**1. Introduction**

Wireless communication paradigm has evolved from single-user single-input single-output (SISO) and multiple-input multiple-output (MIMO) systems to multi-user (MU) MIMO counterparts, which are shown greatly improving the rate performance by transmitting to multiple users simultaneously. The sum-capacity and the capacity region of a single-cell MU MIMO downlink system or the so-called MIMO broadcast channel (MIMO-BC) can be attained by the nonlinear "Dirty Paper Coding (DPC)" scheme [1] [2] [3]. However, DPC requires a high implementation complexity due to the non-linear successive encoding/decoding at the transmitter/receiver, and is thus not suitable for real-time applications. Other studies have proposed to use alternative linear precoding schemes for the MIMO-BC, e.g., the block-diagonalization scheme [4], to reduce the complexity. More information on the key developments of single-cell MIMO communication can be found in, for example, [5] [6] [7].

The performance of a multi-cell MIMO-BC setup, however, is not well understood. It is worth noting that the multi-cell downlink system can be modelled in general as an interference channel (IC) setup. Characterization of the capacity region of the Gaussian IC is still an open problem even for the two-user case [8]. An important development recently is the so-called "interference alignment (IA)" transmission scheme (see, e.g., [9] and references therein). With the aid of IA, the maximum achievable degrees of freedom (DoF), in which the DoF is defined as the sum-rate normalized by the logarithm of the signal-to-noise ratio (SNR) as the SNR goes to infinity or the so-called "pre-log" factor, has been obtained for various ICs to provide useful insights on designing the optimal transmission for interference-limited MU systems. Besides IA-based studies for the high-SNR regime, there is a vast body of works in the literature which investigated the multi-cell cooperative downlink precoding/beamforming at a given finite user's SNR. These works are typically categorized based on two different assumptions on the cooperation level of base stations (BSs). For the case of "fully cooperative" multi-cell systems with global

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transmit message sharing across all the BSs, a virtual MIMO-BC channel is equivalently formed. Therefore, existing single-cell downlink precoding techniques can be applied (see, e.g., [10] [11] [12] and references therein) with a non-trivial modification to deal with the per-BS power constraints as compared to the conventional sum-power constraint for the single-cell MIMO-BC case. In contrast, if transmit messages are only locally known at each BS, coordinated precoding/beamforming can be implemented among BSs to control the inter-cell interference (ICI) to their best effort [13] [14]. In [15] [16] [17], various parametrical characterizations of the Pareto boundary of the achievable rate region have been obtained for the multiple-input single-output (MISO) IC with coordinated transmit beamforming and single-user detection.

10.5772/57131

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http://dx.doi.org/10.5772/57131

Random Beamforming in Multi – User MIMO Systems

The remainder of this chapter is organized as follows. Section 2 describes the multi-cell MIMO-BC system model and the MISO/MIMO RBF schemes. Section 3 presents a literature survey on the single-cell OBF/RBF. Section 4 investigates the multi-cell RBF under finite-SNR regime. The high-SNR/DoF analysis is introduced in Section 5, based on which we study the interplay between the multi-user diversity, spatial receive diversity and spatial multiplexing gains achievable in RBF. Finally, Section 6 ends the chapter with some concluding remarks. *Notations*: Scalars, vectors, and matrices are denoted by lower-case, bold-face lower-case, and bold-face higher-case letters, respectively. The transpose and conjugate transpose operators are denoted as (·)*<sup>T</sup>* and (·)*H*, respectively. **E**[·] denotes the statistical expectation. **Tr**(·) represents the trace of a matrix. The distribution of a circularly symmetric complex Gaussian random variable with zero mean and covariance *σ*<sup>2</sup> is denoted by CN (0, *σ*2); and ∼ stands

This section introduces the multi-cell system model and RBF schemes used throughout this chapter. Particularly, we consider a *C*-cell MIMO-BC system with *Kc* mobile stations (MSs) in the *c*-th cell, *c* = 1, ··· , *C*. For the ease of analysis, we assume that all BSs/MSs have the same number of transmit/receive antennas, denoted as *NT* and *NR*, respectively. We also assume a "homogeneous" channel setup, in which the signal and ICI powers between users of one cell are identical. At each communication time, the *<sup>c</sup>*-th BS transmits *Mc* ≤ *NT* orthonormal beams with *Mc* antennas and selects *Mc* from *Kc* users for transmission. Suppose that the channels are flat-fading and constant over each transmission period of interest. The received

> *C* ∑ *l*=1, *l*�=*c*

the *c*-th cell, which consists of independent and identically distributed (i.i.d.) ∼ CN (0, 1)

unit norm and transmitted data symbol from the *c*-th BS, respectively; it is assumed that

*<sup>γ</sup>l*,*<sup>c</sup>* < 1 stands for the signal attenuation from the *<sup>l</sup>*-th BS to any user of the *<sup>c</sup>*-th cell, *<sup>l</sup>* �=

element is ∼ CN (0, *σ*2), ∀*k*, *c*. In the *c*-th cell, the total SNR, the SNR per beam, and the interference-to-noise ratio (INR) per beam from the *<sup>l</sup>*-th cell, *<sup>l</sup>* �= *<sup>c</sup>*, are denoted as *<sup>ρ</sup>* = *PT*/*σ*2,

<sup>√</sup>*γl*,*cH*(*l*,*c*) *k*

*<sup>k</sup>* <sup>∈</sup> **<sup>C</sup>***NR*×*Ml* denotes the channel matrix between the *<sup>l</sup>*-th BS and the *<sup>k</sup>*-th user of

