Large multi-antenna stochastic geometry

Large Multi-Antenna Stochastic Geometry

Christo Kurisummoottil Thomas,Dirk Slock

Communication Systems Department, EURECOM

IEEE ITA Workshop,

February 14, 2019, San Diego, CA, USA

Large Multi-Antenna Stochastic Geometry

Outline

1 Introduction

2 Asymptotic Analysis for Massive MISO Partial CSIT Model

Optimal EWSR BF

Large System Analysis for Optimal Beamforming

3 Numerical Results, Conclusion and Future Work

Large Multi-Antenna Stochastic Geometry

Fake News

Original title:

Optimizing CSIT for Multi-User MIMO - Beyond LMMSE

-20 -10 0 10 20 30 40 50 60

Transmit SNR in dB 0

20 40 60 80 100 120 140

Sum Spectral Efficiency

iCSIT

Optimal EWSR BF: Subspace Channel Estimate Optimal EWSR BF: Unbiased Subspace Channel Estimate naive BF: Subspace Channel Estimate

Optimal EWSR BF: LMMSE Channel Estimate naive BF: LMMSE Channel Estimate EWSMSE BF: LMMSE Channel Estimate naive BF: LS Channel Estimate Optimal EWSR BF: LS Channel Estimate Optimal EWSR BF: Unbiased LS Channel Estimate

Large Multi-Antenna Stochastic Geometry

Issues

MaMIMO systems may benefit from exploiting channel covariance information (eg coCSIT ZF)

Large System Analysis of MaMIMO involves covariance matrices of all channels [isit’17]: a mess, no insights

UseStochastic Geometry extended to multi-antenna systems to randomize covariance subspaces, simplifying large system results.

utility exploiting partial CSIT: naive, Expected Weighted Sum MSE, MaMIMO limit of Expected Weighted Sum Rate which channel estimate: least-squares, LMMSE, subspace projected LS

Large Multi-Antenna Stochastic Geometry

Motivation

In stochastic geometry based methods, the location of the users is assumed to be random, leading to randomized channel attenuations.

The multipath propagation for the various users leads to randomized angles of arrival at the BS which can be translated into spatial channel response contributions that depend on the antenna array response

In Massive MIMO systems, the channel covariance matrix tends to be low rank (few significant scattering clusters).

Exploiting these, we propose to model the user channel subspaces as isotropically randomly oriented. This allows us to assume the eigen vectors of the channel covariance matrix to be Haar distributed, and this identically and independently for all users.

Large system analysis gives analytical insights into the performance of the various BFs.

Large Multi-Antenna Stochastic Geometry

State of the Art

In [Wagner:TIT12], the authors obtain deterministic (instead of channel realization dependent) expressions for various scalar quantities.

Helps to evaluate the system performance without resorting to heavy monte-carlo simulations.

In [Lakshminarayana:TIT15] or [Muller:TSP15], MISO IBC is considered with perfect CSIT and weighted Regularized Zero-Forcing (R-ZF) BF, with two optimized weight levels, for intracell or intercell interference.

Large system analysis under partial CSIT for optimal BF [Tabikh:ISIT17], but analysis quite cumbersome.

In [Thomas:SAM18], we applied large system analysis to reduced order ZF beamforming for the simplified case of differently attenuated channel covariance matrices of the users.

Large Multi-Antenna Stochastic Geometry

Outline

1 Introduction

2 Asymptotic Analysis for Massive MISO Partial CSIT Model

Optimal EWSR BF

Large System Analysis for Optimal Beamforming

3 Numerical Results, Conclusion and Future Work

Large Multi-Antenna Stochastic Geometry

Channel and CSIT Model

We start from a deterministic Least-Squares (LS) channel estimate

bhLS =h+eh, eh∼ CN(0,eσ²I)

Exploitation of channel covariance information: LMMSE, LS and Subspace channel estimators considered.

