In this thesis, we provide a concise framework that directly extends and general-izes the results in [10,22,25–27] by accounting for per-user correlation and imper-fect CSIT. Furthermore, we apply our SINR approximations to several limited-feedback scenarios that have been previously analyzed by applying bounds on the ergodic rate of finite dimensional systems. In addition, we study the CUBF for finite dimensions and propose an optimization algorithm to parametrize the CUBF.
This dissertation is organized as follows: Chapter 1 motivates the work, pro-vides a detailed outline and introduces the system model. Chapter 2 gives a brief introduction to the field of large dimensional random matrices and pro-vides the results required for the subsequent analysis. In Chapter 3, we derive deterministic equivalents of the SINR under various linear precoding schemes and correlation assumptions. In Chapter 4, we study several applications of the large system approximations. Chapter 5 analyzes a practical linear precoding scheme and presents an optimization algorithm to parametrize the precoder.
Finally, in Chapter 6, we conclude the thesis and give an outlook.
The detailed contributions are summarized in the following sections.
1.2.1 Chapter 2: Mathematical Prerequisites: Large Di-mensional Random Matrix Theory
In this chapter, we provide the mathematical basis for the subsequent large system analysis of the MISO-BC under linear precoding. In the course of the derivations of the SINR approximations, we have to deal with terms of the form m(z) = M1tr(M1HHH−zIM)−1, whereH= [h1, . . . ,hK]H∈CK×M is the compound channel andz <0. The termm(z) is the so calledStieltjes transform of the empirical spectral distribution (e.s.d). of matrixM1HHHatz. To compute a deterministic equivalent of the SINR, we first require a deterministic equivalent
¯
m(z) ofm(z), such thatm(z)−m(z)¯ M→∞−→ 0, almost surely. Depending on the assumptions for the entries ofH, many results exist. The most general result is provided in [31], whereH has a certain variance profile, i.e., √1
M[H]ij =σijzij
with zij i.i.d., zero mean and of variance 1/M. However, the variance profile does not account for per-user correlation and is thus insufficient to study realistic MU-MISO channels. The channel model considered in this thesis goes further and assumes per-user correlation, i.e., M1HHH = PK
i=1Θ1/2i zizHiΘ1/2i , where zi = [zi,1, . . . , zi,M]T which includes the variance profile as a special case, i.e., ifΘi= diag(σ2i1, . . . , σiM2 ), we obtain the variance profile in [31]. Therefore, we need to extend the result in [31] to account for the channel model with per-user correlation also referred to as a channel with generalized variance profile. A deterministic equivalent of the empirical Stieltjes transform for channels with generalized variance profile is provided in Section 2.2. Furthermore, in Section 2.3, we derive theShannon transformV(x) = M1 log det(IM+xM1 HHH),x >0, of the e.s.d. of M1HHHalso extending the results in [31]. Note, that the proof for
the generalized variance profile of both Stieltjes and Shannon transform partly relies on adapting the techniques provided in [31]. These results are partly published in
• S. Wagner, R. Couillet, M. Debbah, D. T. M. Slock “Large System Anal-ysis of Linear Precoding in Correlated MISO Broadcast Channels under Limited Feedback”,to appear in IEEE Trans. Inf. Theory, arXiv Preprint 0906.3682.
1.2.2 Chapter 3: MISO BC under Linear Precoding: A Large System Analysis
Based on the deterministic equivalent of the Stieltjes transform with general-ized variance profile provided in Chapter 2, we derive deterministic equivalents, i.e., approximations that are almost surely exact asM, K→ ∞, of the random SINR under MF (Section 3.2), optimal linear precoding (Section 3.3), RZF (Sec-tion 3.4) and ZF precoding (Sec(Sec-tion 3.5). In general, the deterministic SINR approximations are functions of fixed-point equations but are explicit for uncor-related channels (Θk =IM ∀k). Unlike previous work, we derive deterministic equivalents and not asymptotic results, since for our channel model the e.s.d.
of M1HHHdoes not converge to a limiting spectral distribution. In Section 3.6, we provide rate, sum rate and rate gap (between perfect and imperfect CSIT) approximations which are based on the SINR approximations. Simulation re-sults show that the deterministic SINR approximations are very accurate even for small M, K. Hence, applied to practical optimization problems in Chap-ter 4, they yield close-to-optimal solutions, where the objective is typically to maximize the system sum rate. Those results have partially been published in
• R. Couillet, S. Wagner, M. Debbah, A. Silva, “The Space Frontier: Phys-ical Limits of Multiple Antenna Information Transfer”, Proc. of the 3rd ACM International Conference on Performance Evaluation Methodologies and Tools (VALUETOOLS’08), Athens, Greece, 20-24 Oct. 2008, no. 84.
