Linear Algebra and Stochastic Processes 3-0-0-3 


  • 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.


Text Books

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” 4th edition Tata- Mc Hill (4/e) ,2001

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


Reference Books

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.