'&

$%

Codebook

Desig

nfo

rN

on-c

ohere

nt

Com

munic

atio

nin

Multip

le-a

nte

nna

Syste

ms

Marko

Beko

,Jo

aoX

avieran

dV

ictorB

arroso

Institu

tode

Sistem

ase

Robotica

(ISR)

–In

stituto

Superior

Tecn

ico

Av.

Rovisco

Pais,

1049–001

Lisb

oa,

Portu

gal

{marko,jxavier,vab}@isr.ist.utl.pt

'&

$%

Pro

ble

mForm

ula

tion

.D

ata

model:

Y=

XH

H+

E

PSfra

grep

lacem

entsX

1M

Tx

h1

N

h11

h12

hM

1

hM

N

12N

Rx

Y

Fig

ure

1:

MIM

Osystem

.Y

,E

:T×

N,X

:T×

M,H

:N×

M

.Codeb

ook

:X={X

1,X

2,...,

XK}

isa

poin

tin

the

manifo

ld

M={(X

1,...,X

K)

:tr(X

HkX

k)

=1}

.Contrib

utio

n:

desig

nco

deb

ook

when

Hdeterm

inistic,

unknow

nand

vec(E

)∼CN

(0,Υ

)(co

lored

noise)

'&

$%

Pro

ble

mForm

ula

tion

.G

LRT

receiver:�k=

argm

ax

p(y|X

k, �g

k)

k=

1,2

,...,K

=arg

min

||y− �X

k �gk || 2Υ

−1

k=

1,2

,...,K

�Xk

=I

N⊗

Xk, �g

k=

( �Xk

H �Xk)−

1 �Xk

HΥ−

12y

(ML

channel

estimate),

�Xk

=Υ−

12 �Xk,||z|| 2A

=z

HA

z,y

=vec

(Y)

.PEP

analysis:

itca

nbe

show

nth

at

(see[5

])fo

rhig

hSN

R

PX

i→

Xj

=Q�

1√2 �

gH

Lij

g�≤Q�

1√2||g||�

λm

in(L

ij )�

(1)

where

g=

vec(HH

),L

ij (X)

= �

XiH�

IT− �X

j� �XjH �X

j�

−1�X

jH�

�

�

Π⊥j

�

Xi

'&

$%

Pro

ble

mForm

ula

tion

.O

ptim

izatio

npro

blem

:resu

lt(1

)su

ggests

the

codeb

ook

merit

functio

n

X∗

=arg

max

X∈M

min{

λm

in(L

ij (X))

:1≤

i6=j≤

K}

(2)

.T

he

pro

blem

in(2

)is

ahig

h-d

imen

sional,

non-lin

earand

non-sm

ooth

optim

izatio

npro

blem

!

e.g.

for

K=

256,T

=8,M

=2:

K(K

−1)

=65280

Lij (X

)fu

nctio

ns

and

2K

TM

=8192

realvaria

bles

tooptim

ize

'&

$%

Codebook

desig

n:

geom

etric

alin

terp

reta

tion

. �

Xi

should

liein

the

orth

ogonalco

mplem

ent

ofsp

an{ �X

j }

PSfra

grep

lacem

ents

Π⊥j �

Xi

�Xj

�X

i

.f(X

1,...,X

K)

=f(X

1eiθ

1,...,X

Keiθ

K)

:pack

ing

inco

mplex

pro

jectivesp

ace

'&

$%

Codebook

Constru

ctio

n

.T

wo-p

hase

meth

odolo

gy

tota

ckle

the

optim

izatio

npro

blem

in(2

)

.Phase

I:so

lvesa

convex

semi-d

efinite

pro

gra

mm

ing

(SD

P)

relaxa

tion

.In

cremen

talappro

ach

:LetX

∗k−

1={X∗1,...,

X∗k−

1 }be

the

codeb

ook

at

the

k−

1th

stage.

The

new

codew

ord

isfo

und

by

solvin

g

X∗k

=arg

max

tr(XHk

Xk)

=1

min

1≤

i≤k−

1 {λm

in(L

ik),

λm

in(L

ki )}

(3)

for

k=

2,...,

K

'&

$%

Codebook

Constru

ctio

n-

Phase

I

.T

he

optim

izatio

npro

blem

(3)

iseq

uiva

lent

to(see

[5])

( �Y∗,

X∗,t∗)

=arg

max

t(4

)

with

the

follo

win

gco

nstra

ints

����� tr(Ni A

1 �Y

B1)−

t···

tr(Ni A

MN �

YB

1)

......

tr(Ni A

1 �Y

BM

N)

···

tr(Ni A

MN �

YB

MN

)−

t �����

�0,

�

MZ

i

ZHi

Pi�

�0∀1≤

i≤

k−

1,K �

YK

H= �

X,tr(

�

X)

=1,

f �

Yf

H=

1, �Y

= �Y

H, �Y�

0,ra

nk( �

Y)

=1

and

X=

vec(Xk)vec

H(X

k),

b2

=1, �Y

=zz

H,z

=� vecT

( �Xk)

b�

T,

�Xk

=I

N⊗

Xk.

