The WSMSE algorithm: A Large System Analysis
2.4 The MIMO single stream case
In the section, we will switch to the MIMO single stream case and we will show that the SINR in that case is amenable as well to large system analysis. We provide a large system analysis of the performance of the WSMSE algorithm for MIMO IBC single stream scenario. The received signalyc,k at thekth user
in cellcreads XHi,c,k is anN×M matrix with i.i.d. complex entries of zero mean and variance
1
N M and the Θ1/2i,c,k and Θ1/2r,i,c,k are the Hermitian square-root of Θi,c,k and Θr,i,c,k respectively. Treating interference as noise, user k of cellcwill apply a linear receive filterfc,k of dimensions N ×1 to maximize the signal power (diversity) while reducing any residual interference that would not have been (sufficiently) suppressed by the precoder. The achievable rate of thekth user
of cell c is given by
The WSMSE solution for the MIMO single stream case is as follows:
fc,k=gHc,kHHc,c,k(σ2IN+ Performance analysis is conducted for the proposed precoder. A deterministic equivalent of the SINR is provided in the following theorem.
Theorem 2.3: Letγc,k be the SINR of thekth user of cell c with the precoder defined in (2.46). Then, a deterministic equivalentγ(j)c,k at iteration j >0 is given by
γ(j)c,k = w(j)c,k(m(j)c,k)2 Υ(j)c,k+ ˆΥ(j)c,k+d(j)c,kΨρ(j)c
c (1 +m(j)c,k)2
(2.47)
where
whereac,k denotes the module of the linear receive filterfc,k. Denoting Vc = (Tc+αcIM)−1 (2.56) with α(j)c = trD
(j) c
M ρc , three systems of coupled equations have to be solved. First, we need to introduce em,c,k∀{m, c, k} ∈ {C,C,Kc}, where Kc is the set of all users of cellc, which form the unique positive solutions of
em,c,k = 1
ec,c,k and mc,c,k denote ec,k and mc,k respectively.
Secondly, we givee′1,1, ..., e′1,K, ...e′C,1, ..., e′C,K which form the unique positive
solutions of end of iteration j is given by
g(j)c,k= ξ(j)c
M (Γ(j)c )−1HHc,c,ka(j)c,kf(j)0 w(j)c,k (2.63) for each user k in the cell c,
wheref0,c,k is the the normalized linear receive filter such thatfc,k=ac,kf0,c,k, andξ(j)c is given by
and , i.e., using the same logic and mathematical approach, but for a more complex problem. We will show that in the following.
Proof: We writefc,k as fc,k = ac,kf0,c,k with ac,k = q
fHc,kfc,k and |f0,c,k| = 1.
LetPc,k(j) =|fH,(j)0,c,kHc,c,kg(j)c,k|2 =|Hc,c,kg(j)c,k|2. We have signal power which motivates our choice for the channel correlation matrix at the receiver side as an identity matrix for the rest of the proof. Then,
Ψ(j)c = 1
fM Fc,k =gHc,kHHc,c,k(σ2IN +Hc,c,kgc,kgHc,kHHc,c,k)−1 (2.68) The good performance of the MF filters is demonstrated in Figure 2.4 and in Chapter 1 when we proposed the WSMSE-SR precoder. In fact, whenf(j)c,k is a MF, the WSMSE precoder becomes the WSMSE-SR precoder introduced in 1.5.2. We have proved by simulations that using a MF Rx is correct especially when we have large system dimensions. Finally, the interference power can be
Figure 2.4: Sum rate comparisons between the IBC WSMSE with MMSE filters and the IBC WSMSE with MF filters for C=1, M=10, K=4, N=2
given by ( ξ(j)c
N M)2fH,(j)0,c,k Hm,c,k X
m,l;(m,l)6=(c,k)
Hm,c,kg(j)m,lgH,(j)m,l HHm,c,k (2.69)
×HHm,c,kf(j)0,c,k (2.70)
= (ξ(j)c )2zHm,c,kΘ1/2m,c,k (2.71)
× X
m,l;(m,l)6=(c,k)
Hm,c,kg(j)m,lgH,(j)m,l HHm,c,kΘ1/2m,c,kzm,c,k (2.72)
= ξc2,(j)
d(j)c,k zHm,c,kΘ1/2m,c,k X
m,l;(m,l)6=(c,k)
w(j)m,l(Γ(j)m )−1 (2.73)
×Θ1/2m,lzm,lzHm,lΘ1/2m,l(Γ(j)m )−1Θ1/2m,c,kzm,c,k (2.74) ...→ ξc2,(j)
d(j)c,k
Υ(j)c,k+ ˆΥ(j)c,k
(1 +m(j)c,c,k)2. (2.75)
as in section 2.3.2. The filters are MF filters as denoted previously which completes the proof.
