Comparaison : canal gaussien et canal de Rayleigh
Tse and Viswanath: Fundamentals of Wireless Communication 71
0 10 20 30 40
Non Coherent AWGN
Coherent BPSK Orthogonal 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 over Rayleigh 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<SNR1 .
P{deep fade}≈SNR1 .
We conclude that high-SNR error events most often occur because the channel is in deep fade and not as a result of the additive noise being large. In contrast, in the AWGN 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 the typical error event at high SNR. We can in fact turn the table around and use it as a basis for an approximate analysis of the high-SNR performance (Exercises 3.2 and 3.3). Even though the error probabilitypecan 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 improve it. Moreover, the approximate analysis gives some hints as to how robust the conclu-
Comment communiquer? On utilise la diversité des "chemins indépendants" multiples, i.e. des instances de transmission ayant les attenuations indépendantes.
1 / 24
Comparaison : canal gaussien et canal de Rayleigh
Tse and Viswanath: Fundamentals of Wireless Communication 71
0 10 20 30 40
Non Coherent AWGN
Coherent BPSK Orthogonal 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 over Rayleigh 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<SNR1 .
P{deep fade}≈SNR1 .
We conclude that high-SNR error events most often occur because the channel is in deep fade and not as a result of the additive noise being large. In contrast, in the AWGN 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 the typical error event at high SNR. We can in fact turn the table around and use it as a basis for an approximate analysis of the high-SNR performance (Exercises 3.2 and 3.3). Even though the error probabilitypecan 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 improve it. Moreover, the approximate analysis gives some hints as to how robust the conclu-
Comment communiquer?
On utilise la diversité des "chemins indépendants" multiples, i.e. des instances de transmission ayant les attenuations indépendantes.
1 / 24
Comparaison : canal gaussien et canal de Rayleigh
Tse and Viswanath: Fundamentals of Wireless Communication 71
0 10 20 30 40
Non Coherent AWGN
Coherent BPSK Orthogonal 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 over Rayleigh 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<SNR1 .
P{deep fade}≈SNR1 .
We conclude that high-SNR error events most often occur because the channel is in deep fade and not as a result of the additive noise being large. In contrast, in the AWGN 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 the typical error event at high SNR. We can in fact turn the table around and use it as a basis for an approximate analysis of the high-SNR performance (Exercises 3.2 and 3.3). Even though the error probabilitypecan 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 improve it. Moreover, the approximate analysis gives some hints as to how robust the conclu-
Comment communiquer? On utilise la diversité des "chemins indépendants" multiples, i.e. des instances de transmission ayant les attenuations indépendantes.
1 / 24
Types de diversité
en temps
si les transmissions sont espacés en temps au moins parTS sec.
en espace
les transmissions entre les émetteurs et les récepteurs ayant les positions différentes dans l’espace
en fréquence
si les transmissions sont espacés en fréquence au moins par WC Hz
macro
stations de base différentes,...
par codage
codes orthogonaux ...
2 / 24
Diversité en temps
un de premiers types de diversité exploités utilisé dans 2G (GSM)
idée principale: codage + entrelacement
Tse and Viswanath: Fundamentals of Wireless Communication 77
Interleaving
x2
Codeword x3 Codeword
x0
Codeword x1
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 case but only one coded symbol from each codeword in the latter. In the latter case, each codeword can still be recovered from the other three unfaded symbols.
the channel is highly correlated across consecutive symbols. To ensure that the coded symbols are transmitted through independent or nearly independent fading gains,in- terleavingof codewords is required (Figure 3.5). For simplicity, let us consider a flat fading channel. We transmit a codewordx= [x1, . . . , xL]tof lengthLsymbols and the received signal is given by
y!=h!x!+w!, != 1, . . . , L. (3.31) Assuming ideal interleaving so that consecutive symbolsx!are transmitted sufficiently far apart in time, we can assume that theh!’s are independent. The parameterLis commonly called the number ofdiversity branches. The additive noisesw1, . . . , wLare i.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, pourLbranches de diversité, Perr ≈ 1
L! 1 SNRL. On appeleLle 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, pourLbranches de diversité, Perr ≈ 1
L! 1 SNRL. On appeleLle 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, pourLbranches de diversité, Perr ≈ 1
L!
1 SNRL. On appeleLle gain en diversité.
4 / 24
Gain en performance
Tse and Viswanath: Fundamentals of Wireless Communication 79
Figure 3.6: Error probability as a function ofSNR for different numbers of diversity branchesL.
where
µ:=
! SNR
1 +SNR. (3.38)
The error probability as a function of the SNR for different numbers of diversity branches L is plotted in Figure 3.6. Increasing L dramatically decreases the error probability.
