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Ergodic and Outage Capacity of Narrowband MIMO Gaussian Channels
Yang Wen LiangDepartment of Electrical and Computer Engineering
The University of British Columbia
April 19th, 2005
4/19/2005 Yang 2
Outline of Presentation
w Introduction of MIMOw MIMO system modelw Capacity for channels with fixed coefficientsw Capacity of MIMO fast and block Rayleigh fading
channelsw Capacity of MIMO slow Rayleigh fading channelsw Summary
4/19/2005 Yang 3
Introduction of MIMO
w MIMO is multi-input and multi-output system
w MIMO systems provide significant capacity gains over conventional single antenna array based solutions.
w Hot research topic within academia and industry.
4/19/2005 Yang 4
MIMO system model
A single user multi-input multi-output system with t Txantennas and r Rx antennas
Space-time encoder
Space-time decoder
h 11
x 1
x 2
x t
y 1
y 2
y r
h 21
h r1
h 12
h 1t
h 22
h r2 h 2t
h rt
.
.
.
.
.
.
4/19/2005 Yang 5
MIMO system model – cont’
w The receive signal is given by
Where
Hy x n= +
2
::
::
C received vectorH C channel matrix
C transmited vectorC complex Gaussian noise with zero mean
and covariance matrix
r
r t
t
r
r
y
xn
Iσ
×∈∈
∈∈
4/19/2005 Yang 6
MIMO system model - cont’
w The total power of the complex transmit signal x is constrained to P regardless of the number of transmit antennas
w Assuming the realization of H is known at the receiver, but not always at the transmitter
† †[ ] ( [ ])x x tr xx Pε ε= =
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MIMO system model – cont’
w What is the capacity of this channel- H is a deterministic matrix- H is a ergodic random matrix- H is random, but fixed once it is chosen (non-ergodic).
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Capacity for channels with fixed coefficients
w H is deterministicw Decorrelating H by Singular Value Decomposition
(SVD)
w U and V are rxr and txt unitary matrices respectively.w D is a rxr diagonal matrix with nonnegative square
roots of the eigenvalues of , denoted by
†H UDV=
†HH
, 1,2, ,i i rλ =
4/19/2005 Yang 9
Capacity for channels with fixed coefficients – cont’
w LetThen
w Then
Where is the rank of H
0
0
11
i i ii
i
x n i ryn r i r
λ + ≤ ≤= + ≤ ≤
0r
Hy x n y Dx n= + ⇒ = +
† † †, ,y U y x V x n U n= = =
4/19/2005 Yang 10
Capacity for channels with fixed coefficients – cont’
w The overall channel capacity C is the sum of the subchannels capacities
Where is the received signal power at the ith
subchannel.
0
21
ln 1 / /r
ri
i
PC nats s Hzσ=
= +
∑riP
4/19/2005 Yang 11
Equal Transmit Power Allocation
w The power allocated to subchannel i is given by and is given by
w Thus the channel capacity can be written as
/ , 1, 2,...,iP P t i t= = riP
, 1, 2,...,iri
PP i rt
λ= =
00
2 21 1
ln 1 ln 1 / /rr
ri i
i i
P PC nats s Hztλ
σ σ= =
= + = +
∑ ∏
4/19/2005 Yang 12
Adaptive Transmit Power Allocation
w For the case when the CSI is known at the transmitter, the capacity can be increased by “water-filling” method
where denotes and is chosen to meet the power constraint so that
w The received signal power at the ith subchannel is
2
0, 1,2,...,ii
P i rσµ
λ
+
= − =
µ0
1
rii
P P=
=∑a+ max( , 0)a
( )2ri iP λ µ σ
+= −
4/19/2005 Yang 13
Adaptive Transmit Power Allocation – cont’
w Thus the channel capacity is
( )0
0
0
2
21
21
21
ln 1
ln 1 1
ln / /
ri
i
ri
i
ri
i
C
nats s Hz
λ µ σ
σ
λ µσ
λ µσ
+
=
+
=
+
=
− = + = + −
=
∑
∑
∑
4/19/2005 Yang 14
Capacity of MIMO fast and block Rayleigh fading channels
w The mean (ergodic) capacity of a random MIMO channel with power constraint can be expressed as
where denotes the expectation over all channel realizations and represents the mutual informationbetween x and y.
w The capacity of the channel is defined as the maximum of the mutual information between input and output over all statistical distributions, p(x), on the input satisfy the power constraint.
