As noted in the introduction, we resort to large system analysis in which the number of antennas at each base station and the number of mobiles in each cell grow to infinity, their ratio NK
t, which we also refer to as the cell loading, converging to a constant β. For all cases, we consider the dual of the original (primal) problem [55]: the optimal beamforming vectors in each case can be shown (details are given in Appendix 7.A, 7.B and 7.C for single cell processing, coordinated beamforming and joint transmission) to be of the form:
where the µj’s and λk,j’s correspond to the Lagrange coefficients in the dual problems and may be interpreted as noise powers and UL transmit powers on dual ULs, respectively. Large system analysis is undertaken to obtain a simplification, while still reaching meaningful conclusions for the finite-size system. We thus solve the asymptotic dual problems, obtaining asymptotically optimal noise and UL power levels. This is done in Appendix
7.4 Large system analysis 147 7.D, 7.E and 7.F. For feasible γ, we find the optimal dual parameters in terms of the system parameters, namely and the cell loading parameter β. We then obtain the corresponding maximum feasibleγ for each scheme.
Finally, we characterize the optimumβin terms of total network throughput maximization while maintaining a given SINR across all users. This is done in the following series of theorems.
7.4.1 Asymptotically Optimal Beamformers
Theorem 6 (Asymptotically optimal beamforming for SCP). Assume β
γ
1+γ +γ
< 1. To asymptotically achieve SINR γ at each mobile ter-minal in the two-cell network as Nt → ∞ with NK
t → β > 0, beamforming vectors of the following form are asymptotically optimal:
wSCPkj =
Proof. See Appendix 7.D.
Corollary 2. Subject to per base station power constraintP, the maximum asymptotic network-wide achievable SINR for a given cell loading factor β is the unique solution to the following fixed point equation:
γ∗SCP = 1
148 Chapter 7 Large System Analysis for Beamforming Design
Note 2 (Relation to regularized zero-forcing). Note that the above optimal beamformers asymptotically correspond to a precoding matrix at each of the two transmitters given by
WjSCP =cj where cj is such that the power constraint is met with equality at BSj, and Hj,j is the concatenation of the channels between the users in cellj and their serving base station. This is the regularized zero-forcing scheme proposed in [85] and studied in the asymptotic regime in [86].
Theorem 7 (Asymptotically optimal beamforming for CBf). Assume β
γ
1 +γ + γ 1 +γ
<1. To asymptotically achieve SINR γ at each mobile terminal in the two-cell network asNt→ ∞with NK
t →β >0, beamforming vectors of the following form are asymptotically optimal:
wkjCoord= Proof. See Appendix 7.E.
Corollary 3. Subject to per base station power constraintP, the maximum asymptotic network-wide achievable SINR for a given cell loading factor β is the unique solution to the following fixed point equation:
γCoord∗ = 1
7.4 Large system analysis 149
Theorem 8 (Asymptotically optimal beamforming for MCP). Suppose
βγ
1+γ <1. To asymptotically achieve SINR γ at each mobile terminal in the two-cell network as Nt→ ∞ with NK
t → β >0, beamforming vectors of the following form are asymptotically optimal:
wM CPkj =
Proof. See Appendix 7.F.
Corollary 4. Subject to per base station power constraintP, the maximum asymptotic network-wide achievable SINR for a given cell loading factor β is the unique solution to the following fixed point equation:
γM CP∗ = 1
The optimal SINR expressions above are striking in how they capture the effect of interference in the three different beamformers. Indeed, they sup-ply a simple “effective interference” characterization, which can be used to directly check if a particular target SINR can be achieved.
It is natural to compare the schemes directly using the limiting SINR ex-pressions. This is accomplished in the following theorem, whereSN R, σP2.
150 Chapter 7 Large System Analysis for Beamforming Design
Theorem 9. Let γSCP∗ , γ∗Coord, γM CP∗ denote the SINR’s under SCP, coor-dinated beamforming, and MCP, respectively. Then
γSCP∗ < γCoord∗ < γM CP∗ . (7.30) At signal to noise ratio,SN R, and interference level, , denote the effective interference at target SINR,γ, by:
Ieff(SN R, , γ) =
β
1 +SN R1+γ +SN R
SCP β
1 +SN R1+γ +SN R1+γ
CBf β
1 +SN R1+γ +SN R1+γ
MCP
(7.31)
Then the feasibility of γ in the case of SCP, or CBf, is equivalent to sat-isfaction of the inequality SN R
Ieff(SN R, , γ) > γ, and in the MCP case, it is equivalent to (1 +)SN R
Ieff(SN R, , γ) > γ.
Proof. Follows closely the proof of Proposition 3.2 in [75].
We note the close parallel with the notion of effective interference that arises in the large system analysis of linear UL multiuser receivers [75].
7.4.3 Asymptotically optimal cell loading
The above theorems characterized the optimal SINR under the various schemes considered for fixed cell loadingβ. Figure 7.2 shows, for=.5, the achievable rates per cell per antenna for different values of the SNR for the coordinated beamforming scheme. Clearly there are system-level gains to be attained by selecting the optimal β. This conclusion holds for the other schemes as well. Our next step is to characterize this optimum loading. This corresponds to finding theβ that maximizes the normalized (by the number of antennas) rate per cellr, i.e. solving the following problem:
maximizeβ r=βlog(1 +γ∗) (7.32) with γ∗ characterized by the appropriate fixed-point equation (cf. Eqs (7.18), (7.24) and (7.29)).
Theorem 10 (Characterization of the optimumβ for SCP). If +σ2
P ≥1, (7.33)
7.4 Large system analysis 151
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0 1 2 3 4 5 6
β
Rcell (bits/sec/Hz)
PdB = 0 PdB = 10 PdB = 20 PdB = 30 PdB = 40
Increasing SNR
Figure 7.2: Achievable rates by asymptotically optimal coordinated beam-forming for =.5 and different SNR values.
β∗ → ∞. Otherwise, the optimum occurs at a finite β∗ which may be found by a line search.
Proof. Refer to Appendix 7.G.
Theorem 11 (Optimal cell loading for CBf). If σP2 +−22−1≥0, one can increase β indefinitely. Otherwise, there is a finite value at which r is maximized.
Proof. The proof follows along similar lines to that of Theorem 10, although the algebra is more tedious. Refer to Appendix 7.H.
Theorem 12 (Characterization of the optimum β for MCP). If σ2
P ≥(1 +), (7.34)
β∗ → ∞. Otherwise, the optimum occurs at a finite β∗ that may be found by a line search.
Proof. (7.29) is the same as (7.18) with+ σP2 ← P(1+)σ2 . Performing this substitution in (7.33) yields the result.
The above results define for each scheme a noise-limited region, where cell loading can be increased indefinitely; however, this leads to ever decreasing rates per user, not to mention that more user channels need to be learned.
152 Chapter 7 Large System Analysis for Beamforming Design