Comparaison : canal gaussien et canal de RayleighTse and Viswanath: Fundamentals of Wireless Communication 71

0 10 20 30 40

Non CoherentAWGN

Coherent BPSKOrthogonal

BPSK over

pe

SNR (dB)

10!8

-10-20

1

10!2

10!4

10!6

10!10

10!12

10!14

10!16

Figure 3.2: Performance of coherent BPSK vs noncoherent orthogonal signaling overRayleigh fading channel vs BPSK over AWGN channel.

=1

SNR+ O

!

1

SNR2

"

. (3.23)

This probability has the same order of magnitude as the error probability itself (c.f.(3.21)). Thus, we can define a ”deep fade” via an order-of-magnitude approximation

Deep fade event: |h|2 < 1SNR

.

P {deep fade} ! 1SNR

.

We conclude that high-SNR error events most often occur because the channel is indeep fade and not as a result of the additive noise being large. In contrast, in theAWGN channel the only possible error mechanism is for the additive noise to be large.Thus, the error probability performance over the AWGN channel is much better.

We have used the explicit error probability expression (3.19) to help identify thetypical error event at high SNR. We can in fact turn the table around and use it asa basis for an approximate analysis of the high-SNR performance (Exercises 3.2 and3.3). Even though the error probability pe can be directly computed in this case,the approximate analysis provides much insight as to how typical errors occur. Under-standing typical error events in a communication system often suggests how to improveit. Moreover, the approximate analysis gives some hints as to how robust the conclu-

Comment communiquer? On utilise la diversité des "cheminsindépendants" multiples, i.e. des instances de transmission ayantles attenuations indépendantes.

1 / 24

Comparaison : canal gaussien et canal de RayleighTse and Viswanath: Fundamentals of Wireless Communication 71

0 10 20 30 40

Non CoherentAWGN

Coherent BPSKOrthogonal

BPSK over

pe

SNR (dB)

10!8

-10-20

1

10!2

10!4

10!6

10!10

10!12

10!14

10!16

Figure 3.2: Performance of coherent BPSK vs noncoherent orthogonal signaling overRayleigh fading channel vs BPSK over AWGN channel.

=1

SNR+ O

!

1

SNR2

"

. (3.23)

This probability has the same order of magnitude as the error probability itself (c.f.(3.21)). Thus, we can define a ”deep fade” via an order-of-magnitude approximation

Deep fade event: |h|2 < 1SNR

.

P {deep fade} ! 1SNR

.

We conclude that high-SNR error events most often occur because the channel is indeep fade and not as a result of the additive noise being large. In contrast, in theAWGN channel the only possible error mechanism is for the additive noise to be large.Thus, the error probability performance over the AWGN channel is much better.

We have used the explicit error probability expression (3.19) to help identify thetypical error event at high SNR. We can in fact turn the table around and use it asa basis for an approximate analysis of the high-SNR performance (Exercises 3.2 and3.3). Even though the error probability pe can be directly computed in this case,the approximate analysis provides much insight as to how typical errors occur. Under-standing typical error events in a communication system often suggests how to improveit. Moreover, the approximate analysis gives some hints as to how robust the conclu-

Comment communiquer?

On utilise la diversité des "cheminsindépendants" multiples, i.e. des instances de transmission ayantles attenuations indépendantes.

1 / 24

Comparaison : canal gaussien et canal de RayleighTse and Viswanath: Fundamentals of Wireless Communication 71

0 10 20 30 40

Non CoherentAWGN

Coherent BPSKOrthogonal

BPSK over

pe

SNR (dB)

10!8

-10-20

1

10!2

10!4

10!6

10!10

10!12

10!14

10!16

Figure 3.2: Performance of coherent BPSK vs noncoherent orthogonal signaling overRayleigh fading channel vs BPSK over AWGN channel.

=1

SNR+ O

!

1

SNR2

"

. (3.23)

This probability has the same order of magnitude as the error probability itself (c.f.(3.21)). Thus, we can define a ”deep fade” via an order-of-magnitude approximation

Deep fade event: |h|2 < 1SNR

.

