icassp 2012 tutorialrqiu/teaching/ece7750/...scaling down p with m ⇒ noise will limit performance....

MM

YS

ICASSP 2012 Tutorial

Very Large MIMO Systems

Part I: Theory and Analysis

Erik G. Larsson

March 26, 2012

Div. of Communication SystemsDept. of Electrical Engineering (ISY)

Linkoping UniversityLinkoping, Sweden

www.commsys.isy.liu.se

With thanks to my team and collaborators:

◦ Hien Q. Ngo (LiU, Sweden)◦ Antonios Pitarokoilis (LiU)◦ Saif Mohammed (LiU)◦ Daniel Persson (LiU)

◦ Fredrik Rusek (Lund, Sweden)◦ Ove Edfors (Lund)◦ Buon Kiong Lau (Lund)◦ Fredrik Tufvesson (Lund)

◦ Thomas L. Marzetta (Bell Labs/Alcatel-Lucent, USA)

◦ Christoph Studer (Rice Univ., USA)

1/66

Erik G. LarssonVery Large MIMO Systems

Communication Systems

Linkoping UniversityMM

YS

Why MIMO◮ Array gain (beamforming)

◮ Spatial division mult. access

◮ Spatial multiplexing

◮ Rate ∼ min (nr, nt) log (1 + SNR)

◮ Reliability pe ∼ SNR−nrnt

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Very Large MIMO

M=

x100

antennas!K

terminals

k=1

k=K

◮ We think of very large (multiuser) MIMO as a system with◮ M ≫ K ≫ 1◮ coherent, but simple, processing

◮ Potential to improving rate & reliability dramatically

◮ Potential to scaling down TX power drastically

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Large MIMO Arrays

◮ Reduce bulky items (coax)◮ Each antenna unit simple (low accuracy)◮ Resilience against individual failures (hotswapping)◮ Potential economy of scale in manufacturing◮ Enable for mmWave radio (60-300 GHz)?

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Large MIMO: Some Known Facts(Notation: M antennas, K terminals, power per terminal P )

◮ Linear processing (MRC/MRT, ZF, ... ) nearly optimal asM ≫ K ≫ 1

◮ P and M “large enough” ⇒ pilot contamination limitsperformance

◮ Scaling down P with M ⇒ noise will limit performance.

◮ Perfect CSI & optimal processing ⇒ P can be scaled as 1/M

◮ Given linear processing and imperfect CSI, in a MU system,P can be scaled as 1/

√M

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Large MIMO: Some Known Facts, cont.

◮ System performance can be limited by◮ pilot contamination (M → ∞, P 9 0)◮ thermal noise (P → 0 too fast as M increased)◮ intracell interference (e.g., MRC and M ≯> K)◮ intercell interference

so there are several possible operating points depending on◮ number of antennas◮ available TX power◮ choice of receiver/precoder algorithm◮ coherence time (dictates ultimately the number of users served)

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Large MIMO: Some Beliefs/Speculation◮ Not enough time for CSI feedback, so must operate in TDD mode.

◮ Not enough pilots (optimal training is orthogonal)◮ Not enough resources for fast CSI feedback

◮ System will operate in nearly-noise limited regime (∼1 bpcu/term)◮ Very aggressive spatial multiplexing; aggregate efficiency ∼ K bpcu◮ Each user could get the full bandwidth

⇒ simple MAC, little or no control signaling◮ Impairments, e.g., multiuser interference (almost) drown in noise

⇒ linear or nearly-linear receivers◮ May even get away with equalization-free (matched filter only) SC

transmission

◮ Vast excess (M −K) of degrees of freedom:⇒ use for HW-friendly signal shaping and smart receiver algorithms

◮ Per-antenna constant envelope or low-PAR multiuser precoding◮ Channel estimation exploiting subspaces

◮ Channel will harden (random matrix theory)

◮ Larger array reveals new propagation phenomena

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Large MIMO: Some of the Most Important Questions

◮ Processing will have to be simple (linear). How good is this?

◮ Non-CSI@TX operation: STBC more or less optimal

◮ Acquisition

◮ Hardware imperfections: phase noise, I/Q imbalance, A/D, PAs

◮ Synchronization at low SNR

◮ TDD will bring us pilot contamination in the downlink.How bad is this really in practice?

◮ TDD will require reciprocity calibration.How, when and at what cost?