Each MISO channel h follows Karhunen-Loeve representation, h=C c, c=D^1/2c⁰,

wherec⁰ ∼ CN(0,I_L) andDis diagonal. HereC is the M×L eigen vector matrix of the reduced rank channel covariance R_hh =CDC^H.

The total sum rank across all users Np =

K

X

k=1

Lk,c is assumed to be less than M_c, where L_k,c is the channel rank between userk and BS c.

Large Multi-Antenna Stochastic Geometry

Different Channel Estimates

In the MaMIMO limit, BF design with partial CSIT will depend on the quantities S= E_h|bh hh^H

=bhbh^H+Θ.e LS Channel Estimate: bhLS =h+ehwhereh andeh are independent. In the LS case,Θe =σe²I.

LMMSE Channel Estimate: h=bh+ehin whichhbandeh are decorrelated and hence independent in the Gaussian case. In the LMMSE case,Θe =C

ehehis the posterior covariance.

S=CWC^H, whereW=Db^1/2bcbc^HDb^1/2+D.e

Subspace Projection based Channel Estimate: Effect of limiting channel estimation error to the covariance subspace (without the LMMSE weighting, this is a simplification of the LMMSE estimate).

bhS =PCbhLS =h+PCehLS, C

eh_Seh_S =eσ²PC,

wherePC =C(C^HC)⁻¹C^H represents the projection onto the covariance subspace.

Large Multi-Antenna Stochastic Geometry

Optimality of LMMSE

In the MaMIMO limit, BF design with partial CSIT will depend on the quantities S= E_h|bh hh^H

=bhbh^H+Θ.e LMMSE Channel Estimate: h=bh+ehin whichhbandeh are decorrelated and hence independent in the Gaussian case. In the LMMSE case,Θe =C

ehehis the posterior covariance.

S=CWC^H, whereW=Db^1/2bcbc^HDb^1/2+D.e Gaussian LMMSE bh= E_h|bhh

Nonlinear MMSEestimate : S= E_h|bh hh^H

=bhbh^H+C

eheh

(Bayesian) unbiased: E

bhS= Ehh^H =CDC^H Unbiased and MMSE ⇒ minimum variance.

Also unbiased minimum variance estimate for

|g^Hh|²= (g^T⊗g^H)vec(hh^H)

Large Multi-Antenna Stochastic Geometry

BF with Partial CSIT: Expected WSR

Rx signal at user k,

yk =h^H_k,bkgkxk +

K

X

i=1,6=k

h^H_k,bi gixi+vk. MI Gaussian signaling, Expected Weighted Sum Rate (EWSR):

EWSR = Eh K

X

k=1

uk ln(1 + |h^H_k,b

kg_k|² 1 +

K

X

i=1,6=k

|h^H_k,b

ig_i|² )

= Eh K

X

k=1

uk (ln(sk)−ln(s_k)) s_k = 1 +

K

X

i=1,6=k

|h^H_k,b

igi|², sk =s_k+|h^H_k,b

kgk|²

Large Multi-Antenna Stochastic Geometry

BF with Partial CSIT: Expected WSR (2)

Naive Massive MISO (K → ∞): s_k,s_k →E_hs_k,E_hs_k. ESEI-WSR.

More precisely: upper bound:

EWSR = Eh K

X

k=1

uk ln(1 + |h^H_k,b

kg_k|² s_k )

≈Eh K

X

k=1

uk ln(1 +|h^H_k,b

kg_k|² E_hs_k )

≤

K

X

k=1

uk ln(1 +E_h|h^H_k,b

kg_k|²

E_hs_k ) =ESEI −WSR Eh|h^H_k,b

kgi|²=g^H_i (Ehhk,bkh^H_k,b

k)gi

Large Multi-Antenna Stochastic Geometry

ESEI-WSR/EWSR gap

We will need: E_h|bh|h^H_k,b

kg_i|² =g^H_i S_k,bkg_i,r_k = E_h|bhs_k

gap= ln(1 +

E_h|bh|h^H_k,b

kg_k|²

r_k )−E_h|bhln(1 + |h^H_k,b

kg_k|² r_k )≤γ gap(snr) increases monotonically, gap(0) = 0,

gap(∞)≤γ = 0,577 [Gopala:icassp18]:

gap=γ−gap_c

wheregap_c is the gap fom EWSR to EWSR in whichbh gets replaced by abhwhere a∼ CN(0,1) : degrading Rice to Rayleigh with same channel correlation matrix.

