Ergodic and Outage Capacity of Narrowband MIMO Gaussian Channels

Transcription

1 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, 005

2 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 4/19/005 Yang

3 Introduction of MIMO MIMO is multi-input and multi-output system MIMO systems provide significant capacity gains over conventional single antenna array based solutions. Hot research topic within academia and industry. 4/19/005 Yang 3

4 MIMO system model A single user multi-input multi-output system with t Tx antennas and r Rx antennas h 11 h 1 h r1 h 1 x 1 h y 1 Space-time encoder x... h 1t h r h t... y Space-time decoder x t y r h rt 4/19/005 Yang 4

5 MIMO system model cont The receive signal is given by y = Hx+ n Where r y C : received vector r t H C : channelmatrix t x C : transmitedvector r n C : complex Gaussian noise with zero mean and covariance matrix σ I r 4/19/005 Yang 5

6 MIMO system model - cont The total power of the complex transmit signal x is constrained to P regardless of the number of transmit antennas ε[ xx] = tr([ ε xx ]) = P Assuming the realization of H is known at the receiver, but not always at the transmitter 4/19/005 Yang 6

7 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 (nonergodic). 4/19/005 Yang 7

8 Capacity for channels with fixed coefficients H is deterministic Decorrelating H by Singular Value Decomposition (SVD) H = UDV U and V are rxr and txt unitary matrices respectively. D is a rxr diagonal matrix with nonnegative square roots of the eigenvalues of HH, denoted by λ i, i= 1,,, r 4/19/005 Yang 8

9 Capacity for channels with fixed coefficients cont Let y = U y, x = V xn, = Un Then y = Hx+ n y = Dx+ n Then Where r 0 y i λixi + ni 1 i r0 = ni r0 + 1 i r is the rank of H 4/19/005 Yang 9

10 Capacity for channels with fixed coefficients cont The overall channel capacity C is the sum of the subchannels capacities r0 Pri C = ln1 + nats/ s/ Hz i= 1 σ Where P is the received signal power at the i th ri subchannel. 4/19/005 Yang 10

11 Equal Transmit Power Allocation The power allocated to subchannel i is given by and is given by P = P/, t i = 1,,..., t Pri i λi P Pri =, i = 1,,..., r t Thus the channel capacity can be written as r0 r0 Pri λip C = ln1+ ln 1 nats/ s/ Hz = + i= 1 σ i= 1 tσ 4/19/005 Yang 11

12 Adaptive Transmit Power Allocation For the case when the CSI is known at the transmitter, the capacity can be increased by water-filling method σ Pi = µ, i = 1,,..., r λi where a + denotes max( a,0) and µ is chosen to meet the r0 power constraint so that P = P The received signal power at the i th subchannel is ri i= 1 + i ( ) i P = λµ σ + 0 4/19/005 Yang 1

13 4/19/005 Yang 13 Adaptive Transmit Power Allocation cont Thus the channel capacity is ( ) ln 1 ln 1 1 ln / / r i i r i i r i i C nats s Hz λµ σ σ λµ σ λµ σ + = + = + = = + = + =

14 Capacity of MIMO fast and block Rayleigh fading channels The mean (ergodic) capacity of a random MIMO channel with power constraint tr ε xx = P can be expressed as ε H C = ε ( ) where denotes the expectation over all channel realizations and I ( xy ; ) 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. { ( )} max I xy ; H p ( x): tr ( ε[ xx ]) = P 4/19/005 Yang 14

15 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 C = max I xyh ;, p( x): tr( ε [ xx ]) = P ( ) Definition: A Gaussian random vector x is circularly symmetric, if for x = Re( x) Im ( x) T ( Q) ( Q) ( Q) Re( Q) 1Re Im cov ( x) =, where Q= cov Im ( x) 4/19/005 Yang 15