At high SNR, we can see the role of L analytically: consider the leading term in the 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 in our analysis in Section 3.1.2. The typical error event at high SNR is when the overall
5 / 24
Diversité en temps
Inconvenients:
débit faible
il faut s’assurer que le temps de cohérence TS <Nts, ou ts et le 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 partagé entre 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
SiTS >Nts, on ne peut pas utiliser la diversité en temps.
S’il y a plusieurs antennes à l’émetteur/récepteur, suffisamment espacées l’un de l’autre, alors on peut utiliser ladiversité 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]
aveci =1...Let ni[m]∼ CN(0,N0)
On estimex[m]en ayant reçu y1[m], ...,yL[m].
C’est le code à répétition de longueurL.
Les branches de diversité sont distribués pas en temps mais à l’espace. Leshi indép. et distr. à la loi de Rayleigh, alors le gain de diversité estL.
Comme avant,
Perr ∼ 1 SNRL
10 / 24
SIMO avec L antennes à la réception
yi[m] =hi[m]x[m] +ni[m]
aveci =1...Let ni[m]∼ CN(0,N0)
On estimex[m]en ayant reçu y1[m], ...,yL[m].
C’est le code à répétition de longueurL.
Les branches de diversité sont distribués pas en temps mais à l’espace.
Leshi indép. et distr. à la loi de Rayleigh, alors le gain de diversité estL.
Comme avant,
Perr ∼ 1 SNRL
10 / 24
SIMO avec L antennes à la réception
yi[m] =hi[m]x[m] +ni[m]
aveci =1...Let ni[m]∼ CN(0,N0)
On estimex[m]en ayant reçu y1[m], ...,yL[m].
C’est le code à répétition de longueurL.
Les branches de diversité sont distribués pas en temps mais à l’espace.
Leshi indép. et distr. à la loi de Rayleigh, alors le gain de diversité estL.
Comme avant,
Perr ∼ 1 SNRL
10 / 24
MISO avec L antennes à l’émission
Comment gérer la transmission?
Solution naive:
L’antennei est active au tempsits et transmet x. La diversité? Egale àL.
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’antennei est active au tempsits et transmet x. La diversité? Egale àL.
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’antennei est active au tempsits et transmet x. La diversité? Egale àL.
Le débit? 1/Lts.
Approche sophistiquée: codage espace-temps
11 / 24
MISO 2x 1 et le canal invariant durant 2t
sNous avons
y[m] =h1[m]x1[m] +h2[m]x2[m] +n[m]
et
h1[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 h2
hH2 −hH1
· a1
a2
+ n[1]
n[2]
La matriceH =
h1 h2 h2H −h1H
contient les colonnes orthogonales, alors c’est équivalent à2 canaux parallèles:
˜
y[i] =||h||ai + ˜n[i], i =1,2 oun[i]˜ ∼ CN(0,N0)et ||h||=
q
h12+h22. Alors, la diversité est 2 et le débit est 2 symboles.2t
s .
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 h2
hH2 −hH1
· a1
a2
+ n[1]
n[2]
La matriceH =
h1 h2 h2H −h1H
contient les colonnes orthogonales, alors c’est équivalent à2 canaux parallèles:
˜
y[i] =||h||ai + ˜n[i], i =1,2 oun[i]˜ ∼ CN(0,N0)et ||h||=
q
h12+h22.
Alors, la diversité est 2 et le débit est 2 symboles.2t
s .
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 h2
hH2 −hH1
· a1
a2
+ n[1]
n[2]
La matriceH =
h1 h2 h2H −h1H
contient les colonnes orthogonales, alors c’est équivalent à2 canaux parallèles:
˜
y[i] =||h||ai + ˜n[i], i =1,2 oun[i]˜ ∼ CN(0,N0)et ||h||=
q
h12+h22. Alors, la diversité est 2 et le débit est 2 symboles.2t
s .
13 / 24
MISO Lx1 et le canal invariant durant Mt
sCodage espace-temps avec la matriceX:
X =
x11 x12 . . . x1M x21 x22 . . . x2M ... ... . .. ... xL1 xL2 . . . xLM
Lignes = "temps"
Colonnes = "espace"
Le modèle du canal:
yT =hHX +nT , avec
y =
y[1]
... y[M]
; hH =
h1H
... hLH
; n=
n[1]
... n[M]
.
14 / 24
MISO Lx1 et le canal invariant durant Mt
sPerformances (sous le design approprié et siM ≥L):
Pe ∼ 1 SNRL, c’est-à-dire la diversité estL.