Large Multi-Antenna Stochastic Geometry

Different BFs with Partial CSIT

In the case of partial CSIT we get for the Rx signal, y_k =bh^H_k,b

kg_kx_k+ eh^H_k,bkg_kx_k

| {z } sig. ch. error +

K

X

i=1,6=k

(bh^H_k,bi g_ix_i + eh^H_k,bi g_ix_i

| {z } interf. ch. error

) +v_k.

Naive EWSR BF : just replaceh bybh in a perfect CSIT approach. Ignore eh everywhere.

Optimal ESEI-WSR BF:accounts for covariance CSIT in the signal and interference terms.

EWSMSE BF (Expected Weighted Sum MSE)

[NegroSlock:ISWCS2012]: accounts for covariance CSIT in the interference terms, but also associates the signal eh term with the interference !

Large Multi-Antenna Stochastic Geometry

Max EWSR BF in the MaMISO Limit

EWSR = E_bhmaxgEWSR(g), EWSR(g) =

K

X

k=1

uk E_h|bhln(sk/s_k)^(a)=

K

X

k=1

uk ln((E_h|bhsk)/(E_h|bhs_k)), (a) =⇒ the MaMISO limit which isESEI-WSR (Expected Signal Expected Interference WSR),s_k is the interference plus noise power and s_k is the signal plus interference plus noise power.

Note that we definer_k =E_h|bh(s_k),r_k =E_h|bh(s_k).

Optimizing this leads to thegeneralized eigen vector solution along with ILA-WF user powers,

g⁰_k ∝[X

i6=k

β_iS_i,bk+µ_bkI]⁻¹S_k,bkg⁰_k, β_k=u_k(1 r_k − 1

r_k), g_k =g_k⁰p_k^1/2.

Large Multi-Antenna Stochastic Geometry

Max EWSR BF in the MaMISO Limit Continued ...

ILA-WF for user powers, p_k =

uk

σ⁽²⁾_k +µ_bk − ¹

σ_k⁽¹⁾

+

,

where (x)⁺= max{0,x}and the Lagrange multipliers µ_c are adjusted (e.g. by bisection) to satisfy the power constraints.

Also, σ⁽¹⁾_k =r⁻¹

k g⁰_k^HSk,b_kg⁰_k (direct link power) σ⁽²⁾_k =g⁰_k^HP

i6=kβiSi,b_kg⁰_k (cross links leakage power).

Under stochastic geometry MaMIMO regime, optimal EWSR BF =⇒

g⁰_k ∝ [X

i6=k

β_iS_i,bk+µ_bkI]⁻¹C_k,bkVmax(W_k,bk),

Large Multi-Antenna Stochastic Geometry

Stochastic Geometry MaMIMO Regime

An appropriate model for the covariance subspaces C (independent across different user channels) is a Haar distribution (randomly oriented semi-unitary matrices).

However, as we shall consider that the rank Lremains finite, whereas M grows unboundedly, for the large system analysis we may equivalently consider the elements ofC as i.i.d. with zero mean and variance 1/M so that the expected squared norm (and asymptotically the actual squared norm) of the columns of C is normalized to 1.

From [Wagner:TIT12] that any terms of the form

1

Ntr{(A_N−zI_N)⁻¹}, whereA_N is the sum of independent rank one matrices with covariance matrix Θ_i is equal to the unique positive solution ofej = _N¹tr{(PK

i=1 Θi

1+ei −zI_N)⁻¹}.

