Multi-user communication systems : from interference management to network coding

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Multi-user communication systems : from interference

management to network coding

Asma Mejri

To cite this version:

Asma Mejri. Multi-user communication systems : from interference management to network coding. Information Theory [cs.IT]. Télécom ParisTech, 2013. English. �NNT : 2013ENST0086�. �tel-01183684�

A bst r act

Cont ent s

A ck now l edgm ent s i v

A bst r act v i

T abl e of cont ent s x

L i st of gu r es x i i L i st of t abl es x i i L i st of abbr ev i at i ons x i v L i st of not at i ons x v i R esum e D et ai l l e de l a T hese x l i i i I nt r oduct i on 7 1 N et wor k C odi ng 9

CONT ENT S

2 T he C om pu t e-and-For war d p r ot ocol 29

CONT ENT S

4 T he M ul t i -Sour ce M ul t i -R el ay channel 87

5 D i st r i but ed M I M O ch an nel 105

C oncl usi on and p er sp ect i ves 125

CONT ENT S

B i b l i ogr aphy 142

L ist of F igur es

n ; N ; P n ; N ; P : n ; N ; P Sm Z2

n ; N ; P

n ; N ; P :

n ; N ; P

Sm

L ist of not at ions

0n n Mn m n m In n Fp p R Z C Rn n : t : ? : : x x E x x x x

Fp

Fp

L

R esum e D et aille de la T hese

N1 N2 R N1 N2 N1 _R N2 N N D

S1 SN R1 RN D N D M N S1 SN N D N1 N2

Chapi t r e 2: L e pr ot ocole C om put e-and-For war d

L e m odele canal N w1 x1 R n w2 x2 R n wN xN R n y P N_{i = 1}hixi z y F C

F C C wi Fkp k Fp p p xi nE xi 2 _P P > y N X i = 1 hixi z hi R Si z Rn 2 h h1; :::; hN t h P2

Schem a de decodage p our le Com put e-and-For war d

" N X i = 1 aixi # C ai Z; i ; :::; N a a1; :::; aN t ZN Rn R a y y N X i = 1 hixi z N X i = 1 aixi | { z } t N X i = 1 hi ai xi z | { z } Bruit E ect if t

F t Q F y

C

t C

D ebi t de calcul p our le Com put e-and-For war d

Rcomp

Rcomp + ₂

h a 2

R + x max x ;

Sel ect ion des par am et r es et a

a ; a opt ( 2 R?_{;a2 Z}N_n0) + 2 _{h a} 2 a opt hta h 2 aopt a6= 0 atG a G IN h 2H H Hi j Hi j hihj i ; j N G N

G

G

D ecodeur s e caces p our l e CF : cas du canal G aussien

hi ; i ; :::; N y N X i = 1 xi z " N X i = 1 xi # C s P _N i = 1xi s e s s s s P _N i = 1xi F s s s F s

(a) Hist ogramme pour N= 2. (b) Hist ogramme pour N= 5. M et r i que de D ecodage M A P map s2 s p sy s2 s p s p y s s2 s p s _n y s 2 2 s2 s p s y s 2 2 e X s2 s X ^_{s2 sn s} p s A B A A d2m i n 8 2 B 1 4 p( s) p( ^s) dmin F

2 s map s2 s y s 2 2 s 2 s aug Maug M M t R2n n map xaug aug= xaug Maug s yaug xaug 2 yaug y 0n t map s2 s F y B s 2 F Rn n B Rn n y s z BtB 2 In FtB In

R esul t at s de si mul at i ons

n N n N P N 1 N n I4 N

3 −4 −2 0 2 4 6 8 10 12 14 16 10−3 10−2 10−1 100

SNR (dB)

E

rr

o

r

P

ro

b

a

b

il

it

y

9.5 10 10.5 10−1

Min. dist. decoding MAP decoding−Exhaustive MAP decoding−proposed algorithm

−4 −2 0 2 4 6 8 10 12 10−3 10−2 10−1 100

SNR (dB)

E

rr

o

r

P

ro

b

a

b

il

it

y

9.7 9.8 9.9 10 10.1 0.0631 0.0794

Min. dist. decoding MAP decoding−Exhaustive MAP decoding−proposed algorithm

n ; N ; P : −5 0 5 10 10−4 10−3 10−2 10−1 100

SNR (dB)

E

rr

o

r

P

ro

b

a

b

il

it

y

Min. dist. decoding MAP decoding−Exhaustive MAP decoding−proposed algorithm

D ecodeur s e caces p our l e CF : cas du canal a evanouissem ent t P N_{i = 1}aixi f M et r i que de decodage M L t t 2 f p y t ^t t 2 f X ( x1;:::;xN) 2 N= PN i = 1aixi= t 2 y N X i = 1 hixi 2 ! ' X (x1;:::;xN)2 N= P_N i = 1aixi= t 2 y N X i = 1 hixi 2 ! ' n t t 2 f; q 2 AL ! q 2 L P _N i = 1hiM Ui a P ' t

t t t ' 4 4.5 5 5.5 6 6.5 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

t

ϕ

(t

)

t k2 Z;t 2 At 0 k t y0 k Z 0 y0 y~ h1u1 h2u2 a1 h2 a2 h1 u1; u2 u1a1 u2a2 pgcd a1; a2

R esul t at s de si mul at i ons

n

0 10 20 30 40 50 60 70 80 10−5 10−4 10−3 10−2 10−1 100

SNR (dB)

E

rr

o

r

P

ro

b

a

b

il

it

y

IDA

Min. dist. decoding

Sm

Chapi t r e 3: L e canal a r elai s bidir ect ionnel

N1 N2 w1 Fp w2 Fp N1 N2 w1 w2 x1 x2 F C nE xi 2 _{P; i ;} R N1 N2 xR w2 N1 w1 N2

w1 w1 x1 x2 _w2 w2 yR xR y1 y2 e;sum e;sum 4 w1 w1 w2 w2 ex N1! N2 N2! N1 N1! R; R ! N2

Cas du Canal G aussien

yR x1 x2 zR

zR Rn 2R

xR f x1; x2 N1 N2

Schem a A nal og N et wor k C odi ng

q 1+ 2 xR yR Ni i ; yi xR zi x1 x2 zR zi Ni i ; wj j ;

yi yi xi xj zR zi xj xj 2 yi 2 1 _w j 1 xj ex;A NC eq 2

Schem a C om put e-and-For war d

xR x1 x2 C Ni i ; xR ;j 2 yi 2 ; j ; 1 _u j 1 xR;j w1 w2 wj uj wi j ; ex;CF P 2

Schem a D enoi se-an d-For war d

yR

N1 N2

ex;D oF

P

2

0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 3.5

SNR (dB)

A

ve

ra

g

e

a

ch

ie

va

b

le

r

a

te

i

n

b

it

s/

c.

u

Upper Bound

Analog Network Coding Compute−and−Forward Denoise−and−Forward 4−TS Routing 3−TS Network Coding 0 2 4 6 8 10 12 14 16 18 20 10−3 10−2 10−1 100 101

SNR (dB)

S

u

m

M

es

sa

g

e

E

rr

o

r

ra

te

Compute−and−Forward Analog Network Coding Denoise−and−Forward

2

Cas du Canal a evanouissem ent s

yR h1x1 h2x2 zR

h1; h2 R

N1 R N2 R

Schem a A nal og N et wor k C odi n g

q 1+ kh k2 y1 h1xR z1 h21x1 h1h2x2 h1zR z1 y2 h2xR z2 h22x2 h1h2x1 h2zR z2 N1 N2 N1 y1 y1 h21x1 h1h2x2 h1zR z1 N2 y2 y2 h22x2 h1h2x1 h2zR z2 x2 2 y1 h1h2 2 x1 2 y2 2 w2 1 x2 w1 1 x1 ex;A NC m = 1;2 h2 m 2 h 2

Schem a C om put e-an d-For war d xR a1x1 a2x2 C xR ;i 2 yi hi 2 ; i ; ui 1 xR ;i q1w1 q2w2 ; i ; n1 u1 q1w1 q2w2; n2 u2 q2w2 q1w1 qi w2 n_q21 ; w1 n_q12 q2 N1 q1 N2 a1 p a2 p

R esul t at s de si mul at i ons

0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3

SNR (dB)

A

ve

ra

g

e

a

ch

ie

va

b

le

r

a

te

i

n

b

it

s/

c.

u

Upper bound Compute−and−Forward+loc.opt Compute−and−Forward+non−zero entries Analog Network Coding

a 0 5 10 15 20 25 30 35 40 45 50 10−2 10−1 100

SNR (dB)

S

u

m

M

es

sa

g

e

E

rr

o

r

ra

te

Compute−and−Forward+loc.opt

Compute−and−Forward+non zero entries Analog Network Coding

Chapi t r e 4: L e canal a sour ces et r elai s mul t ipl es

S1 S2 SN R1 R2 RN D w1; :::; wN Si; i ; :::; N wi D xi Rm ym N X i = 1 hi mxi zm hi m R Si Rm xi zm Rn 2 Rm hm h1m ::: hN m t h1; :::; hN m; m ; :::; N 1; :::; N w1; :::; wN D N [ i = 1 wi wi !

