Multicell Massive MIMO Systems
6.2 Average Achievable Rate
In this section, we propose a tractable approach to analyze the average achievable rate of the two-tier HetNet with massive MIMO and non-uniformly small-cell deployment. We define Xm, Zc = (X, Y ), and Ys as the corresponding distances between the typical user and its
Chapter 6: Massive MIMO Heterogeneous Networks
respectively. The probability density functions (PDFs) of Xm, Zc, and Ys are defined as [127, 128]
When massive MIMO is adopted at the MBS, by using the concept of stochastic geometry, a conventional method for evaluating the achievable rate on the macro-cell is to use the moment generating functions (MGFs) [144]. However, this technique requires a high complexity which consists of several integrations [138, 144, 145]. To simplify the analysis, a lower bound on the achievable rate has been recently derived in [69] and [139]. On the other hand, we present a tractable approximation of the achievable rate according to Lemma 1 of [146]. In particular, if both Ω = Pq
i=1Ωi and Φ =Pp
j=1Φj are sums of non-negative random variables (RVs), then we have [146, Lemma 1]
Note that (6.15) does not require both Ω and Φ to be independent of each other. This approx-imation is appropriate to massive MIMO systems because it converges to the exact one when q and p are large. Based on the random theory matrix, this method has been adopted to analyze the achievable rates for linear massive MIMO precoders in the presence of phase noise [147], multi-cell massive MIMO systems with downlink training and pilot contamination precoding
Chapter 6: Massive MIMO Heterogeneous Networks
scheme [148], and massive MIMO systems with low-resolution analog-digital converter [149].
Herein, we apply it in a different network topology. Based on concept from stochastic geometry, the achievable rates for the three association cases are shown in the following lemmas.
Lemma 1 When the typical user is served by MBS0, an approximation of the achievable rate is defined as holds for Imin the denominator of (6.4). Hence, by using (6.15), we approximate the achievable rate of the typical user as [147, Eq. (21)], [148, Eq. (31)], [149, Eq. (14)] this case, the interference of MBSs Im→m comes from all the MBSs which are located outside the circle B(0, x). It is given by
Chapter 6: Massive MIMO Heterogeneous Networks
where (a1) is obtained due to ho,l ∼ Γ(K, 1), 1(.) is the indicator function and (a2) is defined by applying Campbell’s Formula with polar coordinates [66]. Moreover, the SBSs are distributed according to homogeneous PPP with the density λ2, but we only use the SBSs in the outer region. Hence, the small cell tier process Φs becomes a Poisson Hole Process (PHP) with the density λPHP = λ2exp(−πλ1R2) which makes performance analysis more challenging. In the literature, there are four main approaches to characterize the interference of PHP model [151, 152]. The first approach is to ignore the holes and approximate the PHP by the baseline PPP with the density λ2. The second approach is to approximate the PHP by a PPP with the density λPHP. Note that that the PPP in the second approach is independently thinned due to λPHP < λ2 which distorts the local neighbouring SBSs around the typical node. On the other hand, the PPP in the first approach still preserves the local neighbouring SBSs in the outer region. Therefore, it has been shown that the first approach can provide a tighter approximation of the interference [151]. Furthermore, the third approach is to approximate the PHP by a Poisson Cluster Process (PCP) (such as Thomas or Mat´ern Cluster Processes) based on matching the first and second order statistics [152]. The difference between the PHP and the PCP is the higher-order statistics. As compared with these three approaches, the fourth one is the most fitting-based approach, where the holes are dissolved in such a way that the PHP is reduced to an equivalent non-homogeneous PHP. Thus, the upper and lower bounds on the PHP interference can be analyzed [151]. However, the third and fourth approaches require high computational complexities. To achieve the tractable analysis and reasonable performance, we use the first approach to characterize the interference of SBSs in this paper. Specifically, we assume that the small cell interference Is→m approximately comes from whole region outside the circle B(0, R), thus we can obtain [127, Eq. (34)], [128, Eq. (18)]
Chapter 6: Massive MIMO Heterogeneous Networks
Similar assumption was applied and accurately proved in [127] and [128], in which the coverage probabilities of the conventional MIMO HetNets with the non-uniform small cell deployment were analyzed. Moreover, since ho,0 = |hHo,x0wo,x0|2 ∼ Γ(N − K + 1, 1), we have E{ho,0} = N − K + 1. By substituting (6.12), (6.18) and (6.19) into (6.17), we can obtain (6.16).