.T

he

matrices

M,Z

i—

linear

in �Y

.T

he

matrices

Ni ,

Pi ,

K,f

,A

iand

Bi

—co

nsta

nts,

som

edep

end

on

Υ

'&

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Codebook

Constru

ctio

n-

Phase

1

.D

esign

ofth

eco

dew

ord

s:hig

h-d

imen

sionaldiffi

cult

nonlin

earoptim

izatio

n

pro

blem

(rank

conditio

nin

(4))

.Rela

xing

the

rank

constra

int

leads

toan

SD

P[6

]

.T

he

kth

codew

ord

isextra

ctedfro

mth

eoutp

ut

variable

Xw

itha

techniq

ue

similar

to[7

]

.In

itializa

tion

X∗1:

random

lygen

erated

,fillin

gco

lum

ns

ofth

em

atrix

with

eigen

vectors

asso

ciated

toth

esm

allest

eigen

valu

esofth

enoise

covaria

nce

matrix,etc.

'&

$%

Codebook

Constru

ctio

n-

Phase

2

.Phase

II:optim

izesa

non-sm

ooth

functio

non

am

anifo

ld

PSfra

grep

lacem

ents

Xk

γk(t)X

k+

1

dk

M

'&

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Codebook

Constru

ctio

n-

Phase

2

.Itera

tivealg

orith

m,ca

lledG

DA

(geo

desic

descen

talg

orith

m)

.Id

entify

”active”

pairs

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that

atta

inm

inim

um

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ifth

ereis

an

ascen

tdirectio

nd

k∈

TX

k Mfo

rall

active

(i,j)

(consists

ofso

lving

LP)

.W

hen

dk

isfo

und,perfo

rmA

rmijo

rule

alo

ng

geo

desic

γk(t)

.If

no

dk

isfo

und,th

ealg

orith

msto

ps

'&

$%

Com

pute

rSim

ula

tions

�Exam

ple

:

05

1015

2025

10−

5

10−

4

10−

3

10−

2

10−

1

100

SN

R (dB

)

SER

Fig

ure

2:Cate

gory

1-sp

atio

-tem

pora

llyw

hite

obse

rvatio

nnoise

:T

=8,M

=3,N

=1,

K=

256,Υ

=I

NT

.Plu

s-solid

curve-o

ur

codes,

circle-dash

edcu

rve-unitary

codes.

'&

$%

�Exam

ple

:

05

1015

2025

3010

−5

10−

4

10−

3

10−

2

10−

1

100

SN

R (dB

)

SER

Fig

ure

3:Cate

gory

1-sp

atio

-tem

pora

llyw

hite

obse

rvatio

nnoise

:T

=8,M

=2,N

=1,

K=

256,Υ

=I

NT

.Plu

s-solid

curve-o

ur

codes,

circle-dash

edcu

rve-unitary

codes.

'&

$%

PACK

ING

RA

DII

(DEG

REES)

TK

MB

JAT

Rankin

45

75.5

275.5

275.5

2

46

70.8

970.8

871.5

7

47

69.2

969.2

969.3

0

48

67.7

967.7

867.7

9

49

66.3

166.2

166.7

2

410

65.7

465.7

165.9

1

411

64.7

964.6

465.2

7

412

64.6

864.2

464.7

6

413

64.3

464.3

464.3

4

414

63.4

363.4

363.9

9

415

63.4

363.4

363.6

9

416

63.4

363.4

363.4

3

Table

1:

PACK

ING

INCO

MPLEX

PRO

JECT

IVE

SPACE:W

eco

mpare

our

best

con-

figura

tions

(MB

)of

Kpoin

tsin

PT−

1(C

)again

stth

eTro

pp

codes

(JAT

)and

Rankin

bound.

The

pack

ing

radiu

sofan

ensem

ble

ism

easu

redas

the

acu

teangle

betw

eenth

e

closest

pair

oflin

es.