Schem a A nalog N et wor k Coding Rm m mym m N X i = 1 hi mxi zm ! Rm m r hm 2

Schem a Com put e-and-For war d

Rm m " N X i = 1 am ixi # C am 1; :::; am N Z am am 1 ::: am N t Rm L 0 B @ t 1 t N 1 C A A X C A ZN N at₁; :::; at_N A 0 B @ at₁ at_N 1 C A 0 B B B @ a11 a12 a1N a21 a22 a2N aN 1 aN 2 aN N 1 C C C A a1; :::; aN A A p

0 5 10 15 20 25 30 35 40 45 10−3 10−2 10−1 100 SNR (dB) M es sa g e E rr o r R a te a t th e d es ti n a ti o n Compute−and−Forward+loc.opt

Compute−and−Forward+full rank matrix Q Analog Network Coding

0 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 SNR (dB) A ve ra g e A ch ie va b le R a te p er u se r (b it s/ c. u ) Compute−and−Forward+loc.opt

Compute−and−Forward+full rank matrix Q DF with interference as noise

A

A

Chapi t r e 5: L e canal M I M O di st r ibue

M N M S1 S2 SN N D w1; :::; wM w1; :::; wM Y H X Z X 2 6 4 xt 1 xt_M 3 7 5 RM n ; H 2 6 4 ht 1 ht_N 3 7 5 RN M ; Z 2 6 4 zt 1 zt_N 3 7 5 RN n zm m ; :::; N ; 2_I n

A

A

R esul t at s de simulat i ons

0 5 10 15 20 25 30 35 40 10−4 10−3 10−2 10−1 100 SNR(dB)

M

es

sa

g

e

E

rr

o

r

R

a

te

Joint ML ZF MMSE LLL+ ZF LLL+ MMSE IF

0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 4 5 6 7 SNR(dB) A ve ra ge A ch ie va b le R at e in bi ts pe r ch an n el u se Joint ML ZF MMSE LLL+ZF LLL+MMSE I F Upper bound I F

Per spect ives

D ecod eur s I nt eger For ci ng com bi nes avec l e codage Espace-t em ps:

C odage de r eseaux p our l es com muni cat i ons opt i ques:

I nt r oduct ion

N1 N2 R N1 _R N2 N N D S1 SN R1 RN D N D M N S1 SN N D

T hesis Out line and C ont r i but ions

Ch ap t er 1: N et wor k C odi ng

Ch ap t er 4: T he M ul t i -Sour ce M ul t i -R el ay C hannel

Chapt er 1

N et wor k Coding

1.1

N et wor k Coding: b ene t s and chal lenges

D e ni t i on 1.1. G

i j i ; j

D e ni t i on 1.2. D e ni t i on 1.3.

Ex am pl e 1.1.

S

(a) A wireline comput ers net work.

D1 D2 T2 T1 X W S (b) Graph model. 1.1.1 T hr oughput i ncr ease S b1 b2 D1 D2

D1 b1 b2 ST1D1 ST2W X D1 D2 b1 ST1W X D2 b2 ST2D2 D1 D2 T2 T1 X W S b1 b2 b2 b1 b2 b2

(a) Rout ing t o D1.

D1 D2 T2 T1 X W S b1 b2 b1 b2 b1 b1 (b) Rout ing t o D2. D1 D2 W b1 b2 b1 D1 b1 D2 b1 b2 b2 : W b1 b2 X W b1 b2 W X X b1 b2 D1 b1 b1 b2 b2 D2 b1 b2 b1 b2

D1 D2 D1 D2 T2 T1 X W S b1 b2 b1 b2 b2 b1 b1 b2 b1 b2

Ex am pl e 1.2. S1 S2 R S1 S2 b1 b2 R S1 R S2 b2 S1 R S2 b2 S1 R S2 b1 S1 R S2 b1

(a) Tradit ional rout ing.

S1 R S2 b1 S1 R S2 b2 S1 R S2 b1 b2 b1 b2

(b) Net work Coding.

S1 b1 b1 b2 b2 R 1.1.3 Secur it y W X W b1 b2 b1 b2 1.1.4 Com plexit y 1.1.5 Challenges

1.2

A ppli cat i ons of N et wor k Codi ng

1.2.1 W ir eless N et wor k s

1.2.3 D ist r ibut ed St or age

1.3.1 T he M ax-F low M in-Cut T heor em G V; E V E V V S V D V i ; j E Ri j D e ni t i on 1.4. S D B S B D = B B S D D e ni t i on 1.5. EB i ; j E_P i B ; j = B (i ;j )2 EB Ri j T heor em 1.1. G V; E S V D B S D EB S D B X (i ;j )2 EB Ri j S D S D

1.3.2 T he M ain N et wor k Coding T heor em

S D di; i N N

S U

s1; :::; sR; si Fq q Fq

S N

Fq

T heor em 1.2. G V; E S

N

Fq

1.3.3 A n A lgebr aic St at em ent of t he N et wor k Coding T heor em

Fq P s1; :::; sR S N e c e c1 e ::: cR e F1 Rq e p e R X i = 1 ci e si c1 e c2 e ::: cR e 2 6 6 6 4 s1 s2 sR 3 7 7 7 5

Dj pj_i it h Aj Dj Dj 2 6 6 6 4 pj₁ pj₂ pj_R 3 7 7 7 5 Aj 2 6 6 6 4 s1 s2 sR 3 7 7 7 5 Aj Aj j N D1 D2 A1 A2 A1 4 1 3 2 4 ; A2 3 1 4 2 4 F E D2 D1 A B C S1 S2 s1 s2 1s1 2s2 3 1s1 2s2 4s1

k;

k A1 A2

T heor em 1.3. Fq

k

Aj; j N

1.4

P hysical-L ayer N et wor k Codi ng

1.4.2 I llust r at ive exam ple S1 S2 S1 S2 Si si t si t i j i j ! t i ! t i ! t i ; i ; i i i S1 R S2 s1 s2 S1 R S2 sR sR

yR t s1 t s2 t 1 ! t 1 ! t 2 ! t 2 ! t 1 2 ! t 1 2 ! t y(I )_R 1 2 y_R(Q) 1 2 I Q yR t sR t S1 S2 sR t 1 2 j 1 2 R j R R 1 2 R 1 2 yR(I ) y (Q) R S1 1 S2 2 y(I )_R 1 2 R R ( y_R(I ) y_R(I ) R ( y(Q)_R y(Q)_R R R sR S1 S2

Chapt er 2

T he Com put e-and-For war d

pr ot ocol

T

2.1

N est ed L at t ice codes

2.1.1 M ot ivat ion F; C F C C C F Fp p

Fp C F Fp C F wi xi 1 _a 1x1 ::: aMxM C q1w1 ::: qMxM ai Z; qi Fp; i ; :::; M : qi ai qi g 1 ai p ; i ; :::; M g Fp ; :::; p Z+

2.1.2 Const r uct ion of nest ed lat t ice codes

C Fp L Fk np uL ; u Fk_p C g : p Zn _p 1_{g Z}n MC F MC Z=pZ Zp p p p g Zn p C pZn Ex am pl e 2.1. Z11 k

n C Z2 11 G u: ; u Z2₁₁ g i d F Z211 C Z2 C F −10 −5 0 5 10 15 20 −10 −5 0 5 10 15 20 Z2 F; C F C

2.2

Com put e-and-For war d in r eal-valued C hannels

N S1; :::; SN R

Si; i ; :::; N

w1 x1 R n w2 x2 R n wN xN R n y P N_{i = 1}hixi z y 2.2.1 Encoding schem e wi n xi Fk_p Rn wi xi nE xi 2 _P P > P r r n F C n p k k n p 2.2.2 D ecoding schem e x1; :::; xN y N X i = 1 hixi z hi R Si z Rn 2 _{z ;} 2_I n

h h1; :::; hN t h P 2 " _N X i = 1 aixi # C ai Z; i ; :::; N a a1; :::; aN t ZN x1; :::; xN 1 u u 1 N M i = 1 qiwi qi Fp qi g 1 ai p Rn a 4 y < h; a > n h RN a ZN r r < h; a

R em ar k 2.2. Q F t zeq P _N i = 1 hi ai xi z zeq 2 e 2 2 P h a 2 R em ar k 2.3.