( )†xxtr Pε =
( ){ }†( ): ( [ ])max ;
x xxx yH
p tr PC I
εε
==
Hε( );x yI
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Capacity of MIMO fast and block Rayleigh fading channels – cont’
w By the assumption that realization of H is known at the receiver, the output of the channel is the pair (y, H).
w Then the capacity is equivalent to
w Definition: A Gaussian random vector x is circularly symmetric, if for
( )†( ): ( [ ])
max ; ,x xx
x y Hp tr P
C Iε =
=
( ) ( )Re Imx x xT
=
( ) ( ) ( )( ) ( ) ( )1 Re Imcov , covIm Re2
x xQ Q where QQ Q− = =
4/19/2005 Yang 16
Capacity of MIMO fast and block Rayleigh fading channels – cont’
w Given covariance matrix Q, circularly symmetric Gaussian random vector is entropy maximizer.
w The covariance matrix of y with realization of H=H is
w The mutual information is
( )ˆ ( ) ln detxH eQπ=
( )( )† † † † † 2[ ]yy x n x n rH H HQH Iε ε σ = + + = +
( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )
†2
; , ; ; ; ;ˆ ˆ ,ˆ ˆ
1ln det( )
H
H H
H
H
x y H x H x y|H x y|H x y|H
y|H y|x H
y|H n
H Hr
I I I I H
H H H H
H H H
I Q
ε
ε ε
ε
εσ
= + = = = = = − = = = − = +
4/19/2005 Yang 17
Capacity of MIMO fast and block Rayleigh fading channels – cont’
w Telatar proved that it is optimal to use equal power allocation if no knowledge of CSI in the transmitter. Then
w Let and . The random matrix for , or for has the Wishartdistribution with parameters m, n and the unordered eigenvalue have the joint density
Where K is a normalizing factor
† †2 2ln det( ) ln det( )H HHH H Hr t
P PC I It t
ε εσ σ
= + = +
m ax( , )n r t= m in( , )m r t=†HH †H Hr t< r t≥
( )2
1,
1( ,..., )!
i
mn m
m i i ji i jm n
p em K
λλ λ λ λ λ−−
<
= −∏ ∏
4/19/2005 Yang 18
Capacity of MIMO fast and block Rayleigh fading channels – cont’
w Anyone of the unordered eigenvalues has the distribution
where is the associated Laguerre polynomial of order k, and it is given by
( ) ( ) ( )1 2
0
1 !!
mn m n mk
k
kp L em k n m
λλ λ λ−
− − −
=
= + −∑
( )n mkL λ−
( ) ( ) ( )( ) ( )0
!1
! ! !
kln m l
kl
k n mL
k l n m l lλ λ−
=
+ −= −
− − +∑
4/19/2005 Yang 19
Capacity of MIMO fast and block Rayleigh fading channels – cont’
w Then the mean channel capacity is given by
( )
( ) ( )
1 22
21
2
1 2
200
ln det , , ...,
ln 1
ln 1
!ln 1!
