The pathwise MIMO interfering broadcast channel

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The Pathwise MIMO Interfering Broadcast Channel

Wassim Tabikh, Dirk Slock& Yi-Yuan-Wu^#

Mobile Communications Department, EURECOM

#Orange Labs, Issy-les-Moulineaux, France

Information Theory and Applications (ITA) Workshop San Diego, CA, Feb. 2, 2015

Dirk Slock The Pathwise MIMO Interfering Broadcast Channel, ITA2015, Feb. 2 1/51

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MIMO Interfering Broadcast Channel (IBC)

. .

BS1 .

. .

. ^U^1,1

. .

. ^U^1,K1

. . .

. .

BS_𝐶 .

. .

. ^U^𝐶,1

. .

. ^U^C,KC

. . . . .

.

𝐇1,1,1

𝐇1,K1,1

𝐇_𝐶,1,1 𝐇_𝐶,𝐾_𝐶_,1

𝐇1,1,𝐶

𝐇_1,K₁_,𝐶 𝐇𝐶,1,C

𝐇𝐶,K𝐶,𝐶

Cell 1

Cell C

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Possible Application Scenarios

Multi-cell cellular systems, modeling intercell

interference.

Difference from Network MIMO: no exchange of signals, ”only” of channel impulse responses.

HetNets: Coexistence of macrocells and small cells, especially when small cells are considered part of the cellular solution.

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Weighted Sum Rate w Partial CSIT: Motivation

Interference Alignment (IA): showed that interference is not a fatality, essentially half of the interference-free rate can be attained requires perfect global CSIT, focuses on high SNR

various IA flavors:

asymptotic IA(infinite symbol extension & symbol extension), ergodic IA[NazerGastparJafarVishwanath:IT1012] explains the factor 2 loss in DoF of SISO IA w.r.t. an interference-free Tx scenario by transmitting the same signal twice at two paired channel uses in which all cross channel links cancel out each other: group channel realizationsH1,H2s.t. offdiag(H2) =−offdiag(H1).

Linear IAorSignal Space IA orMIMO IA: use MIMO dimensions:

no latency, but DoF factor 2

K+ 1: considered here.

MIMO IA: joint Tx/Rx ZF

at finite SNR: Tx & Rx not ZF but MMSE-style

CSIT in practice: never perfect, partial due to feedback (estimation, quantization, noisy FB) or worse: e.g. covariance CSIT, location based, etc. (esp. for intercell CSIT)

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Introduction

Linear IA for MIMO (I)(B)C:design of any BS Tx filter depends on all Rx filters whereas in turn each Rx filter depends on all Tx filters

[Negro:ita11]. As a result, all Tx/Rx filters are globally coupled and their design requires global CSIT.

Massive MIMO from a DoF point of view it may seem like a suboptimal use of antennas. However [Wagner:IT0712, section V, Fig. 6],K^opt decreases as SNR decreases (also: Net DoF considerations and CSI acquisition).

[Caire:IT1013],[Caire:mmWave]: MISO single cell. Statistical CSIT between user groups, instantaneous CSIT within user groups. Hypothesis:

some users overlap strongly in terms of covariance subspaces but not in terms of instantaneous CSIT. However, users can be differentiated with instantaneous CSIT iff they can be differentiated pathwise. Also, we advocate the use of 2D antenna arrays for 3D beamforming (BF) which should enhance path differentiability.

[Lau:Hierarchical:SP0413] MISO hierarchical approach : Intercell statistical CSIT ZF BF, treating interfering links in a binary fashion (either ZF or ignore). Intracell BF is based on instantaneous CSIT and performs Regularized-ZF.

”Massive MIMO” = ”MU Massive MISO”. Here: actual MU MC Massive MIMO.

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Introduction (2)

Whereas path CSIT by itself may allow ZF, which is of interest at high SNR, we are particularly concerned here with maximum Weighted Sum Rate (WSR) designs accounting for finite SNR.

Massive MIMO makes the pathwise approach viable: the (cross-link) BF can be updated at a reduced (slow fading) rate, parsimonious channel representation facilitates not only uplink but especially downlink channel estimation, the cross-link BF can be used to significantly improve the downlink direct link channel estimates, minimal feedback can be introduced to perform meaningful WSR optimization at a finite SNR (whereas ZF requires much less coordination).

