**3.3. Reduced and quantized feedback in OBF/RBF**

In practical systems, only a limited number of bits representing the quantized channel gain/SINR can be sent from each user to the corresponding BS. Note that the feedback schemes in Section 2.1 and [20] require the transmission of 2*MK* and 2*K* scalar values from *K* users, respectively, i.e., a linear increase with respect to the number of users. It is thus of great interest to develop schemes which can reduce the numbers of users and/or bits to be fed back. The idea of using only one-bit feedback is introduced in several works, e.g., [36] [37] [38]. In this scheme, the user sends "1" when the SINR value is above a pre-determined threshold, and "0" vice-versa. Since the performance of OBF/RBF only depends on the favourable channels, one bit of feedback per user can capture almost all gain available due to the multi-user diversity. Optimal quantization strategy for OBF systems with more than one bit feedback is proposed in [36]. It is also worth noting the group random access-based feedback scheme in [39] and the multi-user diversity/throughput tradeoff analysis in [40]. The main tool to study the performance of OBF/RBF under reduced feedback schemes is the large-number-of-users analysis.

#### **3.4. Non-orthogonal RBF and Grassmanian line packing problem**

Denote the space of unit-norm transmit beamforming vectors in **<sup>C</sup>***NT*×<sup>1</sup> as **<sup>O</sup>**(*NT*, 1). A distance function of *<sup>v</sup>*1, *<sup>v</sup>*<sup>2</sup> ∈ **<sup>O</sup>**(*NT*, 1) can be defined to be the sine of the angle between them [41]

$$d(\mathfrak{v}\_1, \mathfrak{v}\_2) = \sin(\mathfrak{v}\_1, \mathfrak{v}\_2) = \sqrt{1 - |\mathfrak{v}\_1^H \mathfrak{v}\_2|^2} \,\,\,\,\tag{27}$$

which is known as the chordal distance. The problem of finding the packing of *M* unit-norm vectors in **<sup>C</sup>***NT*×<sup>1</sup> that has the maximum minimum distance between any pair of them is called the *Grassmannian line packing problem* (GLPP). The GLPP appears in the problem of designing beamforming codebook for space-time codes [42] [43].

Given that the number of transmit beams is less than or equal to the number of transmit antennas, i.e., *<sup>M</sup>* <sup>≤</sup> *NT*, any orthonormal set {*φm*}*<sup>M</sup> <sup>m</sup>*=<sup>1</sup> is a solution of the GLPP. However, assuming that the BS sends *M* > *NT* beams to serve more users simultaneously and improve the fairness of the system, finding {*φm*}*<sup>M</sup> <sup>m</sup>*=<sup>1</sup> is a nontrivial GLPP. The idea of using more than *NT* beams is first proposed in [44] with *M* = *NT* + 1, and further studied in [45]. Zorba *et. al.* argue that the scaling law *M* log2 log *K* is still true for *M* = *NT* + 1 case [44], while the results in [45] imply that non-orthogonal beamforming matrix induces an interference-limited effect on the sum rate, and the multi-user diversity vanishes. Since both studies are based on approximated derivations, more rigorous investigations are necessary before any conclusion is drawn.