2.3

Com put e-and-for war d in Com pl ex-valued channels

k w(Re)_i w(I m)_i

n

Si xi Cn xi x(Re)i j x (I m) i r 2k_p p y N X i = 1 hixi z h CN z Cn ; 2_I n y(Re) N X i = 1 h h(Re)_i x(Re)_i h(I m)_i x(I m)_i i z(Re) y(I m) N X i = 1 h h(I m)_i x(Re)_i h(Re)_i x(I m)_i i z(I m) h(Re)_i ; h(I m)_i ; z(Re); z(I m) N a Z j Z N y(Re) (Re) h_a(Re) i x (Re) i a (I m) i x (I m) i i C y(I m) (I m) h_a(I m) i x (Re) i a (Re) i x (I m) i i C 1 u u(Re) N M i = 1 q_i(Re)w(Re)_i q(I m)_i w(I m)_i u(I m) N M i = 1 q_i(I m)w(Re)_i q(Re)_i w(I m)_i

i ; :::; N q(Re)_i g 1 h a(Re)_i i p q_i(I m) g 1 h a(I m)_i i p y Cn C : : y(Re) y(I m) Q _F : C Q _F : C (Re) (I m) 1 _: 1 _: u(Re) u(I m)

2.4

Com put at ion R at e

Rcomp T heor em 2.1. h CN a Z j Z N Rcomp Rcomp + P 2 e + 2 _{h a} 2 C + x max x ; a; j a a? 1 2

R em ar k 2.4. 2 e Rcomp;M L Rcomp;M L ₂ h a 2 R em ar k 2.5.

2.5

Select ion of r ecei ver par am et er s

a

; a opt

( 2 C?_{;a2 Z[i ]}N_n0)

+

; a opt ( 2 C?_{;a2 Z[i ]}N_n0) K d 2 2 2 _{h a} 2 d K F C opt h?a h 2 Rcomp h; a + a 2 h?a 2 h 2 1! aopt a2 Z[i ]N_n0 + _a 2 h?a 2 h 2 1! R em ar k 2.6. Rcomp h 2 a 2 h 2 a 2 h?a 2 h a h a 2 N N T heor em 2.2. h CN a Z j Z N aopt a6= 0 a?G a

G IN h 2H H Hi j Hi j hih?j i ; j N G N Q a Q a a 2 h ?_a 2 h 2 a?a h 2 X i ;j h?_ihja?iaj a?a h 2 N X i = 1 h?_iai ! 0 @ N X j = 1 hja?j 1 A a?a h 2a ?_hh?_a a? I h 2H a a ?_{G a} G i; i ; :::; N H G i _{1+ kh k}2 i H 1 h 2; i ; i ; :::; N G N 1 kh k 2 1+ kh k2; i ; i ; :::; N P r op osi t i on 2.1. a G G G G M G M?M Q Q a a?M?M a M a 2 Q a Z j Z N ZN

M G Mred M U U u u N41 vol _G 1 N vol G p G q _Q N i = 1 i h 2 1=2 G Q a 2 h 2 R em ar k 2.7.

2.6

Fast fadi ng channels: Er godi c R at e

R h; a a R h; a 4 a2 Z[i ]N_n0 + _a 2 h?a 2 h 2 1! D e ni t i on 2.1. Re Re 4 Eh R h ; a 2.6.2 L ower B ound a T heor em 2.3. Re NEh Ch c a Ch h 2 h c N₂1 h ; IN E Ch 4 Z ₁ 0 t e tt N N dt Re Eh R h; a Eh a2 Z[i ]N_n0 + _a 2 h?a 2 h 2 1! ! Eh a2 Z[i ]N_n0 + _a?_{G a} 1 Eh + a2 Z[i ]N_n0 a ?_{G a} 1! ! aopt R h; a a2 Z[i ]N_n0a ?_{G a M a} opt 2

aopt M aopt 2 N 1 2 h 2 1 N Re Eh + h 2 1=N N 1 2 ! ! Eh N + _h 2 N NEh + _h 2 N c N₂1 Ch h 2

2.7

Sl ow fading channels: Out age P r obabil it y A nalysi s

h 2.7.1 D e nit i on N R0 D e ni t i on 2.2. Rcomp R0 Pout R0 Rcomp h; a < R0 2.7.2 U pp er B ound T heor em 2.4. R0 Pout R0 h 2< N R 0_{+ c}

N R0 c R h; a R h; a + a2 Z[i ]N_n0 a ?_{G a} 1! + _d 2 min dmin G Pout R0 + dmin2 < R0 dmin> R 0₌₂ Pout R0 N 1 4 h 2 1=2N > R 0₌₂ h 2 1=2N < R0=2+N41 h 2< N R0+ c

2.8.1 Syst em M odel n F; C M F N n x1; :::; xN 1 nE xi 2 _{P; i ; :::; N} y N X i = 1 xi z z Rn ; 2In " _N X i = 1 xi # C s P _N i = 1xi e s s s s P _N i = 1xi s F s s s n o m or e uni for m s P _N i = 1xi s s

2.8.2 D iscr et e G aussian D ist r ibut ion of t he Sum Codeb ook

x 2

x n1E xi

s P _N i = 1xi s s N x 2 s N x2 N s; s2In s

(a) Hist ogram for N= 2. (b) Hist ogram for N= 5.

f _s x 2 s s> x Rn f s x s ne kxk 2 2 2_s F f s F f _s F X s2 F f _s s s n X s2 F e k sk2 2 2_s F

p s

f s s f _s F

N N

2.8.3 M A P decoder : Er r or P r obabil it y and D esign Cr it er ion

map s2 s p sy s2 s p s p y s s2 s p s _n y s 2 2 s2 s p s y s 2 2 T heor em 2.5. F; C N e X s2 s X ^_{s2 sn s} p s A B A A d2m i n 8 2 B 1₄ p( s) p( ^s) dmin F s s s

s p s y s 2 2 < p s y s 2 2 ! p s p s ! y s 2 2 y s 2 2 < ! 2 p s p s ! s s 2 D s s; z E < ! G < G G s s s s p s p s ! ! : G 2 p s p s ! s s 2 < s s; z > G 2G G s s 2 2 p s p s ! 2 G 2 s s 2 e X s2 s p s X ^_{s2 sn s} s s X s2 s X ^_{s2 sn s} p s s s s s p s p s ! ! x 1₂ px 2 e X s2 s X ^_{s2 sn s} p s s s s s p s p s ! ! s s dmin s; s s F

x _x ; R x A B P r op osi t i on 2.2. F; C A F F dmin

2.8.4 P r act ical M A P decodi ng A lgor i t hm s

map s2 s f s F n s ₂ s s 2 ₂ y s 2 s s map s2 s y s 2 2 s 2 s P r op osi t i on 2.3. aug Maug M M t R2n n yaug y 0n t map xaug aug= xaug Maug s yaug xaug 2

map s2 s ( y 0n s s 2) s2 s yaug Iaug s 2 Iaug In In t R2n n s s s M u u s Zn s s P map u 2 As= s= M u yaug IaugM u 2 u 2 As= s= M u yaug Maugu 2 Maug u

xaug Maugu yaug n aug

Maug uopt map M uopt R em ar k 2.8. 2 2 2 s 2 N 2 x 2 x P N _x2 2 N 2_x s

P r op osi t i on 2.4. map s2 s F y B s 2 F Rn n B Rn n y s z BtB 2 In FtB In N s N s y s 2 2 s 2 yty yt s ts s 2 ts s 2 t s s yty yt s t sBtB s yty ytFtB s t sBtB s ytFtF y ytFtB s | { z } kF y B sk2 yt In FtF y | { z } (y ) F Rn n B Rn n BtB 2 In FtB In y > s N s F y B s 2 F B y s z s z F y B M s F B Iaug In In Iaug QR Q1 Q2 R Q R2n n R Rn n F Qt₁ ; B R It_augIaug 2 In RtR BtB