16 Capacity of MIMO fast and block Rayleigh fading channels cont Given covariance matrix Q, circularly symmetric Gaussian random vector is entropy maximizer. H ˆ ( x) = ln det ( πeq ) The covariance matrix of y with realization of H=H is ( )( ) ε[ yy ] = ε Hx n x H n + + = HQH + σ Ir The mutual information is ( xyh ;, ) = ( xh ; ) + ( xy H ; ) = ( xy H ; ) = εh ( xy H ; = ) = ε ˆ ( ) ˆ H H y H H ε H H( y x, H H) = = = ε Hˆ ( y H = H) Hˆ ( n) I I I I H H 1 = ε lndet( I + HQH ) H r 4/19/005 Yang σ 16

17 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 P P C = εh lndet( Ir + HH ) = ε lndet( ) H It + HH σ t σ t Let n = max( r, t) and m = min( r, t).the random matrix HH for r < t, or HHfor r t has the Wishart distribution with parameters m, n and the unordered eigenvalue have the joint density m 1 n m λ ( ) i p( λ1,..., λm) = λi e λi λj mk! mn, i i< j Where K is a normalizing factor 4/19/005 Yang 17

18 Capacity of MIMO fast and block Rayleigh fading channels cont Anyone of the unordered eigenvalues has the distribution m 1 1 k! n m n m λ p( λ) = Lk ( λ) λ e m k+ n m! ( ) k= 0 ( ) n m where L is the associated Laguerrepolynomial of k λ order k, and it is given by L n m k ( λ) = ( 1) k l= 0 l ( k + n m)! ( )( + ) l λ k l! n m l!! l 4/19/005 Yang 18

19 Capacity of MIMO fast and block Rayleigh fading channels cont Then the mean channel capacity is given by P C = ελ lndet Im + diag ( λ1, λ,..., λm) σ t m P = ελ ln1+ λ i i= 1 σ t P = mελ ln1+ λ σ t m 1 P k! n m n m λ = ln1+ λ L k ( λ) λ e dλ σ t k= 0 k n m! 0 ( + ) 4/19/005 Yang 19

20 Capacity of MIMO fast and block Rayleigh fading channels cont The number of receive antenna is 1 Channel Capacity (nats/s/hz) The value of the capacity for r = 1 vs. Number of Tx Antennas(t) SNR = 35dB SNR = 30dB SNR = 5dB SNR = 0dB SNR = 15dB SNR = 10dB SNR = 5dB SNR = 0dB The asymptotic value is P limc = ln1 + nats/ s/ Hz t σ Number of Tx Antennas (t) 4/19/005 Yang 0

21 Capacity of MIMO fast and block Rayleigh fading channels cont The number of transmit antenna is 1 Channel Capacity (nats/s/hz) The value of the capacity for t = 1 vs. Number of Rx Antennas(r) SNR = 35dB SNR = 30dB SNR = 5dB SNR = 0dB SNR = 15dB SNR = 10dB SNR = 5dB SNR = 0dB The asymptotic value is rp limc = ln1 + nats/ s/ Hz t σ Number of Rx Antennas (r) 4/19/005 Yang 1

22 Capacity of MIMO fast and block Rayleigh fading channels cont The number of receive antenna equals the number of transmit antenna Channel Capacity (nats/s/hz) SNR = 0dB SNR = 5dB SNR = 10dB SNR = 15dB SNR = 0dB SNR = 5dB SNR = 30dB SNR = 35dB The value of the capacity for t = r vs. Number of Rx Antennas(r) The approximate asymptotic value is: P lim C = r ln 1 nats/ s/ Hz t= r σ ( ) Number of Rx Antennas (r) 4/19/005 Yang

23 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 ( ( ) ) P = inf p lndet I + HQH < C out Q:Q 0 r outage tr(q) P 4/19/005 Yang 3

