Mini Seminar on

Massive MIMO and Random Matrix

Department of Electronics & Communication Engineering

National Institute of Technology, Rourkela

Varun Kumar (514EC1005)

Under the supervision of Prof . Sarat Kumar Patra

Massive MIMO and Random Matrix (RMT)

Massive MIMO: Massive MIMO makes a clean break with current practice through the use of a very large number of service antennas (e.g., hundreds or thousands) that are operated fully coherently and adaptively. Extra antennas help by focusing the transmission and reception of signal energy into ever-smaller regions of space.

Random Matrix Theory:A wide range of existing mathematical results that are relevant to the analysis of the statistics of random matrices arising in wireless communications. • Complex Gaussian random variables are always circularly symmetric, i.e., with

uncorrelated real and imaginary parts, and complex Gaussian vectors are always proper complex.

Most Important Class of Random Matrices:

• Gaussian• Wigner• Wishart• Haar matricesWe also collect a number of results that hold for arbitrary matrix size.

Wireless Channel Model:

Where ρ is signal-to-noise ratio, is the channel gain matrix in uplink scenario, is the transmitted symbol vector of K user and is the noise vector. is diagonal matrix

is the lognormal distributed random variable. is the kth user location form base station whereas is the hole distance and n is the path loss exponent. All user in a given cell are distributed like Poisson point distributed.

𝑟h

H is fast faded channel matrix. The primary assumption to make mathematical modal are as follows

Capacity Formulation:

Shannon Channel Capacity:

Gamma Distribution

Density function

Extension of Gamma distribution:• Chi-Squared Distribution (d= Degree of freedom)• Exponential Distribution

Role of the Singular Values:

With transmitted signal-to-noise ratio (SNR)

Some important property of random vector and random matrices

• Let andbe mutually independent vectors whose elements are i.i.d zero mean random variables (RVs) with and i=1,2…n Then from the large numbers we have

and as n From the Lindeberg-Levy central limit theorem, we have CN(0, ) as n denotes the convergence of distribution

• Gaussian Matrix:A standard real/complex Gaussian m × n matrix H has i.i.d. real/complex zero-mean Gaussian entries with identical variance . The p.d.f. of a complex Gaussian matrix with i.i.d. zero-mean Gaussian entries with variance is

• Wigner MatricesLet W be an n × n matrix whose (diagonal and upper-triangle) entries are i.i.d. zero-mean Gaussian with unit variance. Then, its p.d.f. is

while the joint p.d.f. of its ordered eigenvalues λ1 ≥ ... ≥ λn is

Wishart Matrices:The m × m random matrix A = HH† is a (central) real/complex Wishart matrix with n degrees of freedom and covariance matrix Σ, (A Wm(n, ∼ Σ)), if the columns of the m × n matrix H are zero-mean independent real/complex Gaussian vectors with covariance matrix Σ. The p.d.f. of a complex Wishart matrix A Wm(n, ∼ Σ) for n ≥ m is

Some Useful Property of Whishart Matrix:

For a central Wishart matrix with n>m,

Other Application:• Single-user matched filter • De-correlator • MMSE • Optimum • Iterative nonlinear

Conclusion:• Random matrix results have been used to characterize the fundamental limits of the

various channels that arise in wireless communications.• Random matrix theory is very useful to converge the very large matrix size. In

stead of solving point to point multiplication addition subtraction and large number of channel realization it easily converge to expected value.

References:1. A. M. Tulino and S. Verdú, “Random matrix theory and wireless communications,” Foundations Trends

Communication. Inf. Theory, vol. 1, no. 1, pp. 1–182, June 20042. H. Q. Ngo, E. G. Larsson, and T. L. Marzetta, “Energy and spectral efficiency of very large multiuser MIMO

systems,” IEEE Trans. Commun., vol. 61, no. 4, pp. 1436–1449, Apr. 2013.3. A Goldsmith, Wireless Communication, Cambridge university press, 20054. Andrew Gelman  , John B. Carlin , Hal S. Stern , David B. Dunson, Aki Vehtari, Donald B. Rubin ,`` Bayesian Data

Analysis’’ (Chapman & Hall/CRC Texts in Statistical Science)  3rd Edition 20135. M. Matthaiou, M. R. MacKay, P. J. Smith, and J. A. Nossek, “On the condition number distribution of complex

Wishart matrices,” IEEE Trans. Commun., vol. 58, no. 6, pp. 1705–1717, Jun. 20106. G. Stewart, Matrix Algorithms: Basic Decompositions. Philadelphia, PA: SIAM, 1998.