Emerging techniques for spectrally-efﬁcient MIMO-enabled ...gesbert/presentations/MIMOS_seminar_final.pdf · Emerging techniques for spectrally-efﬁcient MIMO-enabled wireless

Emerging techniques for spectrally-efficientMIMO-enabled wireless networks

A seminar given MIMOS Center, Kuala LumpurMarch 5-14, 2007

Prof. David Gesbert

Mobile Communications Dept., Eurecom [email protected]

www.eurecom.fr/∼ gesbert

2

General introduction to the seminar

• General information and philosophy of the seminar

• Supporting material for the seminar

• Some useful definitions

• Some useful acronyms

Gesbert - MIMOS Seminar c© Eurecom March 2008

3

General information

This seminar aims at giving understanding in

• Someadvanced wireless design concepts for high spectrum efficiency

• Fundamental MIMO and MU-MIMO theory

• Cross-layer design for multiuser communications

• Someconcepts in cooperative communications in wireless networks

References

• many..partial list at end of the slides


4

Pre-requisites

Pre-requisites/advisable:

• Matrix linear algebra

• Digital Communications

• Statistical Signal Processing

• Signals, Probabilities and Stochastic Processes

• Information theory (basic elements)


5

Supporting material for the seminar

Seminar mainly based on:

• A. Paulraj, R. Nabar and D. Gore, Introduction to Space-Time WirelessCommunications, Cambridge Univ. Press, May 2003

• Multi-antenna Transceiver Techniques for 3G and Beyond by Ari Hottinen,Olav Tirkkonen, Risto Wichman. John Wiley and Sons, March 2003.

• Tutorial reading on single-user MIMO systems: D. Gesbert, M. Shafi, D.Shiu, P. Smith, ”From theory to practice: An overview of space-time codedMIMO wireless systems ”, IEEE Journal on Selected Areas on Communi-cations (JSAC). April 2003.

• Tutorial reading on multi-user MIMO systems: D. Gesbert, M. Kountouris,R. Heath, C-B. Chae, T. Salzer, "From Single User to Multiuser Communi-cations: Shifting the MIMO paradigm", IEEE Signal Processing Magazine,Sept. 2007.

• Tutorial reading on cooperative resource allocation: D. Gesbert, G. Oien,S. Kiani, A. Gjendemsjo, "Adaptation, Coordination and Distributed Re-


6

source Allocation in Interference-Limited Wireless Networks", Proceed-ings of the IEEE, Dec. 2007.

• Space-Time Wireless Systems: From Array Processing to MIMO Commu-nications, by H. Boelcskei, D. Gesbert, C. Papadias, A. J. van der Veen,Editors. Cambridge University Press (Spring 2006) .


7

Seminar philosophy


8

Seminar philosophy


9

Seminar outline Part 1

MIMO and multi-user communications

• Fundamentals

– Key definitions, math backgrounder

– Reminders in wireless network design

– Principles of smart Antenna technology

– Fundamentals of MIMO systems theory

– Channel models

• Single-user MIMO transmission

– Multi-antenna: system goals

– MIMO transmission techniques (space time codes, BLAST etc.)

– MIMO Receiver design and trade-offs

– MIMO-OFDM

– Adaptive MIMO transmission techniquesGesbert - MIMOS Seminar c© Eurecom March 2008

10


• Multiuser communication concepts

– multiuser information theory (what you need to know)

– MIMO multiuser information theory

– Multiuser scheduling techniques - the rate vs. fairness trade-off

• Multiuser MIMO techniques

– MIMO-OFDMA

– Multiuser MIMO in the uplink

– Multiuser MIMO in the downlink

– Linear beamforming vs. Non linear techniques


11


Emerging topics in multiuser, cooperative, MIMO networks

• Limited feedback techniques for Multiuser MIMO

– Practical techniques in standards (3GPP-LTE+, etc.)

– Advanced feedback reduction techniques

– Opportunistic MIMO communications

• Multiuser multicell cooperation

– MIMO in cellular networks

– Multicell MIMO ("‘network MIMO"’)

– Distributed network MIMO

• Resource allocation methods for interference mitigation

– Dynamic resource allocation

– Game theory for resource allocation

– Distributed allocation ih large networksGesbert - MIMOS Seminar c© Eurecom March 2008

12

• Distributed implementation of multicell multiuser mimo

– Distributed beamforming

– Implementation issues


13

Some definitions

• ”Smart Antennas”: Spatially distributed antenna elements combining. Thesignal mapping to the antennas and the combining is smart. Not the an-tennas themselves...

• ”Space-time processing”: Smart antennas with antenna combining in spaceAND time.

• ”Diversity gain”: SNR gain given by extracting the same information fromtwo independently fading channels.

• ”Space-time coding”: Form of smart antenna at the transmitter involvingcoding in addition to linear combining.

• ”Spatial multiplexing (SM)”: transmission approach where several transmitantennas carry multiple signals independent from each other to increaserate.


14

Some more definitions

• "Multi-user diversity": Gain obtained by selecting the "best" user in packetscheduling over fading channels

• "Rate": transmission speed offered at instant t by the channel

• "Throughput": transmission speed experienced by a user, averaged overa window T .

• "Capacity scaling": rate at which the capacity scales with increasing num-ber of antennas or users.


15

Acronyms

BTS: Base Station Transceiver

BER; Bit Error Rate

3G: Third Generation Wireless GSM: Global System for Mobile

extension of GSM to data

CCI: Co−channel interference

GPRS: General Packet RadioServices

MMSE: Minimum mean square error

MIMO: Multiple input multiple output

MISO: Multiple input Single output

ISI: Intersymbol interference

i.i.d.: identically independent distribution

ML: Maximum likelihood estimation

MOS: Mean opnion score (voice quality)

EIRP: Equivalent Isotropic Radiated Power

RX: Receive

SIMO: Single input multiple output

SINR: Signal to noise+interference ratio

SNR: Signal to noise ratio

SU: subscriber unit (Mobile)

TX: Transmit

ZF: Zero Forcing inversion

BWA: Broadband wireless access

FDD: Frequency division duplex TDD: Time division duplex


16

Quick math backgrounder• Notations

• Orthogonality, special matrices

• Matrix rank

• Eigenvalue decomposition

• Singular value decomposition


17

Mathematical notations

• u: scalar

• u: vector

• U: Matrix

• UT : Transpose operator

• U∗: Transpose conjugate operator

• U#: Pseudo-inverse operator

• E(): Expectation operator

• IN : identity matrix of size N ×N

• (x)+: x if x is positive, zero otherwise.


18

OrthogonalityVector Orthogonality: Let u = [u1, .., uN ] and v = [v1, .., vN ] be complex vec-tors of size N . u and v are orthogonal iff:

u∗v =N∑

i=1

u∗ivi = 0


19

Special matricesHermitian: Matrix U is hermitian iff

U = U∗

Unitary: Matrix U = [u1,u2, ..,uN ] is unitary (or ”orthogonal”) iff

U∗U = IN

which means that vectors u1,u2, ..,uN are unit norm and orthogonal to eachother.


20

Matrix rankThe rank r of matrix U = [u1,u2, ..,uK] of size N×K is defined by the dimen-sion spanned by its columns u1,u2, ..,uK (or by its rows).

r = dimension(span(u1,u2, ..,uK))

Therefore r ≤ min(N,K).

If r = min(N,K) the matrix is said to be ”full rank”.

The matrix is left (resp. right) invertible iff it is full column (resp. row) rank .


21

Matrix eigenvalue-decompositionLet the N × N matrix A. The EVD of this matrix is the set of unit-normeigenvectors u1,..,uN and eigenvalues λ1,.., λN such that:

Aui = λiui

in other terms:

A = [u1, ..,uN ]

λ1 0

. . .0 λN

[u1, ..,uN ]−1

i.e.:A = UΛU−1


22

EVD of hermitian matrix• Hermitian matrices are important. All covariance matrices, of the typeE(xx∗), are hermitian.

• The eigenvalues of complex hermitian matrices are positive real. Theeigenvectors are orthogonal.

A = A∗

Then:

A = UΛU∗

with λi ≥ 0 and U∗U = IN


23

Singular value decompositionLet the N ×K matrix A. The SVD of this matrix is given by:

A = UΣV∗

where

• U = [u1,..,uN ] is the N × N , unitary matrix, containing the left singularvectors

• V = [v1,..,vK] is the K × K, unitary matrix, containing the right singularvectors

• Σ is the N×K matrix containing the singular values. Example for N ≥ K:

Σ =

σ1 0. . .

0 σK0 0


24

SVD versus EVDLet the N ×K matrix A. The SVD relates to the EVD by the relation:

AA∗ = UΣ2U∗

andA∗A = VΣ2V∗

The left singular vectors of A are the eigen-vectors of hermitian matrix AA∗.The right singular vectors of A are the eigen-vectors of hermitian matrix A∗A.

Finally |σi| =√

(λi(A∗A)) =√

(λi(AA∗)).


25

Seminar Outline

• General introduction to the seminar

• Challenges of wireless network design

• Introduction to MIMO networks

• Going Multi-user!

• Perspectives


26

Challenges of wireless network design

• A palette of wireless networks

• Challenges of the wireless channel

• Perspectives on "wireless performance"

• Some solutions


27

Current wireless standardsKey system design parameters

• Rates (Bits/Sec)

• Range (kms)

• Mobility support (km/h)

Key wireless standards (Average rates/Range/Moblity)

• GSM (very low/high/high)

• UMTS (low/high/high)

• LTE / WiMax 802.16 (medium/high/low)

• WiFi 802.11x (high/medium/low)

• UWB (very high/very low/low)


28

A palette of networks

How wireless "kills you":

• Attenuation, noise limit the maximum communication rate and range.

• Fading limits the communication reliability of any point to point link.

• Interference limits the reusability of spectral resource in space.


29

A palette of networks

How clever ideas "save you":

• Diversity: Obtaining independent observations of same information

• Cooperation: Pooling degrees of Freedom of many transceivers into asingle basket.

• Adaptivity: Exploiting the best of what the channel has to offer.


30

Design goals (beyond 4G)V. HIGH Performance

• TCP perf comparable to DSL, Cable

• Rates much better than 3G (20-100 Mbs)

• Spectrum efficiency >5x better than 3G (>3 Bits/Sec/Hz/BTS)

• Coverage/cell radius: >20x better than WiFi (several kms)


31

Perspective on Wireless PerformanceTwo Key Metrics:

• Coverage in km (early stages)

• spectral efficiency in Bit/Sec/Hz/Cell (mature deployments):

SE =E(rM)

K(1)

M =average modulation order (bits/symbol), r = code rate (1/2, 3/4, ..), K = Frequency reuse


32

Wireless research a la Carte

SYSTEM ARCHITECTURE

PROTOCOLS

Transport (TCP/IP)Cross Layer 1/2

scheduling, multi−user MIMO,cooperative coding. relaying,multicell MIMO, resource alloc.,..

NETWORK OPTIMIZATION

Bits/Sec/Hz/NOK

point−to−pointCoding/Sig Processing

TRANSMISSION


33

Improving wireless performance?

Definition: set point= required SNR for minimum acceptable QoS

Ad. Modulation MIMO

Incremental coding turbo−coding

SIMO/MISO Diversity

Intf Canceling

ARQ−Fragmentation

space−time coding

ACHIEVABLE 1/SETPOINT

AC

HIE

VA

BLE

TH

RO

UG

HP

UT

Reference

Space−Time Processing

MIMO is unique in the ability to both increase reliability (using MIMO diversity)and extend data rates (using spatial muxing)


34

Challenges of the wireless channelWireless transmission introduces:

• Fading: multiple paths with different phases add up at the receiver, givinga random (Rayleigh/Ricean) amplitude signal.

• ISI: multiple paths come with various delays, causing intersymbol interfer-ence.

• CCI: Co-channel users create interference to the target user

• Noise: electronics suffer from thermal noise, limiting the SNR.

• Doppler: The channel varies over time, needs to be tracked.


35

Multipath Propagation

xxxx

xxxx

Scatterers local to mobile

Scatterers local to base

Remote scatterers


36

Summary diagram

ISI

Rx CCI

Tx CCI

fading

noise

Tx Rx

co-channel Tx user co-channel Rx user


37

Some solutions

Tricking the wireless channel...

• Advanced coding and filtering (turbo)

• Hybid retransmission protocols

• Fast link adaptation

• multi-antenna (MIMO) techniques

• multi-user

– filtering

– scheduling

– inter-cell coordination (for interference control)

– cooperation (multi-user, multi-cell, relays,..)


38

Seminar Outline





• Perspectives


39

Introduction to MIMO networksWe first focus on the single-user scenario:

• Principles of smart antennas

• Smart antenna techniques:

– Receive side

– Transmit side (without feedback)

– Joint transmit-receive (without feedback)

• Joint transmit-receive with feedback


40

Smart Antenna BTS


41

MIMO devices


42

Purposes of Smart Antennas

Using the space dimension (i.e. space-time processing)

• Using antenna arrays permits to process radio signals in space, not onlytime.

...to improve performance in presence of fading/interference

• coverage

• quality (BER, MOS, outage)

• capacity: Bit/sec/Hz/BTS or # users/Hz/BTS

• peak data rates: Bit/sec


43

MIMO Configurations

CO

OP

ER

AT

ION

INTERFERENCE

cell 1

cell 2

COOPERATIVE MULTICELL MU−MIMO SINGLE CELL MU−MIMO

SINGLE CELL MIMOSINGLE CELL SIMO/MISOSINGLE CELL SISO

INTERFERENCE MU−MIMO


44

Zooming on one MIMO TX-RX link

M RXersN TXers

’H’CHANNEL0010110 0010110

codingmodulation

weighting/mapping

weighting/demapping

demodulation

decoding


45

Qualitative MIMO gains


46

Mapping data to multi-antennasKey approaches

• 1 - Beamforming algorithms: To increase received SNR for desired direc-tions/signatures.

• 2 - Diversity algorithms: To combat fading in order to work at less SNR.