P {deep fade} ! 1SNR

.

We conclude that high-SNR error events most often occur because the channel is indeep fade and not as a result of the additive noise being large. In contrast, in theAWGN channel the only possible error mechanism is for the additive noise to be large.Thus, the error probability performance over the AWGN channel is much better.

We have used the explicit error probability expression (3.19) to help identify thetypical error event at high SNR. We can in fact turn the table around and use it asa basis for an approximate analysis of the high-SNR performance (Exercises 3.2 and3.3). Even though the error probability pe can be directly computed in this case,the approximate analysis provides much insight as to how typical errors occur. Under-standing typical error events in a communication system often suggests how to improveit. Moreover, the approximate analysis gives some hints as to how robust the conclu-

Comment communiquer? On utilise la diversité des "cheminsindépendants" multiples, i.e. des instances de transmission ayantles attenuations indépendantes.

1 / 24

Types de diversité

en tempssi les transmissions sont espacés en temps au moins par TS sec.en espaceles transmissions entre les émetteurs et les récepteurs ayant lespositions différentes dans l’espaceen fréquencesi les transmissions sont espacés en fréquence au moins parWC Hzmacrostations de base différentes,...par codagecodes orthogonaux...

2 / 24

Diversité en temps

un de premiers types de diversité exploitésutilisé dans 2G (GSM)idée principale: codage + entrelacement

Tse and Viswanath: Fundamentals of Wireless Communication 77

Interleaving

x2

Codewordx3

Codewordx0

Codewordx1

Codeword

|hl|L = 4

l

No Interleaving

Figure 3.5: The codewords are transmitted over consecutive symbols (top) and inter-leaved (bottom). A deep fade will wipe out the entire codeword in the former casebut only one coded symbol from each codeword in the latter. In the latter case, eachcodeword can still be recovered from the other three unfaded symbols.

the channel is highly correlated across consecutive symbols. To ensure that the codedsymbols are transmitted through independent or nearly independent fading gains, in-terleaving of codewords is required (Figure 3.5). For simplicity, let us consider a flatfading channel. We transmit a codeword x = [x1, . . . , xL]t of length L symbols and thereceived signal is given by

y! = h!x! + w!, ! = 1, . . . , L. (3.31)

Assuming ideal interleaving so that consecutive symbols x! are transmitted su!cientlyfar apart in time, we can assume that the h!’s are independent. The parameter L iscommonly called the number of diversity branches. The additive noises w1, . . . , wL arei.i.d. CN (0, N0) random variables.

3 / 24

Diversité en temps

Un code le plus simple?

Le code à répétition de rendement 1/L.

On peut démontrer que, pour L branches de diversité,

Perr ≈1L!

1SNRL .

On appele L le gain en diversité.

4 / 24

Diversité en temps

Un code le plus simple? Le code à répétition de rendement 1/L.

On peut démontrer que, pour L branches de diversité,

Perr ≈1L!

1SNRL .

On appele L le gain en diversité.

4 / 24

Diversité en temps

Un code le plus simple? Le code à répétition de rendement 1/L.

On peut démontrer que, pour L branches de diversité,

Perr ≈1L!

1SNRL .

On appele L le gain en diversité.

4 / 24

Gain en performanceTse and Viswanath: Fundamentals of Wireless Communication 79

Figure 3.6: Error probability as a function of SNR for di!erent numbers of diversitybranches L.

where

µ :=

!

SNR

1 + SNR. (3.38)

The error probability as a function of the SNR for di!erent numbers of diversitybranches L is plotted in Figure 3.6. Increasing L dramatically decreases the errorprobability.

At high SNR, we can see the role of L analytically: consider the leading term inthe Taylor series expansion in 1/SNR to arrive at the approximations

1 + µ

2! 1, and

1 " µ

2! 1

4SNR. (3.39)

Furthermore,L!1"

!=0

#

L " 1 + !

!

