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On the downlink throughput capacity of hybrid wireless networks with massive MIMO
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 110 (2018)
Abstract
In this paper, the entire network model is a hybrid wireless network model in which each base station is connected to each other via a wired link. On this basis, we place a large number of antennas (massive MIMO) at each base station to serve a single terminal in the downlink scenario, and we research the outage capacity and ergodic capacity in this scenario. The result of this paper is that the expressions for ergodic throughput capacity, outage probability, and outage throughput capacity have been derived under the favorable propagation condition. Through simulation, we can see the trend of outage throughput capacity and ergodic throughput capacity.
Introduction
Because of the decentralized nature of ad hoc networks, nodes can join the network flexibly, but the problem with ad hoc networks have become apparent over long distances transmission due to lack of the infrastructure support. In this regard, in order to better combine both ad hoc and infrastructure advantages, hybrid wireless networks came into being [1]. In [2], Gupta and Kumar initiated the research on scaling law of large selforganizing networks and proposed the concept of scaling law. This research also promoted the research on the throughput capacity of hybrid wireless networks. In [3], T. L. Marzetta initiated the research of massive MIMO and found that using a wireless antenna system at the base station can completely eliminate the effects of uncorrelated receiver noise and fast fading, and the transmission of terminals within its own cell is not affected. In [4], the scaling law of throughput capacity of hybrid wireless network based on Nakagamim fading channel is studied. In the uplink transmission phase, an optimal multiple access technique allows opportunistic sources to transmit concurrently with the scheduled source, and successive interference cancellation (SIC) strategy is then applied at the receiver to limit the intracell interference. However, in the downlink transmission phase, neither of the two technologies in the uplink transmission phase can play a role, only one antenna is located at the base station and the receiving terminal respectively. In order to break the bottleneck of throughput capacity in the downlink transmission scenario in [4], the authors placed a pointtopoint MIMO system on the base station [5]. In [6], the author analyzed the uplink outage throughput capacity of hybrid wireless networks with massive MIMO under the favorable propagation condition, where the K terminals wrap around the base station at the same distance. According to [5, 6], Massive MIMO can fully address the shortcomings of pointtopoint MIMO, and under the favorable propagation condition, massive MIMO greatly mitigates the small scalefading effect. So we think of throughput capacity change when the base station is equipped with a large number of antennas (massive MIMO), downlink services to a single terminal under the favorable propagation condition.
The rest of the paper is arranged as follows. In Section 2, we introduce the established hybrid wireless network model. In Section 3, we get the approximate rate of downlink transmission with massive MIMO under the favorable propagation condition. In Section 4 and Section 5, we obtain the expressions of outage probability, outage capacity, and ergodic capacity, respectively. In Section 6, the whole paper is summarized.
Hybrid wireless network model
In order to facilitate the following analysis, we introduce the hybrid wireless network model as follows:

1.
The entire network consists of b base stations and n nodes, each cell is assigned a base station, and each base station is placed in the center of the cell; n nodes are evenly distributed in b cells; the distance between parallel edges of each cell is \( c=\sqrt{\frac{2\sqrt{3}}{3}\frac{n}{b}} \).

2.
The paper [7] shows that in order to obtain the gains provided by the infrastructure, \( b=O\left(\frac{n}{\log n}\right) \) is the most basic condition.

3.
We assume that the number of nodes in each cell is limited to K, and the proof method of K in the square model is provided in [8]; by the approximation of this method, we can also prove that the number of nodes in the hexagon model cell is \( K=\varTheta \left(\frac{n}{b}\right) \).

4.
We analyze the throughput capacity in the downlink transmission scenario; a large number of antennas (massive MIMO) at the base station send data to a single receiving terminal in the cell at the same time.
The hybrid wireless network model is shown in Fig. 1.
Downlink asymptotic rate with massive MIMO
In [9], when the number of antennas M on the base station is much larger than the number of receiving terminals K, we call such a transmission mode the most favorable propagation condition. Under the most favorable propagation condition, the column vectors of the propagation matrix are asymptotically orthogonal,
where G represents the M × K propagation matrix, which consists of a smallscale fading matrix H and a largescale fading diagonal matrix \( {D}_{\beta}^{\frac{1}{2}} \) (the H in the upper right corner represents the conjugate transpose).