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Favorable Propagation andIts Implications

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“Favorable propagation” and Rate◮ M ×K, MIMO link, channel H, M ≥ K, no CSI@TX. Rate:

R =

K∑

k=1

log2

(

1 +SNR

Kλ2k

)

, SNR =PtotN0

◮ If |Hij | ∼ 1,∑K

k=1 λ2k = ‖H‖2 ≈ MK, so

log2 (1 +MSNR)︸︷︷︸

rank-1 channel (LoS)

λ21=MK, λ2

2=···=λ2

K=0

≤ R ≤ K · log2(

1 +M

KSNR

)

︸︷︷︸

HHH∝I (full rank channel)

λ21=···=λ2

K=M

Favorable propagation

◮ H i.i.d. and M ≫ K ⇒ favorable propagation.◮ In MU-MIMO (H ⇒ G), “favorable propagation” if

GHG

M≈

β1 0 · · · 0

0 β2. . .

......

. . .. . . 0

0 · · · 0 βK

, D, M ≫ K

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Favorable propagation and “ideal channels”

−40 −30 −20 −10 0 10 20 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b(σ

≤ ab

scis

sa)

ordered singular values [dB]

i.i.d 6x128i.i.d. 6x6

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Favorable propagation and detection

◮ Optimal (coherent) uplink detector:

minx,xk∈X

‖y −Gx‖ (∗)

has complexity ∼ exp(K)

◮ With favorable propagation and M ≫ K,

1

MGHG ≈ D

so

(∗) ⇔ minx,xk∈X

∥∥∥∥

1

MGHy −Dx

∥∥∥∥

⇔ minxk∈X

∣∣∣∣xk − GH

k y

Mβk

∣∣∣∣

2

◮ Expect simple (linear) detectors to be good enough: MRC, ZF

◮ Complexity ∼ MK2

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Favorable propagation and linear precoding◮ MRC precoding, x ∝ G∗s, is essentially time-reversal “in space”◮ In rich scattering, focus power not in a direction but at a point

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Do we have “favorable propagation” in practice?

◮ Our partners at Lund Univ., Sweden have conducted uniquemeasurements [RPL2011+,GERT2011].

◮ Indoor 128-ant. (4x16 dual-pol.) array. 3 users indoor, 3 outdoor.

◮ 2.6 GHz CF, 50 MHz BW, 100 snapshots (10m).

◮ Normalized to retain only small-scale fading.

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Lund measurements, example of results (more in part II)

−40 −30 −20 −10 0 10 20 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b(σ

≤ ab

scis

sa)

ordered singular values [dB]

meas 6x128meas 6x6

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Spectral and Energy Efficiency with LinearReceivers:

Analysis of a Single-Cell System

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Cellular UL, (BT )coh. = 14× 14, M = 50 [NLM2011]

0 10 20 30 40 50 60 70 80 9010

-1

100

101

102

103

104

K=1, M=1

MRC

20 dB

10 dB

0 dB

-10 dB

-20 dB

Rel

ativ

e E

nerg

y-E

ffic

ienc

y (b

its/J

)/(b

its/J

)

Spectral-Efficiency (bits/s/Hz)

K=1, M=100

M = 100

ZF

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Derivation of uplink spectral/efficiency tradeoff

◮ gk =√βkhk.

◮ hk: small scale fading;E[|hmk|2] = 1; zero mean.

◮ βk: path loss+shadowing◮ SNR for kth terminal: Pβk

◮ RX signal:

y =√P

K∑

k=1

gk︸︷︷︸

M×1

xk + n, E[|xk|2] = 1, ni ∼ CN(0, 1)

◮ Write as

y =√P G︸︷︷︸

M×K

x︸︷︷︸

K×1

+n, G = HD1/2, H , [h1, . . . ,hK ], D ,

β1

. . .

βK

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Uplink Power Efficiency with MRC - Perfect CSI◮ Maximum-Ratio-Combining

rmrc =GGGHyyy =GGGH(√

PGGGxxx+nnn)

⇒ rk,mrc =√P‖gggk‖2xk

︸︷︷︸

desired signal

+√P

K∑

i6=k

gggHk gggixi

︸︷︷︸

interference

+gggHk nnn︸︷︷︸

noise

◮ Observe that: gi ,gggHk gggi

‖gggk‖ ∼ CN (0, βi), indep. of gk

⇒ SINRk =P‖gggk‖4

P∑K

i6=k |gggHk gggi|2 + ‖gggk‖2=

P‖gggk‖2P∑K

i6=k |gi|2 + 1

M≫1≈ PMβk

P∑K

i6=k |gi|2 + 1

M→∞→{

∞, P fixed

P0βk, P = P0/M

⇒ TX power can be scaled as ∝ 1/M

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Uplink Performance with MRC - Perfect CSI◮ Achievable ergodic rate

Rk,mrc = E

{

log2

(

1 +P‖gggk‖2

P∑K

i6=k |gi|2 + 1

)}

≥ log2

1 +

(

E

{

P∑K

i6=k |gi|2 + 1

P‖gggk‖2

})−1

= log2

(

1 +P (M − 1)βk

P∑K

i6=k βi + 1

)

(use E[log(1 + 1/x)] ≥ log(1 + 1/E[x]))

◮ Here: E{

1

‖gk‖2

}

= 1

M−1

1

βk

◮ Special case of E{

tr

(

WWW−1)}

= m

n−m, if WWW ∼ Wm (n,IIIn).