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Key Points

state of the art:

MIMO IA requires global MIMO channel CSIT recent works focus on intercell exchange of only scalar quantities, at fast fading rate

2-stage approaches with intercell interference zero-forced Massive MISO →Massive MIMO

Pathwise: allows updating at slow fading rate Pathwise: may allow IA at reduced DoF

don’t waste resources for ZF but do max weighted sum rate Key contributions:

pathwise beamformer via uplink-downlink LMMSE duality not only for data Tx but also for channel estimation

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Max WSR Tx BF Design Outline

max WSR Tx BF design with perfect CSIT using WSR - WSMSE relation

from difference of concave to linearized concave MIMO BC: local optima, deterministic annealing Gaussian partial CSIT

max EWSR Tx design w partial CSIT Line of Sight (LoS) based partial CSIT max EWSR Tx design with LoS based CSIT

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MIMO IBC with Linear Tx/Rx, single stream

IBC withC cells with a total ofK users. System-wide user numbering:

theNk×1 Rx signal at userkin cellbk is y_k=Hk,b_kgkxk

| {z } signal

+X

i6=k b_i=b_k

Hk,b_kgixi

| {z } intracell interf.

+X

j6=b_k

X

i:b_i=j

Hk,jgixi

| {z }

intercell interf.

+vk

wherexk = intended (white, unit variance) scalar signal stream,Hk,b_k = Nk×Mb_k channel from BSbk to userk. BSbkservesKb_k=P

i:b_i=b_k1 users. Noise whitened signal representation⇒ vk∼ CN(0,IN_k).

TheMb_k×1 spatialTx filterorbeamformer (BF) isgk.

Treating interference as noise, userk will apply a linearRx filterfk to maximize the signal power (diversity) while reducing any residual interference that would not have been (sufficiently) suppressed by the BS Tx. The Rx filter output isbxk=f_k^Hy_k

xbk = fk^HHk,b_kgkxk+

K

X

i=1,6=k

fk^HHk,b_igixi+fk^Hvk

= f_k^Hhk,kxk+X

i6=k

fk^Hhk,ixi+fk^Hvk

wherehk,i =Hk,b_igi is the channel-Tx cascade vector.

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Max Weighted Sum Rate (WSR)

Weighted sum rate (WSR)

WSR =WSR(g) =

K

X

k=1

uk ln 1 e_k whereg={g_k}, theu_k are rate weights MMSEsek =ek(g)

1

e_k = 1 +g^H_kH^H_kR⁻¹

k H_kg_k = (1−g_k^HH^H_kR⁻¹_k H_kg_k)⁻¹ R_k =R_k+H_kg_kg_k^HH^H_k , R_k =P

i6=kH_kg_ig^H_i H^H_k +I_N_k , R_k,R_k = total, interference plus noise Rx cov. matrices resp.

MMSEek obtained at the outputxbk =f_k^Hy_k of the optimal (MMSE) linear Rx

f_k =R⁻¹_k H_kg_k .

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From max WSR to min WSMSE

For a general Rx filter f_k we have the MSE e_k(f_k,g)

= (1−f_k^HH_kg_k)(1−g^H_kH^H_kf_k) +P

i6=kf_k^HH_kg_ig_i^HH^H_kf_k+||f_k||²

= 1−f_k^HH_kg_k−g^H_kH^H_kf_k+P

if_k^HH_kgig_i^HH^H_kf_k+||f_k||². The WSR(g) is a non-convex and complicated function of g.

Inspired by [Christensen:TW1208], we introduced

[Negro:ita10],[Negro:ita11] an augmented cost function, the Weighted Sum MSE,WSMSE(g,f,w)

=

K

X

k=1

uk(wk ek(fk,g)−lnwk) +λ(

K

X

k=1

||g_k||²−P) whereλ= Lagrange multiplier and P = Tx power constraint.