In Q1R FtB In 2.8.5 N um er ical r esult s p s s s −4 −2 0 2 4 6 8 10 12 14 16 10−3 10−2 10−1 100

SNR (dB)

E

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9.5 10 10.5 10−1

Min. dist. decoding MAP decoding−Exhaustive MAP decoding−proposed algorithm

n ; N ; P

Ex am pl e 1: 2-D i m ensi onal l at t i ce n

N N

−4 −2 0 2 4 6 8 10 12 10−3 10−2 10−1 100

SNR (dB)

E

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9.7 9.8 9.9 10 10.1 0.0631 0.0794

Min. dist. decoding MAP decoding−Exhaustive MAP decoding−proposed algorithm

n ; N ; P : N : 1 _N N 2 x Ex am pl e 2: 4-D i m en si onal l at t i ce n I4 P 3

N 2 x N _x2 −5 0 5 10 10−4 10−3 10−2 10−1 100

SNR (dB)

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Min. dist. decoding MAP decoding−Exhaustive MAP decoding−proposed algorithm

n ; N ; P

2.9

Opt im al D ecoder s for t he CF: fadi ng channels

n Zn F Zn M C Zn M 2.9.1 Syst em M odel y N X i = 1 hixi z xi Zn hi R z Rn

" N X i = 1 aixi # C a a1; :::; aN t ZN t P N_{i = 1}aixi e t t t R a ZN a t f F t f h a a Rcomp;M L G a y N X i = 1 aixi N X i = 1 hi ai xi z hi hi; i ; :::; N z z t P _N i = 1aixi 2.9.2 M L D ecoding M et r ic p y t t f t t 2 f p y t t P N_{i = 1}aixi ^t t 2 f X ( x1;:::;xN) 2 N= PN i = 1aixi= t p y x1; :::; xN p x1; :::; xN

x1; :::; xN p y x1; :::; xN ₂ y N X i = 1 hixi 2 ! 2 2 2 ^t t 2 f X ( x1;:::;xN) 2 N= P_N i = 1aixi= t 2 y N X i = 1 hixi 2 ! ' X (x1;:::;xN)2 N= P_N i = 1aixi= t 2 y N X i = 1 hixi 2 ! ' t xi; i ; :::; N t a xi t P N_{i = 1}aixi n xi; i ; :::; N t

2.9.3 D iophant ine Equat ions: H er m it e N or m al For m

M Zn n N M a1M a2M ::: aNM M M U h 0n (N 1)n B i U Zn N n N B Zn n U U 2 6 6 6 4 U1 V1 U2 V2 UN VN 3 7 7 7 5 Vi Zn n Ui Zn n (N 1) xi di vi vi M ViB 1t di M Ui i ; :::; N

2.9.4 L ikelihood Funct ion xi ^t t 2 f X q 2 L 2 ! t q 2 q P N_{i = 1}hidi P _N i = 1hiM Ui ! t y P N_{i = 1}hiM ViB 1t ' t X q 2 L 2 ! t q 2 ' t D e ni t i on 2.3. n ' y ; X q 2 L 2 y q 2 L 4 Ey ' y ; y 2 Rn ' y ; L Ey ' y ; Ey ' y ; vol Z V(L ) ' y ; dy

t

t 2 f; q 2 AL

! q 2

L

2.9.5 Case st udy: 1-dim ensional lat t ices

Z x1 x2 Z Sm Sm Sm; Sm ; :::; Sm Sm Z+ Z F Z C SmZ y h1x1 h2x2 z hi R z ; 2 a a1 a2t t a1x1 a2x2 t Sm a1 a2 y a1x1 a2x2 h1 a1 x1 h2 a2 x2 z hi hi; i ; z z t t 2 At X (x1;x2)2 A2= a1x1+ a2x2= t 2 y h1x1 h2x2 2 ' t X (x1;x2)2 A2= a1x1+ a2x2= t 2 y h1x1 h2x2 2

' t t a1x1 a2x2 g a1 a2 a1 a2 t a1 a2 ( x1 u_g1t a_g2k x2 u_g2t a_g1k k Z u1; u2 a1x1 a2x2 g t g a a1 a2 g N > t t 2 At + 1 X k= 1 exp ₂ y t k 2 | { z } ' (t ) h1u1 h2u2 a1h2 a2h1 k Z

2.9.5.1 P r op er t i es of t he l i kel i h ood funct i on ' m y p _2~2 w _2~2 ' a Sm Sm x1 x2 h : : t a t t t ' t t

−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 0 2 4 6 8 10 12 14 16 18 20

t

ϕ ( t ) - I m pact of h and a: a h p a1~h1 a2h~2 2~2 ' t Sm ; x1 ; x2 ; h : : t; a t t1 t2 t x1 x2 Rcomp;M L a h

- I m pact of t he const el l at i on si ze: Sm t t ' t Sm ; ; x1 ; x2 ; h : : t; a t t t t

4 4.5 5 5.5 6 6.5 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

t

ϕ ( t ) (a) Sm = 5; SNR = 60dB −6 −4 −2 0 2 4 6 8 0 2 4 6 8 10 12 14 16 18 20

t

ϕ ( t ) (b) Sm = 10; SNR = 10dB 2.9.5.2 D i ophant i ne A ppr ox i m at i on t t R ' t y t k t y k t y Z t Z ' t y t k y t k t k2 Z;t 2 At y t k 0 y0 y~ t k2 Z;t 2 At 0 k t y0

F t; k F t; k 0k t y0 t k; k Z 0 y0 F t; k t=k t0=k0 k0 k F k0; t0 F k; t F k0; t0 F k; t k0 k t t; k t t 2.9.5.3 Si m ul at i on r esul t s x1 x2 Sm Sm t a1x1 a2x2 t t a a a1x1 a2x2 g Sm

0 10 20 30 40 50 60 70 80 10−5 10−4 10−3 10−2 10−1 100

SNR (dB)

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IDA

Min. dist. decoding

Sm Sm ; ; Sm Sm Sm = F t; k

0 10 20 30 40 50 60 70 10−4 10−3 10−2 10−1 100

SNR (dB)

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S_m=5 S m=7 S m=10

2.10

Conclusi on

Chapt er 3

T he T wo-way R elay channel

I

N1 N2 R N1 w1 w2 R N2 w2 w1

w1 w1 x1 x2 _w2 w2 yR xR y1 y2 n

F; C F Rn C Rn

3.1

G aussian T wo-W ay R el ay Channels

3.1.1 Syst em M odel and A ssum pt ions

N1 N2 k w1 Fkp w2 Fkp Fp p Fp n x1 Rn x2 Rn nE xi 2 _{P; i ;} yR x1 x2 zR

zR Rn ; 2RIn xR f x1; x2 P N1 N2 Ni i ; yi xR zi zi Rn i ; i2 2 R i2 2 P 2 N1 N2 w2 w1 Ni i ; i wj j ; wj w1 w1 w2 w2 e 4 w1 w1 w2 w2 e;sum 4 w1 w1 w2 w2 e;sum e ex;scheme N1 N2 Ni! R R ! Ni ex;scheme N1! N2 N2! N1 N1! R; R ! N2 Cex ex;scheme Cex;UB Cex;UB P 2

xR

3.1.2 A nal og N et wor k Coding Schem e

R xR yR xR P r xR Ni i ; yi xR zi x1 x2 zR zi N1 y1 y1 x1 x2 zR z1 N2 y2 y2 x2 x1 zR z2 1 2 x1 x2 x2 2 y1 2 x1 2 y2 2 w2 1 x2 ; w1 1 x1 eq eq 2_P 2 2 2

ex;A NC eq

2

R em ar k 3.1.