**E**[*s***c***s***<sup>H</sup> c** ]

*<sup>k</sup>* <sup>∈</sup> **<sup>C</sup>***NR*×<sup>1</sup> is the additive white Gaussian noise (AWGN) vector, of which each

*Ml* ∑ *m*=1

*<sup>m</sup>* are the *m*-th randomly generated beamforming vector of

*<sup>φ</sup>*(*l*) *<sup>m</sup> s* (*l*) *<sup>m</sup>* + *z* (*c*)

≤ *PT*, where *<sup>s</sup><sup>c</sup>* = [*<sup>s</sup>*

*<sup>k</sup>* , (1)

(*c*) *<sup>1</sup>* , ··· ,*<sup>s</sup>*

(*c*) *Mc* ] *T*;

for "distributed as". **<sup>C</sup>***x*×*<sup>y</sup>* denotes the space of *<sup>x</sup>* × *<sup>y</sup>* complex matrices.

**2. System model**

signal of user *k* in the *c*-th cell is

*<sup>y</sup>*(*c*)

where *<sup>H</sup>*(*l*,*c*)

elements; *<sup>φ</sup>*(*c*)

(*c*)

**2.1. Multi-cell MISO RBF**

*c*; and *z*

*<sup>k</sup>* <sup>=</sup> *<sup>H</sup>*(*c*,*c*) *k*

*<sup>m</sup>* <sup>∈</sup> **<sup>C</sup>***Mc*×<sup>1</sup> and *<sup>s</sup>*

each BS has the total sum power, *PT*, i.e., **Tr**

*η<sup>c</sup>* = *PT*/(*Mcσ*2), and *µl*,*<sup>c</sup>* = *γl*,*cPT*/(*Mlσ*2), respectively.

In this case, *NR* = 1 and the system model in (1) reduces to

*Mc* ∑ *m*=1

*<sup>φ</sup>*(*c*) *<sup>m</sup> s* (*c*) *<sup>m</sup>* +

(*c*)

More important, most of the proposed precoding schemes, in single- or multi-cell case, rely on the assumption of perfect channel state information (CSI) for all the intra- and inter-cell links at the transmitter, which may not be realistic in practical cellular systems with a large number of users. Consequently, the study of quantized channel feedback for the MIMO BC has been recently a very active area of research (see, e.g., [18] and references therein).

The single-beam "opportunistic beamforming (OBF)" and multi-beam "random beamforming (RBF)" schemes for the single-cell MISO-BC, introduced in [19] and [20], respectively, therefore attract a lot of attention since they require only *partial* CSI feedback to the BS. The fundamental idea is to exploit the multi-user diversity gain, by employing *opportunistic user scheduling*, to combat the inter-beam interferences. The achievable sum-rate with RBF in a single-cell system has been shown to scale identically to that with the optimal DPC scheme assuming perfect CSI as the number of users goes to infinity, for any given user's signal-to-noise ratio (SNR) [20] [21]. Essentially, the result implies that the intra-cell interference in a single-cell RBF system can be completely eliminated when the number of users are sufficiently large, and an "interference-free" MU broadcast system is attainable. This thus shows the optimality of the single-cell RBF and motives other studies on opportunistic communication.

Although substantial subsequent investigations and/or extensions of the single-cell RBF have been pursued, there are very few works on the performance of the RBF scheme in a more realistic multi-cell setup, where the ICI becomes a dominant factor. It is worth noting that as the universal frequency reuse becomes more favourable in future generation cellular systems, ICI becomes a more serious issue as compared to the traditional case with only a fractional frequency reuse.

One objective of this chapter is to present a literature survey on the vast body of works studying the single-cell OBF/RBF. The main purpose, however, is to introduce the recent investigations on multi-cell RBF systems. In this chapter, we first review the achievable rates of multi-cell RBF in finite-SNR regime. Such results, albeit important, do not provide any insight to the impact of the interferences on the system throughput. This motivates us to introduce the high-SNR/DoF analysis proposed in [22] and [23], which is useful in characterizing the performance of RBF under multi-user diversity and interference effects. Furthermore, it provides new insights on the role of *spatial receive diversity* in RBF, which is not well understood so far. It is revealed that receive diversity is significantly beneficial to the rate performance of multi-cell RBF systems [24]. This conclusion, interestingly, sharply contrasts with one based on the traditional asymptotic analysis, i.e., assuming that the number of users goes to infinity for any given user's SNR.

The remainder of this chapter is organized as follows. Section 2 describes the multi-cell MIMO-BC system model and the MISO/MIMO RBF schemes. Section 3 presents a literature survey on the single-cell OBF/RBF. Section 4 investigates the multi-cell RBF under finite-SNR regime. The high-SNR/DoF analysis is introduced in Section 5, based on which we study the interplay between the multi-user diversity, spatial receive diversity and spatial multiplexing gains achievable in RBF. Finally, Section 6 ends the chapter with some concluding remarks.

*Notations*: Scalars, vectors, and matrices are denoted by lower-case, bold-face lower-case, and bold-face higher-case letters, respectively. The transpose and conjugate transpose operators are denoted as (·)*<sup>T</sup>* and (·)*H*, respectively. **E**[·] denotes the statistical expectation. **Tr**(·) represents the trace of a matrix. The distribution of a circularly symmetric complex Gaussian random variable with zero mean and covariance *σ*<sup>2</sup> is denoted by CN (0, *σ*2); and ∼ stands for "distributed as". **<sup>C</sup>***x*×*<sup>y</sup>* denotes the space of *<sup>x</sup>* × *<sup>y</sup>* complex matrices.