'&

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PACK

ING

RAD

II(D

EGREES)

TK

MB

JAT

Rankin

56

78

.46

78

.46

78

.46

57

74

.55

74

.52

75

.04

58

72

.83

72

.81

72

.98

59

71

.33

71

.24

71

.57

510

70

.53

70

.51

70

.53

511

69

.73

69

.71

69

.73

512

69

.04

68

.89

69

.10

513

68

.38

68

.19

68

.58

514

67

.92

67

.66

68

.15

515

67

.48

67

.37

67

.79

516

67

.08

66

.68

67

.48

517

66

.82

66

.53

67

.21

518

66

.57

65

.87

66

.98

519

66

.57

65

.75

66

.77

Table

2:

PACK

ING

INCO

MPLEX

PRO

JECT

IVE

SPACE:W

eco

mpare

our

best

con-

figura

tions

(MB

)of

Kpoin

tsin

PT−

1(C

)again

stth

eTro

pp

codes

(JAT

)and

Rankin

bound.

The

pack

ing

radiu

sofan

ensem

ble

ism

easu

redas

the

acu

teangle

betw

eenth

e

closest

pair

oflin

es.

'&

$%

PACK

ING

RA

DII

(DEG

REES)

TK

MB

Rankin

67

80.4

180.4

1

68

77.0

677.4

0

69

75.5

275.5

2

610

74.2

074.2

1

611

73.2

273.2

2

612

72.4

572.4

5

613

71.8

271.8

3

614

71.3

171.3

2

615

70.8

770.8

9

616

70.5

370.5

3

617

70.1

070.2

1

618

69.7

369.9

4

619

69.4

069.7

0

Table

3:

PACK

ING

INCO

MPLEX

PRO

JECT

IVE

SPACE:W

eco

mpare

our

best

con-

figura

tions

(MB

)of

Kpoin

tsin

PT−

1(C

)again

stRankin

bound.

The

pack

ing

radiu

s

ofan

ensem

ble

ism

easu

redas

the

acu

teangle

betw

eenth

eclo

sestpair

oflin

es.

'&

$%

�Exam

ple

:

02

46

810

1210

−4

10−

3

10−

2

10−

1

100

SN

R (dB

)

SER

Fig

ure

4:

Cate

gory

2-

spatia

llyw

hite

-te

mpora

llycolo

ure

d:

T=

8,

M=

2,

N=

1,

K=

67,Υ

=I

NT⊗

Σ(ρ

),ρ=

[1;0.8

5;0.6

;0.3

5;0.1

;zero

s(3,1

)].

Solid

curves-o

ur

codes,

dash

edcu

rves-unitary

codes,

plu

ssig

ned

curves-G

LRT

receiver,sq

uare

signed

curves-B

ayesian

receiver.

'&

$%

�Exam

ple

:

02

46

810

1214

1618

10−

4

10−

3

10−

2

10−

1

100

SN

R

SER

Fig

ure

5:

Cate

gory

2-

spatia

llyw

hite

-te

mpora

llycolo

ure

d:

T=

8,

M=

2,

N=

1,

K=

256,Υ

=I

NT⊗

Σ(ρ

),ρ=

[1;

0.8

;0.5

;0.1

5;

zeros(4

,1)

].Solid

curves-o

ur

codes,

dash

edcu

rves-unitary

codes,

plu

ssig

ned

curves-G

LRT

receiver,sq

uare

signed

curves-B

ayesian

receiver.

'&

$%

�Exam

ple

:

05

1015

10−

5

10−

4

10−

3

10−

2

10−

1

SN

R

SER

Fig

ure

6:

Cate

gory

2-

spatia

llyw

hite

-te

mpora

llycolo

ure

d:

T=

8,

M=

2,

N=

1,

K=

32,

Υ=

IN

T⊗

Σ(ρ

),ρ=

[1;

0.8

;0.5

;0.1

5;

zeros(4

,1)

].Solid

curves-o

ur

codes,

dash

edcu

rves-unitary

codes,

plu

ssig

ned

curves-G

LRT

receiver,sq

uare

signed

curves-B

ayesian

receiver.

'&

$%

�Exam

ple

:

−2

−1.5

−1

−0.5

00.5

11.5

22.5

310

−4

10−

3

10−

2

10−

1

SN

R (dB

)

SER

Fig

ure

7:

Cate

gory

3-

Υ=

αα

H⊗

Υs

+I

NT⊗

Σ(ρ

):T

=8,

M=

2,

N=

2,

K=

32,

s=

[1;0

.7;0

.4;0

.15;zero

s(4,1

)],ρ

=[1

;0.8

;0.5

;0.1

5;zero

s(4,1

)],α

=[-1

.146

+

1.1

89i;1

.191-0.0

38i].