In the downlink transmission, the propagation matrix G^{T} is the transpose of the uplink propagation matrix G, and the downlink model can be expressed as
where s_{ d } is an M × 1 vector transmitted by the M antennas, x_{ d } is a K × 1 vector received by K terminals, N_{ d } is the noise vector whose components are independent and distributed with N_{ d }~CN(0, 1), and ρ_{ d } is proportional to SNR. The total power in the downlink transmission is independent of the number of antennas; to facilitate the analysis, we assume the total power is one
When both the base station and the receiving terminals have known the information of channel state, the base station serves K terminals in the downlink transmission. In order to obtain the sum capacity, we need to make a constrained optimization first,
where D_{ p } is a diagonal matrix whose diagonal elements constitute a K × 1 vector p. The power limit is given by Eq. (5), and the sum power is limited to 1. In the case of satisfying the best power of each terminal, we can get the sum rate under the favorable propagation condition as
where p_{ k } represents the optimal power allocated for each node after power constraint optimization, andβ_{ k } represents the largescale fading coefficient. According to [10], we can define β_{ k } with the following equation
where φ is a constant related to the antenna gain and carrier frequency, d_{ k } is the distance between the base station and k^{th} terminal, α is the path loss exponent, and ζ_{ k } is the lognormal shadowing with \( 10{\log}_{10}{\zeta}_k\sim N\left(0,{\sigma}_k^2\right) \).
According to the above analysis, in order to achieve the purpose of this paper, we can get the throughput capacity expression by combining Eq. (7) and Eq. (8). So, the throughput capacity expression is
Outage probability and outage capacity
The outage capacity is the maximum transmission rate that the system can achieve for a given outage probability [11]. In [4], in the downlink transmission phase, neither the optimal multiple access technique nor successive interference cancellation (SIC) strategy is allowed, each node outage capacity over Nakagamim fading channel scales as \( \varTheta \left(\frac{b}{n}\log \left({\varepsilon}^{\frac{1}{m}}\frac{n}{b}\right){W}_2\right) \) under infrastructure mode. Applying massive MIMO to hybrid wireless networks, the channel capacity under infrastructure mode will be significantly different from that in [4, 5].
According to our above analysis, we can define the outage probability as
In the case of low SNR, we use the approximation ln(1 + x) ≈ x to calculate the outage probability as follows
We define a new variable T, where the variable T = d^{−α}ζ
The shadow fading ζ satisfies the lognormal distribution with 10log_{10}ζ~N(0, σ^{2}), so the variable T we defined is also satisfied with the lognormal distribution with 10log_{10}T~N(μ, σ^{2}) and μ = 10log_{10}d^{−α}. According to the lognormal probability density function (pdf), we can get the outage probability as
According to the nature of the lognormal distribution, the outage probability is represented by the complementary error function as
In [12], we can get a tight exponential upper bound of the complementary error function and a pure exponential approximation for the complementary error function, as follows
The precondition for obtaining the outage capacity is P_{out} = ε. In the approximation by Eq. (16), we create a new variable S^{'} and \( {S}^{\hbox{'}}={e}^{\left(\frac{10\log_{10}S\mu }{\sigma \sqrt{2}}\right)2} \). Substituting S^{'} to Eq. (17), the upper bound of the outage probability we can get is
When \( \varepsilon <\frac{1}{2} \), Eq. (19) is satisfied. Based on the above analysis, we can get the outage throughput capacity expression at low SNR as follows
Since the scalar ρ_{ d } is proportional to SNR, in the following simulation, we consider the change of outage capacity with ρ_{ d }.
Simulation results are shown in Fig. 2.
In the case of high SNR, we use the approximation log(1 + x) = log(x) to calculate the outage probability as follows
Here, we define a new variable X and X = log_{2}(Mρ_{ d }φd^{−α}ζ). The variable ζ obeys the lognormal distribution with 10log_{10}ζ~N(0, σ^{2}). By changing the formula for logarithms and a series of calculations, we find that the new variable X is a random variable that satisfies the normal distribution with \( X\sim N\left({\mu}_X,{\sigma}_X^2\right) \). The values of μ_{ X } and σ_{ X } are respectively as follows
Similarly, in the way we analyzed before, we can get the following equations of outage probability
Equation (25) is satisfied when \( \varepsilon <\frac{1}{2} \), so we can get the outage throughput capacity expression at high SNR:
By combining Eq. (22), Eq. (23), and Eq. (26), we can get
Simulation results are shown in Fig. 3.
