Ergodic and Outage Capacity of Narrowband MIMO Gaussian Channels - - PowerPoint PPT Presentation

Ergodic and Outage Capacity of Narrowband MIMO Gaussian Channels

Yang Wen Liang Department of Electrical and Computer Engineering The University of British Columbia April 19th, 2005

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Outline of Presentation

Introduction of MIMO MIMO system model Capacity for channels with fixed coefficients Capacity of MIMO fast and block Rayleigh fading channels Capacity of MIMO slow Rayleigh fading channels Summary

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Introduction of MIMO

MIMO is multi-input and multi-output system MIMO systems provide significant capacity gains

• ver conventional single antenna array based

solutions. Hot research topic within academia and industry.

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MIMO system model

A single user multi-input multi-output system with t Tx antennas 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

1 2

h

1t

h

2 2

h

r2

h

2t

h

rt . . . . . .

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MIMO system model – cont’

The receive signal is given by

Where

H y x n = +

2

: : : : C received vector H C channel matrix C transmited vector C complex Gaussian noise with zero mean and covariance matrix

r r t t r r

y x n I σ

×

∈ ∈ ∈ ∈

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MIMO system model - cont’

The total power of the complex transmit signal x is constrained to P regardless of the number of transmit antennas Assuming the realization of H is known at the receiver, but not always at the transmitter

† †

[ ] ( [ ]) x x tr xx P ε ε = =

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MIMO system model – cont’

• 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

H is deterministic Decorrelating H by Singular Value Decomposition (SVD)

U and V are rxr and txt unitary matrices respectively. D is a rxr diagonal matrix with nonnegative square roots of the eigenvalues of , denoted by

†

H UDV =

†

HH

, 1,2, ,

i i

r λ = 

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Capacity for channels with fixed coefficients – cont’

Let Then Then Where is the rank of H

1 1

i i i i i

x n i r y n r i r λ  + ≤ ≤ =  + ≤ ≤     

r

H y x n y Dx n = + ⇒ = +   

† † †

, , y U y x V x n U n = = =   

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Capacity for channels with fixed coefficients – cont’

The overall channel capacity C is the sum of the subchannels capacities Where is the received signal power at the ith subchannel.

2 1

ln 1 / /

r ri i

P C nats s Hz σ

=

  = +    

∑

ri

P

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Equal Transmit Power Allocation

The power allocated to subchannel i is given by and is given by Thus the channel capacity can be written as

/ , 1,2,...,

i

P P t i t = =

ri

P

, 1,2,...,

i ri

P P i r t λ = =

2 2 1 1

ln 1 ln 1 / /

r r ri i i i

P P C nats s Hz t λ σ σ

= =

    = + = +        

∑ ∏

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Adaptive Transmit Power Allocation

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 The received signal power at the ith subchannel is

2

, 1,2,...,

i i

P i r σ µ λ

+

  = − =    

µ

1 r i i P

P

=

=

∑

a+

max( , 0) a

( )

2 ri i

P λ µ σ

+

= −

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Adaptive Transmit Power Allocation – cont’

Thus the channel capacity is

( )

2 2 1 2 1 2 1

ln 1 ln 1 1 ln / /

r i i r i i r i i

C nats s Hz λ µ σ σ λ µ σ λ µ σ

+ = + = + =

  −   = +         = + −               =        

∑ ∑ ∑

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Capacity of MIMO fast and block Rayleigh fading channels

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 information between x and y. 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.

( )

†

xx tr P ε   =  

( )

{ }

†

( ): ( [ ])

max ;

x xx

x y

H p tr P

C I

ε

ε

=

=

H

ε

( )

; x y I

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Capacity of MIMO fast and block Rayleigh fading channels – cont’

By the assumption that realization of H is known at the receiver, the output of the channel is the pair (y, H). Then the capacity is equivalent to Definition: A Gaussian random vector x is circularly symmetric, if for

( )

†

( ): ( [ ])

max ; ,

x xx

x y H

p tr P

C I

ε =

=

( ) ( )

Re Im x x x

T

=     

( ) ( ) ( ) ( ) ( ) ( )

1 Re Im cov , cov Im Re 2 x x Q Q where Q Q Q −   = =     

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Capacity of MIMO fast and block Rayleigh fading channels – cont’