• 3 - Interference mitigation: To maximally reuse the channel frequencies

• 4 - Spatial Multiplexing algorithms:

– single-user multiplexing: to increase data speeds

– multi-user multiplexing: to increase cell user capacity

Want to combine all of these? Not possible!

Antennas are dimensions in a finite dimensional vector space, there is only asmall finite number of ways of combining them.


47

Smart Antennas techniques at receive• Steering vectors and signal formulation

• Information theoretic bounds

• Beamforming

• Diversity


48

Steering vector

−2 j sin( )(i−1)d/θπ λe

N sensors

source

propagating field

θplane wavefront

1 2 3 id

phase on the i−th sensor

h(θ) = [1, e−2πj sin(θ)dλ, e−2πj sin(θ)2dλ , .., e−2πj sin(θ)

(N−1)dλ ]T


49

Math formulation• h = [h1, h2, .., hN ]T is the spatial signature of the sent signal s(t)

• w = [w1, w2, .., wN ]T is the vector of antenna weights

• y(t) is the (noisy) measured signal at the array output

x(t) = hs(t) + n(t)

y(t) = wTx(t) = wThs(t) + wTn(t)

Note: if receiver or transmitter moving, then h → h(t).


50

Capacity of SISO Systems (1 by 1)At fixed time t, the SISO channel is an additive white Gaussian noise (AWGN)channel with capacity:

C(t) = log2(1 + SNRsiso(t)) Bit/Sec/Hz (2)

where SNRsiso(t) is the received signal to noise ratio at time t:

SNRsiso(t) =|h(t)|2σ2

s

σ2n

⇒ +3dB of extra power needed for one extra bit per transmission!


51

Capacity of SIMO systems (1 by M)

C(t) = log2(1 + SNRsimo(t)) Bit/Sec/Hz (3)

With

SNRsimo(t) =(|h1(t)|2 + .. + |hM(t)|2)σ2

s

σ2n

• Array gain (gain in SNR mean): ¯SNRsimo = M ¯SNRsiso

• Diversity gain in Rayleigh channel (gain in distribution of SNR):

Probability(error) ≈(

1

4Mσ2

sσ2n

)M


52

The beamforming/Interference canceling concept

w1 w2 w3 wi wN

N sensorsh1 h2 h3 hi hN

measured signal

+

y=W HT

source

Vector space analogy

HW’ W

propagating field

Choosing W’ nulls the source out (interference nulling)Choosing W enhances the source (beamforming)

(for two sensors)


53

Basic algebra explains it all

All key MIMO and MU-MIMO schemes (except diversity-oriented) can be in-terpreted from previous drawing:

• A N -antenna beamformer can amplify one source (no interference) by afactor N in the average SNR: Beamforming

• A N -antenna beamformer can extract one source and cancel out N − 1interferers simultaneously: Interference canceling

• Transmit beamforming realizes the same benefits/gains at receive beam-forming if CSIT is given: Transmit beamforming and interference nulling

• All N sources can be simultaneously extracted (assuming the other N − 1are viewed as interferers) by beamformer superposition: Spatial multi-plexing

• N sources can be assigned to N distinct users: MU-MIMO, SDMA

• Some of the N sources may belong to different cells: cooperative multicellMIMO


54

Representing the channelThe weights depends on the form of parameterization of the channel:

• Channel impulse response: Weights are based on the complex fadingcoefficient

• Geometry of multipaths: weights are based upon knowledge of # multi-paths, and angles of arrival/departure (AOA) (θ1,..θL).

For broadband channels (delay spread>symbol period Ts):

• Scalar weights replaced by filters (single carrier)

• Different weights are applied to different subbands (OFDM)


55

Angles versus channel IRTrade-off depends on:

• Richness of scattering (number of multipaths)

• Power distribution across multipaths

Performance/complexity:

• In near-LOS: Angle-based beamformer is very simple (only one or twoangles to track).

• In rich scattering (e.g. urban): Impulse response-based is more robust(does not assume anything on micro-structure of multipath)

In rest of this seminar we assume the latter...


56

Channel models in MIMOSingle-user MIMO channel composed of line-of-sight and fading components

H =1√K + 1

Hf +

√K√

K + 1Hlos (4)

Fading component: General correlated model

vec(Hf) = R1/2vec(Hiidf ) (5)

where Hiidf is an i.i.d. complex Gaussian matrix of size M × N and R is an

arbitrary correlation matrix.

Particular model: The Kronecker model

Hf = R1/2r Hiid

f R1/2t (6)

where Rt, Rr, are the transmit and receive correlation matrices.

Line-of-sight component:

Hlos is a deterministic ill-conditioned matrix.Gesbert - MIMOS Seminar c© Eurecom March 2008

57

Line-of-Sight MIMO channel

several km

fewλfewλ

The system

Hlos ≈ α

1 1 11 1 11 1 1

is near rank one!


58

Beamforming PatternRegardless of whether the beamformer relies on angle of channel IR infor-mation, it receives (resp. transmits) energy on the best path combinationpossible.

xxxxxxxx

Bea

mfo

rmer


59

On spatial sampling for B/FHow much spacing is necessary?

Answer:

• if beamformer relies on AOA information, critical sampling (spacing lessthan λ/2) is required to avoid spatial aliasing.

• if beamformer does not use AOA modeling, then wider-than-critical spac-ing should be used to maximize decorrelation.


60

Applying Interference Canceling Algorithms?A difficult task..

• Concept similar to beamforming: except one tries to use weights orthog-onal to interferer’s channel vector.

• Tricky to implement from a system point of view because knowledge ofinterfering signature not usually available

• Performance highly dependent on specular nature of interference (a fewstrong better than many weak).

• Future systems, carrying bursty IP data, make interference less predictable.


61

The diversity principle• In the presence of multipath, |h(t)| can follow a Rayleigh distribution (i.e.h(t) = hr(t) + jhi(t) is complex random Gaussian)

• |h(t)| fades with non-negligible probability. But

SNR(t) =(|h1(t)|2 + .. + |hN(t)|2)σ2

s

σ2n

is zero iff hi(t) is zero for all i-s! (7)

⇒ Probability that SNR(t) < ǫ, where ǫ is a chosen small threshold decreasesexponentially with N!


62

The Power of DiversityDiversity gain: reduction in SNR acceptable for the same BER target.

0 50 100 150 200 250 300 350 400 450 500−125

−120

−115

−110

−105

−100

−95

−90

Time in Milliseconds

Sig

nal L

evel

in d

B


63

Diversity GainDiversity gain: reduction in SNR acceptable for the same BER target.


64

Diversity vs. beamforming gain• Diversity gain can be 10dB-20dB with just 2 antennas (much larger than

beamforming (array) gain: 3dB)

• Diversity gain can be used to extend the range or increase data rate ifadaptive modulation is enabled.

At the transmitter (to be seen later)

• Like beamforming, diversity can be used on receiver and transmitter.

• On transmitter, if channel is unknown, transformed domain techniques orspace time coding are used.


65

Receive Diversity AlgorithmsRobust because receiver can estimate the channel h accurately throughtraining sequences.

x = h(t)s + n(t) (8)

• Optimum techniques

– Linear: MRC (max ratio or max SNR) combining:

y(t) = w(t)Tx (9)w(t) = h(t)∗/|h(t)| (10)

– Linear: MMSE (minimum mean square error) combining (solution de-velopped in class)

– Non Linear: max likelihood detection

• Sub-optimum techniques: Equal gain combining, switching (see next page).


66

Receive Diversity Algorithms

Weightsphases

estimation

*s y

1

2

h

h 1

2

w

w

=h

=h

1

2*

Phasesestimation

(1)

Antennaselection

(2)


67

Smart Antennas techniques at transmit (MISO)• Information theoretic bound

• Remarks on algorithms:

– Without some channel knowledge transmit beamforming is not possi-ble!

– Must use diversity oriented techniques!


68

Capacity of MISO systems (unknown channel at TX)Channel unknown at transmitter, hence power split equally across antennas!

C(t) = log2(1 + SNRmiso,unknown(t)) Bit/Sec/Hz (11)

With

SNRmiso,unknown(t) =(|h1(t)|2 + .. + |hN(t)|2)σ2

s

Nσ2n

• Array gain (gain in SNR mean): ¯SNRmiso = ¯SNRsiso (No array gain!)

• Diversity gain in Rayleigh channel (gain in distribution of SNR): Pr(error)

≈(

1

4σ2sσ2n

)N


69

TX Diversity without channel knowledge• Transformed Domain Techniques

– The particularity of all these techniques is to rely on coding to work!

– Idea is to create random fast fading in a domain where interleaved-coded information can recover from it.

Other important approach:

• Space-Time Coding.


70

Transformed Domain Techniques

Z

equalizer

f

power

..

power

DeInt/FEC

t

space-timeencoder

source

source

source

decoder........

.. ....


71

Space-time coding techniquesOrigins

• Invented 1996 at ATT Labs, NJ. (treillis space time codes) (Tarokh,Seshadri,Calderbank)

• Low complexity space-time block codes invented at Lucent Labs, NJ (Alam-outi).

M RXersN TXers

^CC ^^^

S1,S2,..SP

input symbols

S1,S2,..SP

spac

e−tim

e

Space−time CHANNEL’H’mapper

space−time

demapperlinea

rbe

amfo

rmer

dete

ctor


72

General Model1) P input (QAM) modulation symbols are mapped to a codeword matrix ofsize N ×K.

s1, s2, .., sP ⇒ C =

c1(1) c1(2) ... c1(K)

... ... ...cN(1) cN(2) ... cN(K)

2) The codeword (possibly linearly transformed by some matrix V) is launchedinto the channel. We receive:

Y =

h11 ... h1N... ...

hM1 hM2 hMN

V

c1(1) c1(2) ... c1(K)

... ... ...cN(1) cN(2) ... cN(K)

+ Noise.

3) For conventional space-time code, pick P = K. Said to be "rate one" (onesymbol per channel use).


73

Space-Time Block Coding: Alamouti scheme (V = I)2 × 1 case:

constellationmapper

*

*

time:

2T 2T-1

*

*

*

*

c1 -c2

c1c2

information source

time:

r1

2T2T-1

r2 receiverc1 c2

h1

h2


74

Space-Time Block Coding: Receiver

Linear Combiner

soft decision for c2

soft decision for c1

c 1

ML Decision

c 2

ML Decision

H2

H1

r1

~r1

$c1

$c2

~r2r

2

~ * *r H H

r

r=

L

NM

O

QP1 2

1

2


75

Space-Time block coding equations

[r1r2

]=

[−c∗2 c∗1c1 c2

] [h1

h2

]+ n (12)

[r∗1r2

]=

[h∗2 −h∗1h1 h2

] [c1c2

]+ n (13)

[h2h∗1]

[r∗1r2

]= (|h2|2 + |h1|2)c1 + n1 → c1 (14)

[−h1h∗2]

[r∗1r2

]= (|h2|2 + |h1|2)c2 + n2 → c2 (15)

(16)


76

Space-Time Block Coding: Arbitrary N

Theorem (Tarokh et al.):

• There exist full-rate orthogonal block codes for real-valued modulationsfor N = 2, 4, and 8 only. Beyond: loss of rate!

• For complex (e.g. QAM), there exist full-rate orthogonal block codes forN = 2 only.

Orthogonal codes can be developed for N > 2 but with rate P/K<1 only!Example for N = 4, rate is 4/8 = 1/2:

C =

c1 −c2 −c3 −c4 c∗1 −c∗2 −c∗3 −c∗4c2 c1 c4 −c3 c∗2 c∗1 c∗4 −c∗3c3 −c4 c1 c2 c∗3 −c∗4 c∗1 c∗2c4 c3 −c2 c1 c∗4 c∗3 −c∗2 c∗1

(17)


77

Single-user MIMO-OFDMMIMO-OFDM key component of 3GPP-LTE, 802.16 WiMax, 802.11n WiFi

• hij −→ hij(τ ) (τ is delay parameter in IR).

• Single carrier systems: antenna weights are replaced with filters.

– Complexity increases very quickly with IR length! (unpractical at veryhigh data rates)

• Multi-carrier systems:

– Each subband is seen as approximately flat-fading channel

– MIMO algorithms can be used on top of each carrier.

– Complexity only proportional to number of carriers, remains reasonableat very high data rates.


78

MIMO-OFDM PrincipleRayleigh, Cioffi (1998)

(f)H s1,s2,...s1,s2,...

M RXersN TXers

ST

dec

oder

/sep

arat

or

ST

Enc

oder DEMOD

DEMODOFDM

OFDM

OFDM MOD

OFDM MOD


79

MIMO-OFDM Mod and DeMod

IFF

T

FF

Tprefixremoval

Cyclic prefix

c0c1

cK−1

c0

cK−1

DemodulatorModulator

^

^

c1^


80

MIMO-OFDM Characteristics• Cyclic Prefix (CP) chosen as function of channel length to avoid ISI.

• No MIMO-equalization needed.

• FFT is computationally efficient.

• AssumingK orthogonal carriers, f0,f1,..fK−1, the frequency domain MIMOchannel is written as a M ×N matrix on each carrier:

H(k) =

H11(k) H12(k) ..H21(k) H22(k) ..

: : :

where Hij(k) is frequency response at fk of broadband channel hij(τ ).


81

OFDM Spatial Multiplexing

seria

l to

pare

llel

para

llel t

o se

rial

OFDM

^^.....

2S1S2 .....S1S

DEMOD

OFDM MOD

OFDM MOD

OFDMN+11

DEMOD

N TXers M RXers

H (f)

SM

DE

CO

DE

RP

ER

TO

NE

SSS 2N+1 .....

SN S2N S3N .....