$

=

#

2L " 1

L

$

. (3.40)

Hence,

pe !#

2L " 1

L

$

1

(4SNR)L(3.41)

at high SNR. In particular, the error probability decreases as the Lth power of SNR,corresponding to a slope of "L in the error probability curve (in dB/dB scale).

To understand this better, we examine the probability of the deep fade event, as inour analysis in Section 3.1.2. The typical error event at high SNR is when the overall

5 / 24

Diversité en temps

Inconvenients:débit faibleil faut s’assurer que le temps de cohérence TS < Nts , ou ts etle temps de transmission d’un symbole

6 / 24

Exemple du GSM

890-915 MHz (uplink) et 935-960 MHz (downlink);125 sous-canaux de 200 kHz chaque, un sous-canal partagéentre 8 utilisateurs;temps-symbole pour un utilisateur = 577 µs;le rendement de code est 1/2;l’intrelecement est effectué entre les slots voisins.

7 / 24

Diversité en espace

Si TS > Nts , on ne peut pas utiliser la diversité en temps.S’il y a plusieurs antennes à l’émetteur/récepteur, suffisammentespacées l’un de l’autre, alors on peut utiliser la diversité en espace.

? Pourquoi "suffisamment espacées"?? De quoi dépendra l’espacement entre les antennes?

8 / 24

Diversité en espace

1 la diversité au récepteur(Single Input Multiple Output)

2 la diversité à l’émetteur(Multiple Input Single Output)

3 canaux MIMO(Multiple Input Multiple Output)

9 / 24

SIMO avec L antennes à la réception

yi [m] = hi [m]x [m] + ni [m]

avec i = 1...L et ni [m] ∼ CN (0,N0)

On estime x [m] en ayant reçu y1[m], ..., yL[m].

C’est le code à répétition de longueur L.Les branches de diversité sont distribués pas en temps mais à l’espace.

Les hi indép. et distr. à la loi de Rayleigh, alors le gain de diversité est L.Comme avant,

Perr ∼1

SNRL

10 / 24

SIMO avec L antennes à la réception

yi [m] = hi [m]x [m] + ni [m]

avec i = 1...L et ni [m] ∼ CN (0,N0)

On estime x [m] en ayant reçu y1[m], ..., yL[m].C’est le code à répétition de longueur L.Les branches de diversité sont distribués pas en temps mais à l’espace.

Les hi indép. et distr. à la loi de Rayleigh, alors le gain de diversité est L.Comme avant,

Perr ∼1

SNRL

10 / 24

SIMO avec L antennes à la réception

yi [m] = hi [m]x [m] + ni [m]

avec i = 1...L et ni [m] ∼ CN (0,N0)

On estime x [m] en ayant reçu y1[m], ..., yL[m].C’est le code à répétition de longueur L.Les branches de diversité sont distribués pas en temps mais à l’espace.

Les hi indép. et distr. à la loi de Rayleigh, alors le gain de diversité est L.Comme avant,

Perr ∼1

SNRL

10 / 24

MISO avec L antennes à l’émission

Comment gérer la transmission?

Solution naive:L’antenne i est active au temps its et transmet x .La diversité? Egale à L.Le débit? 1/Lts .

Approche sophistiquée: codage espace-temps

11 / 24

MISO avec L antennes à l’émission

Comment gérer la transmission?

Solution naive:L’antenne i est active au temps its et transmet x .La diversité? Egale à L.Le débit? 1/Lts .

Approche sophistiquée: codage espace-temps

11 / 24

MISO avec L antennes à l’émission

Comment gérer la transmission?

Solution naive:L’antenne i est active au temps its et transmet x .La diversité? Egale à L.Le débit? 1/Lts .

Approche sophistiquée: codage espace-temps

11 / 24

MISO 2x1 et le canal invariant durant 2ts

Nous avons

y [m] = h1[m]x1[m] + h2[m]x2[m] + n[m]

eth1[1] = h1[2] = h1; h2[1] = h2[2] = h2.

