Iterative Optimization Algorithm

(5.16) whereγk is defined in (5.1). The diagonal unitary matrixDrcontainsM−1 an-gles. The optimal permutation matricesP^?_r,P^?_c have to be found by exhaustive search. Similarly, all equivalence classes QM ∈ GM have to be tried to deter-mine the optimal one Q^?_M. Consequently, the optimal CUBF matrix G^?_cubf is given by

G^?_cubf = 1

√MD^?_rP^?_rQ^?_MP^?_c. (5.17) Denote A = {ϕ1, . . . , ϕM−1, θ} the set of angles to be optimized. Note that only forM = 4 the setAcontains the additional angleθ. After some algebraic manipulation (5.16) takes the form

{D^?_r,Q^?_M,P^?_c,P^?_r}= arg min

Dr,QM∈GM,Pc,Pr

( _K Y

k=1

1 +ξk−ν_k² )

. (5.18)

This is still a non-convex optimization and the global optimum can only be found by exhaustive search. Subsequently, we present an iterative algorithm to calculate the optimal set of angles A^?. However, this algorithm cannot be guaranteed to converge to the global optimum.

5.4.1 Iterative Optimization Algorithm

A joint optimization of the angles in A is too involved, therefore we optimize the angles one by one while the others are fixed (i.e., alternating maximization)

for given permutation matrices Pc and Pr and equivalence class QM ∈ GM. DenoteW, ^√1_MPrQMPc= [w1, . . . ,wM]. We write

νk(ϕm) =|h¯^H_kgk|=|h¯^H_kDrwk| (5.19)

=|h¯^∗_k1wk1+ ¯h^∗_k2wk2e^iϕ1+· · ·+ ¯h^∗_kMwkMe^iϕM−1| (5.20)

=|akm+bkme^iϕm|, (5.21)

whereϕm∈ Aand bothbkm= ¯h^∗_km+1wkm+1 andakm=νk−bkm are indepen-dent ofϕm. With the auxiliary variables in Table 5.2, we obtain

ν_k²(ϕm) =d₀(k, m) +d₁(k, m) cos(δkm−ϕm). (5.22) With the substitution

sm= tanϕm

2 , (5.23)

we have cosϕ_m = ^1−s_1+s^2m₂

m and sinϕ_m= _1+s^2sm₂

m. Together with cos(δ_km−ϕ_m) = cosδkmcosϕm−sinδkmsinϕm, (5.22) takes the form

ν_k²(sm) =d0(k, m)− 1 1 +s²_m

d2(k, m)s²_m−d3(k, m)sm−d2(k, m)

. (5.24) From (5.18) we have the objective function

F_m(s_m) =

M

Y

k=1

1 +ξ_k−ν_k²(s_m)

(5.25)

=

M

Y

k=1

c2(k, m)s²_m+c1(k, m)sm+c0(k, m)

1 +s²_m , (5.26)

where

c2(k, m) = 1 +ξk−d2(k, m)−d0(k, m) (5.27)

c1(k, m) =d3(k, m) (5.28)

c₀(k, m) = 1 +ξ_k+d₂(k, m)−d₀(k, m). (5.29) Define P1(sm) =QM

k=1c2(k, m)s²_m+c1(k, m)sm+c0(k, m) andP2(sm) = (1 + s²_m)^M. To solve ^dFm_ds^(sm)

m = 0, we have to find the real roots of the polynomial G(sm) of order 2(2M−1) given as

Gm= dP1(sm) dsm

P2(sm)−P1(sm)dP2(sm) dsm

. (5.30)

Once the real roots have been found, we undo the substitution in (5.23), i.e., sm= 2 arctanϕm, and evaluate (5.18) for every solution to obtain the optimal solutionϕ^?_m. IfM = 4, there is an additional angleθinQ^o₄. To find the optimal angleθ^?, we can apply the same approach as forϕm. The algorithm to compute

INPUT: H,σ²,Pr,Pc,QM, AinitNiter

OUTPUT: A^? W=^√1

MP_rQ_MP_c A=A_init

forn= 1,2, . . . , Niter do form= 1,2, . . . , M−1do

ComputeGm in (5.30).

Compute real roots ofGm and findϕ^?_m. ϕm∈ A=ϕ^?_m

end for end for A^?=A

Table 5.3: Alternating optimization to compute the set of optimal anglesA^?.

the optimal set of anglesA^?is given in Table 5.3. The number of iterationsNiter

has to be chosen large enough for the algorithm to converge, which is usually the case after a few iterations. In addition to the exhaustive search over the permutation matrices Pr and Pc as well as QM ∈ GM, the algorithm can be carried out for multiple sets of initial angles Ainitto increase the probability of finding the global optimum.