After optimizing over the aggregate auxiliary Rx filters f and weightsw, we get the WSR back:

min

f,w WSMSE(g,f,w) =−WSR(g) +

constant z }| {

K

X

k=1

u_k

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From max WSR to min WSMSE (2)

Advantage augmented cost function: alternating optimization

⇒ solving simple quadratic or convex functions minwk

WSMSE ⇒ w_k = 1/e_k minfk

WSMSE ⇒ f_k= (X

i

H_kgig^H_i H^H_k+I_N_k)⁻¹H_kg_k ming_k WSMSE ⇒

gk= (P

iuiwiH^H_i fif_i^HHi+λIM)⁻¹H^H_kfkukwk

UL/DL duality: optimal Tx filter g_k of the form of a MMSE linear Rx for the dual UL in which λplays the role of Rx noise variance andukwk plays the role of stream variance.

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Optimal Lagrange Multiplier λ

(bisection) line search onP_K

k=1||g_k||²−P = 0 [Luo:SP0911].

Orupdated analytically as in [Negro:ita10],[Negro:ita11] by exploitingP

kg_k^H^∂WSMSE_∂g∗

k = 0.

This leads to the same result as in [Hassibi:TWC0906]: λ avoided byreparameterizing the BF to satisfy the power constraint: g_k =q _P

PK

i=1||g⁰_i||²g⁰_k with g_k⁰ now unconstrained SINRk = |f_kH_kg⁰_k|²

PK

i=1,6=k|f_kHkg⁰_i|²+_P¹||f_k||²PK

i=1||g⁰_i||² . This leads to the same Lagrange multiplier expression obtained in [Christensen:TW1208] on the basis of aheuristic that was introduced in [Joham:isssta02] as was pointed out in [Negro:ita10].

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From WSR to WSSINR

The WSR can be rewritten as WSR =WSR(g) =

K

X

k=1

u_k ln(1 + SINR_k) where 1 + SINR_k = 1/e_k or for general f_k :

SINR_k = |f_kH_kg_k|² PK

i=1,6=k|f_kH_kg_i|²+||f_k||² . WSR variation

∂WSR =

K

X

k=1

u_k

1 + SINR_k ∂SINR_k

interpretation: variation of a weighted sum SINR (WSSINR) The BFs obtained: same as for WSR or WSMSE criteria.

But this interpretation shows: WSR = optimal approach to the SLNR or SJNR heuristics.

WSSINR approach = [KimGiannakis:IT0511] below.

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[KimGiannakis:IT0511] Difference of Convex fns (DC)

Let Q_k =g_kg^H_k be the transmit covariance for stream k ⇒ WSR =

K

X

k=1

u_k[ln det(R_k)−ln det(R_k)]

w R_k =H_k(P

iQ_i)H^H_k +I_N_k,R_k =H_k(P

i6=kQ_i)H^H_k +I_N_k. Consider the dependence of WSR onQk alone:

WSR =ukln det(R⁻¹

k Rk)+WSR_k, WSR_k =

K

X

i=1,6=k

uiln det(R⁻¹_i Ri) where ln det(R⁻¹

k R_k) is concave in Q_k and WSR_k is convex in Qk. Since a linear function is simultaneously convex and concave, consider the first order Taylor series expansion inQ_k aroundQb (i.e. allQb_i) with e.g. Rb_i =R_i(Q), thenb

WSR_k(Q_k,Qb)≈WSR_k(Qb_k,Q)b −tr{(Q_k−Qb_k)Ab_k} Ab_k =− ∂WSR_k(Qk,Q)b

∂Qk

Qbk,Qb

=

K

X

i=1,6=k

u_iH^H_i (bR⁻¹_i −Rb⁻¹_i )H_i

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[KimGiannakis:IT0511] DC (2)

Note that the linearized (tangent) expression forWSR_k constitutes a lower bound for it.

Now, dropping constant terms, reparameterizing Q_k =g_kg^H_k and performing this linearization for all users,

WSR(g,g) =b

K

X

k=1

u_kln(1+g^H_kH^H_kRb⁻¹_k H_kg_k)−g_k^H(Ab_k+λI)g_k+λP. The gradient of this concave WSR lower bound is actually still the same as that of the original WSR or of the WSMSE criteria! Allows generalized eigenvector interpretation:

H^H_kRb⁻¹_k H_kg_k = 1 +g^H_kH^H_kRb⁻¹_k H_kg_k uk

(Ab_k +λI)g_k or henceg⁰_k =V_max(H^H_kRb⁻¹_k H_k,Ab_k+λI)

which is proportional to the ”LMMSE” g_k,

with max eigenvalue σ_k =σ_max(H^H_kRb⁻¹_k H_k,Ab_k +λI).

