### Detailed Program: Random Matrices and Communication Systems

1. Matrix Analysis “Review” (2~3 weeks)

2. Finite-Size Analysis of Random Matrices (~3 weeks)

3. Applications to Communications (2~3 weeks)

3.a) Capacity of MIMO Systems

3.b) Diversity-Multiplexing Tradeoff

4. Asymptotic Analysis of Random Matrices (3~4 weeks)

4.a) Moments Method

4.b) Stieltjes Transform Method

5. Back to Applications in Communications (1-2 weeks)

6. Free Probability and CDMA systems (1 week)

### Projects: Random Matrices and Communication Systems

#### List of papers

1. Joint Singular Value Distribution of Two Correlated Rectangular Gaussian Matrices and its Application

Related paper: Statistical Properties of EigenModes and Instantaneous Mutual Information in MIMO Time-Varying Rayleigh Channels

(project taken by Mine Alsan)

2. The Empirical Eigenvalue Distribution of a Gram Matrix: From Independence to Stationarity (project taken by Shirin Saeedi)

3. A New Approach for Capacity Analysis of Large Dimensional Multi-Antenna Channels (project taken by Rethnakaran Pulikkoonattu)

4. The Smallest Eigenvalue of a Large Dimensional Wishart Matrix

Related papers: Decoding by Linear Programming

Limit of the Smallest Eigenvalue of a Large Dimensional Sample Covariance Matrix

(project taken by Amin Karbasi)

5. On Certain Large Random Hermitian Jacobi Matrices with Applications to Wireless Communications (project taken by Mahdi Jafari Siavoshani)

6. Financial Applications of Random Matrix Theory: Old Laces and New Pieces

Related paper: Distribution of Eigenvalues for Some Sets of Random Matrices

(project taken by David Morton)

7. Bandlimited Field Reconstruction for Wireless Sensor Networks

Related paper: Reconstruction of Multidimensional Signals from Irregular Noisy Samples

(project taken by Patrick Denantes)

8. Random Vandermonde Matrices – Part I: Fundamental Results

Related paper: Random Vandermonde Matrices – Part II: Applications

(project taken by Ali Hormati)

9. Eigenvalues of Euclidean Random Matrices (project taken by Juraj Sarinay)

10. On the Concentration of Eigenvalues of Random Symmetric Matrices

(project taken by Mohammad Golbabaei)

#### Papers not chosen

Spectrum Estimation for Large Dimensional Covariance Matrices using Random Matrix Theory

Statistical Ensembles of Complex, Quaternion, and Real Matrices

Fluctuations of Eigenvalues and Second Order Poincare Inequalities

Spectral Measure of Large Random Hankel, Markov and Toeplitz Matrices

On Slow Fading Non-separable Correlation MIMO Systems

Untangling the SVD’s of Random Matrix Sample Paths

Capacity of the Gaussian Erasure Channel

Almost Sure Limit of the Smallest Eigenvalue of the Sample Correlation Matrix

### Lecture Notes: Random Matrices and Communication Systems

Lecture 1: overview and history

Lecture 2: finite-size analysis: GOE

Lecture 3: finite-size analysis: Real Wishart Ensemble

Lecture 4: finite-size analysis: computation of marginals

Lecture 5: finite-size analysis: COE, CUE and “physical” interpretation

Lecture 6: capacity of multi-antenna channels

Lecture 7: capacity of multi-antenna channels (cont’d)

Lecture 8: rate-diversity tradeoff in multi-antenna channels

Lecture 9: asymptotic analysis: first approach

Lecture 10: asymptotic analysis of (Toeplitz) deterministic matrices

Lecture 11: probablity “review”

Lecture 12: probablity “review” (cont’d)

Lecture 13: asymptotic analysis: moments

Lecture 14: asymptotic analysis: moments (cont’d)

Lecture 15: asymptotic analysis: Stieltjes transform

Lecture 16: asymptotic analysis: Stieltjes transform (cont’d)

Lecture 17: largest eigenvalue of Wigner’s matrices

Lecture 18: capacity scaling of multi-antenna channels and ad hoc networks

Lecture 19: positive definite matrices and matrix inequalities

Lecture 20: matrix inequalities and information theory

Lecture 21: Gaussian random matrices and free probability

Lecture 22: free probability: sums of random matrices and R-transform

Lecture 23: free probability: additivity of the R-transform

Lecture 24: free probability: products of random matrices and S-transform

### Bibliography: Random Matrices and Communication Systems

Note that there is some overlap between the lists below

(which is normal, otherwise the present class would not exist 🙂

and also that the lists are by far incomplete!

Finite-size analysis of random matrices

Asymptotic analysis of random matrices