5.5 Numerical Results

In this section, we compare the CUBF with the codebooks of CUBF matrices (called CB-CUBF) defined in 3GPP LTE [75]. In case ofM = 2 the codebook contains the identity matrix and two rotations of the DFT matrix according to (5.9) withϕ₁={0,^π₂}andP_r=P_c=I₂. The codebook for 4 transmit antennas is a subset ofV defined in (5.15) generated by 16 vectors uwhere the elements ofuare taken from a 8-PSK constellation andu₁= 1. The optimal CB-CUBF is computed at the transmitter by exhaustive search based on the available CSIT. The performance metric is the ergodic sum rate E[Rsum]. Throughout this section we average our results over 10 000 independent uncorrelated (Θk= IK ∀k) Rayleigh fading channel realizations.

Figure 5.1 shows the sensitivity of CUBF, CB-CUBF and ZF precoding to erroneous CSIT ˆhk which is model similar to (1.14) as ˆhk =√

1−τ²zk+τek, where bothzkandekhave i.i.d. Gaussian entries of zero mean and unit variance.

From Figure 5.1 it can be observed that CUBF and CB-CUBF outperform ZF precoding starting fromτ²≈0.22 andτ²≈0.5, respectively. ZF precoding, that achieves high sum rates under the assumption of perfect CSIT, experiences a severe performance loss as soon as the CSIT is erroneous. In practical systems such a scheme is not attractive since it requires highly accurate CSIT which entails an enormous feedback overhead.

Figures 5.2 and 5.3 present the ergodic sum rate performance forM =K= 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2

3 4 5 6 7

τ²

ergodicsumrate[Bits/s/Hz]

ZF CUBF CB-CUBF

Figure 5.1: M =K= 2, impact of erroneous CSIT on ergodic sum rate, SNR

= 15 dB.

andM =K= 4 system, respectively. We observe that the CUBF significantly outperforms the CB-CUBF in both configurations. At an SNR of 20 dB the gain is about 40 % and 30 %, respectively. Furthermore, it can be observed that UBF significantly outperforms CUBF which suggests that the constant modulus constraint is a severe restriction resulting in a large performance loss.

Indeed, the sum rate performance under UBF does not saturate, since under UBF the precoding vector gk can always be perfectly aligned to the channel hk, i.e.,νk = 1 and therefore the interference for user kis zero. Moreover, the sum rate saturation level of CUBF is lower for M = 4 than forM = 2. This is due to the fact that the number of degrees of freedom (the number of angles to be optimized) of CUBF scales significantly slower with M than the number of channel coefficients. Thus, the CUBF is increasingly maladjusted to the channel and consequently the interference level is rising resulting in a lower sum rate saturation level. Also note that the rather poor performance of the CUBF scheme may be significantly increased if it is applied in conjunction with an appropriate scheduling techniques or an advanced (interference-aware) receive algorithm. However, to cancel the multi-user interference, the receiver requires knowledge of the precoding vectors and modulation scheme of the interfering users. In the current LTE-A standard, this information is not signaled to the users, which makes it difficult to obtain. Under unitary precoding, the multi-user interference power can be computed without the knowledge of the other precoding vectors but still the modulation scheme has to be estimated. Indeed, the results in [74] show that the system performance increases significantly if the receiver accounts for the interference even under false assumptions on the

−5 0 5 10 15 20 25 30 35 0

1 2 3 4 5 6 7 8 9 10 11 12 13

SNR [dB]

ergodicsumrate[Bits/s/Hz]

UBF CUBF CB-CUBF

Figure 5.2: M =K= 2, ergodic sum rate vs. SNR.

interfering modulation scheme.

−5 0 5 10 15 20 25 30 35 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

SNR [dB]

ergodicsumrate[Bits/s/Hz]

UBF CUBF CB-CUBF

Figure 5.3: M =K= 4, ergodic sum rate vs. SNR.

Conclusion and Perspectives

This dissertation studied large MU-MISO systems under various linear precod-ing techniques and imperfect CSIT.

Conclusion

In this thesis, we considered the MU-MISO downlink under linear precoding and imperfect CSIT and analyzed the system behavior for large numbers of transmit antennas and users. We presented a consistent framework for the study of several linear precoding schemes based on the theory of large dimen-sional random matrices. The tools from large RMT allowed us to consider a very realistic channel model accounting for per-user channel correlation as well as individual channel gains for each link. The system performance under this general type of channel is extremely difficult to study for finite dimensions but becomes feasible by assuming large system dimensions. Thanks to large RMT, the system performance (i.e., SINR or rate) can be approximated by a deterministic equivalent (independent of the channel realizations) that is almost surely exact as the system dimensions grow asymptotically large with bounded ratio. Simulation results showed that these approximations are very accurate even for small system dimensions and reveal the deterministic dependence of the system performance on several important system parameters, such as the transmit correlation, signal powers, SNR and CSIT quality. Applied to practical optimization problems, the deterministic approximations lead to important in-sights into the system behavior, which are consistent with previous results, but

go further and extend them to more realistic channel models and other linear precoding techniques. Furthermore, the proposed channel-independent perfor-mance approximations can be used to simulate the system behavior without having to carry out extensive Monte-carlo simulations. Although not yet prac-tical, large numbers of transmit antennas are expected to be widely deployed in the future, at which point the results in this dissertation may prove all the more relevant. Moreover, since practical considerations often limit the choices for a precoding technique, we studied the precoding scheme adopted in the 3GPP LTE standard and developed an algorithm to evaluate the performance of such precoding schemes.