eq

3.1.3 Com put e-and-For war d Schem e

xR x1 x2 C R yR yR xs x1 x2 xs Q F yR xR xs C xR N1 N2 y1 y2 xR xR;i 2 yi 2 ; i ;

ui 1 xR;i w1 w2; i ; w2 u2 w1; w1 u1 w2 ex;CF P 2 1 2 1 2 1 2 P 2 R em ar k 3.2. xR ;i z1 z2 R em ar k 3.3. xR;i Ni i ; xi xR;i xj xR;i xi C; j ; wj 1 xj

3.1.4 D enoise-and-For war d Scheme xR x1 x2 C xs x1 x2 xs Q F yR xR xs C ex;D oF P 2

ex;rout ing ex;net cod ex;rout ing P 2 ; ex;net cod P 2 1 4 1 3 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 3.5

SNR (dB)

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Upper Bound

Analog Network Coding Compute−and−Forward Denoise−and−Forward 4−TS Routing 3−TS Network Coding

: 2 0 2 4 6 8 10 12 14 16 18 20 10−3 10−2 10−1 100 101

SNR (dB)

S

u

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sa

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Compute−and−Forward Analog Network Coding Denoise−and−Forward

3.2

Fadi ng T wo-W ay R el ay C hannels

3.2.1 Syst em M odel and A ssum pt ions

w1 w2 x1 x2 yR h1x1 h2x2 zR h1; h2 R N1 N2 R zR Rn ; R2In h h1 h2t h

xR f x1; x2 N1 N2 Ni i ; yi hixR zi zi Rn i2 2 R i2 2 N1 N2 h1 h2 P 2 ex;UB m = 1;2 h 2 m P 2

3.2.2 A nal og N et wor k Coding Schem e

xR yR P r h 2 xR N1 N2 y1 h1xR z1 h21x1 h1h2x2 h1zR z1 y2 h2xR z2 h22x2 h1h2x1 h2zR z2 h1 N1 h2 N2 N1 y1 y1 h21x1 h1h2x2 h1zR z1

N2 y2 y2 h22x2 h1h2x1 h2zR z2 1 2 N1 N2 x2 2 y1 h1h2 2 x1 2 y2 2 w2 1 x2 w1 1 x1 ex;A NC m = 1;2 h2 m 2 h 2

3.2.3 Com put e-and-For war d Schem e

3.2.3.1 P r ocessi ng at t he r el ay xR xR a1x1 a2x2 C a1; a2 Z a a1 a2t a1x1 a2x2 F xR u u 1 xR q1x1 q2x2 qi g 1 ai p ; i ; xR a yR yR xc a1x1 a2x2 xc Q F yR xR xc C

< ht; a > h 2 Ni! R Ni! R + _a 2 hta 2 h 2 1! aopt a6= 0 atG a G I2 _{1+ kh k}2hht R2 2 G G

3.2.3.2 P r ocessi ng at end nod es and decodabi l i t y condi t i on

a xR N1 N2 xR xR ;i 2 yi hi 2 ; i ; N1 N2 1 ui 1 xR;i q1w1 q2w2 ; i ; q1 g 1 a1 p ; q2 g 1 a2 p a N1 n1 u1 q1w1 q2w2 N2 n2 u2 q2w2 q1w1 Fp w2 n1 q2 ; w1 n2 q1

N1 N2 q1 q2 a a1 p a2 p a1 t a2t L em m a 3.1. aopt a1 p a2 p atG a G G p ex;CF + a 2 ht_a 2 h 2 1! a 2 h 2 a Z2 _{p a}t_{G a} G RtR G R aopt a1 p a2 p R a 2

a a Z2; a 2 h 2

a R a 2

C > R a 2 _C

a

3.2.4 M odi ed F incke-Pohst for Opt im al N et wor k Codes Sear ch

Ri j; i ; j ; R atG a R a 2 R11a1 R12a2 2 R22a2 2 u11 a1 u12a2 2 u22a22 ui i Ri i; i ; u12 R_R12₁₁ R a 2 C ( u22a22 C u11 a1 u12a2 2 u22a22 C a1 a2 _r C u22 a2 r C u22 s C u22a22 u11 u12a2 a1 s C u22a22 u11 u12a2 C a2 a2 L B2 a2 UB2 L B2 $ r C u22 % ; UB2 &r C u22 ' a2 a2 p

a2 a1 L B1 a1 UB1 L B1 6 6 6 4 s C u22a22 u11 u12a2 7 7 7 5 ; UB1 2 6 6 6 s C u22a22 u11 u12a2 3 7 7 7 a1 a1 p a1 a2 a2 a2 p a1 a1 p a2 a1 a a atG a R em ar k 3.4. C a C C G

3.2.5 Simulat ion R esult s

0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

SNR (dB)

P

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o

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tr

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s

:

0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3

SNR (dB)

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Upper bound Compute−and−Forward+loc.opt Compute−and−Forward+non−zero entries Analog Network Coding

0 5 10 15 20 25 30 35 40 45 50 10−2 10−1 100

SNR (dB)

S

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sa

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Compute−and−Forward+loc.opt

Compute−and−Forward+non zero entries Analog Network Coding

Chapt er 4

T he M ult i-Sour ce M ult i-R elay

channel

I

N S1; :::; SN D N R1; :::; RN S1 w1 S2 w2 SN wN R1 R2 RN D w1; :::; wN

4.1

Syst em M odel and A ssum pt ions

S1; :::; SN N Si k wi Fp wi n xi F; C F Rn C Rn F C nE xi 2 _P P > i ; :::; N r k_n p Rm

ym N X i = 1 hi mxi zm hi m R Si Rm zm Rn ; 2_I n Rm hm h1m ::: hN m t h1; :::; hN P 2 ym m f x1; :::; xN P 1; :::; N D Rn Fp w1; :::; wN wi wi; i ; :::; N wi ri > n wi 4 1; :::; N wi wi < D N [ i = 1 wi wi !

4.2

A nalog N et wor k Coding Schem e

Rm m m m mym m N X i = 1 hi mxi zm ! m P Rm m r hm 2 ; m ; :::; N

4.2.2 P r ocessing at t he dest i nat ion and decodabilit y condit ion

1; :::; N 1; :::; N 0 B B B @ t 1 t 2 t N 1 C C C A | { z } L 0 B B B @ 1h11 1h21 1hN 1 2h12 2h22 2hN 2 Nh1N Nh2N NhN N 1 C C C A | { z } B 0 B B B @ xt₁ xt₂ xt_N 1 C C C A | { z } X 0 B B B @ 1zt1 2zt2 NztN 1 C C C A | { z } Z L RN n X RN n Z RN n B RN N L B X Z z1; :::; zN Z 0 B B B @ 2 1 2In 0n 0n 0n 22 2In 0n 0n _N2 2In 1 C C C A m; m ; :::; N h1; :::; hN B

X x1; :::; xN 1 wi 1 xi i ; :::; N B X B 1L B R Bi ;j; i ; j ; :::; N B B X s2 Sn " s N Y i = 1 Bs(i );i Sn ; :::; N " s s B Bi ;j ihj ;i; i ; j ; :::; N B X s2 Sn " s N Y i = 1 ihs(i );i X s2 Sn " s N Y i = 1 i ! N Y i = 1 hs(i );i ! N Y i = 1 i ! X s2 Sn " s N Y i = 1 h_{s(i );i} N Y i = 1 i ! H H RN N ht_i Ri i ; :::; N H i ; i ; :::; N B B Z

4.3

Com put e-and-For war d Schem e

4.3.1 P r ocessing at t he r elays Rm ym m " _N X i = 1 am ixi # C am 1; :::; am N Z am am 1 ::: am N t Rm m ;F P _N i = 1am ixi F P m um um 1 m N M i = 1 qm iwi qm i Fp qm i g 1 am i p Rm m m m am ZN ym mym m ;F m ;F Q F ym m m ;F C Rm m m Rm Rm m m am m Rm m + ₂ m mhm am 2

Rm m am m Rm m < ht_m; a > hm 2 m m am + am 2 ht_mam 2 hm 2 1! Rm m ; :::; N am am2 ZN;am6= 0N at_mGmam Gm IN hm 2 Hm ; Hm hmhtm m am