82

OFDM Space time coding (Alamouti)

S1*− S0

S1

S0

S1

=[S ,....,S ]0K−100S

0

=[S ,....,S ]1K−110S1

K= OFDM Block size

deco

der

Ala

mou

ti

^

^

ante

nna

time

OFDM block 2 OFDM block 1

*0

OFDM MOD

OFDM MOD

OFDM

OFDMDEMOD

DEMOD

M RXers

H (f)

2 TXers

S


83

Diversity in OFDM-STC• The previous scheme (simple OFDM Alamouti) provides 2M orders of di-

versity

• Does not have coding across subcarriers

• Does not yield Frequency diversity (only space diversity)

Solution:

• Suboptimal: Make sure symbols spanning one OFDM block are codedacross. This is equivalent to frequency coding, which will give frequencydiversity in case of rich delay spread channel.

• Optimal: Code simultaneously across subcarriers AND across antennas.This gives rise to space-frequency codes (see Bolcskei, Giannakis, etc.).

• Performance should improve with the number of multipath! (added fre-quency diversity)


84

Space-Time-Frequency codes

TIME

SPACE

FREQUENCY

STF Codewords


85

MIMO-OFDM Performance

N=2,M=1,K=16,64QAM (Space-time-frequency code from Bolcskei and Paulraj2001)


86

MIMO systemsMotivations

• Convergence between PCs and phones in future wireless systems (e.g.PDAs)

• Computing power and form factor compatible with installation of multipleantennas at handset

• Data access requires higher transfer speed and QoS, making MIMO use-ful.

• Practical antenna separation: Half a wavelength is enough at the mobile(1).

(1) measurement campaign using Eurecom Platform, 2007


87

MIMO Laptop (Lucent prototype)16-element array


88

MIMO system: Digital camera

Based on Airgo’s MIMO WIFI technology


89

SIMO, MISO versus MIMO• Stream multiplexing first introduced at Stanford Univ. (1994, Multi-user

MISO ), then Lucent (1996, MIMO point to point)

• Exploit space dimension at both transmitter and receiver: Subscriber unitis also equipped with multiple antennas.

• Leads to hyper-diversity gains (joint TX-RX)

• Channel becomes matrix. Possibility to transmit on several so-called’eigen-modes’ simultaneously (spatial multiplexing gain).

• Exploits multipath instead of mitigating it.

• Can be combined with conventional strategies for beamforming and inter-ference reduction.


90

Quantitative MIMO Gains• Array gain: log(M) dB on receive side, log(N) dB on transmit side if trans-

mit channel known.

• Diversity gain: Up to M ×N orders of diversity.

• Spatial Multiplexing gain: Up to min(M,N) factor of data rate increase!

Caution:

• Not all these gains can be achieved at the same time!

• Specific algorithm will extract specific gains

• Channel models have a direct impact on the gains


91

MIMO channel matrix

hji

MIMO ReceiverMIMO Transmitter

M

1

N TXers M RXers

i

j

1

N

0010110 0010110

Example for 3 × 4 system:

H =

h11 h12 h13

h21 h22 h23

h31 h32 h33

h41 h42 h43


92

Capacity of MIMO systemsNote: we assume channel unknown at transmitter!

Cerg = EH(log2

[det(IM +

ρ

NHH∗

)])≈ αmin(M,N), (18)

where H is the M × N random channel matrix and ρ is the average signal-to-noise ratio (SNR) at each receiver branch.

⇒ Capacity proportional to min of # TX and # RX antennas!


93

Average capacity of ideal MIMO systemideal=i.i.d. Rayleigh distributed


94

Distribution of capacity of ideal MIMO system

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Capacity in Bits/Sec/Hz

Pro

b C

apac

ity <

abs

ciss

a

Capacity of i.i.d Rayleigh diversity channels at 10dB SNR

1x1

3x3

10x10

1x8

1x19

8x1

19x1

MIMO Diversity

SIMO Diversity

MISO Diversity


95

MIMO techniques• Beamforming oriented, requires channel feedback (see later)

• Diversity oriented (MIMO space time codes)

• Rate maximization oriented (MIMO spatial multiplexing)

• Linear dispersion space time codes


96

MIMO Space-time codingPrinciples

• Transmitter design works independent of number of RX antennas!

• Space-time code picks up M ∗N orders of diversity

• Example: Alamouti code for 2 × 2 system (developped in class)


97

MIMO Spatial multiplexingSpatial Multiplexing: We send multiple signals, the receiver learns the chan-nel matrix and inverts it to separate the data.

* *

* *

* *

* *

* *

* *

b1 b4 ...

b2 b5 ...

b3 b6 ...

b3 b4 b5 b6b2 ...b1

b1 b4 ...

b2 b5 ...

b3 b6 ...

b3 b4 b5 b6b2 ...b1

* *

* *

* * *

****

****

*

**

* **

*

***

****

*** ***

*

*

* *

* *

* * *

**

*

****

**

* **

*

***

* ** ***

*

*

* * *

****

***

*

**

* *

*

***

****

** **

*

*

*

* **

***

**

*

** **

***

**

*

*

* *

**

**

*

* *

*

*

*

*

* **** *****

**********

*** * **

* ****

***** ***

* ***

****

**

*

**

**

*

* *

**

*

* **

*

**

*

A1

A3

A1

A2

A3

B1

B2

B3

C2

C3

C1

C1

C2

C3

B1

B2

B3

A2

SIG

NA

L P

RO

CE

SS

ING

Mod

ulat

ion

and

map

ping


98

Spatial multiplexing seen as a space-time codeSpatial multiplexing is equivalent to a space-time code with P = KN :

P = KN input (QAM) modulation symbols are mapped to a codeword matrixof size N ×K.

s1, s2, .., sP ⇒ C =

c1(1) c1(2) ... c1(K)

... ... ...cN(1) cN(2) ... cN(K)

We receive:

Y =

h11 ... h1N... ...

hM1 hM2 hMN

c1(1) c1(2) ... c1(K)

... ... ...cN(1) cN(2) ... cN(K)

+ Noise.


99

A simple mapping exampleThe simplest mapping for spatial multiplexing is given by: cn(k) = snk

Y =

h11 h12 ..h21 h22 ..: : :

s1 sN+1 ..s2 sN+2 ..: : ..sN sN+2 ..

+ n (19)


100

How it worksExample for 3x3:

x1

x2

x3

=

h11 h12 h13

h21 h22 h23

h31 h32 h33

︸︷︷︸H

b1b2b3

+ Noise (20)

b1b2b3

= H−1

x1

x2

x3

(21)


101

Impact of channel modelMIMO Performance very sensitive to channel matrix invertibility

The following degrades the conditioning of the channel matrix:

• Antenna correlation caused by:

– small antenna spacing, or

– small angle spread

Line of sight component compared with multipath fading component:

– multipath fading component, close to i.i.d. random, is well conditioned

– Line of sight component is very poorly conditioned.

All of this fixed in the multi-user scenario!


102

MIMO-SM in Line-of-Sight

several km

fewλfewλ

The system

H ≈ α

1 1 11 1 11 1 1

is near rank one (non invertible)!

⇒ Spatial multiplexing in single-user requires multipath to work!!!

Not so in multi-user MIMO :)


103

Spatial multiplexing receiver design1. Linear receivers for BLAST (Zero-Forcing, MMSE)

2. Non linear receiver (SIC, ML)

3. Performance and complexity trade-offs


104

Zero-Forcing receiver

y1

y2

:

=

h11 h12 ..h21 h22 ..: : :

s1

s2

:

+ n (22)

Zero Forcing implements matrix (pseudo)-inverse (ignores noise enhance-ment problems):

s = H#y (23)

where:

H# = (H∗H)−1H∗


105

MMSE receiverThe MMSE receiver optimizes the following criterion:

W := argminE|W∗y − s|2We find:

s = H∗(HH∗ + Rn)−1y (24)

where Rn is the noise/intf covariance.

This offers a compromise between residual interference between input sig-nals and noise enhancement.


106

Non linear MIMO receiversMaximum likelihood receiver:

• Optimum detection

• Exhaustive search. No iterative procedure for MIMO.

• Complexity exponential in QAM order and N .

y1

y2

:

=

h11 h12 ..h21 h22 ..: : :

s1

s2

:

+ n (25)

Maximum Likelihood Solution:

s = argmin|y − Hs|2where s is searched over the modulation alphabet (e.g. 4QAM, 16QAM..)


107

Iterative procedure: V-BLAST’Onion peeling’ Solution (or ’Successive Interference Canceling’):

s1 = wH1 y (26)

ˆs1 = Slicer(s1) (27)

y1 = y − h1ˆs1 (28)

s2 = wH2 y1 etc

• Offers complexity linear in N

• Disadvantage: performance unequal on various stages (best on s1..worston sN ).

• input signals are ranked in power for better performance


108

Signal ordering for V-BLASTIdea: Decode first the streams which exhibit highest SNR so as to minimizethe propagation of errors in later stages.

Example, at first stage: Let G = H# be the N ×M pseudo-inverse of thechannel matrix.

Let gi be the i-th row of G. The SNR for si is inversely proportional to ‖gi‖2.Therefore we start with si0 where

i0 = argmini=1:N‖gi‖2and continue with si1 (second smallest row norm), etc.


109

Performance comparisonBLAST zero-forcing vs. V-BLAST (SIC) vs BLAST-ML (2x2)

0 5 10 15 20 25 3010

−5

10−4

10−3

10−2

10−1

100

SNR

BE

R

V−BLAST, 2 tx, 2 rx, 4−QAM

ZFSICML


110

Performance comparisonBLAST zero-forcing vs. V-BLAST (SIC) vs BLAST-ML (4x4)

0 5 10 15 20 25 3010

−5

10−4

10−3

10−2

10−1

100

SNR

BE

R

V−BLAST, 4 tx, 4 rx, 4−QAM

ZFSICML


111

SM receiver trade-offs• Linear MMSE receiver is simplest to compute and performs better than

ZF receiver.

• MMSE receiver provides a diversity order only equal to M − N + 1. Verylimiting for square systems.

• ML provides optimal performance and diversity for a SM receiver but com-plexity exponential with QAM order and N .

• V-BLAST provides good compromise between performance of ML andlow complexity of a linear receiver.

• Issue with V-BLAST: Performance very unequal between different stages(last stage much better than first stage because of added diversity).


112

MIMO multiplexing-diversity trade-offThe Loading of the MIMO system is given by the rate P

K .

Rate (# independent bits over MIMO system)

Efficiency/Goodput

over loaded MIMOunder loaded MIMO

K0

Space−time coding Spatial multiplexing

• Overloaded MIMO systems do not provide enough diversity gain.

• Under-loading gives underutilization of MIMO capacity.

• There exists a sweet spot [Tse et al 02].


113

Bridging between STC and Spatial Multiplexing• Rate of general MIMO space time code: P

K .

• By varying the rate, one varies the level of independence between therows of C.

– Maximizing dependence ⇒ maximizing diversity

– Minimizing dependence ⇒ maximizing rate

• Symbol mapping can be linear (to simplify design). Such codes are calledLinear Dispersion ST codes

• Linear mapping optimized to maximize capacity or minimize BER


114

Linear Dispersion STCP input (QAM) modulation symbols and their conjugates are linearly mappedto a codeword matrix of size N ×K (code rate is P/K).

s1, s2, .., sP ⇒ C =

c1(1) c1(2) ... c1(K)

... ... ...cN(1) cN(2) ... cN(K)

LD-STC:

C =P∑

p=1

(Apsp + Bps∗p)


115

Design/decoding of LD-STC• Ap,Bp, p = 1..P are the designed dispersion matrices

• Can be designed to maximize capacity, minimize BER, etc.

• In general, the orthogonal structure is lost

• ML receivers are needed (complex for large N and high order QAM!)


116

Performance comparisonSame rate comparison: Alamouti (1x16 QAM), BLAST zero-forcing, BLASTML, STBC-BLAST-ML (all 2x4QAM).

0 5 10 15 20 2510

−5

10−4

10−3

10−2

10−1

SNR (dB) per receive antenna

Bit

Err

or R

ate

2 transmitters − 2 receivers

Alamouti − Linear (16QAM)SM − ZF (4QAM)SM − ML (4QAM)STBC − ML (4QAM)


117

Smart antenna techniques with feedback• Considerations on channel feedback

• Information theoretic bounds

• Techniques


118

Channel knowledge via feedbackIdeal case:

• Channel is fully known: The same gain can be achieved on either transmitor receive beamforming.

Practical case (channel estimated):

• Receive side: Channel estimated from training sequence

• Transmit side:

– FDD system with dedicated feedback channel: quality depends onfeedback rate.

– FDD system without feedback: partial knowldge obtained reciprocalchannel component (angles, delays). However non faded paths ondownlink can be faded on uplink. Robustness issues.

– TDD system: Beamformer uses uplink channel IR estimates. Qualitydepends on ”ping-pong time”.


119

Capacity of MISO systems (N by 1)Transmit channel is known: The total power is split optimally across the Ntransmit antennas.

C(t) = log2(1 + SNRmiso(t)) Bit/Sec/Hz (29)

With

SNRmiso(t) =(|h1(t)|2 + .. + |hN(t)|2)σ2

s

σ2n

• Array gain (gain in SNR mean): ¯SNRmiso = N ¯SNRsiso.

• Diversity gain in Rayleigh channel (gain in distribution of SNR):

Probability(error) ≈(

1

4Nσ2

sσ2n

)N


120

MIMO capacity with known TX channelNote: we assume channel known at transmitter through some form of feed-back so the power can be optimally distributed on the eigenmodes.

The MIMO capacity is the sum of individual subchannel capacities:

CWF =m∑

i=1

log2(µλi) with ρ =m∑

i=1

(µ− λ−1i )+

λi are linked to the eigenvalues of HH∗ if M ≤ N , resp. of H∗H if N ≤M .

more power is put on the subchannels with larger eigenvalues. Less poweron others.


121

Gain of feedback vs. Rice K factor

2 4 6 8 10 12

010

2030

4050

Number of Antennas (r=t)

Rel

ativ

e ca

paci

ty g

ains

due

to fe

edba

ck (

%)

rayleigh

K=2dB

K=10dB

rayleigh

K=2dB

K=10dB

SNR=3dBSNR=18dB


122

MISO and MIMO techniques with feedback• MISO case: gains become identical to SIMO!

