Asymptotic Analysis: Stochastic Geometry MaMISO Regime

In this section, we analyze the asymptotic SE behaviour of MaMISO systems using ESIP-WSR BFs solved using ergodic capacity formulation under partial CSIT. Our analysis is based on the following technical assumptions.

Assumption 5. ∀k,c, lim

M→∞inftr{Dk,c} Mc >0,

Assumption 6. 0<

KcP

k=1

Lk,c

Mc

=∆αc≤1,∀c.

Assumption 7. The long term channel energy captured by tr{D_k,c}(representative of the large scaling fading factors such as path loss and shadowing) is a constant for a fixed number of BS antennnas M_c, i.e. tr{D_k,c}=ηk,c(M_k,c),∀k,c. Also, we assume that lim

M→∞

tr{Dk,c} Mc

=∆δk,c< ∞.

The first assumption (uniform boundedness on the spatial covariance matrix) is essential for the computation of deterministic equivalents using large system analysis results from [14]. This assumption also means that the antenna array gathers an amount of energy which is propor-tional to the number of antennas and moreover they come from different spatial directions. In the following sections, we use an abuse of notation for convenience, when we refer to−−−−→^M→∞

a.s , we refer to the almost sure convergence in the large system limit whereM,L,K→ ∞at a fixed ratio (Assumption 6).

The channel model (7.2) results from multipath propagation and the use of BS side antenna arrays. An example of a geometry based stochastic channel model is provided by the COST2100 channel model [128], and can be depicted as in Figure 7.1. This model represents the propaga-tion channel between a BS and user through multipath components (MPCs) arriving at the user terminals and are resulting from the interaction of the transmitted waveform with a set of objects (also called as scatterers). More recent studies with valid measurement data to justify the signif-icance of the COST2100 channel model can be found in [129, 130]. One particular application of this model for the user covariance matrices is considered in [131]. Wherein, the authors con-sider the scenario in which the support of the multipath angle of arrival or departure (AoA/AoD) for any desired user does not overlap with that of the interfering users. The authors show that the multipath components with AoA/AoD outside the angular support of the desired user tend

to fall in the null space of its covariance matrix in the large antenna limit, leading to orthogo-nal subspacesCin the MaMISO regime. Here we add a stochastic geometry regime, in which the random positions of users and scatterers lead to antenna array responses at random angles.

In reality, the antenna array responses will be more complex than the Vandermonde vectors for Uniform Linear Arrays considered in [131] due to mutual antenna coupling and various other effects. As a result of this randomness of angles and antenna array responses, and due to limited angular support, the multipath channels live in subspaces that are of limited dimension and uni-formly randomly oriented in array response space. As a result, an appropriate random model for the semi-unitary matricesCspanning these subspaces is a Haar distribution. We shall consider that as the number of antennasM grows unboundedly, the subspace dimensionsLalso go to infinity (leading to hardening of the signal power), but slower thanM. As a result, for the large system analysis we may equivalently consider the elements ofCas i.i.d. with zero mean and vari-ance 1/Mso that asymptotically such aCis still semi-unitary:C^HC^M−→^→∞IL. The subspacesCof different channels will be considered independent.

Before we proceed further, in this section, we recall some of the large system results from [14]

we use.

Theorem 8( [14, Theorem 1]). LetQN∈C^N×N be a deterministic matrix andAN=XNX^H_N+SN, withXN contains n independent columns with covariance matrixΘi for i^{t h} column andSN ∈ C^N×N is a Hermitian non-negative definite matrix. Also, assume thatQN,Θi have uniformily bounded spectral norms. Then, for any z>0,

(7.27)

We briefly summarize the Lemma’s here.

Lemma 3( [14, Lemma 4, Appendix VI]). x_N^HANxN−_N¹tr{AN}−−−−→^N→∞ 0when the elements ofxN

are i.i.d with zero mean and variance1/N and independent ofAN, and similarly whenyNis inde-pendent ofxN, thatx^H_NANyN

N→∞

−−−−→

a.s 0.

Lemma 4( [14, Lemma 6, Appendix VI]). LetANbe a deterministic matrix with uniformly bounded spectral norm andB₁, ...,BN be random Hermitian matrices with BN ∈C^N×N and eigenvalues λ1≤....≤λN. Then rank1perturbation lemma states that forv∈C^N,

Lemma 5( [14, Lemma 1, Appendix VI]). We also use the matrix inversion lemma (MIL) through-out the paper. LetA,Care invertible matrices of size N×N and K×K , withBbeing of size N×K , then MIL states that,

(7.29) (A+BCD)⁻¹=A⁻¹−A⁻¹B¡

C⁻¹+DA⁻¹B¢ DA⁻¹.