◮ Spectral efficiency: S =∑K

k=1 Rk

◮ Small-scale fading goes away in the limit.

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Uplink performance with, ZF, perfect CSI

◮ Zero-forcing:

√P · xzf = (GHG)−1GHy =

√Pxxx+ (GHG)−1GHnnn

◮ Capacity lower bound:

Rk,zf = E

log2

1 +

P[(

GGGHGGG)−1

]

kk

M≫1≈ log2

(

1 +P

1/(Mβk)

)

M→∞→{

∞, P fixed

log2(1 + βkP0), P = P0/M

◮ MRC and ZF are equivalent as M ≫ 1 since

xzf = (GHG)−1GHy ≈ 1

MD−1GHy = xmrc

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Uplink performance with MMSE - perfect CSI◮ Minimum mean-squared error detector:

xxxmmse =

(

GGGHGGG+1

PIIIK

)−1

GGGHyyy


Rk,mmse = E

log2

1[(

IIIK + PGGGHGGG)−1

]

kk

M≫1≈ log2

(1

1/(1+MβkP )

)M→∞→

{∞, P fixedlog2(1+βkP0) , P = P0/M

◮ As M ≫ 1, MRC, ZF, and MMSE are equivalent, i.e.,

xxxmmse =

(

GGGHGGG+1

PIIIK

)−1

GGGHyyy ≈(

MDDD +1

PIIIK

)−1

GGGHyyy

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Optimal linear detector◮ Let AAA be an M ×K linear detector matrix. Then

r = AAAHyyy =√PAAAHGGGxxx+AAAHnnn.

rk =√PaaaHk gggkxk +

√P

K∑

i=1,i6=k

aaaHk gggixi + aaaHk nnn


Rk = E

{

log2

(

1 +P |aaaHk gggk|2

P∑K

i=1,i6=k |aaaHk gggi|2 + ‖aaak‖2

)}

= E

{

log2

(

1+|aaaHk gggk|2aaaHk ΛΛΛkaaak

)}

≤E

{

log2

(

1+‖aaaHk ΛΛΛ

1/2k ‖2‖ΛΛΛ−1/2

k gggk‖2aaaHk ΛΛΛkaaak

)}

= E{log2

(1 + gggHk ΛΛΛ−1

k gggk)}

where ΛΛΛk ,∑K

i=1,i6=k gggigggHi + 1

P IIIM◮ Equality when aaak ∝ ΛΛΛ−1

k gggk: MMSE detector!

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YS

Uplink Power Efficiency with MRC - Estimated CSI◮ CSI from pilots: GGG = GGG + EEE, where εεεi ∼ CN

(0,

βiPpβi+1

IIIM

),

gggi ∼ CN(0,

Ppβ2i

Ppβi+1IIIM

), Pp = τP , τ=#pilots

◮ MRC

rmrc = GGGHyyy = GGG

H(√

PGGGxxx −√PEEExxx +nnn

)

⇒ rk,mrc =√P‖gggk‖2

xk +√P

K∑

i 6=k

gggHk gggixi −

√P

K∑

i=1

gggHk εεεixi + ggg

Hk nnn

◮ Signal-to-interference-plus-noise ratio:

SINRk =P‖gggk‖2

P∑K

i 6=k |ˆgi|2 + P∑K

i=1

βiτPβi+1

+ 1, ˆgi ,

gggHk gggi

‖gggk‖∼ CN

(0,

Ppβ2i

Ppβi + 1

)

M≫1≈PM

τPβ2k

τPβk+1

P∑K

i 6=k |ˆgi|2 + P∑K

i=1

βiτPβi+1

+ 1

M→∞→

∞, P = P0

0, P = P0/M

τP 20 β

2k, P = P0/

√M

⇒ TX power can be scaled as ∝ 1/√M

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YS

Uplink Performance with MRC - Estimated CSI◮ Achievable ergodic rate

Rk,mrc = E

{

log2

(

1 +P‖gggk‖2

P∑K

i6=k |ˆgi|2 + P∑K

i=1βi

τPβi+1 + 1

)}

≥ log2

1 +

(

E

{

P∑K

i6=k |ˆgi|2 + P∑K

i=1βi

τPβi+1 + 1

P‖gggk‖2

})−1

= log2

(

1 +τP 2 (M − 1)β2

k

P (τPβk + 1)∑K

i6=k βi + (τ + 1)Pβk + 1

)