4.3.2 P r ocessing at t he dest inat ion and decodabilit y condit ion

1; :::; N a1; :::; aN L 0 B @ t 1 t N 1 C A A X C A ZN N at₁; :::; at_N A 0 B @ at₁ at_N 1 C A 0 B B B @ a11 a12 a1N a21 a22 a2N aN 1 aN 2 aN N 1 C C C A L A

1 U QW U 1 L FN_p n Q FN_p N qi j g 1 ai j p i ; j ; :::; N W 1 X FNp n wt₁; :::; w_Nt Q wt₁ ::: w_Nt t W Q 1U Q 1 Q Fp Q Fp Q A A R p D R1; :::; RN D 1; :::; N D A a1; :::; aN a1; :::; aN a1; :::; aN ZN A p D a1; :::; aN ZN A p 1; :::; N

1; :::; N L em m a 4.1. a1; :::; aN a1; :::; aN ZN A p m = 1;:::;Na t mGmam Gm

4.3.3 Er r or P r obabili t y A nal ysis at t he D est inat ion

D D S_N i = 1wi wi 1; :::; N Q D Rm; m ; :::; N fr Q Fp D Q [ _[N m = 1 m m ! ! Q N [ m= 1 m m ! Q N X m = 1 m m fr N X m = 1 Rm N N Q p Q N Y i = 1 p i

Rm PR; m ; :::; N D N Y i = 1 p i N PR p N R

4.4

E

cient N et wor k Codes Sear ch for t he CF

am Zn at_mGmam m ; :::; N a1; :::; aN A at1::: atN t p at mGmam Gm St ep 1 : Rm am atmGmam m Nm ax m a(1)m ; a(2)m ; :::; a(Nmm ax) Nmax a(1)m ; :::; a(Nmm ax) Gm Rm Nmax N Nm ax m m ; :::; N St ep 2 : Nm ax 1 ; Nm ax 2 ; :::; Nm ax N a1 1Nm ax a2 2Nm ax aN NNm ax

A at₁ ::: at_N t A p

1; 2; :::; N

Sear ching Candidat e Set for each r elay

Rm mNm ax a (1) m ; a(2)m ; :::; a(Nmm ax) Nmax atmGmam m mNm ax St ep 1 : t ttGmt ttGmt C m t ZN; t 0N; ttGmt C C Nmax max m Nmax St ep 2 : t1; t2; :::; tjTmj m ti i ; :::; m m t1 m t2 ::: m tjTmj St ep 3 : Nmax m mNm ax t ZN 0n Gm RtR Gm R RN N Ri j; i ; j ; :::; N R t t1 t2 ::: tN t ttGmt ttGmt R t 2 N X i = 1 0 @ N X j = i + 1 Ri jtj Ri iti 1 A 2 N X i = 1 pi i 0 @ti N X j = i + 1 pi jtj 1 A 2

pi i R2_{i i}; i ; :::; N pi j R_Ri j i i; j i ; :::; N t t_G mt C C > N X i = 1 pi i 0 @ti N X j = i + 1 pi jtj 1 A 2 C ti; i ; :::; N t Nt h L BN tN UBN L BN $ s C pN N % UBN &s C pN N ' tN N t h tN 1 pN Nt2N pN 1;N 1 tN 1 pN 1;NtN 2 C L BN 1 tN 1 UBN 1 L BN 1 $ s C pN Nt2_N pN 1;N 1 pN 1;NtN % UBN 1 &s C pN Nt2N pN 1;N 1 pN 1;NtN ' tN 1 C tN i N ; :::; ti i N ; ::: v u u u t pi i 0 @C N X l = i + 1 pl l 0 @tl N X j = l + 1 pl jtj 1 A 21 A N X j = i + 1 pi jtj ti v u u u t pi i 0 @C N X l = i + 1 pl l 0 @tl N X j = l + 1 pl jtj 1 A 21 A N X j = i + 1 pi jtj ti

Si N X j = i + 1 pi jtj Ti C N X l = i + 1 pl l 0 @tl N X j = l + 1 pl jtj 1 A 2 Ti 1 pi i Si ti 2 L Bi ti UBi L Bi $ s Ti pi i Si % UBi &s Ti pi i Si ' tN; tN 1; :::; t1 t tN tN 1 tN tN tN 1 tN 2 N tN; tN 1; :::; t1 t tN tN 1 ::: t1t ttGmt C C m C C Gm C t Nm ax m Rm

A l gor i t hm 1 C Gm Nm at hr m m ax g Nm ax m Nmm ax Gm RtR pi i R2_{i i} i ; :::; N pi j R_Ri j_{i i} j i ; :::; N m t ZN; t 0N; ttGmt C i N ; d C; Ti C; Si m ti Z q Ti pi i; UBi Z Si ; L Bi Z Si ti L Bi ti ti ti ti UBi i N m i i i i i i ; Si P _N j = i + 1pi jtj; Ti Ti 1 pi i ti Si 2 t 0N t m m; t m < Nmax C gC t1; t2; :::; tjTmj m ti i ; :::; m m t1 m t2 ::: m tjTmj Nmax m mNm ax Nm ax m Nm ax m t1; t2; :::; tNm ax Nm ax m m t1 ; m t2; :::; m tNm ax

4.5

Si mulat i on R esul t s

N D D 1; 2 R1 y1 h11x1 h21x2 z1 x1 x2 R1 R2 x2 y2 h12x1 h22x2 z2 x1 1 2 1 h2₁₁ h2₂₁ ; 2 h2₂₂ h2₁₂ 0 5 10 15 20 25 30 35 40 45 10−3 10−2 10−1 100 SNR (dB) M es sa g e E rr o r R a te a t th e d es ti n a ti o n Compute−and−Forward+loc.opt

Compute−and−Forward+full rank matrix Q Analog Network Coding

N Q R fr N p 0 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 SNR (dB) A ve ra g e A ch ie va b le R a te p er u se r (b it s/ c. u ) Compute−and−Forward+loc.opt

Compute−and−Forward+full rank matrix Q DF with interference as noise

2

Chapt er 5

D ist r ibut ed M I M O channel

M

M S1; :::; SM D N M S1 w1 S2 w2 SN wM N D w1; :::; wM M N

5.1

Syst em M odel and A ssum pt ions

Si

Si i Fp wi n xi F; C F Rn C Rn nE xi 2 _P P > i ; :::; M r k n p t ot M r Y H X Z X 2 6 4 xt₁ xt_M 3 7 5 RM n ; H 2 6 4 ht₁ ht_N 3 7 5 RN M ; Z 2 6 4 zt₁ zt_N 3 7 5 RN n ht_m hm 1 hm 2 ::: hm N m ; :::; N m ; zm m ; :::; N ; 2_I n P 2 mt h y_mt ht_mX zt_m RN n FM_p Y Y w1; :::; wM

wi wi; i ; :::; M Scheme H 0 0 _{" > n} 1; :::; M w1; :::; wM 4 Y w1; :::; wM w1; :::; wM < t ot Scheme H t ot Scheme H e N [ i = 1 wi wi ! r d e SNR! 1 r ; SNR! 1 e d

5.2

Tr adi t ional M I M O r eceiver s

5.2.1 M L decoder x1; :::; xM X X x1; :::; xM M X xt 1::: xtM Y H X 2

w1 1 x1 Rn wM M xM Rn H z1 y1 zN yN w1; :::; wM wi 1 xi ; i ; :::; M M L H S f 1;2;:::;M g M IS HSHtS HS H ; ; :::; M C M IN H Ht dM L r N r M r ; M

5.2.2 L inear R ecei ver s w1 1 x1 Rn wM M xM Rn H z1 y1 zN yN B y1 1 w1 yM M wM Y B RM N Y B Y B H X B Z y1; :::; yM 1; :::; M ym xm xm wm wm 1 xm BZF BZF HtH 1 Ht BM M SE Ht H Ht IN 1 bt_m mt h

B mt h Rm ;L in H bt mH 2 bt m 2 P i 6= m btiH 2 ! M L in H M m = 1;:::;M Rm ;L in H dL in r r M r ; M