Theorem 9. In Theorem 8, letQ_k=C_kD_kC_k^H∈C^Mbk×Mbk be a Hermitian deterministic matrix and

k, withCi,bkVi,bk contains L_i,bk independent columns with covari-ance matrixΘi,bk = _M¹IM for r^{t h} column. Λi,bk is a diagonal matrix, with r^{t h} diagonal element

Hence,bk is the the eigenvector corresponding to the maximum eigenvalue, or max eigenvec-tor for short, ofWkC_k^H¡P

i6=kβiSi,b_k+µb_kI¢₋1

Ck. Asymptotically C^H_k ¡P

i6=kβiSi,b_k+µb_kI¢₋1

Ck

converges to a deterministic limit which is a multiple of identity,e_bkI, wheree_bk is obtained as follows, by applying Theorem 1 and other Lemmas described above. First, we consider the eigen decomposition ofW_k,bi=V_k,biΛk,biV_k,b^H

i as the product of a Haar matrix and a unitary matrix remains Haar, so Theorem 1 remains applicable. Since the columns ofCk are independent, we use Lemma 3 to obtainC^H_kΓ⁻_k¹C_k −−−−→^M→∞_{a.s M}¹

bktr{Γ⁻_k¹}IL_k,bk (non-diagonal elements converge to zero). Further, we use Lemma 4 to approximate terms of the form Γ⁻¹_k ≈£

Γk+βkC_kW_kC_k^H¤−1

. Further us-ing Theorem 8, we obtain a deterministic equivalent as _M¹

ctr{Γ⁻¹_k }−_M¹_ctr{Tc(z)}−−−−→^M_a.s^→∞ 0 (with as the unique positive solution of the last expression above, i.e.

(7.33) ec=

Computation of analytical solution ofec is not feasible except in a simplified case of multiple of identity eigenvalue matrixDfor all users, as we do in Appendix I. However, we remark that if we consider randomized position of users across the multi-cell system, we can assume a proba-bility distribution for the user location information (possibly poisson distributed as in classical stochastic geometry [110]). Then this randomized location induces different attenuation (rep-resented by the eigenvalues inD) for the UE channels due to different path loss between them and the BS and can be modeled as having a spatial distribution. In this case, in the MaMISO limit, we can replace the summation over users by its expectation, i.e. _M¹

c

, where the expectation is over the probability distribution of the attenuation factor. We remark that it would be of sufficient interest to analyze the SE behaviour with a ran-dom attenuation factor and it is left for future work. Further, this leads tob_k=Vmax(W_k)=^∆v_k,bk. Finally, we write the optimized BF w.r.t. partial CSIT, in the stochastic geometry MaMISO regime as,

It can be intuitively interpreted as follows: The termCkvk,bk represents beamforming (matched filtering) in the covariance subspaceC_k of the channel from BSb_k to userk. The first termΓk

represents a weighting matrix which converges to the projection matrix on the orthogonal com-plement of the covariance subspace of the leakage channels at high SNR. At any intermediate SNR, BF choses to approximately ZF to a subset of leakage channels. A more detailed interpre-tation using the concept of reduced order ZF BF can be found in [99]. At low SNR,Γkreduces to

µ1_bkI. Thus at low SNR, BF reduces toCkvk,bk, which is just the matched filter. Further, we deduce the optimized BF under the special cases such as perfect CSIT and CoCSIT case. For the perfect CSIT case, For the CoCSIT case, we obtain,

(7.36)

whereei,max represents the unit vectorei corresponding toDi,iwhich is the maximum among all the eigenvalues. In fact, (7.35) tells us that BF is along only one of the unitary vectors in the covariance subspaceCk, which leads to a reduction signal power which accounts for the rate offset as in described in the Corollary 12.4.

7.1.7 Computation of eigenvalues of Wk,b_i

To computee_c(z) in the large system expressions as in (7.32), the eigenvalues ofWk,bi need to be known which we discuss here. For the convenience of analysis we omit the user and BS index

here. We representWbyWL,WSfor LMMSE and subspace channel estimators respectively. From