M→∞→{

0 P = P0/M

τP 20 β

2k P = P0/

√M

(gk and ˆgi, i 6= k are independent)

◮ Spectral efficiency: S =T − τ

T

K∑

k=1

Rk, T =coherence BT-product

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Uplink Performance with ZF - Estimated CSI◮ Zero-forcing

√P · xxxzf =

(

GGGHGGG)−1

GGGH(√

PGGGxxx−√PEEExxx+nnn

)

=√Pxxx−

√P(

GGGHGGG)−1

GGGHEEExxx+

(

GGGHGGG)−1

GGGHnnn


Rk,mrc = E

log2

1+

P(∑K

i=1Pβi

Ppβi+1+1)[(

GGGHGGG)−1

]

kk

M≫1≈ log2

1+P

(∑K

i=1Pβi

Ppβi+1+1)

/(

MPpβ2

k

Ppβk+1

)

M→∞→

∞, P fixed

0, P = P0/M

log2(1 + τβ2

kP20

), P = P0/

√M 26/66




YS

Uplink performance with MMSE - estimated CSI◮ Minimum mean-squared error:

xxxmmse = GGGH(

GGGGGGH+

1

PCov

(

−√PEEExxx+nnn

))−1

yyy


Rk,mmse=E

log2

1[(

IIIK+(∑K

i=1βi

Ppβi+1+ 1

P

)−1GGG

HGGG)−1

]

kk

M≫1≈ log2

(

1 +MPpβ

2k/ (Ppβk+1)

∑Ki=1

βi

Ppβi+1+ 1

P

)

M→∞→

∞, P fixed

0, P = P0/M

log2(1 + τβ2

kP20

), P = P0/

√M

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YS

Required power, 1 bit/s/Hz/terminal for K = 10, τ = K

50 100 150 200 250 300 350 400 450 500-9.0

-6.0

-3.0

0.0

3.0

6.0

9.0

12.0

15.0

18.0 MRC ZF MMSE

Perfect CSI

Req

uire

d P

ower

, Nor

mal

ized

(dB

)

Number of Base Station Antennas (M)

Imperfect CSI

1 bit/s/Hz

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Required power, 2 bit/s/Hz/terminal for K = 10, τ = K

50 100 150 200 250 300 350 400 450 500-3.0

0.0

3.0

6.0

9.0

12.0

15.0

18.0

21.0

24.0

27.0

30.0

MRC ZF MMSE

Perfect CSI

Req

uire

d Po

wer

, Nor

mal

ized

(dB

)

Number of Base Station Antennas (M)

Imperfect CSI

2 bits/s/Hz

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Spectral-energy efficiency tradeoff◮ Sum-spectral efficiency (βk = 1 here):

R =

(1− τ

T

)K log2

(

1 + τ(M−1)P 2

τ(K−1)P 2+(τ+K)P+1

)

, for MRC(1− τ

T

)K log2

(

1 + τ(M−K)P 2

(τ+K)P+1

)

, for ZF

◮ Energy efficiency: η =R

P◮ Reference mode: K = 1,M = 1

arg max1≤τ≤T

η

◮ Single-user system: K = 1,M fixed

argmaxP,τ

η, s.t. S = const., 1 ≤ τ ≤ T

◮ Multi-user system: M fixed

arg maxP,K,τ

η

s.t. S = const.,K ≤ τ ≤ T(K ≤ M for ZF) 30/66




YS


0 10 20 30 40 50 60 70 80 9010

-1

100

101

102

103

104

K=1, M=1

20 dB

10 dB

0 dB

-10 dB

Rel

ativ

e E

nerg

y-E

ffic

ienc

y (b

its/J

)/(b

its/

J)

Spectral-Efficiency (bits/s/Hz)31/66




YS


0 10 20 30 40 50 60 70 80 9010

-1

100

101

102

103

104

K=1, M=1

20 dB

10 dB

0 dB

-10 dB

Rel

ativ

e E

nerg

y-E

ffic

ienc

y (b

its/J

)/(b

its/J

)


K=1, M=100

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YS


0 10 20 30 40 50 60 70 80 9010

-1

100

101

102

103

104

K=1, M=1

MRC

20 dB

10 dB

0 dB

-10 dB

-20 dB

Rel

ativ

e E

nerg

y-E

ffic

ienc

y (b

its/J

)/(b

its/J

)


K=1, M=100

M = 100

ZF

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YS

Cellular UL, (BT )coh. = 14× 14, 100 = 50 [NLM2011]