– TX diversity

– TX beamforming

• MIMO case

– eigen-beamforming

– Spatial waterfilling

Conclusion: feedback (CSIT) is useful to permit SNR (array) gain and smartpower allocation at TX. Feedback gain thus vanishes at high SNR.


123

MIMO Eigen-beamforming• Requires transmit channels knowledge

• Relies on singular value decomposition of H (SVD)

• Trade-off between diversity and rate easy to control

SVD of H: H = UΣV∗

1> ...σ 2σ

Η U Vσ1σ2 σ

Ν=

0 0

0

0

*

Σ

Μ

Ν Μ Ν


124

Diversity maximization (I)• We transmit one symbol s on the top right singular vector v1, and receive

on u1:

• Rate is 1. Diversity order is maximum MN .

w

s y

11

12

21

22

1

2 2

(Τ)

h

h

h

(R)

(R)

(T)

1(T)

h

w

w w


125

Diversity maximization (II)Output signal:

y = u∗1Hv1s + u∗

1n.

Output signal-to-noise ratio:

SNRoutput =σ2

1σ2s

σ2n

= σ21SNRinput

SNR gain is given maximum singular value!


126

MIMO Eigen-beamforming with multiplexing• We transmit P symbols on the top P right singular vectors, and receive

on the top P left.

• Rate is P . Diversity is maximum on top singular pair, then decreasing withsingular value index.

Y = [u1, ..,uP ]∗H[v1, ..,vP ][s1, .., sP ] + [u1, ..,uP ]∗n.


127

Capacity-achieving eigen-beamforming• We transmit min(M,N) symbols on each of the min(M,N) right singular

vectors

• The power of each symbol is adjusted separately (with√Pi) according to

the singular values

• Rate is min(M,N)

Assume M ≥ N :

Y = [u1, ..,uN ]∗H[v1, ..,vN ][√P1s1, ..,

√PNsN ] + [u1, ..,uN ]∗n.


128

MIMO Waterfilling

Y = [u1, ..,uN ]∗H[v1, ..,vN ][√P1s1, ..,

√PNsN ] + [u1, ..,uN ]∗n.

1> ...σ 2σ

Η U Vσ1σ2 σ

Ν=

0 0

0

0

*

Σ

Μ

Ν Μ Ν


129

MIMO Receiver limitations• Channel estimation not perfect

– Estimation relies on finite length training blocks and noisy data

– Channel is varying between blocks

– Channel estimation noise ”adds” to the thermal and interference noise.

• Real world noise

– Non white (if caused by other CCI signals)

– Non Gaussian (to some extent)

• RF front-end and other limitations:

– Phase noise

– Residual frequency offset (post lock loop)


130

Seminar Outline





• Perspectives


131

Going Multi-user!

• Motivations

• Introduction to multi-user techniques

• Multi-user MIMO networks:

– Foundations from information theory

– Techniques


132

Motivations

Multi-user makes certain things difficult:

• Dealing with users of unequal channel conditions (fairness issues).

• Mixing antenna filtering and scheduling problems into a harder problem.

• Multiple users can’t cooperate as well as multiple antennas on a singledevice.

• Leads to multiple (rather than single) power constraints.

But others much easier !

• Provides multi-user diversity (less reliance on antenna diversity).

• Provides decorrelation of spatial signatures.

• Allows for user- (in addition to stream-) multiplexing.

• low rank channels no longer a problem but an advantage.


133

Introduction to multi-user MIMO techniques

In practice multi-user techniques involves interplay of:

• Admission control

• Multi-antenna combining (for MIMO case)

• Power control

• User scheduling


134

Multi-user scheduling

Multiple-access (MA) can be classified as either

• Bandwidth expanding (FDMA, TDMA, OFDMA, CDMA, etc..)

• Non bandwidth expanding (superposition coding, spatial division etc.)

In bandwith expanding MA:

• oneuser is selected per unit of spectral resource (time slot, OFDMA slot,CDMA code, etc.).

• This gives multi-user diversity [Knopp et al, 95].


135

TDMA multi-user diversity

κγ is the SNR of user k

C6

C5C4C3

C2

C1

R3=MaxRk, k=1..

poll

pollpoll

poll

poll

poll

data

sleepy

sleepy

sleepy

Incoming traffic

ACCESS POINT+SCHEDULER


136

Scheduling criteria

• Many schemes: proportional rule, LWDF,max rate,min delay, etc.

• Most do not achieve the sum capacity of the stability region (except expo-nential rule) [Shakkottai]

On user fairness:

• For a symmetric network, all schemes are fair with ∞ time horizon

• For finite time horizon, fairness is not guaranteed

– =⇒ There is fundamental fairness-throughput trade-off

• Some schemes try to strike a balance. Example Proportional Fair Schedul-ing (PFS) [Kelly]


137

Proportional Fair Scheduler

At slot k, schedule user i∗(k) among N , with maximum normalized capacity:

i∗(s) = argmaxi=1..NC(i, k)

T (i, k), (30)

where T (i, k) is the throughput up to slot s.

The throughputs are updated according to:

T (i, k + 1) = T (i, k)(1 − 1/tc) i 6= i∗ (31)T (i∗, k + 1) = T (i∗, k) + C(i∗, k)/tc, (32)

This maximizes∑

i log(T (i,∞) NOT∑

i T (i,∞)!!


138

Max rate scheduler

Can be seen as PFS with infinite window tc in a symmetric network. We get:

T (i, k) = T ∀i, k (33)

At slot k, schedule user i∗(k) with maximum capacity:

i∗(s) = argmaxi=1..NC(i, k), (34)

This maximizes∑

i T (i,∞)


139

Non bandwidth-expanding MA

Definition:

• K ≥ 1 user signals are superposed (within the same unit of spectral re-source)

• Ex: Spatial division multiple access (K users are differentiated from theirspatial signature)

Multi-user MIMO can operate with:

• K = 1: at each time/slot, we deal with a single-user MIMO link.

• K > 1: The scheduler selects a group of K users

• With K > 1, the scheduler and antenna technique become coupled so asto maximize capacity


140

User group scheduling

set of Kusers Power allocation

Antenna combining

Computing sum rate

QoS constraints

if not satisfactory

Selection of new


141

Some system issues

• Channel aware scheduling transforms the fading statistics as seen by up-per layer

• Gives less reliance in PHY-layer diversity (e.g. STC)

• Allows for compact antenna spacing at BTS, mobile.

• Multiple antennas at mobile only give a bonus (extra SNR, allow for feed-back reduction)


142

Multi-user MIMO

• Multi-user MIMO information theory

• Multi-user MIMO techniques

– The uplink problem (MAC)

– The downlink problem (BC)

Fundamental idea: The MIMO gains are greater and more easily capturedin the multi-user setting than in the single-user setting!


143

Multi-user MIMO information theory

Assumptions:

(1) We now leave the scheduling aside, assume the group of K users isgiven, study MU-MIMO capacity and techniques for this group. (but K canbe large, larger than the number of antennas!).

(2) We limit ourselves to key types of channels

• The multiple-access channel (MAC)

• The broadcast channel (BC)

• MAC-BC duality

First, some reminders on the SISO case...


144

Multi-user information theory (reminders)

[to be developped on white-board with examples]

The SISO MAC

• The MAC capacity region

• The Gaussian MAC capacity region

• Lessons for system design

The SISO BC

• The BC capacity region (open problem!)

• The Gaussian BC capacity region

• Lessons for system design


145

Multi-user information theory design lessons

The optimal transmit-receive SISO structure uses

• Superposition coding on the downlink (BC), ordering, and successivestream decoding

• Successive stream decoding on the uplink (MAC)


146

The MIMO MAC channel

MK

user 1

user k

user K

M1

Mkbase station (N antennas)

K users (user k has Mk antennas)


147

The MIMO MAC channel: Notations

We have:

• K users transmitting to the base station.

• User k has Mk transmit antennas and peak power constraint Pk.

• User k transmits signal vector Xk with covariance Qk, Tr(Qk) ≤ Pk

• Base has N receive antennas

• Channel between user k and base is matrix H∗k, of size N ×Mk.

• White noise with variance 1.


148

The MIMO MAC capacity region

The capacity region is the set of all achievable rate vectors R1, R2, .., RK ∈R+K

given by

⋃

Tr(Qk)≤Pk

R1, .., RK such that

∑

k∈SRk ≤ log2 det

[I +

∑

k∈SH∗kQkHk

]∀S

where S is any subset of [1, ..,K].


149

The 2-user MIMO MAC capacity region

The 2-user MIMO MAC region is the union of many pentagons, with all ad-missible Q1,Q2

max sum rate points

R2

R1

Rate region given Q1 and Q2

Two users

A

B

CD


150

MIMO MAC optimal decoding

Interpretation of point B (previous plots):

• Base decodes user 2 with interference of user 1:

R2 = log2 det[I + (I + H∗

1Q1H1)−1H∗

2Q2H2

]

• Base then subtracts user 2 data and decodes user 1 with no interference:

R1 = log2 det [I + H∗1Q1H1]

More generally: the base decodes the lowest rate signal first then proceedsto decode all users using successive interference canceling.

The max sum rate point give the best total throughput, obtained by point B,C and all time sharing strategies.


151

The MIMO BC channel

MK

user 1

user k

user K

M1

Mkbase station (N antennas)

K users (user k has Mk antennas)


152

The MIMO BC channel: Notations

We have:

• K users receiving from the base station.

• User k has Mk receive antennas.

• Base has N transmit antennas and peak power constraint P .

• Base transmits signal vector X =∑

k Xk

• Xk is signal intended to user k, with covariance Qk.

• Power constraint ensured by∑

k Tr(Qk) ≤ P .

• Channel between user k and base is matrix Hk, of size Mk ×N .

• White noise with variance 1.


153

The MIMO BC capacity region

Some comments:

• MIMO-BC is not degraded (ranking of users is not possible!).

• Capacity region a difficult and long standing problem.

• Recently breakthrough was done using Dirty Paper Coding as encodingstrategy for the users (Caire et al. 03).

• Capacity region then revealed (Shamai et al.04).


154

Introduction to Dirty Paper Coding (I)

Introduced by M. Costa in 1983 for an interference and noise channel:

received signal = transmit information + interference + noise (35)

Basic idea is remove known interference at the transmit side:

transmit signal = transmit information − interference + noise (36)

new received signal = transmit information + noise (37)

Key result: Interference pre-cancelation can induce power increase, handledby modulo operation. Resulting capacity is same at AWGN channel!


155

Introduction to Dirty Paper Coding (II)

• y is received signal ∈ R

• s is transmitted signal, u is information signal with variance P .

• i is interference, n is noise with variance 1.

fτ(y) = y − floor(y + τ/2

τ)τ (modulo τ operation)

s = fτ(u− i) = u− i− τk

fτ(y) = fτ(s + i + n) = fτ(u− i− τk + i + n) = fτ(u + n) ≈ u (38)

for a proper choice of τ .

The capacity of this channel is same as with no interference log2(1 + P ) !!!


156

The 2-user MIMO BC capacity region

Based on dirty paper approach. Choose user order (any):

• At the base, encode user 2’s signal normally.

• At the base, encode user 1’s signal using DPC by treating user 2 as inter-ference.

• At user 2, interference is present from user 1’s signal.

• At user 1, no interference from user 2’s signal is present.

The following rates are thus obtained:

R1 = log2 det [I + H∗1Q1H1]

R2 = log2 det[I + (I + H∗

2Q1H2)−1H∗

2Q2H2

]= log2

det [I + H∗2(Q1 + Q2)H2]

det [I + H∗2Q1H2]

Repeat for all choices of Q1,Q2 s.t. Tr(Q1 + Q2) ≤ P and of user ordering.


157

MIMO MAC-BC duality (I)

Motivation:

• The BC capacity region is very difficult to obtain due to non-convexity interms of the Qk, k = 1..K

• The MAC region is simpler

• Idea is to exploit simple link between MAC (uplink) and BC (downlink) rateregions for the same H matrix.

• We assume that∑

k Pk = P (same total power constraint).

Cbc =⋃

P1,..PK s.t. ∑k Pk=P

Cmac (39)


158

MIMO MAC-BC duality (II)

From Goldsmith 2005: Duality from MAC to BC


159

MIMO MAC-BC duality (III)

From Goldsmith 2005: Duality from BC to MAC


160

Scaling laws of capacity with CSIT

We study the scaling factor of the sum rate for fixed number Mk = M , N , andlarge K.

It is found that [Hassibi05]:

limK→∞

E(RDPC)

N log log(MK)= 1 (40)

Interpretation: With large K, the base can select and spatially multiplex theN best users out of K with negligible interference loss.


161

Multi-user MIMO techniques

• Introduction

• The uplink problem (MAC)

• The downlink problem (BC)


162

Introduction to multi-user MIMO techniques

Key design parameters

• Choice of objective function:

– max sum rate

– achieving per use rate targets

• Complexity:

– pushed toward the base (rather than at the users!)

– iterative/block based

– linear vs. non linear (optimal)

• Channel state information:

– CSI at RX, CSI at TX

– CSI at RX, no CSI at TX

– CSI at RX, partial CSI at TX

– no CSI at RX, no CSI at TXGesbert - MIMOS Seminar c© Eurecom March 2008

163

Multi-user MIMO: The uplink (MAC)

Received signal model at the base:

y = H∗X + n (41)

with global uplink channel matrix:

H∗ = [H∗1,H

∗2, ..,H

∗K ] (42)

And global user transmit vector:

X =[xT1 ,x

T2 , ..,x

TK

]T(43)

where vector xk carries mk ≤Mk symbols per channel uses.


164

Multi-user MIMO: The uplink with Mk = 1

The traditional SDMA (space division multiple access) setup:

user 1

user k

user K

base station (N antennas)

K users (users have 1 antenna)

Each user transmits xk =√qksk where qk is the power level.


165

Non-linear methods for the MU-MIMO uplink with Mk = 1

We first consider single-antenna users (Mk = 1). i.e. each user transmits onestream of data (mk = 1).

