Presented by: Jia (Jasmine) Meng Advisor: Dr. Zhu Han

1. Capacity Scaling in MIMO Wireless System Under Correlated Fading--by Chen-Nee Chuah, David N. C. Tse, Joseph M. Kahn, and Reinaldo A. ValenzuelaIEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 3, MARCH 2002 Presented by: Jia (Jasmine) Meng Advisor: Dr. Zhu Han Wireless Network, Signal Processing & Security Lab University of Houston, USA Oct. 1, 2009

2. Outline • Introduction • Concepts • Existing results • System Model • Assumptions • Channel models • MEA Capacity and Mutual Information • Asymptotic Analysis • Simulations • Conclusions

3. Introduction -- Concepts • Channel Capacity (Bits/ Channel Use) is the tightest upper bound on the amount of information that can be reliably transmitted over a communications channel. • MIMO & Multiple-element arrays (MEAs) • Single-user, point-to-point links, use multiple (n) antennas at both transmitter and receiver side, (n,n)-MEA system • Increases the channel capacity significantly • Capacity Scaling Normalize channel capacity with respect to the number of transmitter/receiver pair (n)

4. Introduction-- Existing Results • If the fades between pairs of transmit-receive antennas are i.i.d., the average channel capacity of a MEAs system that uses n antennas paires is approximately n times higher than that of a single-antenna pair system for a fixed bandwidth and overall transmitted power. • Recap (1) i.i.d. channel assumption (2) Channel capacity grows linearly in the # of antenna pair n • HOWEVER i.i.d. does not always hold • This paper discusses under a more general case, the channel capacities under the correlated fading

5. is the signal transmitted by the i-th transmitter is the signal received by the i-th receiver is the noise received by the i-th receiver is the path gain from the j-th transmitter to the i-th receiver System Model • Linear and time-invariant channel use the following discrete-time equivalent model:

6. Transmitter Power Allocation Strategies • H is known only to the receiver but not the transmitter. Power is distributed equally over all transmitting antennas in this case. • H is known at both the transmitter and receiver, so that power allocation can be optimized to maximize the achievable rate over the channel.

7. Channel Model-- Assumptions • H is considered as quasi-static, and average total power and noise variancewon’t change during communication; • H changes when the receiver moves; • The associated capacity and mutual information and for each specific realization of H can be viewed as random variables; • We are interested in study the statistics of these random variables, specifically the averages of capacity and mutual information.

8. MEA Capacity and Mutual Information (I) • Capacity with water filling power allocation • MEA capacity with optimal power allocation is

9. MEA Capacity and Mutual Information (II) • Mutual information with equal-power allocation • MEA capacity with optimal power allocation is

10. Asymptotic Analysis--Independent Fading • Both the capacity and the mutual information and depends on H only through the empirical distribution (CDF) of the eigenvalues. • Conclusions: • At high SNR, it is well known that the water-filling and the constant power strategies yield almost the same performance • At low SNR, the water-filling strategy shows a significant performance gain over the constant-power strategy.

11. Asymptotic Analysis -- Correlated Fading (I) • Each of the are assumed to be complex, zero-mean, circular symmetric Gaussian random variables with variance . The are jointly Gaussian with the following covariance structure:

12. Asymptotic Analysis -- Correlated Fading (II)

13. Asymptotic Analysis • Analyze the capacity and mutual information@ high and low SNR • No analytical expression for C @ low SNR • @ high SNR, C I • @ low SNR, mutual information is I_highSNR+ capacity penalty at both sides

14. Simulation • Multipath, Rayleigh fading channel simulation • Verify the correlations @ both sides • Verify the feasibility of multiply the correlations • Show strength of the correlations @ different antenna distances • Show channel capacity @ different antenna distributions

15. SHOW: Channel Correlation

16. Average & Asymptotic Capacity Vs. n for in-line & broadside case Correlated Independent

17. Conclusions • The model of multiply the correlation @ both sides is feasible • Fading correlation can significantly reduce MEA system capacity and mutual information • Capacity and mutual information still scale linearly with n, while the rate of growth is different. • The rate of growth of is reduced by correlation over the entire range of SNRs, while that for is reducedbycorrelation at high SNR but isincreased at low SNR.

19. Empirical distribution function • In statistics, an empirical distribution function is a cumulative probability distribution function that concentrates probability 1/n at each of the n numbers in a sample. • Let X1, …, Xn be iid real random variables with the cdf F(x). The empirical distribution function F̂n(x) is a step function defined by • where I(A) is the indicator of event A. • For fixed x, I(Xi ≤ x) is a Bernoulli random variable with parameter p = F(x), hence nF̂n(x) is a binomial random variable with mean nF(x) and variance nF(x)(1 − F(x)). BACK