0 10 20 30 40 50 600

20

40

60

80

100

120

140

number of users


number of uplink pilots

ZF

MRC

M=100

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YS


0 10 20 30 40 50 60 70 80 9010

-1

100

101

102

103

104

-20 dB

M=50

ZF

MRC

20 dB

10 dB

0 dB

-10 dB

Rel

ativ

e E

nerg

y-E

ffic

ienc

y (b

its/J

)/(b

its/J

)


K=1, M=50

K=1, M=1

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Some Observations: Single-Cell System Uplink

◮ The MRC receiver is limited by intracell interference, but will bevery competitive at ∼1 bpcu/terminal and facilitates decentralizedimplementation

◮ Power efficiency: with large number of BS antennas, the TX powerof each user can be scaled as

◮ 1/M if the BS has perfect CSI◮ 1/

√M if the BS estimates CSI from uplink pilots

◮ Key points remain the same for the downlink

◮ In the downlink, with TDD, training costs only uplink (notdownlink) resources

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Uplink in Interference Limited Multicell System:Limits Dictated by Pilot Contamination

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Interference limited multicell system

cell l

cell j

Glj

◮ RX data: yl︸︷︷︸

M×1

=√P∑L

j=1 Glj︸︷︷︸

=HljD1/2lj

M×K

xj︸︷︷︸

K×1

+nl

◮ CSI from pilots: Gll =

Gll +

L∑

j=1,j 6=l

Glj

+1

√Pp

Wl (LS

estimate)

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Interference limited system - pilot contamination◮ Consider MRC processing in the lth cell:

rrrl = GGGH

ll yyyl =

(∑L

j=1GGGlj +1√Pp

WWW l

)H (√P∑L

i=1GGGlixxxi +nnnl

)

rrrlM

=√P

L∑

i=1

L∑

j=1

GGGHljGGGli

Mxxxi+

L∑

j=1

GGGHljnnnl

M+

√

P

Pp

L∑

i=1

WWWHl GGGli

Mxi+

1√

Pp

WWWHl nnnl

M

≈√P

L∑

i=1

DDDlixxxi, as M ≫ K

◮ The SIR of the uplink transmission for the kth user in the lth cell

SIRlk =β2llk

∑Li6=l β

2lik

indep. of P ⇒ Pilot contamination!

◮ Limited (only) by interfering pilots. No noise, no fast fading.◮ Similar analysis for ZF (a bit more involved)

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Cell. UL, M = ∞, (14×15kHz)×0.5ms, (3 + 3 + 1)/7 T+D+O, 3× 14 = 42 term., [Mar2010]

10−1

100

101

102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

capacity per terminal (megabits/second)

cum

ulat

ive

dist

ribut

ion

7 3 1

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YS

Pilot contamination - does finite dimensional channel help?

1

2

M

Φ1

Φ2

ΦM’

◮ Consider finite-dimensional channel:

G = A︸︷︷︸

M×M ′

G′︸︷︷︸

M ′×K

= AH ′D1/2

where M ′ fixed as M → ∞

◮ Pilot contamination continues to fundamentally limit the SIR[NML2011]

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Interference limited system - general remarks

◮ P > ǫ and M → ∞, thermal noise and fast fading vanish◮ Since pilots need be reused, received pilots will be contaminated and

this effect will persist no matter what channel estimation scheme isused

◮ Using different pilot sequences in different cells does not help - finitedimensionality of pilot signal space

◮ Finite dimensionality of the channel does not eliminate the problem

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Large-Scale MU-MIMODownlink Precoding

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Downlink precoding - general remarks

◮ In the downlink, the are M −K “unused” degrees of freedom.These excess DoF could be used to

◮ Null out interference◮ Shape the transmitted signals in a hardware-friendly way◮ Exploit asymptotic orthogonality

◮ An excess in the number of antennas also means that simpleprecoders (MRT, time-reversal) may be used to combat frequencyselectivity

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Constant Envelope MU-MIMO Downlink Precoding

◮ Constant-envelope transmission [ML2011a,ML2011b]

x =

√

P

M

ejθ1

...ejθM

◮ Insensitive to PA non-linearity.

◮ How well can we approximate a desired wavefield at K locations, byvarying only the phase of the transmitted signals?

◮ Not to be confused with equal-gain transmission (something entirelydifferent)!