Pour 2 symboles consécutifs,

[ y [1] y [2] ] = [ h1 h2 ] ·[

x1[1] x1[2]x2[1] x2[2]

]+ [ n[1] n[2] ]

12 / 24

Schéma d’Alamouti

Soit

x1[1] = a1; x1[2] = −aH2 ;

x2[1] = a2; x2[2] = aH1 .

Alors nous obtenons[y [1]y [2]

]=

[h1 h2hH2 −hH

1

]·[

a1a2

]+

[n[1]n[2]

]

La matrice H =

[h1 h2hH2 −hH

1

]contient les colonnes orthogonales,

alors c’est équivalent à 2 canaux parallèles:

y [i ] = ||h||ai + n[i ], i = 1, 2

ou n[i ] ∼ CN (0,N0) et ||h|| =√

h21 + h2

2. Alors, la diversité est 2

et le débit est 2 symboles.2ts .

13 / 24

Schéma d’Alamouti

Soit

x1[1] = a1; x1[2] = −aH2 ;

x2[1] = a2; x2[2] = aH1 .

Alors nous obtenons[y [1]y [2]

]=

[h1 h2hH2 −hH

1

]·[

a1a2

]+

[n[1]n[2]

]

La matrice H =

[h1 h2hH2 −hH

1

]contient les colonnes orthogonales,

alors c’est équivalent à 2 canaux parallèles:

y [i ] = ||h||ai + n[i ], i = 1, 2

ou n[i ] ∼ CN (0,N0) et ||h|| =√

h21 + h2

2.

Alors, la diversité est 2

et le débit est 2 symboles.2ts .

13 / 24

Schéma d’Alamouti

Soit

x1[1] = a1; x1[2] = −aH2 ;

x2[1] = a2; x2[2] = aH1 .

Alors nous obtenons[y [1]y [2]

]=

[h1 h2hH2 −hH

1

]·[

a1a2

]+

[n[1]n[2]

]

La matrice H =

[h1 h2hH2 −hH

1

]contient les colonnes orthogonales,

alors c’est équivalent à 2 canaux parallèles:

y [i ] = ||h||ai + n[i ], i = 1, 2

ou n[i ] ∼ CN (0,N0) et ||h|| =√

h21 + h2

2. Alors, la diversité est 2

et le débit est 2 symboles.2ts .

13 / 24

MISO Lx1 et le canal invariant durant Mts

Codage espace-temps avec la matrice X :

X =

x11 x12 . . . x1Mx21 x22 . . . x2M...

.... . .

...xL1 xL2 . . . xLM

Lignes = "temps"Colonnes = "espace"

Le modèle du canal:

yT = hHX + nT ,

avec

y =

y [1]...

y [M]

; hH =

hH1...

hHL

; n =

n[1]...

n[M]

.14 / 24

MISO Lx1 et le canal invariant durant Mts

Performances (sous le design approprié et si M ≥ L):

Pe ∼1

SNRL ,

c’est-à-dire la diversité est L.

15 / 24

MIMO avec M antennes à l’émission et N antennes à laréception

La matrice du canal H:

H =

h11 x12 . . . h1Mh21 h22 . . . h2M...

.... . .

...hN1 hN2 . . . hNM

16 / 24

MIMO MxN

Modèle du canal:y = Hx + n

avec

y =

y [1]...

y [N]

; x =

x [1]...

x [M]

; n =

n[1]...

n[N]

;

et n ∼ CN (0,N0INxN).

nombre des chemins indépendantsSoit R le rang de la matrice H. Alors le canal MIMO peut êtredecomposé en R canaux parallèles indépendants.

17 / 24

MIMO MxN

Modèle du canal:y = Hx + n

avec

y =

y [1]...

y [N]

; x =

x [1]...

x [M]

; n =

n[1]...

n[N]

;

et n ∼ CN (0,N0INxN).

nombre des chemins indépendantsSoit R le rang de la matrice H. Alors le canal MIMO peut êtredecomposé en R canaux parallèles indépendants.