Large Multi-Antenna Stochastic Geometry

Stochastic Geometry MaMIMO Regime Continued...

Considering the eigen decomposition ofW_i,bk =V_i,bkΛ_i,bkV_i^H_,b

k, whereΛ_i,bk = diag(ζ_i,b⁽¹⁾

k, ..., ζ_i,b^(Li,bk)

k ). Further terms of the following form in SINR,

C^H_k,b

k[X

i6=k

β_iC_i,bkW_i,bkC_i,b^H

k+µ_bkI]⁻¹C_k,bk =

1

M_bktr{[X

i6=k

βiC_i,bkWi,b_kC_i,b^H

k+µb_kI]⁻¹}I_Lk,

bk =eb_k. Using large system analysis results [Wagner:TIT12] simplifies it to the scalar,e_bk, =⇒

e_bk =



_M¹

bk

K

X

i=1 Li,bk

X

r=1

β_iζ_i,b^(r)

k

1 +β_iζ_i,b^(r)

ke_bk +µ_bk





−1

, Also, we define,B_k,bi = diag(1 +β_kζ_k,b⁽¹⁾

ie_bi, ...,1 +β_kζ_k,b^(L)

ie_bi), Ee_k =e_b⁻¹

k De_k,bk+I,e_b⁰

kis the derivative ofe_bk w.r.tµ_bk.

Large Multi-Antenna Stochastic Geometry

Stochastic Geometry MaMIMO Regime Continued...

For theLMMSE case,

W=F(bcbc^H+σe²I)F^H+ (I−F)D(I−F)^H

whereF = (I+eσ²D⁻¹)⁻¹,bc=C^HbhLS,Fbc=bcLMMSE orbcL for short.

So,W=bcLbc^H_L +σe²D(σe²I+D)⁻¹. Athigh SNR, up to first order inσe², W=bcLbc^H_L +σe²I where the first term contains first-order terms also. At low SNR,F ≈eσ⁻²D and

W=bcLbc^H_L + (I+σe⁻²D)⁻¹D ≈D.

Forany SNR, these two extremes can be connected by the following approximation:

Λ = eσ²D(eσ²I+D)⁻¹+||bcL||²e1e₁^H

= eσ²D(eσ²I+D)⁻¹+ tr{D(I+eσ²D⁻¹)⁻¹}e1e₁^H where the last equality is due to the LLN.

Large Multi-Antenna Stochastic Geometry

Large System Results

In the large system limit (M,K → ∞,with_M^K =c <1), the quantities r_k−r_k −−−−−→^Mbk→∞

a.s 0 and rk−rk

M_bk→∞

−−−−−→

a.s 0. By applying the continuous mapping theorem, it follows from the almost sure convergence of r_k andr_k that,

R_k−R_k −−−−−→^Mbk→∞

a.s 0, whereR_k is the rate of user k, with R_k = ln(^r_r^k

k). The deterministic limits are obtained as,

r_k= 1 + Υ_k,rk= 1 + Υ_k+pk e²

bkλmax(W_k,bk) e⁰

bk ,where, Υ_k=

K

X

i=1, i6=k

pi

1 Mbi

[

Lk,bi

X

r=1

ζ_k,b^(r)

i

(1 +βkζ_k,b^(r)

ieb_i)² ],

β_k=uk

1 r_k−_r¹

k

.

Large Multi-Antenna Stochastic Geometry

Rate Expressions (At High SNR)

R^OptBF_k = ln(1 +p_kλ_max(W_k,bk)), R^EWSMSE_k = ln(1 + ^{pkλmax(Wk,bk(I−eE}

−1 k )) 1+

K

P

i=1,i6=k

p_itr^{Λk,biB−2 k,bi} Li,biMbi

1 (1−2tr^{eE−1

i } Li,bi +tr^{eE−2

i } Li,bi )

).

Conclusion:

The suboptimal EWSMSE BF doesn’t ZF at high SNR and the interference power also increases with SNR

→ ∞ =⇒ large gap in performance between the optimal BF and EWSMSE BF.

