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

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 4/19/2005 Yang 2

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. 4/19/2005 Yang 3

MIMO system model A single user multi-input multi-output system with t Tx antennas and r Rx antennas h 11 h h 21 1 2 h r1 h y x 2 2 1 1 h h r2 2t Space-time Space-time y x . . 2 2 encoder decoder . . h . . 1t x y t r h rt 4/19/2005 Yang 4

MIMO system model – cont ’ � The receive signal is given by = + y H x n ∈ r y C : received vector Where × ∈ r t H C : channel matrix ∈ t x C transmited vector : ∈ r n C : complex Gaussian noise with zero mean σ 2 and covariance matrix I r 4/19/2005 Yang 5

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 = ε = † † [ x x ] tr ( [ xx ]) P � Assuming the realization of H is known at the receiver, but not always at the transmitter 4/19/2005 Yang 6

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). 4/19/2005 Yang 7

Capacity for channels with fixed coefficients � H is deterministic � Decorrelating H by Singular Value Decomposition (SVD) = † H UDV � U and V are r x r and t x t unitary matrices respectively. � D is a r x r diagonal matrix with nonnegative square roots of the eigenvalues of , denoted by † HH λ = i i , 1,2, , r  4/19/2005 Yang 8

Capacity for channels with fixed coefficients – cont ’ = = = � Let † † † y U y x , V x n , U n    Then = + ⇒ = + y H x n y Dx n     λ + ≤ ≤ x n 1 i r   =  y � Then  i i i 0 i + ≤ ≤ n r 1 i r   i 0 r Where is the rank of H 0 4/19/2005 Yang 9

Capacity for channels with fixed coefficients – cont ’ � The overall channel capacity C is the sum of the subchannels capacities r   P 0 ∑ = + C ln 1 ri nats s Hz / /   σ 2   = i 1 P Where is the received signal power at the i th ri subchannel. 4/19/2005 Yang 10

Equal Transmit Power Allocation � The power allocated to subchannel i is given by and is given by = = P P t i / , 1,2,..., t P ri i λ P = = P i , i 1,2,..., r ri t � Thus the channel capacity can be written as λ r r     P P 0 0 ∑ ∏ = + = + C ln 1 ri ln 1 i nats s Hz / /     σ σ 2 2    t  = = i 1 i 1 4/19/2005 Yang 11

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 2 = µ − = P , i 1,2,..., r   i 0 λ   i a + µ where denotes and is chosen to meet the max( , 0) a ∑ r power constraint so that = i P P 0 i = 1 � The received signal power at the i th subchannel is ( ) + = λ µ − σ 2 P ri i 4/19/2005 Yang 12

Adaptive Transmit Power Allocation – cont ’ � Thus the channel capacity is ( )   + λ µ − σ 2 r 0 ∑   i = + C ln 1   σ 2 = i 1     + λ µ r   0 ∑ = + − ln 1  i 1    σ 2       = i 1 +  λ µ  r   0 ∑ = ln i nats s Hz / /     σ 2     = i 1 4/19/2005 Yang 13

Capacity of MIMO fast and block Rayleigh fading channels � The mean (ergodic) capacity of a random MIMO channel ( ) ε   = with power constraint can be expressed as † tr xx P   { } ( ) = ε C max I x y ; H † ε = p ( ): x tr ( [ xx ]) P ε where denotes the expectation over all channel H ( ) I x y ; 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. 4/19/2005 Yang 14

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 x y H ; , † ε = p ( ): x tr ( [ xx ]) P � Definition: A Gaussian random vector x is circularly ( ) ( ) T =   symmetric, if for x Re x Im x    ( ) ( )  −  1 Re Q Im Q ( ) ( ) = = cov x , where Q cov x   ( ) ( )  Im Q Re Q 2   4/19/2005 Yang 15

Capacity of MIMO fast and block Rayleigh fading channels – cont ’ � Given covariance matrix Q, circularly symmetric Gaussian random vector is entropy maximizer. ( ) ˆ ( ) =  π  H x ln det eQ   � The covariance matrix of y with realization of H =H is ) ( ) (   ε = ε + + = + σ † † † † † 2 [ yy ] H x n x H n HQH I   r � The mutual information is ( ) ( ) ( ) ( ) ( ) = + = = ε  =  I x y H ; , I x H ; x y|H ; I x y|H ; I x y|H ; H   H ( ) ( ) ˆ ˆ     = ε = − ε = H y|H H H y|x H , H     H H ( ) ( )  ˆ  ˆ = ε = − H y|H H H n   H   1 = ε + † ln det( I H H Q )   H r σ 2   4/19/2005 Yang 16

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  = ε + = ε + † † C ln det( I HH ) ln det( I H H )     H r H t σ σ 2 2  t   t  = = � Let and . The random n max( , ) r t m m in( , ) r t < ≥ † r t † HH H H r t matrix for , or for has the Wishart distribution with parameters m, n and the unordered eigenvalue have the joint density 1 m ( ) ∏ ∏ 2 λ λ = λ − − λ λ − λ n m p ( ,..., ) e i 1 m i i j m K ! < i i j m n , Where K is a normalizing factor 4/19/2005 Yang 17

Capacity of MIMO fast and block Rayleigh fading channels – cont ’ � Anyone of the unordered eigenvalues has the distribution − 1 m 1 k ! ∑ ( ) ( ) 2   λ = − λ λ − − λ n m n m p L e   ( ) + − k m k n m ! = k 0 ( ) where is the associated Laguerre polynomial of − λ n m L k order k, and it is given by ( ) + − k n m ! k ∑ ( ) ( ) l − λ = − λ n m l L 1 ( ) ( ) k − − + k l ! n m l ! ! l = l 0 4/19/2005 Yang 18

Capacity of MIMO fast and block Rayleigh fading channels – cont ’ � Then the mean channel capacity is given by     P ( ) = ε + λ λ λ C ln det I diag , ,...,     λ m 1 2 m σ 2  t        m P ∑ = ε + λ ln 1     λ i σ 2  t    = i 1     P = ε + λ m ln 1     λ σ 2 t     ∞ −   P m 1 k ! ∑ ( ) 2 ∫   − − − λ = + λ λ λ λ n m n m ln 1 L e d     ( ) k σ + − 2  t  k n m ! = k 0 0 4/19/2005 Yang 19

Capacity of MIMO fast and block Rayleigh fading channels – cont ’ � The number of receive antenna is 1 The value of the capacity for r = 1 vs. Number of Tx Antennas(t) 9 SNR = 35dB 8 The asymptotic value is SNR = 30dB 7   P Channel Capacity (nats/s/Hz) 6 SNR = 25dB = + lim C ln 1 nats s Hz / /   σ 2   →∞ t 5 SNR = 20dB 4 SNR = 15dB 3 SNR = 10dB 2 SNR = 5dB 1 SNR = 0dB 0 0 2 4 6 8 10 12 14 16 18 20 Number of Tx Antennas (t) 4/19/2005 Yang 20

Capacity of MIMO fast and block Rayleigh fading channels – cont ’ � The number of transmit antenna is 1 The value of the capacity for t = 1 vs. Number of Rx Antennas(r) 12 SNR = 35dB The asymptotic value is SNR = 30dB 10   rP SNR = 25dB = + lim C ln 1 nats s Hz / /   σ 2   →∞ t SNR = 20dB Channel Capacity (nats/s/Hz) 8 SNR = 15dB 6 SNR = 10dB SNR = 5dB 4 SNR = 0dB 2 0 0 5 10 15 20 25 30 Number of Rx Antennas (r) 4/19/2005 Yang 21