Solid

curves-o

urco

des,

dash

edcu

rves-unitary

codes,

plu

ssig

ned

curves-G

LRT

receiver,sq

uare

signed

curves-B

ayesian

receiver.

'&

$%

�Exam

ple

:

−6

−4

−2

02

46

10−

4

10−

3

10−

2

10−

1

100

SN

R (dB

)

SER

Fig

ure

8:

Cate

gory

3-

Υ=

αα

H⊗

Υs

+I

NT⊗

Σ(ρ

):T

=8,

M=

2,

N=

2,

K=

67,

s=

[1;

0.8

;0.5

;0.1

5;

zeros(4

,1)

],ρ

=[

1;

0.7

;0.4

;0.1

5;

zeros(4

,1)

],

α=

[−0.4

53+

0.0

07i;0

.4869+

1.9

728i].

Solid

curves-o

urco

des,

dash

edcu

rves-unitary

codes,

plu

ssig

ned

curves-G

LRT

receiver,sq

uare

signed

curves-B

ayesian

receiver.

'&

$%

Conclu

sions

.Codeb

ook

desig

nfo

rnonco

heren

tsetu

p

–H

determ

inistic,

unknow

n

–Colo

rednoise:

vec(E

)∼CN

(0,Υ

)

.Resu

lts

–outp

erform

signifi

cantly

unitary

constella

tions

for

colo

rednoise

case

–pro

vide

good

pack

ings

for

com

plex

pro

jectivesp

ace

(M=

1)

(near

bound

perfo

rmance)

–fo

rso

me

cases

actu

alEquia

ngular

Tig

ht

Fra

mes

(ET

F’s)

'&

$%

Refe

rences

[1]

T.L.M

arzettaand

B.M

.Hoch

wald

,“Capacity

ofa

mobile

multip

le-anten

na

com

munica

tion

link

inRayleig

h

flat

fadin

g,”

IEEE

Tra

ns.

Info

rm.

Theory

,vo

l.45,pp.

139-1

57,Jan.

1999.

[2]

B.M

.H

och

wald

and

T.L.M

arzetta,“U

nitary

space-tim

em

odula

tion

for

multip

le-anten

na

com

munica

tion

in

Rayleig

hflat-fa

din

g,”

IEEE

Tra

ns.

Info

rm.

Theory

,vo

l.46,pp.

543-5

64,M

ar.2000.

[3]

B.M

.H

och

wald

,T

.L.M

arzetta,T

.J.Rich

ardso

n,W

.Sweld

ens,

and

R.U

rbanke,

“System

atic

desig

nof

unitary

space-tim

eco

nstella

tions,”

IEEE

Tra

ns.

Inf.

Theory

,vo

l.46,no.

6,pp.

1962-1

973,Sep

.2000.

[4]

J.A

.Tro

pp,“Topics

insp

arseappro

ximatio

n”,Ph.D

.disse

rtatio

n:

Univ

.Texas

at

Austin

,2004.

[5]

M.B

eko,J.X

avier

and

V.B

arroso

,“N

on-co

heren

tCom

munica

tion

inM

ultip

le-Anten

na

System

s:Receiver

desig

nand

Codeb

ook

constru

ction,”

inpre

para

tion.

[6]

J.F.Stu

rm,“U

sing

SeD

uM

i1.0

2,a

MAT

LA

Bto

olb

ox

for

optim

izatio

nover

symm

etricco

nes

(Updated

for

Versio

n1.0

5),”

http

://sed

um

i.mcm

aster.ca

[7]

M.X

.G

oem

ans,

“Sem

idefi

nite

pro

gra

mm

ing

inco

mbin

ato

rialoptim

izatio

n,”

Math

em

atic

alPro

gra

mm

ing,

Vol.

79,pp.

143-1

61,1997.

[8]

A.Edelm

an,T

.A

.A

rias,

and

S.T

.Sm

ith,“T

he

geo

metry

ofalg

orith

ms

with

orth

ogonality

constra

ints,”

SIA

MJ.M

atrix

Anal.

Appl.,

vol.

20,no.

2,pp.

303-3

53,1998.

[9]

J.H

.M

anto

n,“O

ptim

izatio

nalg

orith

ms

explo

iting

unitary

constra

ints,”

IEEE

Tra

ns.

Sig

nalPro

cess.,

vol.

50,no.

3,pp.

635-6

50,M

ar.2002.