The optimal detection method is given by the maximum likelihood multi-userdetector:

We form s = (s1, s2, .., sK)T

s = arg mins∈AK

‖y − H∗√Qs‖2 (44)

where A is the known symbol alphabet for each transmitted symbol

Q = diag(q1, ..qk) is the diagonal power allocation matrix.


166

Linear methods for the MU-MIMO uplink with Mk = 1

At the base, we perform linear filtering:

z = W∗y = W∗H∗√Qs + W∗n (45)

where W∗ is complex of size K ×N

H∗ = [h∗1,h

∗2, ..,h

∗K] (46)

and h∗k is the user k’s channel vector of size N × 1.


167

Multi-user MIMO uplink SINR

Given W and H, the SINR (signal to noise and interference ratio) for user kis given by

γk =‖φkk‖2qk∑

l 6=k ‖φlk‖2ql + 1(47)

where φ = HW is the combined equivalent channel.

γk =ψkkqk∑

l 6=k ψlkql + 1(48)

Where we define ψlk = ‖φlk‖2


168

MU-MIMO uplink beamforming with SINR target

Problem: Given a beamforming matrix W and a vector of SINR targets(γ1, γ2, ..γK), is there a finite power control strategy q = (q1, q2, ..qK)T thatachieves the target?

Answer:

Theorem of feasibility

Let a = (a1, ..aK) with ak = γk(1+γk)ψii

. Then target is feasible with finite powersiff

[I − diag(a)ΨT

]q ≥ a (49)

or equivalently iff diag(a)Ψ has Perron-Frobenius norm < 1.

Proof: verify as exercise ;-)

The minimum power allocation is then given by:

q =[I − diag(a)ΨT

]−1a


169

Multi-user MIMO uplink: optimization methods

We optimize the following:

• The receive beamforming matrix W (N ×K), given user power levels qk,k=1..K

• The user power levels, given W, to satisfy certain criteria

Different possible criteria for system optimization:

• Zero-interference (ZF), MMSE, given power levels.

• SINR constraint γk ≥ γk for each user k

• Total power minimization min∑

k qk

• Throughput maximization max∑

k log2(1 + γk)

• Maximum minimum throughput: max minklog2(1 + γk)• other..


170

Multi-user MIMO uplink: The zero-forcing solution

Goal is to extract signal of each user, free of interference (SINR=SNR), withhighest SNR:

We take:

W∗ =√

Q−1

(HH∗)−1H (50)

where Q is the power allocation matrix:

Q =

q1 0 00 . . . 00 0 qK

(51)


171

Multi-user MIMO uplink: The MMSE solution

We solve:

W = arg min E‖W∗y − [s1, s2, .., sK ] ‖2 (52)

We obtain:W = (I + H∗QH)−1H∗A (53)

where A is a diagonal power normalization matrix matrix such that

[W∗W]k,k = 1 (54)

Note: the normalization leaves the SINR unchanged.


172

Uplink SINR levels under the MMSE solution

Theorem:

For given power levels q1, ..qK, the SINR for user k’s signal after filtering isgiven by:

γk = qkhkΣ−1k h∗

k (55)

whereΣk = I +

∑

l 6=kqlh

∗lhl (56)

Proof: verify as exercise ;-)


173

Uplink beamforming with SINR constraints

Problem:

minq,W

∑

k

qk (57)

subject toqkhkΣ

−1k h∗

k ≥ γk (58)

Note: we exploit that fact that MMSE beamformer yields the best SINR.

Iterative algorithm (Ulukus and Yates, 98) Converges iff problem feasible:

q(i+1)k =

γk

hk

[I +

∑l 6=k q

(i)l h∗

lhl

]−1

hk

(59)


174

Uplink with multiple antennas per user

• User k has Mk > 1 antennas.

• Each user has three main strategies: (i) Beamforming, (ii) ST coding,and/or (iii) spatial multiplexing

(i) Beamforming:

• User needs CSIT

• Combined channel+beamformer becomes equivalent to Mk = 1 antennachannel

(ii) Space-time Coding:

• To combine user multiplexing with diversity at the level of each user

(iii) Spatial multiplexing

• User k can transmit mk streams, with mk ≤Mk,∑

kmk ≤ N .

• Each stream seen as a "virtual single antenna user"

• Single antenna case applies, with modified power constraint


175

Multi-user MIMO: The downlink (BC)

• Notations

• Fundamental trade-off between channel knowledge (CSIT) and capacityscaling

• Techniques with full CSIT

• Techniques with partial CSIT


176

Multi-user MIMO: The downlink (BC)

Received signal model at user k:

yk = HkX + nk where X =∑

k

Xk (60)

usng the global downlink channel matrix:

H =

H1

H2...

HK

(61)

We have the global receive vector for all users:

y =[yT1 , ..,y

TK

]T= H

∑

k

Xk + n (62)

where Xk is the signal vector designed to reach user k.Gesbert - MIMOS Seminar c© Eurecom March 2008

177

Fundamental CSIT/performance trade-off

There exists an interesting trade-off between

(i) the capacity performance

(ii) the number of antennas at the users

(iii) the need for CSIT.

• Capacity scales with min(K,N) provided the base has CSIT.

• In the absence of CSIT, user multiplexing is generally not possible: Thebase does not know in which "direction" to form beams!

• This is contrast with single user MIMO where CSIT is not necessary toget multiplexing gain.

• One case where multiplexing gain is restored is when at least Mk =min(N,K) antennas are installed at each user (exercise!)


178

Multi-user MIMO downlink with CSIT

Two main classes of techniques

• Linear techniques

– with single antenna users

– with multi-antenna users

• Non-linear techniques

– Based on Dirty Paper Coding (DPC)

– Based on vector perturbation


179

Linear multi-user MIMO downlink

The complexity/performence trade-off:

• Linear solutions favored for their reduced complexity

• Do not generally attain capacity bounds

• However may (or may not) exhibit the optimial capacity scaling (with e.g.nb of antennas)


180

Multi-user MIMO: The downlink with Mk = 1

We first consider single antenna users (Mk = 1)

user 1

user k

user K

base station (N antennas)

K users (users have 1 antenna)


181

Signal model for MU-MIMO downlink beamforming

The base transmits signal vector X = W√

Qs where

• W is the N ×K downlink beamformer and

• s = (s1, .., sK)T contains the symbols.

• Q = diag(q1, .., qK) is the power allocation matrix.

The received signal at all users becomes:

y = HW√

Qs + n (63)


182

Multi-user MIMO downlink SINR

Given W and H, the SINR (signal to noise and interference ratio) received atuser k is given by

γk =‖φkk‖2qk∑

l 6=k ‖φkl‖2ql + 1(64)

where φ = HW is the combined equivalent channel.

γk =ψkkqk∑

l 6=k ψklql + 1(65)

Where we define ψkl = ‖φkl‖2


183

downlink/uplink duality of SINR

There is an interesting analogy with the MAC/BC capacity duality...

If same power levels, same channel matrix and beamforming matrix areused:

UPLINK SINRγULk =

ψkkqk∑l 6=k ψlkql + 1

(66)

DOWNLINK SINRγDLk =

ψkkqk∑l 6=k ψklql + 1

(67)

Note: the UL and DL SINRS are not equal! However there is an interestingduality result on the feasibility.


184

Downlink/uplink duality of feasibility

Question: If a certain SINR vector is feasible (within power constraint) onuplink, is it also feasible on downlink?

Answer: Yes!

• (1) Uplink feasibility:[I − diag(a)ΨT

]q ≥ a

• (2) Downlink feasibility: [I − diag(a)Ψ]q ≥ a

Theorem 1

(1) and (2) are feasible simultaneously iff diag(a)Ψ has Perron-Frobeniusnorm < 1.

Theorem 2

The uplink and downlink solutions provide the same minimum sum power


185

MU-MIMO downlink zero-forcing beamformer

Goal is to send an interference free signal to each user.

We compute, withouth power constraint:

W = H∗(HH∗)−1√

Q−1

risk of explosion! (68)

With total power constraint P :

W =1√λH∗(HH∗)−1

√Q

−1(69)

where λ = 1P Tr

[(HH∗)−1

]


186

MU-MIMO downlink MMSE beamformer

Goal is to trade off interference at each user for power usage at the basestation.

We solve:

W = arg min E‖s− y‖2 = arg minE‖s − HW√

Qs‖2 (70)

under total power constraint.

We obtain:W = H∗(αI + HH∗)−1

√Q

(−1)(71)

where α ∈ R+ is tuned such that Tr(WQW∗) = P

Note: the uplink MMSE and downlink MMSE filters are different in general!


187

MU-MIMO downlink performance: linear methods

From Peel et al. 2005.


188

Downlink beamforming with SINR target

Problem: Find W and Q, such that γk ≥ γk ∀k• A difficult problem in general (:

• But..this problem can be solved using the duality of feasibility in up anddownlink :)

Procedure:

1. Consider the uplink with same H matrix and SINR targets.

2. Compute the UL MMSE beamformer and power vector (using Ulukus etal.)

3. If iterations converge, UL problem is feasible. Then it is DL feasible withsame W and sum power.

4. Plug W into DL feasibility equation, obtain DL power levels.

5. Get MMSE beamformer from DL power levels.


189

Downlink beamforming max min SINR

Goal is to maximize the rate of the weakest user (fairness), subject to totalpower constraint:

maxq,W

minkγk (72)

subject to ∑

k

qk ≤ P (73)

Algorithm (doubly iterative):

• (1) For given γ, solve MMSE power control with target γk = γ ∀k• (2) Increase γ a bit

• (3) Check∑

k qk ≤ P . If yes go back to (1), if no stop.


190

Downlink max throughput beamforming

Goal is to maximize the sum of users’ rates, subject to total power constraint:

maxγ

∑

k

log2(1 + γk) (74)

subject to γ being feasible under sum power P .

Problem is equivalent to (using again duality)..

maxq

∑

k

log2(1 + qkhkΣ−1k h∗

k) (75)

subject to∑

k qk ≤ P .

Problem: Not concave function!!


191

Iterative multi-user waterfilling

An approximation to the previous problem is offered by the iterative MU-WFalgorithm:

• Start with q(0) = (q(0)1 , q

(0)2 , .., q

(0)K )T

• (1) Compute βk = hkΣ−1k h∗

k based on q(i)

• (2) find q(i+1) from

maxq

∑

k

log2(1 + qkβk) (76)

subject to ∑

k

qk ≤ P (77)

That’s a waterfilling solution at each step: q(i+1)k = [µ− 1/βk]+


192

Non-linear multi-user MIMO downlink

Two popular types of approaches:

• Dirty Paper Coding (DPC) with successive user cancelation

• Joint multi-user precoding (e.g. so-called "vector perturbation")


193

Applying Dirty Paper Coding to MU-MIMO

Key idea: The base performs interference cancelation before transmitting!

LQ decomposition of the channel: H = LQ

X = Q∗d (78)

y = HX + n = Ld + n (79)

where d = (d1, ..dK)T is DPC pre-coded data vector.

Example for 2 users:d1 = s1 (80)

d2 = fτ(s2 −l21

l22s1) (81)

No interference at reception at user 1 and user 2!


194

Joint non-linear multiuser precoding

Vector Perturbation Precoding (Peel et al. 2003):

• Does not rely on QR decomposition of the channel.

• Designs transmit signal of all users jointly.

• Modifies the transmit vector X to transmit in the direction of "small" eigen-vectors of (HH∗)−1

• Algorithm admits efficient implementation and is better than QR-DPC


195

Vector Perturbation technique (I)

Pure channel inversion:

y = HX + n (82)

X =1√λH∗(HH∗)−1s (83)

Vector perturbation: we replace s above by s

s = s + τ (a + jb) (84)

where τ is a constant and a, b are vectors K × 1 of arbitrary integers.


196

Vector Perturbation technique (II)

Find s by:

arg mina,b

(s + τ (a + jb))∗(HH∗)−1(s + τ (a + jb)) (85)

Note:

• This is equivalent to finding the minimum of a quadratic function over a2D integer lattice.

• There exists efficient implementation using so-called "sphere encoding"algorithm.


197

MU-MIMO downlink performance comparison

From Peel et al. 2005.


198

MU-MIMO downlink with multiple antennas per user

• User k has Mk > 1 antennas.

• The base has several possible strategies, including :

1. (1) mk = Mk, i.e. base sends as many symbols as receive antennas. Thisassumes N ≥∑kMk.

• Sending one different symbol to each user antenna (e.g. in ZF orMMSE sense).

• Block diagonalization: the base cancels inter-user but not intra-userinterference. Users perform stream separation.

2. (2) mk = 1, i.e. base sends as many symbols as users. This assumesN ≥ K.

• Each user performs MMSE RX beamforming on the combined DL chan-nel+filter


199

On the role of CSIT in MU-MIMO


200

Relative capacity gain with CSIT (SU-MIMO case)

2 4 6 8 10 12

010

2030

4050

Number of Antennas (r=t)

Rel

ativ

e ca

paci

ty g

ains

due

to fe

edba

ck (

%)

rayleigh

K=2dB

K=10dB

rayleigh

K=2dB

K=10dB

SNR=3dBSNR=18dB


201

Role of CSIT in MU-MIMO

Role of CSIT in downlink evidenced by capacity scaling analysis.

With CSIT, it is found that [Hassibi05], with Mk = M ∀k:

limU→∞

E(RDPC)

N log log(MU )= 1 (86)

whereRDPC is the sum rate achieved by dirty paper coding (optimal scheme).

Interpretation:

• CIST allows for transmit beamforming.

• With large U , the base can select and spatially multiplex the N best usersout of U with negligible interference loss.

• Mobile antenna provide extra M diversity factor

• Multiplexing gain is not limited by single-antenna mobiles!


202

Role of CSIT in MU-MIMO (II)

Without CSIT, it is found that:

limU→∞

E(RDPC)

min(M,N) log SNR= 1 (87)

Interpretations:

• In the absence of CSIT, multiuser diversity gain vanishes

• multiplexing gain is limited to min(M,N).