BEST STUDENT PAPER AWARD,
in which we apply tools from large RMT to derive the asymptotic SINR of ZF and RZF precoding, where the regularization term is set to fulfill the transmit power constraint, in MISO broadcast channels.
• R. Couillet, S. Wagner, M. Debbah, “Asymptotic Analysis of Correlated Multi-Antenna Broadcast Channels”, Proc. of the IEEE Wireless Com-munications & Networking Conference (WCNC’09), Budapest, Hungary, 5-8 Apr. 2009,
where we consider ZF and RZF precoding and apply the results to systems with dense antenna packings at the transmitter.
• S. Wagner, R. Couillet, M. Debbah, D. T. M. Slock, “Deterministic Equiv-alent for the SINR of Regularized Zero-forcing Precoding in Correlated
MISO Broadcast Channels with Imperfect CSIT”, Proc. IEEE Interna-tional Conference on Communications (ICC’11), Kyoto, Japan, 5-9 Jun.
2011,
where we derive the deterministic equivalent of the SINR under ZF and RZF precoding accounting for per-user correlation and imperfect CSIT and derive the SINR maximizing regularization term.
• S. Wagner and D. T. M. Slock “Weighted Sum Rate Maximization of Cor-related MISO Broadcast Channels under Linear Precoding: A Large Sys-tem Analysis”, Proc. IEEE 12th International Workshop on Signal Pro-cessing Advances in Wireless Communications (SPAWC’11), San Fran-cisco, USA, 26-29 Jun. 2011,
where we consider the weighted sum rate maximizing precoder in [13] and carry out a large system analysis and derive a deterministic equivalent of the SINR for MU-MISO with per-user correlation and perfect CSIT.
1.2.3 Chapter 4: Large MISO BC under Linear Precod-ing: Applications
In this chapter, we apply the deterministic SINR approximations derived in Chapter 3 to solve various optimization problems. More precisely, in Section 4.1, we compute the sum rate maximizing regularization term ¯α? for the RZF precoder under imperfect CSIT and common correlation (Θk=Θ∀k), extend-ing the results in [10, 25, 27]. In Section 4.2, we consider the problem of com-puting the sum rate maximizing number of users for afixednumber of transmit antennas. Under ZF precoding and uncorrelated channels (Θk =IM ∀k), we obtain a closed form solution which extends the result in [22] to imperfect CSIT.
Furthermore, we solve the problem of optimal power allocation for channels with common correlation under MF, ZF and RZF precoding when the CSIT of the users is unequal. Under this assumption, the optimal power allocation strategy is the solution of a water-filling algorithm.
In Section 4.3, we study the optimal amount of channel training (pilot signal-ing) in a TDD system, where coherent transmission and channel training occur in the same channel coherence interval. This scheme has been studied in [32–34]
for finite dimensional systems. We assume uncorrelated channels, MMSE esti-mation at the transmitter and we neglect the common training phase. The amount of training optimizing the net sum rate, i.e. the sum rate taking into account the reduced interval for coherent downlink data transmission, for MF, ZF and RZF precoding is the solution of a convex optimization problem. We derive approximated but closed form solutions for high downlink SNR. Our re-sults are in line with [32–34] in terms of the scaling in the coherence interval (for a fixed SNR) and the SNR (for a fixed coherence interval) but extend them to RZF and MF precoding.