Note that the expression (6.16) is simpler to implement than the achievable rate [153, Eq. (26)], which is defined by using the complicated F`aa di Bruno’s formula. The tightness of (6.16) will be shown in the Numerical Results and Discussions Section.
Lemma 2 When the typical user is served by both MBS0 and SBS0, an approximation of the achievable rate is defined as
Proof: Similar to the proof of Lemma 1, when BS = {MBS0, SBS0}, the interference of MBSs Im→m comes from the outside circles B(0, x), while the interference of SBSs Is→s
approximately comes from the whole region outside the circle B(0, y) [127, 128]. Hence, an expected value of Ic in the denominator of (6.7) is approximated as
E{Ic} = En X
Chapter 6: Massive MIMO Heterogeneous Networks
while an expected value of the nominator of (6.7) with respect to the fading channel power gain is defined as Ex,y|q
Pmx−α0 /KhHo,x0wo,x0+ q
P2y0−αho,y0|2 = P1x−α+ P2y−α. By combining these results with (6.13) and (6.15), we can define (6.20).
Lemma 3 When the typical user is served by SBS0, the achievable rate is given by
τs= 1
Proof: We do not use (6.15) to approximate the achievable rate of the typical user because it cannot perform well in this case. Hence, we define the achievable rate as
τs= E{log2(1 + γs)} = 1
Chapter 6: Massive MIMO Heterogeneous Networks
where (c1) is obtained based on the fact that the channel power levels and the BS locations are independent, (c2) is defined by applying the probability generating functional (PGFL) of PPP [126], (c3) results from the Laplace transform of ho,l ∼ (K, 1), xmin is the minimum distance between the interfering MBS and the typical user, (c4) results from applying Binomial expansion, and (c5) is obtained by using Eq. (3.194.1) of [90], where2F1(., ., ., .) is the Gaussian hypergeometric function [90, Eq. (9.142)]. Moreover, we define ϕ = (et− 1)yα. Similarly, it has been shown in [127] that the interference of SBSs approximately comes from all the SBSs outside the circle B(0, y). We can approximate the Laplace transform (6.25) as [127, Eq.
Chapter 6: Massive MIMO Heterogeneous Networks
where (e1) is obtained due to g ∼ exp(1), ymin is the minimum distance between the interfering SBS j ∈ Φs\SBS0 and the typical user, (e2) results from adopting the approach provided in [129], (e3) results from replacing ymin= y and ϕ = (et− 1)yα. In addition, if y ≤ R(µPP2
1)1/α, the interference of MBSs comes from all the MBSs outside the circle B(0, R), and if y >
R(µPP2
1)1/α, the macro interference is from all MBSs outside the circle B(0, y(µPP1
2 )1/α) [127,128].
Substituting (6.25), (6.26) and (6.27) into (6.24), we obtain (6.22).
When α = 4, due to R∞ a
1
1+v2dv = π2 − arctan(a), (6.27) can be expressed through elementary functions such as Ω(t, y) = exp(−πλ2y2√
et− 1(π2−arctan(√1
et−1))). Similarly, when K = 1 and α = 4, (6.23) is simplified as Ψ (β, t, y) = exp(−(et−1)y4σ2/P2) exp(−πλ1y2p(et− 1)Pm/P2(π2− arctan(β2y−2/p(et− 1)Pm/P2))). These results help to simplify the expression (6.22) for this specific case.
By using the law of total expectation and the three Lemmas 1 to 3, an average achievable rate of the typical user in the two-tier HetNet with massive MIMO can be approximated as
C ' ∆mτm+ ∆cτc+ ∆sτs. (6.28)
Chapter 6: Massive MIMO Heterogeneous Networks
Note that (6.28) can be efficiently computed by numerical methods. Thus, obtaining analytical results for studying network performance is more effective than using Monte Carlo methods which rely on a large number of realizations to define their results.
Remark 2: As observed from Lemmas 1 to 3, adding more antennas at the MBS has no effect on the existing interference environment, but it is beneficial at the achievable rate.