Perspectives

From a mathematical point of view, the deterministic equivalent of the empirical Stieltjes transform under the generalized variance profile is a general result and may have applications outside the field of mobile communications.

The proposed framework considered a single isolated cell but is the founda-tion for the study of more complex multi-cell systems. For instance, recently, the results in this thesis have been applied to study the effect of pilot contamina-tion on the system performance in large multi-cell networks. But other scenarios can be considered as well, for example the impact of inter-cell interference can be included in the system model. Moreover, our analysis can be extended to multi-cell systems with different levels of base station cooperation (coordinated beamforming), with imperfect CSIT at the base stations due to limited back haul capacity. Furthermore, it is possible to extend this framework to MU-MIMO with a single stream per user by considering linear receive filters. On the contrary, an extension to MU-MIMO with multiple streams per user is very involved and requires the development of more advanced random matrix theory tools. But the application of the presented results is not limited to broadcast channels. The proposed framework can be extended to linearly precoded multi-user MISOamplify-and forward downlink channels, where the signal is linearly processed at the transmitter and the relay before being received by the users.

Lastly, the proposed methodology may also be applied to interference channels with linear transmit and receive filters.

A Proof of Proposition 2.3

The proof is based on the matrix inequalities in Lemmas F.8, F.10, F.11, F.12.

F.9 and F.13.

Applying Lemma F.9,|d⁽¹⁾_i |can be upper-bounded as

|d⁽¹⁾_i |=

We further bound |d⁽¹⁾_i | by applying Lemmas F.10, F.12 and the fact that k B_[i]−zIN⁻¹

where (a) follows from the observation that (A.8) can be written ask(R−zIN+ SN)⁻¹k =A−iB− =zIN, where B is nonnegative-definite, since the Θi are nonnegative-definite and=[_N¹trΘ_i(B_N −zI_N)⁻¹] is always positive as it is the Stieltjes transform of a nonnegative finite measure. Therefore, we apply Lemma F.13 and obtain (A.9). Similarly to (A.8) we have

k(R[i]−zI_N +SN)⁻¹k ≤ 1 Applying Lemma F.12 together with the rank-1 perturbation Lemma F.8, the numerator in (A.11) can be bounded as

is the Stieltjes transform of a nonnegative finite measure, we apply [19, Corollary 3.1] and (2.20) to upper bound (A.11) as

kR_[i]−Rk ≤ T|z|²

Substituting (A.9), (A.13) and (A.11) into (A.7) yields kD⁻¹_[i] −D⁻¹k ≤ TkQNk|z|²

Therefore, with (A.14) the upper bound of|d⁽¹⁾_i |in (A.3) becomes

|d⁽¹⁾_i | ≤ kyik²₂TkΘikkQNk|z|³

Before considering the pth order moment of |d⁽¹⁾_i |, we proceed with bounding the remaining terms|d⁽²⁾_i |,|d⁽³⁾_i | and|d⁽⁴⁾_i |.

Applying Lemma F.9, the term |d⁽²⁾_i | is bounded as

|d⁽²⁾_i | ≤ |z|

Again, using Lemma (F.9), |d⁽³⁾_i |is bounded as Applying the triangle inequality, we obtain

the RHS of (A.18) can be bounded as

Similarly with Lemma F.11, F.12 and (A.5), we have

Applying the Cauchy-Schwarz inequality, Lemmas F.9, F.11 and F.12 we have

|d⁽⁴⁾_i | ≤kΘikkQNk|z|²

from the term in brackets and applying the triangle inequality together with the bound in (A.12) we obtain

|d⁽⁴⁾_i | ≤kΘ_ikkQ_Nk|z|²

Applying the inequality|x+y|^p≤2^p−1(|x|^p+|y|^p) yields

where Cp⁽¹⁾ is some constant depending only on p. Similarly, applying Lemma F.3 yields

B Deterministic Equivalents for Precoders with RZF Structure

This section provides three lemmas that are applied in the derivation of a deter-ministic equivalent of the SINR for WSR maximizing precoding in Section 3.3 and RZF precoding in Section 3.4. We remind that the kth column ˆh_k of ˆH^H

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