- The subject introduces the probability, random process and the linear algebra that are required for the theoretical analysis of the communication systems.

Vector spaces. Four fundamental vector spaces of the matrix. Rank-Nullity theorem. Projection theorem.- Linear transformation matrix with different basis-Gram-Schmidt orthogonalization procedure.QR factorization. Eigen values and Eigen vectors. Diagonalization of the matrix.Schur’s lemma. HermitianMatrices- Unitary Matrices - Normal Matrices. Singular Value Decomposition.

Probability spaces. Random variables and random vectors. Distributions and densities-Conditional distributions and densities. Independent random variables. Transformation of random variables

Expectations. Indicator. Moment generating function. Characteristic function. Multiple random variable. Gaussian random vector. Co-variance matrix. Complex random variables. Sequence of random variable- Central limit theorem.

Strictly stationary random process. Wide sense stationary random process. Complex random process. Jointly strictly and wide sense stationary of two random processes. Correlation matrix obtained from random process .Ergodic process. Independent random process. Uncorrelated random process. Random process as the input and output of the system. Power spectral density.

White random process. Gaussian random process. Cyclo-stationary random process. Wide sense cyclo stationary random process. Sampling and reconstruction of random process. Band pass random process.

*1. R.B.Ash & C.Doleans-Dade, “Probability and Measure Theory (2/e)”, Elsevier, 2005*

*2. A.Papoulis, S.U.Pillai, “Probability, Random variables and Stochastic processes” 4**th **edition Tata- Mc Hill (4/e) ,2001*

*3. G.Strang, “Linear Algebra”, Thomson Brooks/Cole Cengage Hill (4/e), 2006*

*1. Stakgold, I., Green’s “Functions and Boundary value Problems (e)”, Wiley,1998*

*2. E.S.Gopi, “Mathematical summary for digital signal processing applications with Matlab”, Springer,2011.*

*3. E.Wong & B.Hajek, “Stochastic Processes in Engineering systems”, Springer, 1985.*

*4. R.B.Ash & W.A.Gardner, “Topics in stochastic processes”, Academic Press, 1975.*

Students are able to

CO1: solve the problems associated with Linear algebra

CO2: solve the problem associated with transformation of random variables

CO3: summarize the concepts associated with multiple random variables and to solve the problems associated with Multivariate Gaussian random vector

CO4: summarize the concepts associated with random process and to compute the power spectral density of the output of the system.

CO5: recognize the usage of random process in telecommunication engineering and to solve the corresponding problems.