Ergodic capacity
Ergodic capacity refers to the time average of the maximum information rate of random channels in all fading states [13]. In [14], in the downlink transmission phase, the successive interference cancellation strategy (SIC) is not feasible, the pernode ergodic throughput capacity over Nakagamim fading channel at low SINR scales as Θ(W_{2}) under the infrastructure transmission mode, and the pernode ergodic throughput capacity at high SINR is \( \Theta \left(\frac{b}{n}\log \left(\frac{1}{m}\cdot \frac{n}{b}\right){W}_2\right) \) (W_{2} is the bandwidth allocated for the downlink transmission; m is the shape parameter). In [15], the change of ergodic throughput capacity under the infrastructure transmission mode with distributed base stations (DBS) is analyzed. The ergodic capacity gain of the hybrid wireless network with distributed base stations is N × N_{ BS } that compared to the traditional hybrid wireless network (N is the number of distributed base stations; N_{BS} is the number of antennas placed at the base station). Based on the previous analysis, what will happen to the ergodic throughput capacity under the infrastructure transmission mode when a large number of antennas (massive MIMO) are placed at the base station?
According to the above Eq. (9), we define \( \overline{C} \) on behalf of the ergodic capacity; the expression of the downlink ergodic capacity is as follows
Similarly, in the previous analysis, at low SNR, we approximate ln(1 + x) ≈ x, then we can get the expression of the ergodic capacity as
where W is the same as the variable T defined in Eq. (12) above. According to the means of averaging, we can get
By transforming \( \frac{10\lg W\mu }{\sigma } \) to Z, making integral to the expression again, we can get
Combining Eq. (29) and Eq. (31), the ergodic throughput capacity at low SNR is
Simulation results are shown in Fig. 4.
At high SNR, using the approximation log(1 + x) = log(x), we can get the following ergodic throughput capacity expression
As with the method of analyzing the outage capacity at high SNR, we also define log_{2}(Mρ_{ d }φd^{−α}ζ) as the variable X above, we can get
Simulation results are shown in Fig. 5.
Conclusions
In this paper, we mainly analyzed the downlink transmission throughput capacity of the hybrid wireless network with massive MIMO under the favorable propagation condition. Under the favorable propagation condition, the column vectors of the propagation matrix are asymptotically orthogonal, and largescale fading became the dominant factor. From the above analysis, we derived the closed expressions of the outage throughput capacity and the ergodic throughput capacity (low SNR and high SNR) in the downlink transmission scenario when the base station is placed with massive MIMO. With the help of simulation, we can find that when the number of antennas M at the base station increase, the outage throughput capacity and the ergodic throughput capacity also increase, but the extent of growth is different.
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Funding
This paper is supported by the Natural Science Foundation of China (61271411) and Natural Youth Science Foundation of China (61501326, 61401310). It is also supported by the Tianjin Research Program of Application Foundation and Advanced Technology (15JCZDJC31500) and Tianjin Science Foundation (16JCYBJC16500).
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BZ gives the overall research direction and ideas. YT read the relevant literature and books and drafts the article. WW makes the corresponding experimental simulation. All authors read and approved the final manuscript.
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Baoju Zhang is a professor at College of Electronic and Communication Engineering of Tianjin Normal University. She received the B.S. degree from Tianjin Normal University in 1990, M.S. degree from Tianjin Normal University in 1993, and Ph.D. degree from Tianjin University in 2002. Her main research directions involve compressive sensing, audio and video processing, data stream clustering, and so on.
Yipeng Tian is working on his M.S. degree in Electronics and Communication Engineering at Institute of Tianjin Normal University; he received his B.S. degree in Tianjin Normal University. He is studying hybrid wireless network and massive MIMO capacity.
Wei Wang received his Ph.D. degree in Tianjin University; currently, he is an Associate Professor in College of Electronic and Communication Engineering, Tianjin Normal University, Tianjin, China. His main research directions involve compressive sensing, radar signal processing, optical fiber sensing technology and applications, and so on.
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The authors declare that they have no competing interests.
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Zhang, B., Tian, Y. & Wang, W. On the downlink throughput capacity of hybrid wireless networks with massive MIMO. J Wireless Com Network 2018, 110 (2018). https://doi.org/10.1186/s1363801811341
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Keywords
 Hybrid wireless network
 Massive MIMO
 Outage capacity
 Ergodic capacity
 Favorable propagation condition