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[KimGiannakis:IT0511] DC = Optimally Weighted SLNR

Again, [KimGiannakis:IT0511] BF:

g⁰_k =Vmax(H^H_kRb⁻¹_k H_k,

K

X

i=1,6=k

u_iH^H_i (Rb⁻¹_i −Rb⁻¹_i )H_i+λI) This can be viewed as an optimally weighted version ofSLNR (Signal-to-Leakage-plus-Noise-Ratio) [Sayed:SP0507]

SLNR_k = ||H_kg_k||² P

i6=k||H_ig_k||²+P

i||g_i||²/P vs SINR_k = ||H_kg_k||²

P

i6=k||H_kg_i||²+P

i||g_i||²/P SLNR takes as Tx filter

g⁰_k =V_max(H^H_kH_k,X

i6=k

H^H_i H_i+I)

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[KimGiannakis:IT0511] Interference Aware WF

Let σ_k⁽¹⁾=g_k⁰^HH^H_kRb⁻¹_k Hkg⁰_k and σ_k⁽²⁾=g_k⁰^HAbkg⁰_k. The advantage of this formulation is that it allows straightforward power adaptation: substituting g_k =√

p_kg⁰_k yields

WSR =λP+

K

X

k=1

{u_kln(1 +p_kσ_k⁽¹⁾)−p_k(σ_k⁽²⁾+λ)}

which leads to the following interference leakage aware water filling

p_k = u_k σ⁽²⁾_k +λ

− 1 σ⁽¹⁾_k

!+

.

For a given λ,gneeds to be iterated till convergence.

And λcan be found by duality (line search):

minλ≥0max

g λP+X

k

{u_kln det(R⁻¹

k R_k)−λp_k}= min

λ≥0WSR(λ).

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High/Low SNR Behavior

At high SNR, max WSR BF converges to ZF solutions with uniform power

g_k^H =f_kH_kP_(fH)^⊥ H k

/||f_kH_kP_(fH)^⊥ H k

||

whereP_X^⊥=I−PX andPX=X(X^HX)⁻¹X^H projection matrices

(fH)_k denotes the (up-down) stacking of f_iH_i for users i = 1, . . . ,K,i 6=k.

At low SNR, matched filter for user with largest||H_k||₂ (max singular value)

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Simulated Annealing

Athigh SNR:max WSR solutions are ZF. When ZF is possible (IA feasible), multiple ZF solutions typically exist.

These different ZF solutionsare thepossible local optima for max WSR at infinite SNR. By homotopy, this remains the number of max WSR local optima as the SNR decreases from infinity. As the SNR decreases further, a stream for some user may get turned off until only a single stream remains at low SNR. Hence, the number of local optima reduces as streams disappear at finite SNR.

At intermediate SNR, the number of streams may also be larger than the DoF though.

Homotopy for finding global optimum: atlow SNR, noise dominates interference ⇒ optimal: one stream per power constraint,matched filter Tx/Rx. Gradually increasing SNR allows lower SNR solution to be in region of attraction of global optimum at next higher SNR.

Phase transitions: add a stream.

As a corollary, in the MISO case, the max WSR optimum is unique, since there is only one way to perform ZF BF.

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Outline

multi-cell multi-user: Interfering Broadcast Channel (IBC) max (weighted) sum rate (WSR)

UL/DL duality, WSR vs WSMSE WSR as optimal SLNR

Max WSR with Partial CSIT CSIT: perfect, partial, LoS problem of EWSMSE Massive MIMO limit large MIMO asymptotics distributed designs

per cell initialization intercell update

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Mean and Covariance Gaussian CSIT

Mean information about the channel can come from channel feedback or reciprocity, and prediction, or it may correspond to the non fading (e.g. LoS) part of the channel (note that an unknown phase factor e^j^φ in the overall channel mean does not affect the BF design).

