the bounds were rigorously justified. From these results, behavior of the system achievable rate with respect to different parameters such as the size of the antenna array, the location of the antenna array, the pathloss exponent, and the transmit power can be understood. We also derived tight closedform bounds for the average achievable rate per user assuming that users are randomly located in the cell, from which the optimal radius of the distributed antenna array that maximizes the average rate was derived. Numerical results were illustrated to justify our analytical results. Our work has shown that multiuser distributed massive MIMO largely outperforms centralized massive MIMO. Our derived results can assist infrastructure providers in solving the fundamental problems of performance measurement and antennas placement for distributed massive MIMO systems in practice.
Appendix A Proof of Lemma 1
Let . Thus ’s are independent and there exists a finite positive constant such that
(33) 
The variance of the arithmetic mean of satisfies the following:
(34) 
Appendix B Proof of Theorem 3
As shown in Figure 2, we use for the cell center. To help the derivation, we denote the angle of the segments (where is the location of the th BS antenna) and (where is the location of the user) as . The distance between and , denoted as , can be expressed as
(38) 
Without loss of generality, we assume that and the evenly circularly distributed BS antennas are labeled such that for and for . Let .
(39)  
(40)  
(41) 
Employing [39, 2.5.16.38], we have, from (41),
(42) 
By using (42) in (8), Eq. (14) can be proved. Eq. (16) can be subsequently obtained by using Eq. [37, 8.912].
To further illuminate the asymptotic result in (41), we investigate the difference between (41) and (40). To help the presentation, we use the following notation:
For , we can obtain via straightforward calculations that
(43) 
This shows that decreases with when . For the simplicity of presentation, we assume that is even. The proof for odd is similar. The difference between (41) and (40) can be bounded as follows:
(44)  
(45)  
(46)  
(47)  
(48) 
In obtaining (45), we use the symmetry in . This analysis shows that the difference between (41) and (40) is linear in . For large but finite number of antennas, (41) is a tight approximation of (40).
Appendix C Proof of Theorem 4
Define . Notice that