15 / 24
MIMO avec M antennes à l’émission et N antennes à la réception
La matrice du canal H:
H =
h11 x12 . . . h1M h21 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]
;
etn∼ CN(0,N0INxN).
nombre des chemins indépendants
SoitR le rang de la matrice H. Alors le canal MIMO peut être decomposé enR 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]
;
etn∼ CN(0,N0INxN).
nombre des chemins indépendants
SoitR le rang de la matrice H. Alors le canal MIMO peut être decomposé enR canaux parallèles indépendants.
17 / 24
Capacité du canal MIMO
Diversity 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
Capacity (bits/channel use)
Eb/N0 Cawgn
2Cawgn
1x1 2x1,1x2
4x1,1x4
8x1,1x8 2x2
4x2,2x4 8x2,2x8
8x8 4x4
Joseph J. Boutros Journ´ees Codage et Cryptographie, Carcans, Gironde March 2008 17 / 30 18 / 24
Taux d’erreurs du canal MIMO
Diversity 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
Outage Probability
Eb/N0 (dB)
1 2 4 8
1 2 4
4x4 - 1 bpcu 4x4 - 2 bpcu 4x4 - 4 bpcu 4x4 - 8 bpcu 2x2 - 1 bpcu 2x2 - 2 bpcu 2x2 - 4 bpcu
Joseph J. Boutros Journ´ees 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 le pre-codage
CSIR Channel state information at the receiver (au récepteur)- dans le cas des canaux statiques (séquence pilot); bon débit par l’allocation des puissances
no CSI augmentation du nombre des antennes n’augmente pas la capacité
21 / 24
Débit pour un canal MIMO
Conclusion
Avec la CSIR, pour les grands SNRs:
le débit maximalC ≈min(N,M)logSNR 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 pour améliorerPerr, mais aussi pour augmenter le débit (si le canal est assez bon). L’augmentation de débit se décrit par le gain de multiplexage
r =R/logSNR/
The function (10.23) is plotted in Fig. 10.8. Recall that in Chapter 7 we found that transmitter or receiver diversity withM antennas resulted in an error probability proportional toSN R−M. The formula (10.23) implies that in a MIMO system, if we use all transmitandreceive antennas for diversity, we get an error probability proportional to SN R−MtMr and that, moreover, we can use some of these antennas to increase data rate at the expense of diversity gain.
Diversity Gain 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 parametersranddbe as defined in (10.21) and (10.22). Suppose thatr anddapproximately satisfy the diversity/multiplexing tradeoffdopt(r) = (Mt−r)(Mr−r)at any large finite SNR. For anMt=Mr = 8MIMO system with an SNR of 15 dB, if we require a data rate per unit Hertz of R= 15bps, what is the maximum diversity gain the system can provide?
Solution: With SNR=15 dB, to getR = 15we requirerlog2(101.5) = 15which impliesr = 3.01. Thus, three of 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 relationshipy=Hx+n, the symbol transmitted over the channel each symbol 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 aspace-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 ofT 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×Tchannel input matrix withith columnxiequal to the
312
23 / 24
Gain de multiplexage
On peut utiliser les branches de diversité non seulement pour améliorerPerr, mais aussi pour augmenter le débit (si le canal est assez bon). L’augmentation de débit se décrit par le gain de multiplexage
r =R/logSNR/
The function (10.23) is plotted in Fig. 10.8. Recall that in Chapter 7 we found that transmitter or receiver diversity withM antennas resulted in an error probability proportional toSN R−M. The formula (10.23) implies that in a MIMO system, if we use all transmitandreceive antennas for diversity, we get an error probability proportional to SN R−MtMr and that, moreover, we can use some of these antennas to increase data rate at the expense of diversity gain.
Diversity Gain 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 parametersranddbe as defined in (10.21) and (10.22). Suppose thatr anddapproximately satisfy the diversity/multiplexing tradeoffdopt(r) = (Mt−r)(Mr−r)at any large finite SNR. For anMt=Mr= 8MIMO system with an SNR of 15 dB, if we require a data rate per unit Hertz of R= 15bps, what is the maximum diversity gain the system can provide?
Solution: With SNR=15 dB, to getR = 15we requirerlog2(101.5) = 15which impliesr = 3.01. Thus, three of 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 relationshipy=Hx+n, the symbol transmitted over the channel each symbol 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 aspace-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 ofT 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×Tchannel input matrix withith columnxiequal to the
312
23 / 24
MIMO dans les standards sans fil existants
UMTS - HSPA : schéma d’Alamouti 2x1 WI-FI, 802.11n: 2x2
LTE: 2x2 WiMAX
24 / 24