Covariance information may correspond to channel estimation (feedback, prediction) errors and/or to information about spatial correlations. The separable (or Kronecker) correlation model (for the channel itself, as opposed to its estimation error or knowledge) below is acceptable when the number of propagation paths Np becomes large (NpMN) as possibly in indoor propagation.

Given only mean and covariance information, the fitting maximum entropy distribution is Gaussian.

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Mean and Covariance Gaussian CSIT (2)

Hence consider

vec(H)∼ CN(vec H

,C^T_t ⊗Cr) orH=H+C^1/2r HeC^1/2_t whereC^1/2r ,C^1/2_t are Hermitian square-roots of the Rx and Tx side covariance matrices

E (H−H)(H−H)^H = tr{C_t}Cr

E (H−H)^H(H−H) = tr{C_r}Ct

and the elements of He are i.i.d. ∼ CN(0,1). A scale factor needs to be fixed in the product tr{C_r}tr{C_t}for unicity.

In what follows, it will also be of interest to consider the total Tx side correlation matrix

R_t = E H^HH=H^HH+ tr{C_r}C_t . Gaussian CSIT model could be considered an instance of Ricean fading in which the ratio tr{H^HH}/(tr{C_r}tr{C_t}) = Ricean factor.

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Max Expected WSR (EWSR)

scenario of interest: perfect CSIR, partial (LoS) CSIT Imperfect CSIT ⇒ various possible optimization criteria:

outage capacity,.... Here: expected weighted sum rate E_HWSR(g,H) =

EWSR(g) = EH

X

k

ukln(1 +g_k^HH^H_kR⁻¹

k Hkgk) perfect CSIR: optimal Rx filters f_k (fn of aggregate H) have been substituted: WSR(g,H) = maxfP

kuk(−ln(ek(fk,g))).

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EWSR Lower Bound: EWSMSE

EWSR(g) : difficult to compute and to maximize directly.

[Negro:iswcs12] much more attractive to consider

EHek(fk,g,H) sinceek(fk,g,H) is quadratic inH. Hence optimizing E _HWSMSE(g,f,w,H).

min_f,wE_HWSMSE(g,f,w,H)

≥EHminf,wWSMSE(g,f,w,H) =−EWSR(g) or henceEWSR(g)≥ −min_f,wE_HWSMSE(g,f,w,H). So now only a lower boundto the EWSR gets maximized, which corresponds in fact to theCSIR being equally partial as the CSIT.

E _He_k = 1−2<{f_k^HH_kg_k}+P_K

i=1f_k^HH_kg_ig_i^HH^H_kf_k +f_k^HR_r,kf_kPK

i=1g_i^HR_t,kg_i+||f_k||².

⇒ signal term disappears if H_k = 0! Hence theEWSMSE lower bound is (very) looseunless the Rice factor is high, and is useless in the absence of mean CSIT.

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Massive MIMO Limit

We get a convergence for any term of the form

HQH^{H M}−→^→∞EHHQH^H=HQH^H+ tr{QCt}Cr. Go one step further in separable channel correlation model:

Cr,k,b_i =Cr,k,∀bi. This leads us to introduce

Hk= [Hk,1· · ·Hk,C] =Hk+C^1/2_r_,kHekC^1/2_t,k

Q=





 X

i:b_i=1

Qi

. .. X

i:b_i=C

Q_i







=

C

X

j=1

X

i:b_i=j

IjQiI^Hj

Q_k=Q−Ib_iQiI^H_b_i

whereCt,k= blockdiag{Ct,k,1, . . . ,Ct,k,C}, andIjis an all zero block vector except for an identity matrix in blockj. Then we get for theWSR (=EWSR),

WSR=

K

X

k=1

ukln det( ˘R⁻¹_k R˘k) where R˘k=IN_k+HkQH^Hk + tr{QCt,k}Cr,k

R˘_k=IN_k+HkQ_kH^Hk + tr{Q_kCt,k}Cr,k

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Massive MIMO Limit (2)

This leads to

WSR =u_kln det(I+ ˘R⁻¹_k

H_k,b_kg_kg^H_kH^H_k_,b

k+ tr{gkg^H_kC_t,k,b_k}Cr,k

)+WSR_k .