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Constant Envelope versus “Beamforming”

u

√P

h∗1

‖h‖

√P

h∗m

‖h‖

√P

h∗M

‖h‖

Amplitude range = [0 · · ·√P |u|]

ejθu1

ejθum

ejθuM

Constant amplitude =√

PM

√PM

√PM

√PM

Antenna 1 Antenna 1

Antenna m Antenna m

Antenna M Antenna M

Beamforming Constant-envelope transmission

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Constant Envelope MU-MIMO Downlink Precoding◮ Ek: energy per information symbol uk (per user)◮ P : total transmit power◮ Received signals:

yk =

√

P

M

M∑

i=1

hk,iejθi+nk =

√P√

Ekuk+√P

(∑Mi=1 hk,ie

jθi

√M

−√

Ekuk

)

︸︷︷︸

,δk

+nk

◮ Performance bound, with Gaussian symbols per user:

I(yk;uk) = h(uk)− h(uk | yk) = h(uk)− h(

uk − yk√P√Ek

∣∣∣ yk

)

≥ h(uk)− h(

uk − yk√P√Ek

)

≥ h(uk)− h( δk√

Ek

+nk√P√Ek

)

= log2(πe)− h( δk√

Ek

+nk√P√Ek

)

≥ log2(πe)− log2

(

πe var[ δk√

Ek

+nk√P√Ek

])

≥ log2(πe)− log2

(

πeE[ ∣∣∣

δk√Ek

+nk√P√Ek

∣∣∣

2 ])

= log2(πe)− log2

(

πe[E[|δk|2]

Ek+

1

PEk

])

◮ Downlink signal design: arg minθi∈[−π,π) , i=1,...,M

|δk|2

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Constant-env. MU-MIMO precoding, Gauss. symbols

0 50 100 150 200 250 300−12

−9

−6

−3

0

3

6

9

12

1515

No. of Base Station Antennas (M)

Min

. req

d. P

T/σ

2 (dB

) to

ach

ieve

a p

er−

user

rat

e of

2 b

pcu

K = 10, Proposed CE Precoder (CE)K = 10, ZF Phase−only Precoder (CE)K = 10, GBC Sum Cap. Upp. Bou. (APC)K = 40, Proposed CE Precoder (CE)K = 40, ZF Phase−only Precoder (CE)K = 40, GBC Sum Cap. Upp. Bou. (APC)

1.7 dB

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Constant-env. MU-MIMO precoding, Gauss. symbols

0 0.5 1 1.5 2 2.5 31

2

3

4

5

6

7

8

9

Desired per−user information rate (bpcu)

Ext

ra T

rans

mit

Pow

er R

equi

red

w.r

.t. S

um C

ap. U

pp. B

ou. (

dB)

ZF Phase−only Precoder

Proposed CE Precoder

M = 12, N = 48

1.5 dB

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PAR-Aware MU-MIMO OFDM Downlink [SL2012]

◮ Precoder design problem:

(PMP)

minimizea1,...,aN

max{‖a1‖∞ , . . . , ‖aN‖∞

}

subject to sw = Hwfw(a1, . . . , aN ), w ∈ T0N×1 = fw(a1, . . . , aN ), w ∈ T c.

◮ Hw: time-frequency channel

◮ fw(·) linear function; includes OFDM modulation and S/Pconversion

◮ T : used subcarriers; T c: null subcarriers

◮ Convex optimization problem, fast algorithm; relaxation

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PAR-aware MU-MIMO OFDM DL

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PAR-aware MU-MIMO OFDM DL (99% percentiles)

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PAR-aware MU-MIMO OFDM DL (99% percentiles)

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Single-Carrier Downlink in ISI Channels [PML2012]

MU-MISO ISI Channel:

y[i] =

L−1∑

l=0

D1/2l HH

l x[i− l] + n[i]

◮ L independent channel taps

◮ Normalized Power Delay Profile

Dl = diag{dl[1], . . . , dl[K]},

dl[k] ≥ 0,

L−1∑

l=0

dl[k] = 1

Matched Filter Precoder

x[i] =

√ρfMK

L−1∑

l=0

HlD1/2l s[i+ l]

◮ s[i]: White Gaussianinformation symbols

◮ ρf : long-term average totalradiated power by the basestation in a single channel use

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Single Carrier Transmission - Achievable Sum Rate◮ Define vl[k]

∆= HlD

1/2l ek. Then,

yk[i] =

(√ρfMK

L−1∑

l=0

E[vHl [k]vl[k]

]

)

sk[i] + n′k[i],

where n′k[i] is an effective noise term.

n′k[i] ,

√ρfMK

(L−1∑

l=0

vHl [k]vl[k]−

L−1∑

l=0

E[vHl [k]vl[k]

]

)

sk[i]

︸︷︷︸

Additional Interference Term (IF)

+

√ρfMK

L−1∑

a=1−La 6=0

min(L−1+a,L−1)∑

l=max(a,0)

vHl [k]vl−a[k]sk[i− a]

︸︷︷︸

Intersymbol Interference (ISI)