• multiplexing gain vanishes if mobile are equiped with single antenna.


203

Acquiring CSI

• Receive side (easy): Channel estimated from training sequence

• Transmit side (hard):

– TDD system: Base recycle uplink channel estimate. Quality dependson ”ping-pong time” and Doppler.

– FDD systems: Exploit a dedicated feedback channel with quantizing


204

Acquiring CSIT with feedback

A numerical example:

• 4x2 MIMO-OFDM complex channel with 512 OFDM tones.

• 100 Hz Doppler (vehicular application).

• Channel estimation approx 10 times faster than Doppler.

• 8 bits quantizing per real-vaued coefficients.

• Total feeback load: 8x512x16*1000= 65.5 Mb/s per user !!!

→ Feedback reduction techniques are critical

→ Fortunately, a little information yields large gains!


205

Living with incomplete channel knowledge...


206

Feedback reduction techniques

A panorama:

1. Efficient quantizing (Lloyd-max, Grassmanian,..) [Love, Heath, et al.]

2. Quantizing the leading channel eigen directions (rather than the channel)

3. Eliminating users from feedback pool using Selective Multiuser Diversity(SMUD)

4. Dimension reduction (includes concept of random beamforming!)

5. Exploiting redundance (temporal, frequency) to reduce feedback close torate of innovation

6. Exploiting spatial statistics

7. Using hybrid direction/gain information

8. Flexible feedback design

Let us now investigate approaches 3, 4, 5, 6, 7, 8 in greater detail.


207

Selective Multiuser diversity

Principles:

⇒ Proposed in IEEE ICC2004 [Gesbert et al.]

⇒ Selective MUD (SMUD) exploits idea that scheduled user is bound to havea "good" channel.

⇒ By thresholding channel quality, one can reduce feedback dramatically

⇒ SMUD can be analyzed/optimized in closed form (SISO case, ICC 2004).


208

Selective multi-user diversity scheduling

γ1<γth

γ2<γth

γ4<γth

γ5>γth

γ6<γth

γ3>γth

pollpoll

poll

poll

R3=MaxRk, k=1..

C1

C2

C3C4 C5

C6

is the SNR of user kγκ

poll

ACCESS POINT+SCHEDULER

Incoming traffic

sleepy

sleepy

sleepy

data

poll


209

Capacity loss vs. Feedback reduction

We compare SMUD+PFS with full feedback MUD+PFS (SISO case)

• tc (PFS time constant) is 500 slots. Average SNR is 5 dB. Number ofusers is 4, 10, 16, 22, 28 (bottom to top).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.81.5

2

2.5

3

3.5

4

Normalized feedback load

Ave

rage

sys

tem

cap

acity

(po

st−

sche

dulin

g)


210

Dimension reduction techniques

Key idea: Mapping the M × N scalar channel dimensions of CSI down to asmaller number p.

• Projection of the channel matrix/vector onto one or more basis vectorsknown to the Tx and Rx.

• Once the projection is carried out, user k feeds back a metric ξk = f(H)which is typically related to the square magnitude of the projected signal.

• Important example: projection onto a unitary precoder known by both BSand user.


211

Projection on a unitary precoder (I)

• Let Mk = 1, the BS designs an arbitrary unitary precoder QM×p. p ≤M .

• Each terminal identifies the projection of its vector channel onto the pre-coder and reports the SINR on the best precoding column:

ξk = max1≤i≤p

|hHk qi|2σ2 +

∑j 6=i |hHk qj|2

(88)

where qi denotes the i-th column of Q.

• The scheduling algorithm opportunistically assigns to each beamformerqi the user which has selected it and has reported the highest SINR.


212

Projection on a unitary precoder (II)

Some interesting particular cases:

• p = M , Q is fixed and equal to identity. This yields the per-antenna SDMAscheduler.

– This scheduler is optimal with large number of users but unfair in lowDoppler scenarios.

• p = 1, Q is random, unit-norm. This yields opportunistic beamforming[Viswanath et al.’02].

• p = M , Q is random, unitary. This yields opportunistic multiuser beam-forming [Sharif,Hassibi’05].

• In both cases, randomization restores fairness on shorter horizons.


213

Opportunistic multi-user beamforming (I)


214

Opportunistic multi-user beamforming (II)


215

Opportunistic multi-user beamforming: Performance

• Each user reports the SINR observed on his preferred beam.

• Sum rate performance (in the large number of users case):

SR ≈ E

N∑

m=1

log2(1 + max1≤k≤U

SINRk,m)

(89)


216

Opportunistic BF performance

For very large number of users:

• The sum rate converges to sum rate obtained under optimal unitary pre-coder with CSIT.

• The scaling laws (with nb of users) of rate under unitary and optimal pre-coder are identical (N log logU )

• Threfore opportunistic multiuser beamforming is asymptotically optimal inthe number of users U .

For low number of users ("sparse network"):

• Random beams do not reach users precisely

• Severe degradation

This problem can be fixed by monitoring matching between users and beamsand adjusting beam power accordingly.


217

Beam power control (I)

• Denote S the set of selected users and p the beam power vector.

• BS knows gkm =∣∣hHk qm

∣∣2 for k ∈ S, m = 1, . . . , N .

• The sum-rate optimal beam power allocation [Kountouris et al 05]:

maxp

∑

k∈Slog

1 +

Pmgkm

1 +∑

j 6=mPjgkj

subject toN∑

i=1

Pi = P

• Closed-form solution for M = 2 antennas (optimal).

• Iterative WF-like algorithm for M > 2 (optimality is not guaranteed).


218

Beam power control (II)

0 10 20 30 40 501

2

3

4

5

6

Number of users

Sum

rat

e (b

ps/H

z)

Random BF equal power, 5dB M=4Iterative Power Alloc, 5dB M=4Greedy PA (SU BF), 5dB M=4SU Opp. BF, 0dB M=2Random BF equal power 0dB, M=2Closed form Optimal PA, 0dB M=2

Performance of Beam Power Allocation vs. the number of users for N = 2, 4Tx antennas.


219

On-off beam selection

• Turn off the worst beams → reducing inter-user interference.

• Decision based on comparing SINR on each beam with a threshold.

• Power on unallocated beams is reported to active beams.

• Gives discrete transition between TDMA and SDMA.


220

Performance with beam power control

0 10 20 30 401

1.5

2

2.5

3

Number of users

Sum

rat

e (b

ps/H

z)

Random beamformingMulti−Mode schemeSU Opp. BFGreedy power allocation

Sum rate vs. number of users for N = 2 Tx antennas and SNR = 0 dB


221

Exploiting temporal redundance (I)

• Random opportunistic beamforming can be made robust to sparsity thanksto redundance.

• Temporal redundance exists for slow varying channel scenarios.

• Feedback aggregation concept: information derived from low-rate feed-back channel can be cumulated over time to approach the performanceof full CSIT scenario.

• Idea: use successive refinement of random beams (single user [Avidor etal 2004], multiuser [Kountouris et al. 2005])


222

Memory based opportunistic beamformer

First phase (’best’ unitary matrix selection)

Initialize Set with random BF matrices Qi, with sum rate SR(Qi)

At each time slot t,

• Generate a new random Qrand,with sum rate SR(Qrand)

• Select from the Set of ’preferred’ matrices,Qi∗, such that i∗ = argmaxQi SR(Qi)

• Calculate SR(Qi∗) given updated channel

• If (SR(Qi∗) > SR(Qrand)) use Qi∗ , else use Qrand

Second phase (update of the Set)

• Update SR(Qi∗) value of the set

• If (SR(Qrand) > SR(Qimin)), replace Qimin by Qrand, where Qimin is matrixwith minimum sum rate (imin = argminQi SR(Qi))


223

Exploiting temporal redundance: Performance

20 40 60 80 100 120 140 160 180 2003

4

5

6

7

8

9

10

11

Number of users

Sum

rat

e (b

ps/H

z)

Static channel Tc=infRandom BFBlock fading Tc=1200Hz Doppler120Hz Doppler10Hz Doppler


224

Exploiting spatial structure

• Spatial channel statistics reveal a great deal of information on the macro-scopic nature of the channel:

– multipath’s mean AoA

– angular spread

• Spatial statistics have a long coherence time compared with that of fading.

• Several forms of statistical CSI are reciprocal (second-order correlationmatrix, power of Ricean component, etc.) → no additional feedback re-quired.

• Second-order statistical information Rk = E[hHk hk

]can be used to infer

knowledge on users’ average spatial separability.

Previous work: [Hammarwall et al. 06, Kountouris et al 06]


225

Interpretation spatial statistics

The BTS should schedule users which are likely to be away from each otherstatistically.

θMS2

MS2

MS1

BTS

θMS1


226

Using spatial statistical feedback in MU-MIMO downlink

• Consider a correlated Rayleigh MISO channel hk ∼ CN (0,Rk), whereRk ∈ CN×N is the transmit covariance matrix (known to BS).

• Objective: How to combine long-term CSIT with instantaneous scalarCSIT in order to exploit Multiuser Diversity ?

• Instantaneous CSIT given by:

γk = ‖hkQk‖2 (90)

where Qk ∈ CN×L is a training matrix containing L orthonormal vectors

qkiLi=1.

• Key idea: Conditioned on short-term CSIT γk, derive a coarse channelestimate.


227

ML estimation framework

• We estimate a coarsely the channel by maximizes the likelihood of hkunder the scalar constraint γk = |hkqk|2 (L = 1).

• The solution to the optimization problem

maxhk

hkRkhHk

s.t. |hkqk|2 = γk(91)

is given by [Kountouris et al, Eusipco’06]

hk = arg maxhk

hkRkhHk

hk(qkqHk )hHk(92)

which corresponds to the (dominant) generalized eigenvector associatedwith the largest positive generalized eigenvalue of the Hermitian matrixpair (Rk,qkq

Hk ).


228

What to do with a coarse h estimate ?

• h can be used to schedule and form beams to selected users (risky)

• h can be used to form a scheduling metric at BTS, but not form the beams(robust)

We evaluate the second approach..


229

0 0.1 0.2 0.3 0.4 0.57.5

8

8.5

9

9.5

10

Angle spread at the BS, σθ/πS

um R

ate

[bps

/Hz]

MMSE full CSIT − exhaustive searchStatistical feedback methodRandom beamforming

ML estimation framework - approach 2

Sum rate vs. angle spread σθ at the base station (N = 2, SNR = 10 dB andK = 50


230

ML estimation framework - approach 2

Conclusions:

• Performance close to that of full CSIT when angular spread per user issmall enough (less 30.

• ideal for wide area networks in suburban environment.

• Robustness to the case of wide angular spread (worst case performanceis that of random beamforming).


231

Hybrid direction/gain feedback metrics

Consider that the feedback channel is divided into 2 types of information:

• Channel Direction Information (CDI)

• Channel Quality Information (CQI)


232

CDI Finite Rate Feedback Model• Quantization codebook known to both the k-th Rx and Tx:

• Vk = vk1,vk2, . . . ,vkN containing 2B unit norm vectors

• the k-th mobile sends index (using B bits) for following vector:

hk = vkn = arg maxvki∈Vk

|hHk vki|2 = arg maxvki∈Vk

cos2(∠(hk,vki)) (93)

where hk = hk/ ‖hk‖.


233

CQI under Zero Forcing beamforming (1/2)• Let S, be a group of |S| = K ≤ N users selected for transmission.

• The signal model is given by

y(S) = H(S)W(S)Ps(S) + n (94)

where H(S), W(S), s(S) are the concatenated channel vectors, beam-forming vectors, uncorrelated data symbols.

• Assuming ZF beamforming on the quantized channel directions:

W(S) = H(S)H(H(S)H(S)H)−1Λ (95)


234

CQI under Zero Forcing beamforming (2/2)• The SINR at the k-th receiver is

SINRk =Pk|hHk wk|2∑

j∈S−kPj|hHk wj|2 + 1

(96)

where∑

i∈S Pi = P (power constraint)

• Sum rate is measured by:

Rk = E

∑

k∈Slog (1 + SINRk)

(97)

• Key problem: How can the user report the SINR? without knowing thebeamformer? it can’t..


235

Using an upper bound of SINR as CQI• Let φk = ∠(hk, hk) be the angle between the normalized channel vector

and the quantized channel direction.

• Each user feeds back the following scalar metric [Jindal 06, Kountouris06]

ξUBk =P ‖hk‖2 cos2 φk

P ‖hk‖2 sin2 φk +M(98)

• This gives information on the channel gain as well as the CDI quantizationerror (sin2 φk).

• Can be interpreted as an upper bound (UB) on the received SINRk (underequal power allocation)

• Exact for orthogonal user sets (valid for case with many users)


236

Limited feedback MU-MIMO in 3GPP-LTE• MU-MIMO under discussion in LTE+/UMB but not decided yet

• Base will have up to 4 antennas - Mobile has up to 2 antennas (exceptspecial devices)

• Several competing approaches but common point: Linear precoding un-der finite feedback

• Approach 1:

– Each user estimates and quantizes its channel hk.

– User sends back index of quantized channel hk to the base, along withSINR estimate

– Base forms ZFprecoding matrix W based on multi-user channel H


237

Limited feedback MU-MIMO in 3GPP-LTE• Approach 2:

– Base pre-stores a codebook of precoding unitary matrices W1,W2

,..,WL, where L = 2B is the codebooksize with B bits of feedback.

– Each user measured its SINR under each possible precoder, feedsback intex of best precoder and column.

– Base schedules N users on the preferred precoder Wi∗.

– L = 1 gives opportunistic "random" beamforming.


238

Exploiting flexible feedback design in MU-MIMO• Trick 1: Feedback splitting (between scheduling and beamforming)

• Trick 2: Channel-adaptive feedback


239

Trick 1: ”feedback splitting”Key ideas behind feedback splitting:

• MU-MIMO schemes can be decomposed into scheduling and beamform-ing stages

• Both stages require CSIT

• Scheduling requires CSIT from U >> N users, but can live with coarseestimates.