Given covariance matrix Q, circularly symmetric Gaussian random vector is entropy maximizer. The covariance matrix of y with realization of H=H is The mutual information is

( )

ˆ ( ) ln det x H eQ π =    

( )(

)

† † † † † 2

[ ] yy x n x n

r

H H HQH I ε ε σ   = + + = +   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

† 2

; , ; ; ; ; ˆ ˆ , ˆ ˆ 1 ln 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 H

r

I I I I H H H H H H H H I Q ε ε ε ε ε σ = + = = =         = = − =       = = −     = +    

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Capacity of MIMO fast and block Rayleigh fading channels – cont’

Telatar proved that it is optimal to use equal power allocation if no knowledge of CSI in the transmitter. Then Let and . The random matrix for , or for has the Wishart distribution with parameters m, n and the unordered eigenvalue have the joint density Where K is a normalizing factor

† † 2 2

ln det( ) ln det( )

H H

HH H H

r t

P P C I I t t ε ε σ σ     = + = +        

max( , ) n r t = m in( , ) m r t =

†

HH

†

H H

r t < r t ≥

( )

2 1 ,

1 ( ,..., ) !

i

m n m m i i j i i j m n

p e m K

λ

λ λ λ λ λ

− − <

= −

∏ ∏

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Capacity of MIMO fast and block Rayleigh fading channels – cont’

Anyone of the unordered eigenvalues has the distribution where is the associated Laguerre polynomial of

• rder k, and it is given by

( ) ( ) ( )

1 2

1 ! !

m n m n m k k

k p L e m k n m

λ

λ λ λ

− − − − =

  =   + −

∑

( )

n m k

L λ

−

( ) ( ) ( ) ( ) ( )

! 1 ! ! !

k l n m l k l

k n m L k l n m l l λ λ

− =

+ − = − − − +

∑

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Capacity of MIMO fast and block Rayleigh fading channels – cont’

Then the mean channel capacity is given by

( ) ( ) ( )

1 2 2 2 1 2 1 2 2

ln det , ,..., ln 1 ln 1 ! ln 1 !

m m m i i m n m n m k k

P C I diag t P t P m t P k L e d t k n m

λ λ λ λ

ε λ λ λ σ ε λ σ ε λ σ λ λ λ λ σ

= ∞ − − − − =

    = +             = +             = +             = +     + −  

∑ ∑ ∫

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Capacity of MIMO fast and block Rayleigh fading channels – cont’

The number of receive antenna is 1 The asymptotic value is

2

lim ln 1 / /

t

P C nats s Hz σ

→∞

  = +    

2 4 6 8 10 12 14 16 18 20 1 2 3 4 5 6 7 8 9 The value of the capacity for r = 1 vs. Number of Tx Antennas(t) Number of Tx Antennas (t) Channel Capacity (nats/s/Hz) SNR = 35dB SNR = 30dB SNR = 25dB SNR = 20dB SNR = 15dB SNR = 10dB SNR = 5dB SNR = 0dB

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Capacity of MIMO fast and block Rayleigh fading channels – cont’

The number of transmit antenna is 1 The asymptotic value is

2

lim ln 1 / /

t

rP C nats s Hz σ

→∞

  = +    

5 10 15 20 25 30 2 4 6 8 10 12 The value of the capacity for t = 1 vs. Number of Rx Antennas(r) Number of Rx Antennas (r) Channel Capacity (nats/s/Hz) SNR = 35dB SNR = 30dB SNR = 25dB SNR = 20dB SNR = 15dB SNR = 10dB SNR = 5dB SNR = 0dB

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Capacity of MIMO fast and block Rayleigh fading channels – cont’

The number of receive antenna equals the number of transmit antenna The approximate asymptotic value is:

( )

2

lim ln 1 / /

t r

P C r nats s Hz σ

= →∞

    = −        

2 4 6 8 10 12 10 20 30 40 50 60 70 80 90 The value of the capacity for t = r vs. Number of Rx Antennas(r) Number of Rx Antennas (r) Channel Capacity (nats/s/Hz) SNR = 0dB SNR = 5dB SNR = 10dB SNR = 15dB SNR = 20dB SNR = 25dB SNR = 30dB SNR = 35dB

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Capacity of MIMO slow Rayleigh fading channels