+

√ρfMK

K∑

q=1

q 6=k

L−1∑

a=1−L

min(L−1+a,L−1)∑

l=max(a,0)

vHl [k]vl−a[q]sq[i− a]

︸︷︷︸

Multiuser Interference (MUI)

+ nk[i]︸︷︷︸

AWGN

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Single Carrier Transmission - Achievable Sum Rate◮ Achievable Rate for user k:

Rk = log2

1 +

=ρfM/K︷︸︸︷

Esk[i]

∣∣∣∣∣

√ρfMK

L−1∑

l=0

E[vHl [k]vl[k]

]sk[i]

∣∣∣∣∣

2

Var (n′k[i])

︸︷︷︸

=ρf+1

◮ Achievable sum-rate:

Rsum(ρf ,M,K) =

K∑

k=1

Rk = K log2

(

1 +ρfM

Kρf +K

)

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Single Carrier Transmission versus OFDM

40 60 80 100 120 140 160 180 200−14

−12

−10

−8

−6

−4

−2

0

No of Base Station Antennas (M)

Pow

er [d

B]

Proposed PrecoderCo−op Sum−Capacity BoundOFDM Bound T

cp=T

u/4

K=20

K=10

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MIMO Detection in Non-M ≫ K Systems

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General Observations

◮ For favorable propagation and M ≫ K, linear detectors may besufficiently good.

◮ When M ≈ K, the BS must switch from linear detection to moreadvanced algorithms.

◮ Detection: classical algs (SD, FCSD, LLL, ...) too complex!

◮ Envisaged detection methods for very large MIMO:◮ Iterative linear filtering schemes:

- soft information-based methods (MMSE-SIC)- hard information-based method (BI-GDFE)

◮ Random search methods: Tabu Search (TS).◮ Reading: [RPL+2011] and papers cited therein

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Iterative Linear Filtering Scheme (MMSE-SIC)◮ At the BS: x = Gq +wr, where q = [q1, ..., qK ]T is the transmitted

vector from K users.

�

� � ��

��

�

�

� � �

�

0

0

0 �

( )� �

� � �

�

�� !" #�$!%&!%!� !�

��" '"�$! �

( ) ( )*

( + ( ) ( )

, ,

- .

/

0 1

0 2

0

3 3 4

5 3

1

1

6

789: ;< =>

?@AB ?@AB

?@AB ?@AB ?@AB

C C

C C C

DEFGHIGFJKL L MNIOP

D D D DMKJIQOL R R SSSR

T U T U

VT T T W

X X

X X X

Y Z[ \ ]

Y Z[ \ ]^1 2

_ `

abc

def fgh i j h j

j ikl

mn∑o p

1q

◦ wi,k: linear filter.◦ h: rounding-off operation.◦ S: 1D complex signal constellation.

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Likelihood Ascent Search Algorithm

( )

r s

r s r s

r s r s

t uvvwx yz{|}~{y �� t

�|{ }~��|�� t

� �

� �

� �

� �

� �

� ��

0

0 0

0 0

0

�

�

�

�

{ }

� � � � � � � � � � � �

� � � �

� � � �

� � � � � � � ��

�

� ��

�� ¡

�� ¢��

£ £ £ £ £ £¤ ¤ ¤ ¤ ¥

£ £¤ ¤

£ £¤ ¤£ £ £ £

¥ ¤¤

¦ ¦

§ § § § § ¨ ©

¨

ª ª ª ª ªª «

¬

® ¯¬ ° ¬± ²

¬ ³

¬ °

¬ ¬

´

µ ´

¶ ·µ

1 1 1 1 1

1 1 1

1

1

1

¸

¸ ¸ ¸ ¸

¹¹

¹

º » º »¼½¾ ¿À ÀÁÂ 1Ã Ã

Ä ÅÆÇÈÉÊËÌ ÍÎÏ 1Ð

ÑÒÓ

ÔÕÖ×Ø

Ö×Ø

Ù Ú Ù Ú

Ù Ú Ù Ú

Ù Ú Ù Ú

Û ÛÜ

Û ÛÜ

Û Û

Ý

Ý

Ý Þ

ß ß

à à

á âàã

ä ä

ä

◦ φ : q → b: mapping fromQAM vectors to bit vectors.◦ φ : b → q: mapping frombit vectors to QAM vectors.

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Tabu Search Scheme

åæçè

åæçè

éêë

ì íîîïð ñòóôõöò÷ì

øùúö÷ù õûüô óöñõ ì

ýþÿÿ õò í ñùõ ì

ýì

�

�

�

�

� �

�

� � �

� ��

0

�

�

�

� �

��

��

� ��

1 !