Consider simplified case: ”Ricean factor”µ∼SNR, for the direct linksHk,b_k (only) (properly organized (intracell) channel estimation and feedback) ⇒ approximation

WSR=ukln det(I+g_k^HB˘kgk) +WSR_k with B˘k =H^H_k,b_kR˘⁻¹_k Hk,b_k + tr{Cr,kR˘⁻¹_k }Ct,k,b_k

The linearization of WSR_k w.r.t. Q_k now involves A˘k =

K

X

i6=k

ui

h H^H_i,b

k

R˘⁻¹_i −R˘⁻¹_i

Hi,b_k+ tr{

R˘⁻¹_i −R˘⁻¹_i

Cr,i}Ct,i,b_k

i .

The rest of the development is now completely analogous to the case of perfect CSIT.

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WSMSE-DC BF relation

min WSMSE iteration (i+ 1) A⁽ⁱ⁾_k = P

jujw_j⁽ⁱ⁾H^H_i f_i⁽ⁱ⁾f_j^(i)HHj+λ⁽ⁱ⁾IM

g⁽ⁱ⁺¹⁾_k = (A⁽ⁱ⁾_k )⁻¹H^H_kf_k⁽ⁱ⁾ukw_k⁽ⁱ⁾

= (A⁽ⁱ⁾_k )⁻¹B⁽ⁱ⁾_k g⁽ⁱ⁾_k u_kw_k⁽ⁱ⁾ B⁽ⁱ⁾_k = H^H_kR⁻⁽ⁱ⁾_k H_k

WSMSE does one power iteration of DC !!

g⁽ⁱ⁺¹⁾_k =V_max{(A⁽ⁱ⁾_k )⁻¹B⁽ⁱ⁾_k } partial CSIT (or MaMIMO)case: modified WSMSE:

g_k⁽ⁱ⁺¹⁾= (EHA⁽ⁱ⁾_k )⁻¹(EHB⁽ⁱ⁾_k )g⁽ⁱ⁾_k ukw_k⁽ⁱ⁾

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Outline

multi-cell multi-user: Interfering Broadcast Channel (IBC) max (weighted) sum rate (WSR)

UL/DL duality, WSR vs WSMSE WSR as optimal SLNR

Max WSR with Partial CSIT CSIT: perfect, partial, LoS problem of EWSMSE Massive MIMO limit large MIMO asymptotics distributed designs

per cell initialization intercell update

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Reduced CSIT and Decoupled Tx/Rx Design

for IA to apply to cellular: overall Tx/Rx design has to decompose so that the CSIT required is no longer global and remains bounded regardless of the network size.

simplest case : localCSIT : a BS only needs to know the channels from itself to all terminals. In the TDD case : reciprocity. The local CSIT case arises when all ZF work needs to be done by the Tx:

dc,k =Nc,k,∀c,k. The most straightforward such case is of course the MISO case: dc,k =Nc,k = 1. It extends to cases ofNc,k >dc,k

if less than optimal DoF are accepted. One of these cases is that of reduced rank MIMO channels.

reducedCSIT [Lau:SP0913]: variety of approaches w reduced CSIT FB in exchange for DoF reductions.

incompleteCSIT [deKerretGesbert:TWC13]: min some MIMO IC optimal DoF can be attained with less than global CSIT. Only occurs whenM and/or N vary substantially so that subnetworks of a subgroup of BS and another subgroup of terminals arise in which the numbers of antennas available are just enough to handle the interference within the subnetwork.

Massive MIMO leads to exploitingcovarianceCSIT, which will tend to have reduced rank and allows decoupled approaches.

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Distributed IBC Design

Global intracluster CSIT can also be gathered but it takes an overhead that evolves withC² [Negro:isccsp12]. Hence, high quality (high Ricean factor) intracell CSIT and Tx covariance only intercell CSIT may be a more appropriate setting. For what follows we shall assume the LoS Tx intercell CSIT. We shall focus on a MaMIMO setting.

The approach considered here is non-iterative, or could be taken as initialization for further iterations.

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Distributed IBC Design: Initialization

Start with a per cell design.

To simplify design, assume Rx antennas are used to handle intracell interference. Hence all intercell interference needs to be handled by Tx (BS) antennas.