17 / 24

Capacité du canal MIMODiversity MIMO channel MIMO Capacity Space-Time Coding

Capacity of MIMO channels (4)

0

1

2

3

4

5

6

7

8

-2 0 2 4 6 8 10

Capa

city

(bits

/cha

nnel

use

)

Eb/N0

Cawgn

2Cawgn

1x1

2x1,1x2

4x1,1x4

8x1,1x82x

24x2,2x4

8x2,2x8

4x4

8x8

Joseph J. Boutros Journees Codage et Cryptographie, Carcans, Gironde March 2008 17 / 30

18 / 24

Taux d’erreurs du canal MIMODiversity MIMO channel MIMO Capacity Space-Time Coding

Outage probability (3)

10-5

10-4

10-3

10-2

10-1

0 2 4 6 8 10 12 14 16

Outa

ge

Pro

bab

ilit

y

Eb/N0 (dB)

1 2 4 8

1 2 4

4x4 - 1 bpcu4x4 - 2 bpcu4x4 - 4 bpcu4x4 - 8 bpcu2x2 - 1 bpcu2x2 - 2 bpcu2x2 - 4 bpcu

Joseph J. Boutros Journees Codage et Cryptographie, Carcans, Gironde March 2008 19 / 30

19 / 24

Difficultés avec le canal MIMO

Connaissance du canal:CSIT Channel state information at the transmitter (à l’émetteur) -

pour les canaux avec la voie de retour;CSIR Channel state information at the receiver (au récepteur)-

dans le cas des canaux statiques (séquence pilot);no CSI

20 / 24

Difficultés avec le canal MIMO

Connaissance du canal:CSIT Channel state information at the transmitter (à l’émetteur) -

pour les canaux avec la voie de retour; bon débit par lepre-codage

CSIR Channel state information at the receiver (au récepteur)-dans le cas des canaux statiques (séquence pilot); bon débitpar l’allocation des puissances

no CSI augmentation du nombre des antennes n’augmente pas lacapacité

21 / 24

Débit pour un canal MIMO

ConclusionAvec la CSIR, pour les grands SNRs:le débit maximal C ≈ min(N,M) log SNR bits/sec/Hz,pour les petits SNRs:C ≈ M · SNR bits/sec/Hz.

Avec la CSIT : gain supplémentaire par le beamforming(pre-codage).

22 / 24

Gain de multiplexage

On peut utiliser les branches de diversité non seulement pouraméliorer Perr , mais aussi pour augmenter le débit (si le canal estassez bon). L’augmentation de débit se décrit par le gain demultiplexage

r = R/ log SNR/

The function (10.23) is plotted in Fig. 10.8. Recall that in Chapter 7 we found that transmitter or receiver diversity

with M antennas resulted in an error probability proportional to SNR!M . The formula (10.23) implies that in a

MIMO system, if we use all transmit and receive antennas for diversity, we get an error probability proportional to

SNR!MtMr and that, moreover, we can use some of these antennas to increase data rate at the expense of diversity

gain.

Div

ers

ity G

ain

d (

r)*

Multiplexing Gain r=R/log(SNR)

(0,M M )t r

t r(1,(M !1)(M !1))

t r

t r(min(M ,M ),0)

(r,(M !r)(M !r))

Figure 10.8: Diversity-Multiplexing Tradeoff for High SNR Block Fading.

It is also possible to adapt the diversity and multiplexing gains relative to channel conditions. Specifically,

in poor channel states more antennas can be used for diversity gain, whereas in good states more antennas can be

used for multiplexing. Adaptive techniques that change antenna use to trade off diversity and multiplexing based

on channel conditions have been investigated in [39, 40, 41].

Example 10.4: Let the multiplexing and diversity parameters r and d be as defined in (10.21) and (10.22). Supposethat r and d approximately satisfy the diversity/multiplexing tradeoff dopt(r) = (Mt ! r)(Mr ! r) at any largefinite SNR. For anMt = Mr = 8MIMO system with an SNR of 15 dB, if we require a data rate per unit Hertz ofR = 15 bps, what is the maximum diversity gain the system can provide?