2.4.1 Applications of the deterministic equivalent of the SINR In this subsection, the deterministic equivalent of the SINR in (2.47) is used in order to prove a property of the MU communications. We prove that for a BC system, the achievable SINR for a system with N-antennas receivers Rx and where only a single stream (SS) equalsN times the SINR achieved in the case of MISO for identical channel covariance matrices. Thus,
γBC,SS,N Rx=N ×γBC,M ISO (2.76)
Proof: For a BC system withK users where all channel covariances matrices Θk are identical, the equations (2.59) and (2.61) can be written as:
e′,(j)= e′i,(j) di = 1
Ξ(j)
e(j)1
1−c(j)e(j)2 , e′k,(j) = e′i,k,(j)
di = d(j)k Ξ(j)
e(j)2
1−c(j)e(j)2 , (2.77)
wherec(j)= ∆(j)
Furthermore, the equations (2.49), (2.50) and (2.51) can be written as:
Ψ(j)=e′,(j)Ω(j),Υ(j)k =e′k,(j)Ω(j)k (2.81) Thus, the SINR in (2.47) will be equivalent to:
γk(j)=e(j)w(j)k (Ξ(j))2 d(j)k e(j)[1−c(j)e(j)2 ] The interference is diminished after the convergence of the WSMSE:e(j)2 Ω(j)k → 0. Substituting (2.85) in (2.84), we get due to the presence of N in the denominator of (2.82):
γk(j) =N e=N ×γk,SS(j) (2.86) Where γ(j)k,SS is the SINR when N = 1. We can extend the result in (2.76) to IBC systems. Now, we will prove using numerical simulations the double findings of this section. We prove the correctness of the deterministic equivalent
of the SINR of MIMO SS system (2.47) as well as the validity of (2.76). We have seen that the SINR scales with N. Thus, the weighted sum rate function of SNR curve in the case of N antennas Rx must be parallel to the one obtained in the case of MISO. Figure2.5shows the simulation of WSMSE precoder for C= 1, K = 15, M = 30 and its approximation for the both cases of N = 1 and N = 2. For the simulations of the WSMSE algorithm, we have used 200 channel realizations. It can be observed that for i.i.d channels the approximation is accurate and that our asymptotic sum rate follows the simulated one; which validates our asymptotic approach. Although the sum rate expression for the approximation approach (2.47) seems to be complex, however we need to calculate it only once per a given SNR, while we need to run the IBC WSMSE simulations as many times as the number of channel realizations, i.e.
200 times. Moreover, we can observe that the curves of N = 1 and N = 2 are parallel which validates our proposition (2.76). Similarly, Figure 2.6with C = 1, K = 15, M = 30 for both cases of N = 1 and N = 2 validates our results for IBC systems.
Figure 2.5: Sum rate comparisons between the IBC WSMSE and our proposed approximation for C=1,K=20,M=30,N={1,2}
2.5 Conclusion
In order to assess the performance of algorithms like KG and WSMSE, Monte-Carlo simulation of the average rate versus SNR needs extensive averaging
Figure 2.6: Sum rate comparisons between the IBC WSMSE and our proposed approximation for C=2,K=10,M=30,N={1,2}
over many channel realizations. In order to ease this procedure of evaluation, in this chapter, we presented a consistent framework to study the WSMSE precoding for MISO based on the theory of large-dimensional random matrices.
The tools from Random Matrix Theory allowed us to derive a deterministic expression of the rate for MISO. In MISO, the SINR is the ratio of scalar signal variance to scalar (interference + noise) variance. Hence, since we have a ratio of two scalars, Random Matrix Theory can be applied on each of these scalars apart resulting in a deterministic expression of the rate, which depends only on channels statistics and constant system parameters. The advantage of this proposition is that from now on, we do not need to do Monte-Carlo to evaluate performance. We have seen as well in this chapter that the deterministic expression represents well the true rate, especially when the total number of served users is inferior to the number of transmit antennas, which is true in general for Massive MIMO scenarios. Then, we proposed the same deterministic expression for MIMO single stream, i.e. MIMO but only a single stream is allowed to be transmitted. We used the resulting deterministic expression to show that the capacity scales with the number of receive antennas.