• Beamforming to selected users requires CSIT from ≤ N users, but CSITmust be precise.

• Therefore the feedback requirement is clearly not the same for both stages.

⇒ Why not split the feedback load over the two stages?


240

Feedback Splitting Process: Stage I

MT1

MT2

MT3

MTk

MTU

.

.

.

.

.

.

.

Base Station

(N antennas)

.

.

• All U users feedback their CSI → Channel estimate H1 (U ×N )


241

Feedback Splitting Process: Stage I

MT1

MT3

MTU

.

.

.

Base Station

(N antennas)

.

.

.

MTk

MT2

• The base station uses H1 to select users (set A ∈ 1, ..., U, |A| = N )

Example with Zero Forcing:

A = arg maxA

Zero-forcing Sum rate = arg minA

tr(H1,AHH1,A)−1 (99)


242

Feedback Splitting Process: Stage II

MT1

MT3

MTU

.

.

.

Base Station

(N antennas)

.

.

.

MTk

MT2

• Users in A feed back refined CSI → Channel estimate H2,A (N ×N )


243

Performance metric

• The base station designs the ZF precoding matrix WZF :

WZF =H

†2,A√

tr((H2,AHH2,A

)−1)(100)

• The received signal:

y =√PHA

H†2,A√

tr((H2,AHH2,A

)−1)s + n (101)

• Performance metric (HA = H2,A + E2,A):

SRZF–Q2 =N∑

i=1

log2(1 + SINRAi) (102)

whereSINRAi

=P

tr((H2,AHH2,A

)−1) + P‖(E2,AH†2,A

)i‖2(103)


244

Feedback split modelLet 0 ≤ α ≤ 1 be the split factor:

• Let Btotal denote the total number of bits available for feedback

• B1 = αBtotal bits dedicated to the scheduling

• B2 = (1 − α)Btotal bits dedicated to beamforming matrix design

• A user selected in second phase refines his initial B1/U -bit feedback withB2/N -bit feedback

• Achievable distortion at each stage:

σ2e1

= 2−b1/N = 2−αBtotal/(U×N) (104)

σ2e2

= 2−(b1+b2)/N = 2−BtotalN (αU+1−α

N ), (105)


245

Extreme cases1. α = 0: No user selection, σ2

e = 2−Btotal

N2

Using statistics of the minimum eigenvalue of a Wishart matrix, we boundaverage sum rate

NeE1

(N 2

Pc0

)< SRZF–Q,0 < N

[eE1

(N

Pc0

)+ γEM

](106)

2. α = 1: No refinement, σ2e = 2−

BtotalN×U

Average sum rate bounded as follows

N

⌊UNt

⌋

∑

k=1

(−1)k+1

(N2

k

)eE1

(kN 2

Pc0

)< SRZF–Q,1 < N

[γEM + log

(1 + P

c0NH( UNt)

)]

(107)

⇒ c0 = 1−σ2e

1+Pσ2e

quantifies power loss with respect to perfect channel knowl-edge.


246

Extreme cases IllustratedSum rate for N = 2 base antennas, U = 20 single-antenna users, Btotal = 80bits:

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

SNR (dB)

Sum

rat

e (b

its/s

/Hz)

Full CSIT

α = 0

α = 1


247

Feedback split optimization

Lemma: αopt is approximated by the following solution:

αopt ≈ arg maxα∈[0,1]

PL (108)

where

PL = PL1 + PL2 =1 − σ2

e1

1 + Pσ2e2

+σ2e1− σ2

e2

logU (1 + Pσ2e2

)(109)

Interpretation: PL1: Power Loss in MUD due to first stage quantizationerror

PL2: Power Loss due to remaining error in CSIT after refinement


248

Sum rate performanceSum rate for N = 2 base antennas, U = 30 single-antenna users, Btotal = 120bits

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

SNR [dB]

Sum

rat

e (b

its/s

/Hz)

Full CSIT

α = 0

α = 1

Using αheur

Using αopt


249

Trick 2: ”Adaptive feedback rate control”Key ideas:

• Exploit channel time variability, under average feedback rate constraint

• Instantaneous number of quantization bits should be function of:

– the user’s channel quality

– the scheduling probability for this user

Why describe your channel accurately if:

• if you’re unlikely to be selected?

• if you’re not extracting a substantial rate gain?


250

Channel description using CDI + CGIWe assume the channel is described at user side by:

• Channel direction information (CDI): h = h/‖h‖• Channel gain information (CGI) [Jindal 06]

γ ,P/Nα cos2 ǫ

1 + P/Nα sin2 ǫ(110)

where α , ‖h‖2 and ǫ , ∠(h, h) (quantization error angle).

Upper bound on CDF:

Fsin2 ǫ(x) =

δ1−NxN−1 0 ≤ x ≤ δ1 x > δ

(111)

where δ , 2−b/(N−1), b being the number of bits used for quantization.


251

Distributed adaptive quantizingKey ideas:

• User adjusts the quantization precision so as to maximize its expectedrate, as function of α = ‖h‖2.

• Let b(α) specify the instantaneous number of feedback bits.

• Direction information seen by one user alone h is ignored because irrele-vant

Optimization problem: Find b(α) such that

ER =

∫ ∞

0

E[Pr[S]R|α = a]fα(a)da, (112)

is maximized, under average feedback constraint B∫ ∞

0

b(a)fα(a)da = B. (113)


252

Difficulty of the problemWe condition on the CGI γa:

ER ≈∫ ∞

0

dafα(a)

[∫ ∞

0

dγaPr[S|α = a, γa] log2(1 + γa)fγα|α(γa|a)].

Unfortunately Pr[S|α = a, γa] is not mathematically tractable (depends onfeedback bit assignement b(a) is an unknown way)

Can we ignore this term?


253

Adaptive quantizing: Suboptimal solution 1We ignore dependency on scheduling likelihood Pr[S|α = a, γa] = constant

The expected rate becomes:

ER

∫ ∞

0

dafα(a)

[∫ ∞

0

dγa log2(1 + γa)fγα|α(γa|a)]. (114)

The inner integral is given in closed-form by:∫ ∞

0

dγa log2(1 + γa)fγα|α(γa|a)

=1

log2 e

log (1 + ca) −

(1 +

(−1)N

(caδa)N−1

)log (1 + caδa)

+

N−2∑

i=0

(−1)i

(caδa)i1

N − 1 − i

, (115)

where ca = PaN and δa = 2−b(a)/(N−1).


254

Adaptive quantizing: Suboptimal solution 2We assume scheduling probability only function of energy α

We get:

ER =

∫ ∞

0

dafα(a)Pr[S|α = a]

[∫ ∞

0

dγa log2(1 + γa)fγα|α(γa|a)]

(116)

We obtain a waterfiling-type solution:

For a ≤ athres, no bits are allocated for feedback, b(a) = 0

for a > athres, the optimal mapping b∗(.) is such that:

PS(a)∂g(a, b(a))

∂b(a)

∣∣∣∣b(a)=b∗(a)

= λ∗ (117)


255

Sum rate performance for 5 users, N = 2 antennasB = 6 bits per user

0 5 10 15 20 25 302

4

6

8

10

12

14

16

18

20

22N = 5 users

SNR (dB)

Sum

rat

e (b

its/H

z/se

c)

Full CSITScheme 1Scheme 2Uniform CDI quantization


256

Sum rate performance for 20 users, , N = 2 antennasB = 6 bits per user

0 5 10 15 20 25 302

4

6

8

10

12

14

16

18

20

22N = 20 users

SNR (dB)

Sum

rat

e (b

its/H

z/se

c)

Full CSITScheme 1Scheme 2Uniform CDI quantization


257

Sum rate performance with quantized CQI, 10 users

0 5 10 15 20 25 302

4

6

8

10

12

14

16

18

20

22

SNR (dB)

Sum

rat

e (b

its/s

ec/H

z)

Full CSITAdaptive quantization, b

avg = 2 bits/user, 2 bits CQI

Adaptive quantization, bavg

= 2 bits/user, 1 bit CQIAdaptive quantization, b

avg = 4 bits/user, Unquantized CQI

Adaptive quantization, bavg

= 4 bits/user, 2 bits CQIAdaptive quantization, b

avg = 4 bits/user, 1 bit CQI

Uniform quantization, bavg

= 6 bits/user, Unquantized CQI

Adaptive quantizer with 4 bits and quantized CGI does better than fixed quan-tizer with 6 bits and ideal CGI!


258

Conclusion• Scheduling and beamforming carry differences in terms of channel de-

scription requirement.

• Efficient feedback channel design should take into account the differenti-ation.

• Can be done through a simple two-stage feedback concept (Special ses-sion PIMRC07 [Zakhour-Gesbert)]

• Adaptive quantization gives finer channel description that takes schedul-ing likelihood into account (ITA Workshop San Diego Jan 2008 [Zakhour-Gesbert].

Further work:

• Optimizing bits across CQI and CGI

• multiple mobile antennas

• OFDMA


259

Cooperation in multi-cell multi-user networks

• In practice, MU-MIMO cell capacity is limited by co-channel interferencefrom other cells

• Network-level cooperation is a promising avenue to deal with interference

• Cooperation may be a good alternative to building interference-resilientreceivers.


260

Non-cooperating network

R3

R1

T1

R2

T3

T2

T4

R4


261

Cooperating network

R3

R1

T1

R2

T3

T2

T4

R4

Degrees of freedom for cooperation:

Power

Delay

Bandwidth

Antennas

Users (scheduling)


262

Cellular vs. Adhoc

R3

R1

T1

R2

T3

T2

T4

R4

T4

R4

R1

R2

T3

R3

T1

T2

(a) (b)

• "cellular": Users connect to an infrastructure point, close-by.

• "Adhoc": Destinations are other abitrary located users.


263

User-based cooperation (conventional)• Conventional source-relay-destination framework emphasizes diversity gain

for the source user.


264

Mutual cooperation• Mutual cooperation balances benefit of relaying vs. overhead.

• Goal is to maximize Rate user1 + Rate user2.


265

A mutual cooperation protocolAssumptions:

• Non-orthogonal Amplify-Forward protocol (NAF) [1]

• Each mobile divides its power across relay and own transmission tasksover time

• User 1 allocates α Watts for relaying user 2’s data, keeps 1 − α for owntransmission.

• User 2 allocates β Watts for relaying user 1’s data, keeps 1 − β for owntransmission.

[1] [Azarian, El Gamal, Schniter] Trans IT 05.


266

Expression of sum rate (mobile 1 + mobile 2)Lemma: For the Gaussian memoryless multiple-access channel, the sum-rate is such that R1 +R2 ≤ Iα,β where [2]

Iα,β=log2

[1 + γ01 + (1 − α)

K1

l1(β)+ f(βγ02, γ21)

]

+log2

[1 + γ02 + (1 − β)

K2

l2(α)+ f(αγ01, γ12)

]

whereK1 =

[γ2

01 + γ01

][γ21 + 1]

K2 =[γ2

02 + γ02

][γ12 + 1]

l1(β) = 1 + γ21 + βγ02

l2(α) = 1 + γ12 + αγ01

f(x, y) = xyx+y+1

[2] [Tourki, Gesbert, Deneire] ISIT’07


267

Mutual cooperation is selfish!Lemma: Optimal power allocation is given by either [2]

1. α = α∗ 6= 0 and β = 0 if

γ > γ2

02 + γ02

γ01 >(1+γ02)

2(1+γ)

γ−(γ202+γ02)

− 1

2. α = 0 and β = β∗ 6= 0 if

γ > γ2

01 + γ01

γ02 >(1+γ01)

2(1+γ)

γ−(γ201+γ01)

− 1

3. α = 0 and β = 0 if neither condition above is met.

At most one user cooperates with the other one:

(⇒ opportunistically selfish behavior!)

[2] [Tourki, Gesbert, Deneire] ISIT’07


268

Infrastructure-based cooperation


269

Levels of infrastructure cooperationSeveral levels:

• Coding, signal processing level

– Data routed to multiple access points

– Optimum use of the available radio links

– Centralized control required

• Resource allocation level

– Data routed to a single access point

– Interference is a problem but reduced coordinated power control andscheduling

– Scalable with network size

– Distributed solutions?


270

coding, signal processing-based cooperationInterference ⇒ Energy ⇒ Additional data pipe ⇒ good for you!

From competition to cooperation

CO

OP

ER

AT

ION

join

t con

trol

CO

OP

ER

AT

ION

CO

OP

ER

AT

ION

(rel

ay p

roto

col)

(rel

ay p

roto

col)

cell 2

cell 1

INTERFERENCE

Capacity of Multicell MIMO can be reached as regular multi-user MIMO ca-pacity with additional power constraints [3][4]

[3] [Shamai, Zaidel] VTC’01

[4] [Karakayali, Foschini, Valenzuela, Yates] ICC’06


271

MIMO vs. Multicell MIMO

<=>Mk

Pmax

PmaxPmaxPmax

Pmax

Pmax

join

t pro

cess

ing

1 base (with per−antenna power constraint)3 bases (1 antenna each)


272

Cooperative MIMO strategies

Several possible approaches:

1. 1 We are after diversity: distributed space time coding

2. 2 We are after boosting data rates: multicell multiplexing (assuming schedul-ing takes care of diversity)

We go for multiplexing..

• Same algorithms as single-cell MU-MIMO (DPC,linear beamforming, etc.)

• Main challenges: Per-base power constraints and inter-cell signaling over-head.


273

An example

Cooperative spatial multiplexing with two bases

Notations:

• Two bases with N antennas each.

• Two mobile users with single antenna.

• Network wishes to transmit [s0, s1] (one symbol si per user i) via bothbases.

• Symbols are uncorrelated.

• Each base has peak power constraint Pi.

• Channel from base i to all users is Hi ∈ C2×N .