24 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 C ins is needed) for equal power allocation case. Recall the instantaneous channel capacity is C ins m P ln 1 i i 1 σ t λ = + = Recall the mean channel capacity is m 1 P k! n m n m λ ε{ C} = ln1+ λ L k ( λ) λ e dλ σ t k= 0 k+ n m! 0 ( ) 4/19/005 Yang 4

25 Capacity of MIMO slow Rayleigh fading channels cont Smith derive the exact expression of variance of C ins as Pλ Var( Cins ) = m ln 1 p( λ) dλ 0 + σ t m i= 1 j= 1 0 m ( i 1)!( j 1)! ( i 1 n m)!( j 1 n m)! + + n m λ n m n m Pλ λ e Li 1 ( λ) Lj 1 ( λ)ln1+ dλ σ t 4/19/005 Yang 5

26 Capacity of MIMO slow Rayleigh fading channels cont Using Gaussian approximation ε( Cins) C outage Pout = pc ( ins < Coutage) = Q Var( Cins ) where Q-function is tail integral of a unit-gaussian pdfand it is defined as z 1 Qx ( ) = e dz π x 4/19/005 Yang 6

27 Capacity of MIMO slow Rayleigh fading channels cont The probability of outage capacity curve of MIMO channel with Rx= and SNR=15dB for various number of Txantennas 1 The outage capacity of MIMO channel with Rx = and SNR = 15dB and various number of Tx P out = p(c=<r th ) Tx = Tx = 4 Tx = Tx = /19/005 Yang Rate Threshold in (nats/s/hz) 7

28 Capacity of MIMO slow Rayleigh fading channels cont The probability of outage capacity curve of MIMO channel with Tx= and SNR=15dB for various number of Rx antennas 1 The outage capacity of MIMO channel with Tx = and SNR = 15dB and various number of Rx P out = p(c=<r th ) Rx = 1 Rx = Rx = Rx = /19/005 Yang Rate Threshold in (nats/s/hz) 8

29 Capacity of MIMO slow Rayleigh fading channels cont The probability of outage capacity curve of MIMO channel with Tx=Rx=4 for various SNR 1 The outage capacity of MIMO channel with Tx = Rx = 4 for various SNR dB 15dB 5dB P out = p(c=<r th ) dB 10dB 0dB 30dB /19/005 Yang Rate Threshold in (nats/s/hz) 9

30 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 4/19/005 Yang 30

31 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 , Jun 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 , Mar I. E. Telatar, Capacity of multi-antenna Gaussian channels, Technical Report # BL TM, AT \& T Bell Laboratories, G. J. Foschini. Layered space-time architecture for wireless communication in a fading environment when using multiple antennas, Bell Labs Technical Journal, 1():41-59, Autumn R. G. Gallager, Information Theory and Reliable Communication. New York: John Wiley & Sons, M. Dohler, H. Aghvami, A Closed Form Expression of MIMO capacity over Ergodic Narrowband Channels, IEEE Comm. Letter, vol. 8, Issue: 6, pp , June 004. H. Shin, J. H. Lee, Closed-form Formulas for Ergodic Capacity of MIMO Rayleigh Fading Channels, IEEE ICC 003, May 003, pp /19/005 Yang 31

32 Reference cont P. J. Smith, M. Shafi, On a Gaussian approximation to the capacity of wireless MIMO systems, IEEE ICC 00, New York, April 00. 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 BaionaWorkshop on Signal Processing in Communications, Baiona, Spain, September 8-10, 003. 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 , Mar. 00. M. Kang, M. S. Alouini, On the Capacity of MIMO Rician channels, Proc. 40th Annual AlltertonConference on Communication, Control, and Computing (Allerton'00), Monticello, IL, Oct. 00, pp B. Vucetic, J. Yuan, Space-time coding. New York: John Wiley & Sons, 003. M. Dohler, H. Aghvami, On the Approximation of MIMO Capacity, IEEE Letter Wireless Communications, July 003, submitted. 4/19/005 Yang 3

33 !!! Thank You!!! Any questions? 4/19/005 Yang 33