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. The maximum mutual information is in general not equal to the channel capacity because it is not always achievable. Another measure of channel capacity is the outage capacity associated with a outage probability

( )

( )

† Q:Q 0 (Q)

inf ln det HQH

• ut

r

• utage

tr P

P p I C

≥ ≤

= + <

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Capacity of MIMO slow Rayleigh fading channels

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. Recall the instantaneous channel capacity is Recall the mean channel capacity is

ins

C

2 1

ln 1

m ins i i

P C t λ σ

=

  = +    

∑

{ } ( ) ( )

1 2 2

! ln 1 !

m n m n m k k

P k C L e d t k n m

λ

ε λ λ λ λ σ

∞ − − − − =

    = +     + −  ∑

∫

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Capacity of MIMO slow Rayleigh fading channels – cont’

Smith derive the exact expression of variance of as

ins

C

2 2 1 1 2 1 1 2

( ) ln 1 ( ) ( 1)!( 1)! ( 1 )!( 1 )! ( ) ( )ln 1

ins m m i j n m n m n m i j

P Var C m p d t i j i n m j n m P e L L d t

λ

λ λ λ σ λ λ λ λ λ σ

∞ = = ∞ − − − − − −

    = + −           − − ×   − + − − + −       +        

∫ ∑∑ ∫

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Capacity of MIMO slow Rayleigh fading channels – cont’

Using Gaussian approximation where Q-function is tail integral of a unit-Gaussian pdf and it is defined as

( ) ( )

( )

ins

• utage
• ut

ins

• utage

ins

C C P p C C Q Var C ε   −   = < =    

2

2

1 ( ) 2

z x

Q x e dz π

∞ −

= ∫

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Capacity of MIMO slow Rayleigh fading channels – cont’

The probability of outage capacity curve of MIMO channel with Rx=2 and SNR=15dB for various number of Tx antennas

1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 The outage capacity of MIMO channel with Rx = 2 and SNR = 15dB and various number of Tx Pout = p(C=<Rth) Rate Threshold in (nats/s/Hz) Tx = 1 Tx = 4 Tx = 7 Tx = 14

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Capacity of MIMO slow Rayleigh fading channels – cont’

The probability of outage capacity curve of MIMO channel with Tx=2 and SNR=15dB for various number of Rx antennas

2 4 6 8 10 12 14 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 The outage capacity of MIMO channel with Tx = 2 and SNR = 15dB and various number of Rx Pout = p(C=<Rth) Rate Threshold in (nats/s/Hz) Rx = 1 Rx = 4 Rx = 7 Rx = 10

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Capacity of MIMO slow Rayleigh fading channels – cont’

The probability of outage capacity curve of MIMO channel with Tx=Rx=4 for various SNR

5 10 15 20 25 30 35 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 The outage capacity of MIMO channel with Tx = Rx = 4 for various SNR Pout = p(C=<Rth) Rate Threshold in (nats/s/Hz) 0dB 5dB 10dB 15dB 20dB 25dB 30dB

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Summary

The MIMO capacity for H with fixed coefficient is derived Ergodic and Outage capacity of MIMO Rayleigh channel were introduced with some examples MIMO configuration could provide significant capacity gains over conventional single antenna array based system

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Reference

• 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.

• 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.
• I. E. Telatar, Capacity of multi-antenna Gaussian channels, Technical Report #

BL0112170-950615-07TM, AT \& T Bell Laboratories, 1995.

• 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.

• R. G. Gallager, Information Theory and Reliable Communication. New York: John Wiley

& Sons, 1968.

• 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.

• H. Shin, J. H. Lee, “Closed-form Formulas for Ergodic Capacity of MIMO Rayleigh

Fading Channels,” IEEE ICC 2003, May 2003, pp. 2996-3000.

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Reference – cont’

• P. J. Smith, M. Shafi, “On a Gaussian approximation to the capacity of wireless MIMO

systems,” IEEE ICC 2002, New York, April 2002.

• 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.

• 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.

• 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.

• B. Vucetic, J. Yuan, Space-time coding. New York: John Wiley & Sons, 2003.
• M. Dohler, H. Aghvami, “On the Approximation of MIMO Capacity,” IEEE Letter

Wireless Communications, July 2003, submitted.

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!!! Thank You !!!

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