274

Transmit-receive signal model

Base i transmits N × 1 signal vector formed by:

xi = Ais. (118)

where Ai ∈ CN×2 is such that

Tr AiAi = Pi. (119)

The received signal vector, y = [y0, y1]T , y ∈ C

2×1, is then given as

y = H0A0s + H1A1s + v. (120)

Problem: obtain optimal transmit filters under CSIT and power constraint


275

Cooperative spatial multiplexing with full CSIT

We use the MMSE criterion:

arg minA0,A1

MSE = Es,v

[‖y − s‖2

](121)

under constraint:

TrA0A

H0

= P0. (122)

TrA1A

H1

= P1. (123)


276

Cooperative multiplexing with full CSIT (2)

The optimal filters are given by the equation:

[HH

0 H0 + µ0IN HH0 H1

HH1 H0 HH

1 H1 + µ1IN

] [A0

A1

]=

[HH

0

HH1

], (124)

where µ0 and µ1 must be chosen such that the power constraints are satis-fied.


277

Evaluation in a cellular network context


278

Zooming on cooperative subnetwork


279

Evaluation with max rate (greedy) schedulingSingle-cell processing (no cooperation) - 2 antennas per BTS

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.5

1

1.5

2

2.5

3


280

Evaluation with max rate (greedy) schedulingMulti-cell processing (cooperation)

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1

2

3

4

5

6

7

8

9

10


281

Evaluation with max rate (greedy) schedulingSum rate performance vs. number of users (SNR=40)

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

Number of Mobile Users / Cell

Sum

Rat

e [b

its/s

ec/H

z]

System SNR = 40.00 dB and 2 Antennas/Sector

Multi−CellMulti−Cell PSSingle−Cell


282

Evaluation with max rate (greedy) schedulingSum rate performance vs. SNR (30 users per cell)

−10 0 10 20 30 40 50 600

20

40

60

80

100

120

System SNR [dB]

Sum

Rat

e [b

its/s

ec/H

z]

30 MSs/Cell and 2 Antennas/Sector

Multi−CellMulti−Cell PSSingle−Cell


283

Evaluation with round robin scheduling

Rate performance without cooperation (single cell processing)

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1

2

3

4

5

6

7


284

Evaluation with round robin scheduling

Rate performance with cooperation (multi cell processing)

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

4

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

5.8

6


285

Multi-cell multiplexing: Conclusions

• Significant rate gains

• Does not give significant advantage for edge-of-cell users, unless hardfairness is enforced.

• Easy to implement for small subnets (2 cells)

• More than 2 cells cooperating may be difficult due to inter-cell CSI over-head

• Cooperative clustering becomes necessary for many cell case.

• Distributed solutions preferred to get scalability.


286

Cooperative Dynamic clustering [5]Looking for the optimal partitioning of the network into disjoint cooperativeclusters (here of size two)

[5] [Papadogiannis, Gesbert] Proceedings of IEEE ICC’08Gesbert - MIMOS Seminar c© Eurecom March 2008

287

Multicell MU-MIMO: distributed implementations


288

Distributed multicell MU-MIMOIdea 1: The optimization of transmission in cell n is done based on locallyavailable instantaneous information, and external statistical information.

Idea 2: Maximum ratio combining lends itself naturally to distributed imple-mentation.

Proposed scenario [Skjevling 07]:

• BTS performs distributed MRC-based beamforming based on local in-stantaneous channel phase compensation

• Selects exactly one user in the network

• One user may be selected by several bases

• BTS selects best user based on conditional expected system capacity,using statistics of non-local channels.


289

Multi-cell Maximum-Ratio-Combining

BS j

MS i

BS k

BS l

h *ik

h ij*

h *il

h ik

h il

h ij


290

Network seen as a graphN base stations (BS), U mobile stations.

Definition: a scheduling graph, given by the U×N -sized matrix G = [g1g2 . . . gN ],gj = [g1j g2j . . . gUj]

T .

The set of feasible scheduling graphs SG include all G for which each columncontains a single non-zero element.

SG = G : gj ∈ ei, i∈1, 2, . . . , U, j∈1, 2, . . . , N ,such that

gij =

1 if BSj transmits to MSi ,

0 otherwise .


291

Maximizing the sum capacityThe network sum capacity is

C(G,H) =U∑

i=1

log2

(1 + SINRi(G,H)

),

where the SINR of user i is

SINRi(G,H) =

(√P∑N

j=1 gij|hij|)2σ2s√

P∑U

k=1

∣∣∑Nj=1 hijgkjh

∗kj/|hkj|

∣∣2σ2s + σ2

n

.

Distributed B/F but centralized scheduling:

Gpref = arg maxG∈SG

C(G,H)


292

Distributed schedulingStart from an initial graph G.

Next, in a given order, each BSj updates its corresponding vector in thescheduling matrix:

(gj)pref = arg maxgj∈ei

Ehl

C(G,H)

,

where Ehl, l∈1, 2, . . . , N\ j, reflects that BSj only has local, instantaneouschannel state information.


293

Performance vs. number of mobiles

4 6 8 10 12 14 162.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

Number of receiving MS (4 transmitting BS)

Cap

acity

[bits

/sec

/Hz/

cell]

Iterative, capacity−maximizing scheduling with hybrid CSIGreedy user schedulingConventional single−base assignment


294

Performance vs. number of mobiles and cells

4 6 8 10 12 14 162

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

Number of receiving MS (and transmitting BS)

Cap

acity

[bits

/sec

/Hz/

cell]

Iterative, capacity−maximizing scheduling with hybrid CSIGreedy user schedulingConventional single−base assignment


295

Performance vs. SNR

0 10 20 30 40 50 60 701

2

3

4

5

6

7

8

SNR [dB]

Cap

acity

[bits

/sec

/Hz/

cell]

Centralized, capacity−maximzing scheduling with full CSIIterative, capacity−maximizing scheduling with hybrid CSIGreedy user schedulingConventional single−base assignment


296

Multi-cell MIMO in practice

• Gives significant advantage for edge-of-cell users if hard fairness is en-forced.

• Easy to implement for small subnets (2 cells)

• Many cells cooperating may be difficult due to inter-cell CSI overhead

• Routing in backhaul must be optimized

• Dynamic clustering can be a solution


297

Resource allocation-based cooperationMotivations:

• Multicell-MIMO is not scalable.

• Distributed MIMO signal processing hard.

• Broadcast routing of data not always desirable.

• Can we achieve cooperation gains without it?

Remaining degrees of freedom:

• Delay (equivalently user scheduling)

• Power

• bandwidth


298

Centralized resource allocation

Centralized

Resource Controller

BS

BS

BS

BS

UT

UT

UT

UT UT

UT

UT

UT

UTUT

UT

UT

UT

UT


299

Optimal scheduling and power controlSearching over all scheduling vectors U and power vectors P :

(U ∗,P ∗) = arg maxU∈ΥP∈Ω

C(U ,P ), (125)

where:

C(U ,P )∆=

1

N

N∑

n=1

log(1 + Γ([U ]n,P )

). (126)

and the SINR in cell n is:

Γ([U ]n,P ) =Gun,nPun

σ2 +N∑

i 6=nGun,iPui

, (127)


300

A surprising result for two cellsTheorem:

For two cells, the optimum power allocation is ON-OFF:

argmax(P1, P2) ∈ ∆Ω2C(U , (P1, P2)) = argmax(P1, P2) ∈ ΩC(U , (P1, P2))(128)

where ∆Ω2 = (Pmax, 0), (0, Pmax), (Pmax, Pmax)

[6] [Gjendemsjoe, Gesbert, Oien , Kiani] IEEE Trans. Wireless Comm. toappear.


301

But we want...distributed resource allocation

BS

BS

BS

BS

UT

UT

UT

UT UT

UT

UT

UT

UTUT

UT

UT

UT

UT


302

Paths toward distributed allocation [7]

• Game theoretic approaches

• Stastical optimization approaches

• Optimization under ON-OFF power control model

• Optimization in the large number of user case

[7] [Gesbert, Kiani, Gjendemsjoe, Oien] Proceedings of the IEEE, 2007.


303

Game theoretic approachesThe non-cooperative power control game [8][9] writes

max0≤pn≤Pmax

n

fn(pn,p−n) ∀ n.

or with pricing

max0≤pn≤Pmax

n

fn(pn,p−n) − cn(pn) ∀ n.

where fn is selfish utility of user n. Nash equilibrium may not maximize net-work utility.

Cooperative games lead to Nash bargaining equilibrium, socially more opti-mal, but non-distributed.

[8] [Meshkati, Poor, Schwartz] IEEE SP Magazine 2007

[9] [Goodman, Mandayam] IEEE Personal Comm. Mag 2000


304

Optimization under ON-OFF power controlLet N is the number of active cells, assumed large.

Cell m weighs its capapcity contribution to the system against the interfer-ence it generates:

Cell m is activated if (Capacity (with cell m) > Capacity (without cell m)), thatis if

Γ([U ]n,P ) ≥

∏

n∈Nn 6=m

∑

i 6=ni∈N

Pi

∏

n∈Nn 6=m

∑

i 6=n 6=mi∈N

Pi=

(N − 1

N − 2

)(N−1)

≈ e

Leads to opportunistic reuse patterns.


305

Static reuse patterns

Inactive Cell

Active Cell

Cluster size 3 Cluster size 4


306

Opportunistic reuse pattern

Active Cell

Inactive Cell


307


Active Cell

Inactive Cell


308


Active Cell

Inactive Cell


309

Capacity performance vs. number of users

1 2 3 4 5 6 7 8 9 102

3

4

5

6

7

8

No. of Users

Net

wor

k C

apac

ity (

bits

/sec

/Hz/

cell)

Game Theoretic ApproachIterative Binary Power AllocationFull Resue


310

The large number of users case• We let number of users per cell grow asymptotically

• System capacity will grow with number of users

⇒ (multi-user multi-cell diversity!)

• What is the loss due to interference ?

• What can we achieve with a distributed scheme (power control + schedul-ing)?

[Gesbert, Kountouris] IEEE Trans. IT 2007, submitted


311

A bounding approachWe study two bounds on capacity:

• Upper bound obtained with no interference

• Lower bound obtained with full powered interference

In three network scenarios:

1. All users have same average received power (located on circle around thebase)

2. Users uniformly located in the cell

3. Users uniformly located but cannot get too close to the base


312

Upper bound on capacity: No interference

C(U ∗,P ∗) ≤ Cub =1

N

N∑

n=1

log(1 + Γubn

). (129)

where the upper bound on SINR is given by:

Γubn = maxun=1..U

Gun,nPmax/σ2 (130)

The corresponding scheduler is the max SNR scheduler: Fully distributed


313

Lower bound on capacity: Full interference

C(U ∗,P ∗) ≥ Clb = C(U ∗FP ,Pmax) (131)

U ∗FP is the optimal scheduling vector assuming full interference, defined by

[U ∗FP ]n = arg max

U∈Υ

(Γlbn =

Gun,nPmaxσ2 +

∑Ni 6=nGun,iPmax

)(132)

The corresponding scheduler is the max SINR scheduler: Also fully dis-tributed


314

Capacity scaling for symmetric network


315

Capacity scaling with many users (U → ∞)

In the interference-free case (using extreme value theory):

Lemma: For fixed N and U asymptotically large, the upper bound on theSINR in cell n scales like

Γubn ≈ Pmaxγnσ2

logU (133)

Theorem: For fixed N and U asymptotically large, the average of the upperbound on the network capacity scales like

E(Cub) ≈ log logU (134)


316

Capacity scaling with many users (U → ∞)

In the full powered interference case (using extreme value theory):

lemma: For fixed N and U asymptotically large, the lower bound on the SINRin cell n scales like

Γlbn ≈ Pmaxγnσ2

logU (135)

theoremThen for fixedN and U asymptotically large, the average of the lowerbound on the network capacity scales like

E(Clb) ≈ log logU (136)

System with and without interference have same growth rates!


317

Interpretations• Interference creates vanishing loss for large number of users

• Physically, the max-rate resource allocator looks for users which are

– shielded from interference and

– with large SNR

• When number of users is large, interference becomes small comparedwith noise.


318

Capacity scaling for symmetric network

0 50 100 150 2001

2

3

4

5

6

7

8

9

Num

ber

of B

its/S

ec/H

z/C

ell

number of users per cell

Interference−free optimum capacityOptimum capacity assuming full−powered interference

Scaling of upper and lower bounds of capacity, versus U for a symmetricnetwork (N = 4)


319

Capacity scaling for non-symmetric network

Important: Path loss fading has heavy tail behavior while Rayleigh fading hasnot


320

Capacity scaling for non-symmetric networkFrom extreme value theory of heavy-tailed random variables:

Theorem: The upper bound on capacity will behave like:

E(Cub) ≈ ǫ

2logU for large U (137)

Theorem: The lower bound on capacity will behave like:

E(Clb) ≈ ǫ

2logU for large U (138)


321

Capacity scaling for non-symmetric network

0 50 100 150 2000

2

4

6

8

10

12

14

16

18

20

Num

ber

of B

its/S

ec/H

z/C

ell



Scaling of upper and lower bounds of capacity, versus U for a non-symmetricnetwork (N = 4)


322

Capacity scaling for hybrid networkUsers excluded from disk with radius 5 percent of cell radius.

0 50 100 150 2000

2

4

6

8

10

12

14

Num

ber

of B

its/S

ec/H

z/C

ell



Scaling of upper and lower bounds of capacity, versus U for a hyrbid network(N = 4)


323

Conclusions

• Large number of users reveals simple structure of the resource allocationproblem:

– Fully ditributed solution possible

– Price paid due to interference is small

• QoS-oriented scheduling will give different results


324

Open problemsCooperation creates gains and challenges:

• May affect routing

• Easier with infrastructure based cooperation (than user-based)

• In theory, each user receives tiny bits of information through everybodyelse.

• In practice, optimization must be distributed to keep information exchangelocal

• More issues: Synchronization, QoS guarantee issues

• Promising avenue: A two-scale optimization within a single network

– Small scale: coding, signal processing based cooperation (multicellMIMO)

– Large scale: resource allocation-based


325

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