Precoding Scheme in Single-Cell Systems
4.2 Achievable Ergodic Rate
In the following, we study the achievable ergodic rates of the SLNR-PS for both cases of perfect and imperfect CSI, respectively, when N is large. We also analyze the asymptotic analysis of the achievable rates at very high SNR region. In particular, the achievable rate of the kth user is defined as [18, 94]
Rk= E n
log2(1 + γk) o
≈ En
log2(1 + λmax(Bk)) o
. (4.9)
Chapter 4: Massive MIMO with SLNR Precoding Scheme in Single-Cell Systems 4.2.1 Imperfect CSI Case
In the reality, the BS can estimate the channel state information imperfectly, then the achievable ergodic rate of this case is presented in the following theorem.
Theorem 3 If the SLNR-PS is applied, under the imperfect CSI condition, the UB on achiev-able ergodic rate of the kth user is given by
Rˆipk = log2
1 + v +σg2ˆ ρk
(N − K + 1)
, (4.10)
where v = N −K+1K−1 .
Proof: We study the achievable rate based on (4.9). In order to obtain the largest eigenvalue λk of Bk, we employ Weyl’s inequality in matrix theory. In particular, if C, D ∈ Mn are Hermitian matrices and the eigenvalues λi(C), λi(D), and λi(C + D) are arranged in increasing order λmin= λ1 ≤ · · · ≤ λn= λmax, for each l = 1, . . . , n we have [106, Eq. (4.3.2)]
λl(C) + λmin(D) ≤ λl(C + D) ≤ λl(C) + λmax(D). (4.11)
Based on (4.11), from (4.7), we have
γk≈ λmax(Bk) ≤ λmax(Ck) + λmax(Dk), (4.12) Ck = ˆG¯kGˆ¯Hk + ρkI
−1
gˆkgˆHk, (4.13) Dk = ˆG¯kGˆ¯Hk + ρkI
−1
σe2I, (4.14)
where λmax(Ck) and λmax(Dk) are the corresponding maximum eigenvalues of Ck and Dk, respectively. According to [94], λmax(Ck) can be obtained by solving the following characteristic
Chapter 4: Massive MIMO with SLNR Precoding Scheme in Single-Cell Systems By applying the matrix determinant lemma, i.e.,
det
after some mathematical manipulations, λmax(Ck) is obtained as [94]
λmax(Ck) = 1
1 − ˆgHk( ˆGkGˆHk + ρkI)ˆgk − 1
= 1/ρk
[( ˆGHkGˆk+ ρkI)−1]k,k − 1. (4.18)
Moreover, based on the approaches in [94] and [107], (4.18) can also be decomposed as
λmax(Ck) = 1 It has been shown that θad,k is a nondecreasing function of 1/ρk; θzf,k and θad,k are statistically independent [107]. The expected value of θzf,k is given by
E{θzf,k}(a= E1)
Chapter 4: Massive MIMO with SLNR Precoding Scheme in Single-Cell Systems where (a1) is obtained based on the result presented in [107] and (a2) is obtained due to3 1/[(HHkHk)−1]kk ∼ χ22(N −K+1), where Hk = [h1, . . . , hK] [13, 107]. The challenge in deriving
When N is very large, the eigenvalues of ˆG¯HkGˆ¯k converge to a fixed deterministic distribution which is given by the Marchenko-Pastur distribution [6]. We define x = K−1N . Based on the results presented in [109] and [110], as N → ∞, the deterministic approximation of the smallest
3A Chi-square random Z variable, denoted as χ22n, has probability density function fZ(z) = Γ(n)1 zn−1e−z, z ≥ 0 [107].
Chapter 4: Massive MIMO with SLNR Precoding Scheme in Single-Cell Systems
Substituting (4.24) into (4.23), as N is very large, we obtain
λmax(Dk) ≤ σe2 (1 −√
x)2σg2ˆN + ρk. (4.25)
Applying Jensen’s inequality, i.e., E{log2(1 + ζ)} ≤ log2(1 + E{ζ}), and using results in (4.20), (4.22) and (4.25), for the case of imperfect CSI, the UB on rate of the kth user is defined as
Rˆipk = log2
The tightness of (4.26) is presented in Figs. 4.3 and 4.4 of the Numerical Results and Discussion Section. It is shown that the expression (4.26) performs as well as the expression (4.10).