In that case, the crosslinks (cascades of channel and Rx) can be considered as independent from the intracell channels.

In a MaMIMO setting, the ZF by BSj towardsK−Kj crosslink channels (or LoS components in fact) will tend to have a deterministic effect of reducing the effective number of Tx antennas by this amount and hence of reducing the Tx power by a factor _M ^M^j

j−(K−K_j). Hence a per BS design can be carried out with (partial) intracell CSIT, with BS Tx powerPj

replaced by _M ^M^j

j−(K−K_j)Pj, and with all intercell linksHk,b_i = 0,bi 6=bk. This first step (which is itself an iterative design for the scenario

considered with reduced Tx power and no intercell links) leads to BFsg⁽⁰⁾ which lead to

R˘k= (1 + tr{Q⁽⁰⁾_b

kCt,k,b_k})IN_k+Hk,b_kQ⁽⁰⁾_b

kH^H_k,b_k, where Q⁽⁰⁾_b

k = X

i:b_i=b_k

g_i⁽⁰⁾g^(0)H_i and similarly for ˘R_k.

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Distributed IBC Design: Iteration 1

Do one iteration in order to adjust the Tx filters for the intercell interference.

With the initial BFs g⁽⁰⁾, the local intercell CSITCt,i,b_k also, the correct power constraints, and ˘R_k, ˘R_k as above, we get ˘B_k as before, and ˘Ak becomes

A˘k = X

i6=k:bi=bk

ui

h H^H_i,b_k

R˘⁻¹_i −R˘⁻¹_i

Hi,b_k + tr{

R˘⁻¹_i −R˘⁻¹_i

Cr,i}Ct,i,b_k

i

+ X

i:b_i6=bk

u_itr{R˘⁻¹_i −R˘⁻¹_i }

| {z }

=µ_i

C_t,i,b_k.

Hence the only information that needs to be fed back from user i in another cell is the positive scalarµi. This is related to the

interference pricing in game theory [XuWang:JSAC1012].

The normalized BFs are then computed as

g_k⁰ =V_max( ˘B_k,A˘_k+λ_b_kI) where theλ_b_k are taken from the previous iteration.

The stream powers are obtained from the interference-aware WF.

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Massive MIMO: from spatial to spatiotemporal and back

spatial: to null a user, need to null all paths of that user spatiotemporal: # antennas ># users

spatial: # antennas> # paths # users

but: paths are slowlyfading, user channels arefast fading

Keysight ieee comsoc M-MIMO tutorial, mmWave

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Specular Wireless (Massive) MIMO Channel Model

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Mean and Covariance Gaussian CSIT

Dominant Paths Partial CSIT Channel Model

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Specular Wireless MIMO Channel Model

We get for the matrix impulse response of a time-varying MIMO channelH(t, τ)

H(t, τ) =

Np

X

i=1

A_i(t)e^j^2π^fⁱ^th_r(φ_i)h^T_t (θ_i)p(τ −τ_i) . The channel impulse responseH has per path a rank 1

contribution in 4 dimensions (Tx and Rx spatial multi-antenna dimensions, delay spread and Doppler spread); there areNp

(specular) pathwise contributions where Ai: complex attenuation

fi: Doppler shift

θ_i: direction of departure (AoD) φ_i: direction of arrival (AoA) τi: path delay (ToA)

h_t(.),h_r(.): M/N×1 Tx/Rx antenna array response p(.): pulse shape (Tx filter)

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Specular Wireless MIMO Channel Model (2)

The antenna array responses are just functions of angles AoD, AoA in the case of standard antenna arrays with scatterers in the far field. In the case of distributed antenna systems, the array responses become a function of all position parameters of the path scatterers.

The fast variationof the phase in e^j2π^fⁱ^t and possibly the variation of the A_i correspond to thefast fading. All the other parameters(including the Doppler frequency) vary on a slower time scale and correspond to slow fading.

OFDM transmission H=

Np

X

i=1

e^j^ψⁱh_r(φ_i)h^T_t (θ_i)A_i = B A^H (not the same Ai ≥0, path amplitude)

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IA Feasibility Reduced Rank MIMO IBC

The ZF from BS j to MT (i,k) requires

F^H_i,kHi,k,jGj,n=F^H_i,kBi,k,jA^H_i,k,jGj,n= 0

which involves min(di,kdj,n,di,kri,k,j,ri,k,dj,n) constraints to be satisfied by the (Ni,k−di,k)di,k/(Mj−dj,n)dj,n variables parameterizing the column subspaces ofFi,k/Gj,n.