5.2.3 L at t ice R educt ion-aided L inear R ecei ver s

H Hr RN M Hr H T T ZM M Y H X Z H T T 1X Z Hr T 1X Z U T 1X Y HrU Z T U F X U Hr U X T U Hr U X BL R ZF HtrHr 1 Ht_r

BL R M M SE TtT 1 Ht_r Hr TtT 1Htr I 1 mt h Rm ;L R L in H bt_r;mHr 2 bt r;m 2 P i 6= m btr;iHr 2 ! bt_r;m mt h L R L in H M m= 1;:::;M Rm ;L R L in H M N dL R L in N

5.3

I nt eger For cing L inear R eceiver s

5.3.1 A r chi t ect ur e Over view

w1 1 x1 Rn wM M xM Rn H z1 y1 zN yN B y1 1 u1 yM M uM A 1 x1 xM 1 w1 wM

B RM N A ZM M B Y B Y B H X B Z A X B H A X B Z A A mt h y_mt at_mX | { z } ut m bt_mH at_m X bt_mZ at_m bt_m mt h A B am um m ; :::; M F y1; :::; yM 1; :::; M m ut_m at_mX Q _F um um Q F ym u1; u2; :::; uM U A X A

X A 1U xt₁; :::; xt_M wi 1 xi ; i ; :::; M R em ar k 5.1. BZF BL R ZF AZF IN AL R ZF T 1 5.3.2 A chievable R at es T heor em 5.1. H RN M I F H M m = 1;:::;M Rm ;I F H ; bm; am Rm ;I F H ; bm; am + bt m 2 btmH atm 2 B RM N bt_m; m ; :::; M A ZM M at_m; m ; :::; M I F;UB M ₂ max

max H

am 2 max2

5.3.3 D iver sit y M ul t iplexing Tr adeo

T heor em 5.2. M N M dI F r N r M r ; M

5.3.4 D esign cr it er ia for Opt im al I F par am et er s

B A B A I F H B ; A _opt B RM N A ZM M; A I F H bm; a_{m opt} M jA j6= 0 m = 1;:::;M + bt m 2 btmH atm 2 bm am bt_{m ;opt} at_mHt H Ht IN 1

B A Bopt A Ht H Ht IN 1 Rm ;I F H ; bm; am Rm ;I F am atmV D Vtam V RN N H D RM M Di i ( 1 1+ 2 i i r ank H i > r ank H i it h H I F M jA j6= 0 m = 1;:::;M at_mV D Vtam I F A a1; :::; aM L em m a 5.1. a1; :::; aM opt A am 2 max2 m = 1;:::;Na t mG am G V D Vt RM M a1; :::; aM am m ; :::; M _at 1 mV D Vtam at_mV D Vtam

5.4

E

cient I F D esign A lgor it hm s

am; m ; :::; M Q am atmG am St ep 1 : Mmax t1; :::; tMm ax Q Mmax C Mmax M St ep 2 : M am m ; :::; M Q am ord t1; :::; tMm ax Q t1 ::: Q tMm ax ord M t1; :::; tM ai ti; i ; :::; M C M t tt_{G t C C >} G RtR G R RM M Ri j; i ; j ; :::; M R t t1 t2::: tM t ttG t ttG t R t 2 M X i = 1 0 @ M X j = i + 1 Ri jtj Ri iti 1 A 2 M X i = 1 pi i 0 @ti M X j = i + 1 pi jtj 1 A 2

pi i R2_{i i}; i ; :::; M pi j R_Ri j i i; j i ; :::; M t t_{G t C} C > M X i = 1 pi i 0 @ti M X j = i + 1 pi jtj 1 A 2 C ti; i ; :::; M t Mt h L BM tM UBM L BM $ s C pM M % UBM &s C pM M ' tM M t h tM 1 pM Mt2M pM 1;M 1 tM 1 pM 1;MtM 2 C L BM 1 tM 1 UBM 1 L BM 1 $ s C pM Mt2_M pM 1;M 1 pM 1;MtM % UBM 1 &s C pM Mt2M pM 1;M 1 pM 1;MtM ' tM 1 C tM i M ; :::; ti i M ; ::: v u u u t pi i 0 @C M X l = i + 1 pl l 0 @tl M X j = l + 1 pl jtj 1 A 21 A M X j = i + 1 pi jtj ti v u u u t pi i 0 @C M X l = i + 1 pl l 0 @tl M X j = l + 1 pl jtj 1 A 21 A M X j = i + 1 pi jtj ti

Si M X j = i + 1 pi jtj Ti C M X l = i + 1 pl l 0 @tl M X j = l + 1 pl jtj 1 A 2 Ti 1 pi i Si ti 2 L Bi ti UBi L Bi $ s Ti pi i Si % ; UBi &s Ti pi i Si ' tM; tM 1; :::; t1 t tM tM 1 tM tM tM 1 tM 2 M tM; tM 1; :::; t1 t tM tM 1::: t1t ttG t C C C G

5.5

N um er i cal R esult s

M N

0 5 10 15 20 25 30 35 40 10−4 10−3 10−2 10−1 100 SNR(dB)

M

es

sa

g

e

E

rr

o

r

R

a

te

Joint ML ZF MMSE LLL+ ZF LLL+ MMSE IF : A 2 max : 2

0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 4 5 6 7 SNR(dB) A ve ra ge A ch ie va b le R at e in bi ts pe r ch an n el u se Joint ML ZF MMSE LLL+ZF LLL+MMSE I F Upper bound I F

5.6

Conclusion

Conclusion and p er sp ect ives

Per spect ives

I nt eger For ci ng ar chi t ect ur e comb i ned t o Space-T i m e C odi ng

P L N C i n Opt i cal C om muni cat i ons

A pp endices

5.A

L at t ice D e nit ions

D e ni t i on 5.1 . n p p n Rn p v1; :::; vp Rn ( x p X i = 1 aivi; ai Z ) p v1; :::; vp x x M s; s Zn M a; b Z x ; y ax by G Mt_M D e ni t i on 5.2 . 1v1 ::: pvp; i < M p G D e ni t i on 5.3 . x x 2 x

D e ni t i on 5.4 . m od x Rn _{x x x} D e ni t i on 5.5 . F C F C; F F r n F C n F C D e ni t i on 5.6 . dmin dmin x y x y x x 0 D e ni t i on 5.7 . 2 n Z V x 2dx G 2 2=n D e ni t i on 5.8 . (n ) Rn n ! 1 G (n ) e D e ni t i on 5.9 . z n ; 2_I n ; e 2=n 2 2 _{z =} e (n) n ! 1 (n )_; e e; e ; n e

5.B

Com put e-and-For war d

5.B .1 Opt im al scaling fact or for t he CF

opt h?a h 2 f f ₂ h a 2 1 h a 2 Rcomp + =f f a f ? h a ? h a f @f @ h ? _{h a h a} ?_h h?h h?a h?h a?h h?h h?a @f ( op t) @op t opt h?a 1 _h?_h opt h?a h 2

5.B .2 M ax imum Com put at i on R at e

Rcomp h; a + a 2

h?_a 2

h 2 1!

f

Rcomp h; a + =f opt f opt

f opt 2 _h?_a 2 h 2 h?a h 2h a 2 h?a 2 h 2 2 _h?_a 2 h 2 2 h 2 _a 2 h?a 2 h 2 h?a 2 2 h?a 2 h 2 h 2 h?a 2 h 2 2 a 2 a 2 h ?_a 2 h 2 Rcomp h; a + =f opt

