Although we will not apply the result in this thesis, a recent application to random beamforming can be found in [56]. The following theorem is novel and extends the results in [31, 53].
Theorem 2.2 (Shannon transform of BN). Let x > 0 and SN be uniformly bounded spectral norm with respect to N, given the assumptions of Theorem 2.1, denote the Shannon transform of BN byVBN(x), N1 log det (IN+xBN).
Proof of Theorem 2.2. The proof of (2.35) exploits the fact that the Shannon transform can be expressed as a function of the Stieltjes transform as (2.5)
VBN(x) =
First notice that (2.36), we require an expression that, when integrated with respect toz, yields (2.37). Observe that
Subtracting (2.38) and (2.39) from (2.40) yields (2.37). Applying (2.36) along with limN→0VBN(x) = 0 leads to (2.35).
The proof is similar to the one provided in [31, Section C.1]. The dominated convergence Theorem [50, Theorem 16.4] together with (2.8) ensures that for allω >0,
Now, observe that
where FBN and ¯FBN are the distribution functions with respective Stieltjes transforms mBN and ¯mBN. The upper bound (2.46) is integrable in ω over (1/x,∞). Therefore, the dominated convergence theorem ensuresE[VBN(x)]− V¯BN(x)N→∞−→ 0, which completes the proof.
MISO BC under Linear
Precoding: A Large System Analysis
In this chapter, we consider the MISO broadcast channel under the following linear precoding techniques: Matched filter (MF), WSR maximizing (WSRM) linear precoding, regularized zero-forcing (RZF) and zero-forcing (ZF) precod-ing. We suppose that the system is large in the sense that the number of users K and the number of transmit antennasM grow large with finite ratio M/K → β <∞. Under this assumption, we will derive deterministic equiva-lents ¯γk of the random SINRγk of userkfor each of the precoding schemes. The chapter is structured as follows: In Section 3.1, we introduce the problem. In Sections 3.2, 3.3, 3.4 and 3.5, we derive a deterministic equivalent of the SINR under MF, WSRM linear precoding, RZF and ZF precoding, respectively. Sec-tion 3.6 discusses approximaSec-tions of the rate based on the deterministic equiv-alent of the SINR. Finally, Section 3.7 presents simulation results to evaluate the accuracy of the proposed approximations.
3.1 Introduction
To gain valuable insights into the system behavior, it is necessary to determine the fundamental dependence of the system performance measure (e.g. the SINR γk or the rateRk = log(1 +γk) of user k ) on the relevant system parameters (e.g. SNR, imperfect CSIT, correlation,...) through therandom channel. That is, to predict how the change of a certain parameter will most likely change the
system performance. To accomplish this, one can consider the average system performance and approximate or bound it by a deterministic quantity, which then predicts how a particular system parameter affects the average system performance. The derivation of accurate bounds becomes extremely difficult for advanced channel models such as the per-user channel correlation model.
However, another possibility to approximate the system performance is to as-sume that the system dimensions, i.e., the number of usersK and the number of transmit antennas M, grow large with finite ratioM/K → β <∞. Under this assumption one can resort to the advanced tools of large RMT to derive a deterministic equivalent ¯γk such thatγk−γ¯k
M→∞−→ 0 almost surely. Under the per-user channel correlation model (or generalized variance profile) some important tools have been provided in the previous Chapter. In this chapter, we apply those tools to derive ¯γk for each of the considered linear precoding schemes in MISO broadcast channels. As a result, we obtain asymptotically exact (almost surely) approximations of the SINR or per-user rate with deter-ministic dependencies on the important system parameters. The results in this chapter provide powerful tools that can be applied to solve various optimization problems, which is the subject of Chapter 4.
In order to also demonstrate the current limitations of large system analysis, consider the MU-MIMO setting withSk=Nk streams per user. In general, the objective of the large system analysis is to find a deterministic approximation of the communication rate Rk of user k. For MU-MIMO channels, the rate of user k is given in (1.5), which is the scaled Shannon transform of the e.s.d.
of the matrix BNk ,P1/2k GHkHHkRn−1˜
kn˜kHkGkP1/2k . It is quite clear that for a channel-dependent precoding matrix Gk, like MF, ZF or RZF precoding, the matrix model forBNk may be very complicated since it may depend in a non-trivial way on the random channels Hk as opposed to point-to-point MIMO channels [42, 57, 58]. It is complicated to derive a deterministic equivalent of the Shannon transform or equivalently (by exploiting the relation in (2.5)) the Stieltjes transform of the e.s.d. of BNk, assuming that the number of receive antennasNk of userkand the number of transmit antennasM grow large with
M
Nk bounded for a finite number of users K. For instance, under the simple MF precoder and perfect CSIT withGk =ξHHk (where ξis such that (1.3) is Even for this simple case, a deterministic equivalent of the Stieltjes transform of the e.s.d. ofBNkis unknown. In our subsequent analysis we assumeNk= 1, since single-antenna receivers are practical, and letM, Kgrow jointly to infinity with bounded ratio. In this caseRk = log(1 +γk), whereγk is given in (1.12) and it suffices to find a deterministic equivalent ¯γk for the SINR γk, which immediately yields the deterministic equivalent ¯Rk of the rate of user k.
The deterministic equivalents obtained in this chapter are useful
approxi-mations, since applied to practical optimization problems, cf. Chapter 4, they yield very accurate approximate solutions even for small realistic values of M, e.g. M = 5 orM = 10.
In the following, the notation M → ∞ implies that both M and K grow asymptotically large while 1≤limM→∞M
K =β <∞. Subsequently, to evaluate the large system results for finite dimensions, we replace β by its finite value
M K.
3.1.1 Technical Assumptions
Let us first clarify several technical assumptions that are required for the large system analysis.
Assumption 3.1. All correlation matricesΘk have uniformly bounded spectral norm on M, i.e.,
We further require the subsequent assumption on the power allocation ma-trixP.
Assumption 3.2. The powerpmax= max(p1, . . . , pK)is of orderO(1/K), i.e.,
kPk=O(1/K). (3.3)
Furthermore, we have to restrict the matrix containing the rate weights U= diag(u1, . . . , uK).
Assumption 3.3. The rate weight matrix U has uniformly bounded spectral norm on M, i.e.,
lim sup
M→∞
kUk<∞. (3.4)
Moreover, we require the following assumption
Assumption 3.4. The matrix M1HˆHHˆ has uniformly bounded spectral norm on M with probability one, i.e.,
lim sup
∞, where |A| denotes the cardinality of the set A. That is, {Θk} belongs to a finite family [57]. In practice, we typically deal with finite many distinct Θk. For instance consider the channel model for MIMO systems in 3GPP LTE [59], where the transmit correlation Θk only comprises three different matrices corresponding to low, medium and high correlation. In particular, if Θk = Θ ∀k, then Assumption 3.4 is satisfied, since M1kHˆHHk ≤ kΘkkˆ ZˆHZk, whereˆ Zˆ = [ˆz1, . . . ,ˆzK]Hand bothkΘk andkZˆHZkˆ are uniformly bounded for all large M with probability one [60].
In the following, we derive deterministic equivalents of the SINR for all considered precoding schemes. Hereby, apart from the results in Chapter 2, we will apply various results summarized in Appendix B.1 and Appendix F.