Capacity of MIMO Channels: Asymptotic Evaluation Under Correlated Fading

1. Capacity of MIMO Channels: Asymptotic Evaluation Under Correlated Fading Presented by: Zhou Yuan University of Houston 10/22/2009

2. Outline • Introduction • Signal model and assumptions • Asymptotic capacity per antenna in correlated fading model • Analysis of results • Conclusion

3. Introduction • MIMO technology • Previous work • Consider a random model for the channel (such as Rayleigh-fading channel) • Channel capacity can increase as much as linearly with the minimum number of antennas either at transmitter or receiver • Cannot generally obtain a closed form analytical solution

4. Introduction • Analyze asymptotic capacity • Number of transmit and receive antennas are driven to infinity • Ratio between transmit and receive antennas is constant • Free Probability Theory • Useful to describe the distribution of eigenvalues of random matrices when their dimensions increase without bound

5. Signal Model • Signal model: • Channel capacity • This expression does not have a closed analytical form • Need to consider asymptotic expression

6. Signal Model • Asymptotic expression of capacity • Using random matrix theory • As the dimensions of the random matrix are driven to infinity, the empirical distribution function of the eigenvectors of some random matrix models tends to a nonrandom quantity c=M/N

7. Capacity Under Correlated Fading Channel • In nonasymptotic scenario, only based on numerical evaluation or they can only describe the behavior of the capacity in terms of bounds • In asymptotic scenario, be more representative in situations where the number of transmit antennas is of the same order of magnitude as the number of receive ones • Focus on the case where fading correlation arises at the receiver only

8. Correlation at Receive Side • Model the channel matrix as: • C: N*NToeplitz matrix which contains the fading correlation between two receive elements • U: N*M matrix with i.i.d. circular symmetric complex Gaussian entries with zero mean and unit variance • The expression of the asymptotic capacity becomes: • where m(z) is the Stieltjes transform of F(x)

9. Correlation at Receive Side • Stieltjes transform m(z) is derived using results from Free Probability Theory • Describe the eigenvalue distribution function of a product of infinite-dimensional matrices as a function of the eigenvalue distribution of each matrix • When the dimension of the problem increase without bound, the two matrices become asymptotically free • In this case, we can obtain the asymptotic eigenvalue distribution function of the product of the two matrices from the asymptotic eigenvalue distribution of each one • S-transform, which can be obtained from the Stieltjes transform where is the formal inverse of and

10. Correlation at Receive Side • Capacity per antenna: where

11. Correlation at Receive Side Fig 1: Comparative representation of the shape of the proposed and the exponential correlation models. Stems: proposed model. Solid line: exponential model that generates the same eigenvalue spread of the covariance matrix (exponential correlation parameter denoted by rho). Dotted line: exponential model that generates the same correlation between consecutive elements (denoted by lambda).

12. Correlation at Transmit Side • Model the channel matrix as: • Capacity per antenna:

13. Preliminary Analysis Fig 2:Asymptotic capacity per receive antenna.

14. Analysis of Results Fig 3: Asymptotic capacity per receive antenna as a function of Eb/N0 for different values of the correlation parameter u.

15. Analysis of Results Fig 4: Loss in capacity per receive antenna in presence of fading correlation.

16. Analysis of Results Fig 5: Asymptotic capacity per antenna for Eb/N0=10dB.

17. Simulation Results Fig 6: Convergence of the mean value of the capacity per antenna toward its asymptotic value for c=0.5.

18. Conclusion • Presented a closed form of the asymptotic uniform power allocation capacity of MIMO systems • Assume correlated fading at one side • The expression is obtained modeling the asymptotic eigenvalue distribution of the fading correlation matrix as a titled semicircular law depending on a parameter that describes the degree of fading correlation between antennas • Proposed model is close to an exponentially decaying function • Interesting properties: • Fading correlation does not influence the rate of growth with Eb/N0 • Number of transmit antennas increases without bound and number of receive antennas is held constant , the asymptotic spatial efficiency saturates to a constant value that depends on the fading correlation parameter only

19. Questions?