We consider the scenario of the channel estimation error remaining finite with SNR (otherwise the channel estimate becomes perfect at high SNR and thus all the BFs converg to the same solution).

Large Multi-Antenna Stochastic Geometry

Outline

1 Introduction

2 Asymptotic Analysis for Massive MISO Partial CSIT Model

Optimal EWSR BF

Large System Analysis for Optimal Beamforming

3 Numerical Results, Conclusion and Future Work

Large Multi-Antenna Stochastic Geometry

Simulation Setup

C, number of cells. K_c, number of (single-antenna) users in cell c andK =X

c

Kc. M, number of transmit antennas in each cell.

We consider a path-wise or low rank channel model, with L= number of paths = channel covariance rank. d : scale factor in the LS channel estimation error variance eσ²=d/SNR.

The elements of C are generated i.i.d complex Gaussian with variance 1/(2M).

Notations: in the figures, iCSIT refers to the optimal BF design for the instantaneous CSIT case

[Christensen:TWC2008].

Large Multi-Antenna Stochastic Geometry

Spectral Efficiency Results

0 10 20 30 40 50 60 70 80

Transmit SNR in dB 0

20 40 60 80 100 120 140 160

Sum Spectral Efficiency

iCSIT

Optimal EWSR BF: LMMSE Channel Estimate Optimal EWSR BF: Large System Approximation EWSMSE BF: LMMSE Channel Estimate Naive BF: LMMSE Channel Estimate

EWSR forC= 1 cell,K = 10 users,M= 64,L= 4, eσ²= 0.1.

Large Multi-Antenna Stochastic Geometry

Spectral Efficiency Results

-20 -10 0 10 20 30 40 50 60

Transmit SNR in dB

0 20 40 60 80 100 120

Sum Spectral Efficiency

iCSIT

Optimal EWSR BF: LMMSE Chnl. Est.

Optimal EWSR BF: Subspace Chnl. Est.

EWSMSE BF: LMMSE Chnl. Est.

Optimal EWSR naive BF: LMMSE Chnl. Est.

Optimal EWSR naive BF: Subspace Chnl. Est.

Optimal EWSR BF: LS Chnl. Est.

Optimal EWSR naive BF: LS Chnl. Est.

Optimal EWSR BF: unbiased LS Chnl. Est.

EWSR forC = 1 cells,K1=K = 10 users,M = 64,L= 2,eσ²= 1/SNR.

Large Multi-Antenna Stochastic Geometry

Spectral Efficiency Results

-20 -10 0 10 20 30 40 50 60

Transmit SNR in dB

0 20 40 60 80 100 120

Sum Spectral Efficiency

iCSIT

Optimal EWSR BF: LMMSE Chnl. Est.

Optimal EWSR BF: Subspace Chnl. Est.

EWSMSE BF: LMMSE Chnl. Est.

Optimal EWSR naive BF: LMMSE Chnl. Est.

Optimal EWSR naive BF: Subspace Chnl. Est.

Optimal EWSR BF: LS Chnl. Est.

Optimal EWSR naive BF: LS Chnl. Est.

Optimal EWSR BF: unbiased LS Chnl. Est.

EWSR forC = 2 cells,K1=K2= 5 users, M= 32,L= 2, eσ²= 1/SNR.

Large Multi-Antenna Stochastic Geometry

Conclusion and Future Work

We introduced a stochastic geometry inspired randomization of the channel covariance eigen spaces and analyzed the large system behavior with an optimal precoder under partial CSIT.

Moreover, we show theimprovement in performance by using an LMMSE channel estimate compared to just having LS estimates, and by furthermore properly exploiting all covariance information.

Simulation results indicated that large system approximations are very accurate even for small system dimensions and reveal the deterministic dependence of the system performance on several important system parameters, such as the channel attenuation, signal powers and SNR.

Large Multi-Antenna Stochastic Geometry