Section 4.4 considers a system where the CSIT is obtained by direct feedback of the quantized channel direction. In [35], an information-theoretic analysis of
the impact of quantized CSIT on the achievable rate of a ZF-precoded MU-MISO downlink channel withM =Khas been carried out. Hereby, the author derives an upper bound on the ergodic per-user rate gapbetween perfect CSIT and imperfect CSIT under RVQ with B feedback bits per user. Under finite-rate feedback, both [35] and [36] observe a sum finite-rate ceiling for high SNR. In [35, Theorem 3] provides a formula for the minimum scaling of B to maintain an average per-user rate gap of log2b bits/s/Hz and hence to achieve the full multiplexing gain of K. Although derived for ZF, the author claims that for all SNR, [35, Theorem 3] is more accurate for the RZF-CDU precoder proposed in [10]. Similar to [35], we derive the necessary feedback scaling to maintain a target rate gap between perfect and imperfect CSIT. But unlike [35], the derived feedback scaling maintains the rate gap exactly asM, K → ∞, almost surely. We find that our solutions, give more accurate results than [35] for large dimensions. Finally, as in [32], we compare quantized feedback to analog feedback in terms of rate gap between perfect and imperfect CSIT for MF, ZF and RZF precoding. The results have partially been published in
• S. Wagner, R. Couillet, D. T. M. Slock, M. Debbah “Large System Analy-sis of Zero-Forcing Precoding in MISO Broadcast Channels with Limited Feedback”, Proc. IEEE 11th International Workshop on Signal Process-ing Advances in Wireless Communications (SPAWC’10), Marrakech, Mo-rocco, 20-23 Jun. 2010,
where we derive a deterministic equivalent of the SINR under ZF precoding and apply this result to solve the problem of the sum rate maximizing number of users for a fixed number of transmit antennas.
• S. Wagner, R. Couillet, M. Debbah, D. T. M. Slock, “Optimal Train-ing in Large TDD Multi-user Downlink Systems under Zero-forcTrain-ing and Regularized Zero-forcing Precoding”,Proc. IEEE Global Communications Conference (GC’10), Miami, USA, 6-10 Dec. 2010,
where we apply the deterministic equivalents of the SINR under ZF and RZF precoding to large TDD systems and derive the optimal amount of training for both schemes.
• S. Wagner, R. Couillet, M. Debbah, D. T. M. Slock “Large System Anal-ysis of Linear Precoding in Correlated MISO Broadcast Channels under Limited Feedback”,to appear in IEEE Trans. Inf. Theory, arXiv Preprint 0906.3682,
where all applications are summarized and presented in detail.
The optimum linear precoder, MF, ZF and RZF precoders have several draw-backs that prevent their adoption into practical wireless communication stan-dards like 3GPP LTE. First, they increase the PAPR and the power imbalance between the transmit antennas leading to a reduced efficiency of the RF power amplifiers. Secondly, they require significant amount of feedback overhead to provide accurate CSIT. Lastly, the receivers are unable to compute their exact
SINR since they have no knowledge about precoders of the interfering users. The inaccurate SINR results in an inaccurate CQI which translates into a system capacity loss since the CQI is used for scheduling and to adapt the modulation and coding scheme. Therefore, a different precoder is required to alleviate those drawbacks, which is discussed in the next chapter.
1.2.4 Chapter 5: Unitary Precoding with Constant Mod-ulus Constraint
In this chapter, we consider a practical precoding scheme, where the precoding matrices are orthogonal and have entries of constant magnitude. Those pre-coders are attractive, since they do not increase the PAPR at the transmitter and allow for exact SINR computation at the receivers. All current 3GPP LTE standards adopted codebooks (to reduce the feedback overhead) of such ma-trices for the precoding in MU-MIMO mode. We apply the theory of complex Hadamard matrices to describe those constrained unitary beamformers (CUBF) and propose an algorithm for optimal parametrization of the CUBF matrices.
Contrary to practical systems where the receivers feedback their preferred pre-coding matrix, the proposed algorithm assumes CSIT (in the form of a channel estimate) and therefore provides a benchmark for the evaluation of the code-books used in practical systems. The results of this chapter have been published in
• S. Wagner, S. Sesia, D. T. M. Slock, “Unitary Beamforming under Con-stant Modulus Constraint in MIMO Broadcast Channels”, Proc. IEEE 10th Workshop on Signal Processing Advances in Wireless Communica-tions (SPAWC’09), Perugia, Italy, 21-24 Jun. 2009,
where we consider orthogonal precoding matrices with constant modulus entries and apply the description of complex Hadamard matrices to derive an algorithm that optimizes those precoders.
• S. Wagner, S. Sesia, D. T. M. Slock, “On Unitary Beamforming for MIMO Broadcast Channels”, Proc. IEEE International Conference on Commu-nications (ICC’10), Cape Town, South Africa, 23-27 May 2010,
where we compare the sum rate performance of unitary precoding to uni-tary precoding with constant modulus constraint for equal and optimal power allocation.
In the next section, we introduce the general system model of a MU-MIMO communications system.