The analytical results obtained by using (4.10) and (4.26) match well with the corresponding simulation results when the number of BS antennas is large.
Proposition 1 When both the number of users K and the number of BS antennas N grow large while their ratio remains bounded, i.e., N, K → ∞ and NK = c0 > 1, the UB on achievable rate of the kth user is defined as
Rˆipk = log2
1 + 1
c0− 1+ ρdσ2e
1 + ρdσˆg2(c0− 1)
. (4.27)
Chapter 4: Massive MIMO with SLNR Precoding Scheme in Single-Cell Systems Proof: Let us define ρd= σP2
n as the transmit SNR in the downlink channel. From (4.10), we can obtain (4.27).
If c0 1, the result in (4.27) becomes the exact one which is defined by employing ZF-PS in [6, Table I]. It implies that the achievable ergodic rate of SLNR-PS is lower bounded by that of ZF-PS. In addition, at very high SNR region, i.e., log2(1 + ζ) ≈ log2(ζ), from (4.10), the asymptotic analysis of achievable rate for the kth user is given by
ρdlim→∞
When the BS can estimate CSI perfectly, we derive the UB and lower bound (LB) on the achievable rate as the following.
Theorem 4 If the SLNR-PS is applied, under the perfect CSI condition, the UB on achievable ergodic rate of the kth user is given by
Rˆkp = log2
By applying the similar approach, which is used to obtain λmax(Ck) in the previous Section, the largest eigenvalue λk of Bk can be defined as
λpmax(Bk) = 1 1 − gHk(GkGHk + σp2n
kI)−1gk
− 1. (4.31)
Chapter 4: Massive MIMO with SLNR Precoding Scheme in Single-Cell Systems Based on results in (4.19), (4.20) and (4.22), the expected value of λmax(Bk) is given by
E{λpmax(Bk)} = v + pk
σn2βk(N − K + 1). (4.32)
By substituting above result into (4.9), we can obtain (4.29)
We observe that ˆRpk is monotonically increasing with the transmit power of user k, hence Rˆkp −→ ∞ as pk−→ ∞. This result is different from that of the case with imperfect CSI, where the achievable rate of the kth user converges to an asymptotic limit at high SNR region. In addition, under assumption of EPA, as N, K → ∞, but KN = c0< ∞ and c0 1, then the UB expression in (4.29) converges to
Rˆkp → log2(1 + ρdβk(c0− 1)). (4.33)
This result is similar to the one which is obtained by using ZF-PS in [6]. Therefore, the rate of SLNR-PS converges to that of ZF-PS when SNR is high.
Theorem 5 If a SLNR-PS is applied, under the perfect CSI condition, the lower bound on achievable ergodic rate of the kth user can be approximated as
Rˇpk= log2(1 + (ψk− 1)ϑk), (4.34) θ are defined by solving the following equations
Chapter 4: Massive MIMO with SLNR Precoding Scheme in Single-Cell Systems where φk= N βipσk2
n(1 − K−1N +K−1N µ) + 1 and mk= σpk2
nβiµ(K − 1) + 1, respectively.
Proof: We observe that (4.31) can be decomposed as
λpmax(Bk) = 1
By applying Jensen’s inequality and the convexity of log2 1 + 1ζ, the LB on the achievable rate of the kth user is defined as
Rk = En
Following [111], the probability density function of λpmax(Bk) is approximated by a Gamma function, i.e., λpmax(Bk) ∼ Γ(ψk, ϑk), where shape parameter ψk and scale parameter ϑk are defined in (4.34). Based on that, we can obtain (4.34).
Note that it is complicated to derive the closed-form LB on achievable ergodic rate of massive MIMO systems with SLNR-PS and imperfect CSI, hence it will be investigated by Monte-Carlo simulations in Section 4.6. Moreover, for the case of perfect CSI, it is more convenient to obtain (4.29) than (4.34) due to its low complexity. Therefore, the closed-form UBs in (4.10) for the case of imperfect CSI and in (4.29) for the case of perfect CSI are used for further analysis.