"

( )#$%&' ()*+ ,- ./0 0

1234

567896 :;6 6<=>?6@:

<AA6A 6B:=C ?B D

E FGHIJK LMNOPQRS

PQRSPQRS

TUV TUV

WXYZ[YX\]^ X_`abc d Xee \f

g]afh] \i] jb_k\ h]Z\f_ bj h]Z\f_k bc

l lmj d k]\ ^ d Xce ^

n

o

p p

q rst u v

w

u x t t u

yz { |z z

{ |z z z { |z

} }~

~

� �

� �

��

��

��

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Complexity Estimates for Detectors

◮ M ×K system

Detector Complexity for each realization of x Complexity per realization of G

MMSE MK MK2 +K3

MMSE-SIC (M2K +M3)NIter

BI-GDFE MKNIter (M2K +M3)NIter

TS ((M +NTabu)NNeigh +MK)NIter MK2 +K3

FCSD (M2 +K2 + r2)|S|r MK2 +K3

MAP MK|S|K

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Detection in the Uplink: Example, 40× 40 system

105

106

107

108

10−4

10−3

10−2

TS

MMSE-SIC

BI-GDFE

MMSE

FCSD

BER

Number of floating point operations

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LiteratureRPL+2011 F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L.

Marzetta, O. Edfors, and F. Tufvesson, “Scaling up MIMO:Opportunities and Challenges with Large Arrays”,arXiv:1201.3210, 2011.

NLM2011 H.Q. Ngo, E.G. Larsson and T. Marzetta, “Energy andSpectral Efficiency of Very Large Multiuser MIMOSystems”, arXiv:1112.3810, 2011.

LM2011a S.K. Mohammed and E.G. Larsson, “Per-antenna ConstantEnvelope Precoding for Large Multi-User MIMO Systems”,arXiv:1201.1634v1, 2011.

LM2011b S.K. Mohammed and E.G. Larsson, “Single-UserBeamforming in Large-Scale MISO Systems...: TheDoughnut Channel”, arXiv:1111.3752, 2011.

SL2012 C. Studer and E.G. Larsson, “PAR-Aware Large-ScaleMulti-User MIMO-OFDM Downlink”, arXiv:1202.4034,2012.

HBD2011 J. Hoydis, S. ten Brink, M. Debbah, “Massive MIMO: HowMany Antennas do We Need?”, arXiv:1107.1709, 2011.

GERT2011 X. Gao, O. Edfors, F. Rusek, and F. Tufvesson, “Linearpre-coding performance in measured very-large MIMOchannels”, IEEE VTC 2011

Mar2010 T. L. Marzetta, “Noncooperative MU-MIMO with unlimitednumbers of base station antennas,” IEEE Trans. Wireless.Comm. 2010.

JAMV2011 J. Jose, A. Ashikhmin, T. L. Marzetta, and S. Vishwanath,“Pilot contamination and precoding in multi-cell TDDsystems,” IEEE Trans. Wireless Commun., 2011.

GJ2011 B. Gopalakrishnan and N. Jindal, “An Analysis of PilotContamination on Multi-User MIMO Cellular Systems withMany Antennas,” IEEE SPAWC 2011.

NML2011 H. Q. Ngo, T. Marzetta and E. G. Larsson, “Analysis of thepilot contamination effect in very large multicell multiuserMIMO systems for physical channel models,” IEEE ICASSP2011.

PML2012 A. Pitarokoilis, S. K. Mohammed and E. G. Larsson, “Onthe optimality of single-carrier transmission in large scaleantenna systems”, IEEE Wireless Communication Letters,submitted, 2012.

CMT2004 G. Caire, R. Muller and T. Tanaka, “Iterative multiuserjoint decoding: optimal power allocation and low-complexityimplementation,” IEEE Trans. IT, 2004.

ZLW2007 H. Zhao, H. Long and W. Wang, “Tabu search detection forMIMO systems,” IEEE PIMRC 2007.

VMCR2008 K. Vishnu Vardhan, S. Mohammed, A. Chockalingam, andB. Sundar Rajan, “A low-complexity detector for largeMIMO systems and multicarrier CDMA systems,” IEEEJSAC 2008.

Sun2009 Y. Sun, “A family of likelihood ascent search multiuserdetectors: an upper bound of bit error rate and a lowerbound of asymptotic multiuser efficiency,” IEEE JSAC 2009.

LH1999 A. Lampe and J. Huber, “On improved multiuser detectionwith iterated soft decision interference cancellation,” inProc. IEEE Communication Theory Mini-Conference, 1999.

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Thank You

Visit the very large MIMO website

www.commsys.isy.liu.se/ egl/vlm/vlm.html

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icassp 2012 tutorialrqiu/teaching/ece7750/...scaling down p with m ⇒ noise will limit performance....

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