Solution: With SNR=15 dB, to get R = 15 we require r log2(101.5) = 15 which implies r = 3.01. Thus, threeof the antennas are used for multiplexing and the remaining five for diversity. The maximum diversity gain is then

dopt(r) = (Mt ! r)(Mr ! r) = (8 ! 3)(8 ! 3) = 25.

10.6 Space-Time Modulation and Coding

Since a MIMO channel has input-output relationship y = Hx + n, the symbol transmitted over the channel eachsymbol time is a vector rather than a scalar, as in traditional modulation for the SISO channel. Moreover, when

the signal design extends over both space (via the multiple antennas) and time (via multiple symbol times), it is

typically referred to as a space-time code.

Most space-time codes, including all codes discussed in this section, are designed for quasi-static channels

where the channel is constant over a block of T symbol times, and the channel is assumed unknown at the trans-

mitter. Under this model the channel input and output become matrices, with dimensions corresponding to space

(antennas) and time. LetX = [x1, . . . ,xT ] denote theMt"T channel input matrix with ith column xi equal to the

312

23 / 24

Gain de multiplexage

On peut utiliser les branches de diversité non seulement pouraméliorer Perr , mais aussi pour augmenter le débit (si le canal estassez bon). L’augmentation de débit se décrit par le gain demultiplexage

r = R/ log SNR/

The function (10.23) is plotted in Fig. 10.8. Recall that in Chapter 7 we found that transmitter or receiver diversity

with M antennas resulted in an error probability proportional to SNR!M . The formula (10.23) implies that in a

MIMO system, if we use all transmit and receive antennas for diversity, we get an error probability proportional to

SNR!MtMr and that, moreover, we can use some of these antennas to increase data rate at the expense of diversity

gain.

Div

ers

ity G

ain

d (

r)*

Multiplexing Gain r=R/log(SNR)

(0,M M )t r

t r(1,(M !1)(M !1))

t r

t r(min(M ,M ),0)

(r,(M !r)(M !r))

Figure 10.8: Diversity-Multiplexing Tradeoff for High SNR Block Fading.

It is also possible to adapt the diversity and multiplexing gains relative to channel conditions. Specifically,

in poor channel states more antennas can be used for diversity gain, whereas in good states more antennas can be

used for multiplexing. Adaptive techniques that change antenna use to trade off diversity and multiplexing based

on channel conditions have been investigated in [39, 40, 41].

Example 10.4: Let the multiplexing and diversity parameters r and d be as defined in (10.21) and (10.22). Supposethat r and d approximately satisfy the diversity/multiplexing tradeoff dopt(r) = (Mt ! r)(Mr ! r) at any largefinite SNR. For anMt = Mr = 8MIMO system with an SNR of 15 dB, if we require a data rate per unit Hertz ofR = 15 bps, what is the maximum diversity gain the system can provide?

Solution: With SNR=15 dB, to get R = 15 we require r log2(101.5) = 15 which implies r = 3.01. Thus, threeof the antennas are used for multiplexing and the remaining five for diversity. The maximum diversity gain is then

dopt(r) = (Mt ! r)(Mr ! r) = (8 ! 3)(8 ! 3) = 25.

10.6 Space-Time Modulation and Coding

Since a MIMO channel has input-output relationship y = Hx + n, the symbol transmitted over the channel eachsymbol time is a vector rather than a scalar, as in traditional modulation for the SISO channel. Moreover, when

the signal design extends over both space (via the multiple antennas) and time (via multiple symbol times), it is

typically referred to as a space-time code.

Most space-time codes, including all codes discussed in this section, are designed for quasi-static channels

where the channel is constant over a block of T symbol times, and the channel is assumed unknown at the trans-

mitter. Under this model the channel input and output become matrices, with dimensions corresponding to space

(antennas) and time. LetX = [x1, . . . ,xT ] denote theMt"T channel input matrix with ith column xi equal to the

312

23 / 24

MIMO dans les standards sans fil existants

UMTS - HSPA : schéma d’Alamouti 2x1WI-FI, 802.11n: 2x2LTE: 2x2WiMAX

24 / 24