m m
m
ii
mn m n mk
k
PC I diagt
Pt
Pmt
P k L e dt k n m
λ
λ
λ
λ
ε λ λ λσ
ε λσ
ε λσ
λ λ λ λσ
=
∞ −− − −
=
= + = +
= +
= + + −
∑
∑∫
4/19/2005 Yang 20
Capacity of MIMO fast and block Rayleigh fading channels – cont’
w The number of receive antenna is 1
The asymptotic value is
2lim ln 1 / /t
PC nats s Hzσ→∞
= +
0 2 4 6 8 10 12 14 16 18 200
1
2
3
4
5
6
7
8
9The value of the capacity for r = 1 vs. Number of Tx Antennas(t)
Number of Tx Antennas (t)
Cha
nnel
Cap
acity
(nat
s/s/
Hz)
SNR = 35dB
SNR = 30dB
SNR = 25dB
SNR = 20dB
SNR = 15dB
SNR = 10dB
SNR = 5dB
SNR = 0dB
4/19/2005 Yang 21
Capacity of MIMO fast and block Rayleigh fading channels – cont’
w The number of transmit antenna is 1
The asymptotic value is
2lim ln 1 / /t
rPC nats s Hzσ→∞
= +
0 5 10 15 20 25 300
2
4
6
8
10
12The value of the capacity for t = 1 vs. Number of Rx Antennas(r)
Number of Rx Antennas (r)
Cha
nnel
Cap
acity
(nat
s/s/
Hz)
SNR = 35dB
SNR = 30dB
SNR = 25dB
SNR = 20dB
SNR = 15dB
SNR = 10dB
SNR = 5dB
SNR = 0dB
4/19/2005 Yang 22
Capacity of MIMO fast and block Rayleigh fading channels – cont’
w The number of receive antenna equals the number of transmit antenna
The approximate asymptotic value is:
( ) 2lim ln 1 / /t r
PC r nats s Hzσ= →∞
= −
0 2 4 6 8 10 120
10
20
30
40
50
60
70
80
90The value of the capacity for t = r vs. Number of Rx Antennas(r)
Number of Rx Antennas (r)
Cha
nnel
Cap
acity
(nat
s/s/
Hz)
SNR = 0dBSNR = 5dBSNR = 10dBSNR = 15dBSNR = 20dBSNR = 25dBSNR = 30dBSNR = 35dB
4/19/2005 Yang 23
Capacity of MIMO slow Rayleigh fading channels
w H is chosen randomly according to a Rayleigh distribution at the beginning of transmission, and held fixed for all channel uses. The channel is non-ergodic.
w The maximum mutual information is in general not equal to the channel capacity because it is not always achievable.
w Another measure of channel capacity is the outage capacityassociated with a outage probability
( )( )†
Q:Q 0(Q)
inf ln det HQHout r outagetr P
P p I C≥
≤
= + <
4/19/2005 Yang 24
Capacity of MIMO slow Rayleigh fading channels
w Smith demonstrated that the narrowband Rayleigh MIMO channel capacity can be accurately approximated by Gaussian approximation (only mean and variance of is needed) for equal power allocation case.
w Recall the instantaneous channel capacity is
w Recall the mean channel capacity is
insC
21
ln 1m
ins ii
PCt
λσ=
= +
∑
{ } ( ) ( )1 2
200
!ln 1!
mn m n mk
k
P kC L e dt k n m
λε λ λ λ λσ
∞ −− − −
=
= + + − ∑∫
4/19/2005 Yang 25
Capacity of MIMO slow Rayleigh fading channels – cont’
w Smith derive the exact expression of variance of as insC2
20
1 1
2
1 1 20
( ) ln 1 ( )
( 1)!( 1)!( 1 )!( 1 )!