IA feasibility singular MIMO IC with Tx/Rx decoupling

F^H_i,kB_i,k,j= 0 or A^H_i,k,jG_j,n= 0.

This leads to a possibly increased number of ZF constraints r_i,k,jmin(d_i,k,d_j,n) and hence to possibly reduced IA feasibility. ZF of every cross link now needs to be partitioned between all Txs and Rxs, taking into account the limited number of variables each Tx or Rx has. The main goal of this approach however is that it leads to Tx/Rx decoupling and local CSI.

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Covariance CSIT

Averaging over the (uniform) path phasesψ_i leads to Chh =

Np

X

i=1

A²_i hih^H_i =

Np

X

i=1

A²_i (hr(φi)h^H_r (φi))⊗(ht(θi)h^H_t(θi)) whereC_hh = E hh^H,h= vec(H) andh_i =h_t(θ_i)⊗h_r(φ_i).

Note that the rank ofChh can be substantially less than the number of paths. Consider e.g. a cluster of paths with narrow AoD spread, then we have

θi =θ+ ∆θi

whereθis the nominal AoD and ∆θi is small. Hence ht(θi)≈ht(θ) + ∆θi h˙t(θ).

Such a cluster of paths only adds a rank 2 contribution toC_hh. Not of Kronecker form.

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Tx side Covariance CSIT

Tx side covariance matrixC^t, which only explores the channel correlations as they can be seen from the BS side

C^t = E H^HH We can factor the channel response as

H=B A^H , B= [hr(φ1)hr(φ1)· · ·]





 e^jψ¹

e^j^ψ² . ..





,

A^H =





 A1

A2

. ..













h^T_t (θ1) h^T_t (θ2)

...







Averaging of the path phasesψi, we get for the Tx side covariance matrix

C^t =AA^H

since due to the normalization of the antenna array responses, E B^HB= diag{[h_r(φ₁)h_r(φ₁)· · ·]^H[h_r(φ₁)h_r(φ₁)· · ·]}=I.

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IA Feasibility Reduced Rank MIMO IBC

The ZF from BS j to MT (i,k) requires

F^H_i,kHi,k,jGj,n=F^H_i,kBi,k,jA^H_i,k,jGj,n= 0

which involves min(di,kdj,n,di,kri,k,j,ri,k,dj,n) constraints to be satisfied by the (Ni,k−di,k)di,k/(Mj−dj,n)dj,n variables parameterizing the column subspaces ofFi,k/Gj,n.

IA feasibility singular MIMO IC with Tx/Rx decoupling

F^H_i,kB_i,k,j= 0 or A^H_i,k,jG_j,n= 0.

This leads to a possibly increased number of ZF constraints r_i,k,jmin(d_i,k,d_j,n) and hence to possibly reduced IA feasibility. ZF of every cross link now needs to be partitioned between all Txs and Rxs, taking into account the limited number of variables each Tx or Rx has. The main goal of this approach however is that it leads to Tx/Rx decoupling.

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Massive MIMO & Covariance CSIT

In massive MIMO, the Tx side channel covariance matrix is very likely to be (very) singular even though the channel responseHmay not be singular:

rank(C^t_i,k,j =A_i,k,jA^H_i,k_,j) =r_i,k,j, A_i,k,j :M_j×r_i,k_,j LetPX=X(X^HX)^#X^H andP^⊥_X be the projection matrices on the column space ofXand its orthogonal complement resp. Consider now a massive MIMO IBC withC cells containingK_i users each to be served by a single stream. The following result states when this will be possible.

Theorem

Sufficiency of Covariance CSIT for Massive MIMO IBC In the MIMO IBC with (local) covariance CSIT, all BS will be able to perform ZF BF if the following holds

||P^⊥_A

i,k,jA_i,k,j||>0, ∀i,k,j whereA_i,k,j ={An,m,j,(n,m)6= (i,k)}.