5.B .3 M odi ed Spher e D ecoder for M A P D ecoding

uopt s yaug Maugu 2

C yaug yaug Maugu 2 C2 s Maug QR Maug R ri j; i ; j ; :::; n % Maug1yaug % u R 2 C2 n X i = 1 pi i 0 @_i n X j = i + 1 pi j j 1 A 2 C2 pi i r_{i i}2 ; i ; :::; n ; pi j r_ri j_{i i} ; j i ; :::; n ui u nt h C pn n n C pn n % u %n C pn n un %n C pn n

i n ; :::; v u u u t pi i 0 @C2 n X l = i + 1 pl l 0 @_l n X j = l + 1 pl j j 1 A 21 A _%i n X j = i + 1 pi j j ui v u u u t pi i 0 @C2 n X l = i + 1 pl l 0 @_l n X j = l + 1 pl j j 1 A 21 A _%i n X j = i + 1 pi j j ui Si %i n X j = i + 1 pi j j Ti C2 n X l = i + 1 pl l 0 @_l n X j = l + 1 pl j j 1 A 2 Ti 1 pi i Si ui 2 s Ti pi i Si ui s Ti pi i Si u bi n f ;i & s Ti pi i Si ' bsup;i $ s Ti pi i Si % bi n f ;i ui bsup;i; i ; :::; n u s

ui cimin cimax cimin ui cimax

Ii ui

Ii max bi n f ;i; cimin ; mi n bsup;i; cimax

n

In 1 un 1 un n un; un 1; :::; u1 Ii yaug A l gor i t hm 2

yaug; Maug; C; cimin; cimax; i ; :::; n

sopt

Maug QR %

M 1

augyaug d C; Tn C2; Sk %k; k ; :::; n

i n

si bi n f ;i; bsup;i L B ui max bi nf ;i; cimin

UB ui mi n bsup;i; cimax ui L B si ui ui ui UB ui i n uopt u i i i i i i 1 %i 1 Si Ti 1 Ti pi i Si ui 2 Si 1 %i 1 P _n j = i pi 1;j j d2 _T n T1 p11 S1 u1 2 d d ui ui; i ; :::; n; d d; Tn d

5.C

M M SE-G D F E pr epr ocessi ng

lt er s

F B BtB 2 In ; FtB In y s z s _n1E s 2 2s Fm Bm Fmy Fm s Fmz Bm s Fm Bm s Fmz

w Fm Bm s Fmz " " nE w t_w nE w w t n E Fm Bm s t s Fm Bm t E FmzztFtm n 0 B @ Fm Bm E s ts | { z } 2 sIn Fm Bm t FmE zzt | { z } 2_I n Ft_m 1 C A 2 s n Fm Bm Fm Bm t 2_F mFtm 2 s n FmF t m FmBtm BmFtm BmBtm 2FmFtm 2 s n Fm In 2_I n FtmFmBtm BmFtm BmBtm T T Tt 2 In G G FmT " " 2 s n G BmT t _Gt _T 1_Bt m Bm In T Tt 1 Btm 2 s n G BmT t _Gt _T 1_Bt m 2 2BmB t m Bm Fm " G B T t Fm ₂Bm "min 2 s n 2 2 BmB t m 2 s n 2 2 B t mBm Bt_mBm 2 In "min 2s 2 F Fm B Bm F B

F B w "eq "eq 2 s n F B F B t 2_{F F} 2 s n 2 _{F F}t _{F B}t _{B F}t _{B B}t (a) _s2 n 2 _{F F}t _{F B}t _{B F}t _{B B}t (b) _s2 n 2 _Ft_{F B}t_{F F}t_{B B}t_B (c) _s2 n 0 B @ 2 FtF FtB | { z } n BtB | { z } (1+ 2_)n 1 C A 2 s n 2 _Ft_F 2 _n A B B A A At Ft_{B I} n FtF BtB 1 Ft_F n 1+ 2 "eq s2 2 "min

5.D

M odi ed C assel' s A lgor it hm

y0 y~ 0 t t; k t; N 0 y0 t 1: 1 0 0 1 y0 2: t0 t1 T1 3: k0 k1 K1 4: n 5: w hi l e n 1 n 1 Tn 1 t do 6: an _nn ₁2 7: tn tn 2 antn 1 kn kn 2 ankn 1 8: _{n n 2} an n 1 9: i f Kn 1 kn 1 t hen 10: bn n _n1 ₁n 2

11: Tn Tn 1 tn 2 bntn 1 Kn Kn 1 kn 2 bnkn 1 12: n n 1 n 2 bn n 1 13: el se 14: Tn Tn 1 tn 1 Kn Kn 1 kn 1 15: n n 1 n 1 16: end i f 17: n n 18: end w hi l e 19: t Tn 20: k Kn

5.E

Opt im al N et wor k Code Sear ch A l gor i t hm for t he CF

in t he T W RC

A l gor i t hm 3 C G aopt G RtR ui i R2i i i ; u12 R_R12₁₁ a atG a i ; Ti C; Si ai Z q Ti ui i; UB ai Z Si ; L B ai Z Si ai L B ai ai ai ai ai p ai ai UB ai i aopt a1a2t i i i i i i ; S1 u12a2; T1 C u22a22 a a aopt a atG a C C atG a

5.F

L L L R educt ion

B b1; :::; bn Or t hogonal i zat i on B b1; :::; bn B? b?₁; :::; b?_n b?₁ b1 b?_i bi i 1 X j = 1 i jb?j ; i ; :::; n i j bi ; b?j b? j ; b?j :; : B? Si ze r educt i on i j B b1; :::; bn i j ; i < j n i j 1₂ 1 2 i j bi bj j i ; :::; 1: i f k;l > 1₂ t hen

2: q : k;l

3: bk bk qbl

4: _{k;l k;l} q

5: for i l do

6: _{k;i k;i} q k 1;i

7: end for 8: end i f Swap bi bj bi 1: bk bk 1 bk bk 1 2: i f k t hen 3: for j k do 4: _{k;j k 1;j k;j k 1;j} 5: end for 6: end i f 7: _{k;k 1} 8: B Bk 2Bk 1 9: _{k;k 1} Bk_B 1 10: Bk Bk 1B_Bk 11: Bk 1 B b1; b2 b1 b?1 b2 21b?1 b?2 b1 ; b2 b1 2 b1 2 b2 2 b1 2 b2 2 b1; :::; bn bi bi + 1 b1; :::; bi 1 bi i b?i bi + 1 i b?i + 1 i + 1;ib?i

i j ; i < j n

b?_i 2 b?_{i + 1} i + 1;ib?i 2

b?_{i + 1} 2 2_{i + 1;i} b?_i 2

bi; bi + 1 4₃

R equi r e: B b1; :::; bn n

1: I . C om put at i on of t h e G r am Schm i d t coe ci ent s of t he basi s vect or s bi; i ; :::; n 2: b?₁ b1 3: B1 b?1; b?1 4: for i n do 5: b? i bi 6: for j i do 7: _{i j} bi; b?j 8: b? i b?i i jb?j 9: end for 10: Bi b?i; b?i 11: end for 12: I I . V er i cat i on of condi t i on s i ) et i i ) 13: k 14: w hi l e k n do 15: RE D k; k 16: i f Bk 3₄ 2_{k;k 1} Bk 1 t h en 17: for l k do 18: RE D k; l 19: end for 20: k k 21: el se 22: SW AP k 23: k max ; k 24: end i f 25: end w hi l e

5.G

I nt eger For ci ng L inear R eceiver s

5.G .1 Opt im al P r epr ocessi ng I F m at r ix

bt_m at_mHt H Ht IN 1 f bm f bm bm 2 btmH atm 2 Rm ;I F H ; bm; am 1₂ + _{f (b}1_m) bm f am f bm btmbm btmH atm Htbm am bt_mbm btmH Htbm btmH am atmam bt_m H Ht IN bm btmH am atmam f bm @f bm @bm H Ht IN bm H am @f (bm ; op t) @bm ; op t bt_{m ;opt} at_mHt H Ht IN 1

5.G .2 Opt im al I F Coe cient M at r i x

Rm ;I F am

bm

Rm ;I F am atmV D Vtam

Rm ;I F am +

f bm ;opt

f bm ;opt btm ;optbm ;opt btm ;optH Htbm ;opt btm ;optH am atmHtbm ;opt atmam

bt_{m ;opt} IN H Ht bm ;opt btm ;optH am atmHtbm ;opt atmam

f bm ;opt atmam atmHt IN H Ht 1 H am H U Vt H U RN N U 1 Ut RN M i i i2 i it h H i j i j V RM M HtH f bm ;opt atmam atmV tUt IN U tUt 1 U Vtam at_mam atmV tUt U IN t Ut 1 U Vtam at_mam atmV tUt Ut 1 IN t 1 U 1U Vtam at_mIMam atmV t IN t 1 Vtam at_mV Vtam atmV t IN t 1 Vtam at_m " V Vt V t IN t 1 Vt # am at_mV " IM t IN t 1 # Vtam D IM t IN t 1 D RM M Di i ( 1 1+ 2 i i r ank H i > r ank H

f bm ;opt atmV D Vtam

r d

t h

Cur r iculum V it ae

A sm a M ej r i

Educat ion

P r ofessional Exp er ience