( ) ( ) ln 1
ins
m m
i j
n m n m n mi j
PVar C m p dt
i ji n m j n m
Pe L L dt
λ
λλ λ
σ
λλ λ λ λ
σ
∞
= =
∞ − − − −− −
= + − − −
× − + − − + −
+
∫
∑∑
∫
4/19/2005 Yang 26
Capacity of MIMO slow Rayleigh fading channels – cont’
w Using Gaussian approximation
where Q-function is tail integral of a unit-Gaussian pdf and it is defined as
( )( )
( ) ins outageout ins outage
ins
C CP p C C Q
Var C
ε − = < =
2
21( )2
z
x
Q x e dzπ
∞−
= ∫
4/19/2005 Yang 27
Capacity of MIMO slow Rayleigh fading channels – cont’
w The probability of outage capacity curve of MIMO channel with Rx=2 and SNR=15dB for various number of Tx antennas
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1The outage capacity of MIMO channel with Rx = 2 and SNR = 15dB and various number of Tx
P out =
p(C
=<R
th)
Rate Threshold in (nats/s/Hz)
Tx = 1
Tx = 4 Tx = 7
Tx = 14
4/19/2005 Yang 28
Capacity of MIMO slow Rayleigh fading channels – cont’
w The probability of outage capacity curve of MIMO channel with Tx=2 and SNR=15dB for various number of Rx antennas
0 2 4 6 8 10 12 140
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1The outage capacity of MIMO channel with Tx = 2 and SNR = 15dB and various number of Rx
P out =
p(C
=<R
th)
Rate Threshold in (nats/s/Hz)
Rx = 1 Rx = 4
Rx = 7
Rx = 10
4/19/2005 Yang 29
Capacity of MIMO slow Rayleigh fading channels – cont’
w The probability of outage capacity curve of MIMO channel with Tx=Rx=4 for various SNR
0 5 10 15 20 25 30 350
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1The outage capacity of MIMO channel with Tx = Rx = 4 for various SNR
P out =
p(C
=<R
th)
Rate Threshold in (nats/s/Hz)
0dB
5dB
10dB
15dB
20dB
25dB
30dB
4/19/2005 Yang 30
Summary
w The MIMO capacity for H with fixed coefficient is derived
w Ergodic and Outage capacity of MIMO Rayleigh channel were introduced with some examples
w MIMO configuration could provide significant capacity gains over conventional single antenna array based system
4/19/2005 Yang 31
Reference
w J. H. Winters, “On the capacity of radio communications with diversity in a Rayleigh fading environment,” IEEE J. Selected Area Commun., vol. SAC-5, pp. 871-878, Jun. 1987.
w G. J. Foschini and M. J. Gans, “On limits of wireless communication in a fading environment when using multiple antennas,” Wireless Personal Communications, vol. 6, pp. 311-335, Mar. 1998.
w I. E. Telatar, Capacity of multi-antenna Gaussian channels, Technical Report # BL0112170-950615-07TM, AT \& T Bell Laboratories, 1995.
w G. J. Foschini. “Layered space-time architecture for wireless communication in a fading environment when using multiple antennas,” Bell Labs Technical Journal, 1(2):41-59, Autumn 1996.
w R. G. Gallager, Information Theory and Reliable Communication. New York: John Wiley & Sons, 1968.
w M. Dohler, H. Aghvami, “A Closed Form Expression of MIMO capacity over Ergodic Narrowband Channels,” IEEE Comm. Letter, vol. 8, Issue: 6, pp. 365-367, June 2004.
w H. Shin, J. H. Lee, “Closed-form Formulas for Ergodic Capacity of MIMO Rayleigh Fading Channels,” IEEE ICC 2003, May 2003, pp. 2996-3000.
4/19/2005 Yang 32
Reference – cont’
w P. J. Smith, M. Shafi, “On a Gaussian approximation to the capacity of wireless MIMO systems,” IEEE ICC 2002, New York, April 2002.
w M. Kang, L. Yang, M. S. Alouini, G. Oien, “How Accurate are the Gaussian and Gamma Approximations to the Outage Capacity of MIMO Channels?”, 6th Baiona Workshop on Signal Processing in Communications, Baiona, Spain, September 8-10, 2003.
w C. Chuah, D. Tse, J. M. Kahn, R. A. Valenzuela, “Capacity Scaling in MIMO Wireless System Under Correlated Fading”, IEEE Trans. Inform. Theory, vol. 48, pp. 637-650, Mar. 2002.
w M. Kang, M. S. Alouini, “On the Capacity of MIMO Rician channels,” Proc. 40th Annual Allterton Conference on Communication, Control, and Computing (Allerton'2002), Monticello, IL, Oct. 2002, pp. 936-945.
w B. Vucetic, J. Yuan, Space-time coding. New York: John Wiley & Sons, 2003.w M. Dohler, H. Aghvami, “On the Approximation of MIMO Capacity,” IEEE Letter
Wireless Communications, July 2003, submitted.
4/19/2005 Yang 33
!!! Thank You !!!
w Any questions?
