Brief Syllabus
The course covers several analytical methods that are used by electrical and computer engineers in their design and implementation studies. Various mathematical concepts are used in the field of electrical and computer engineering such as, electrical circuit analysis, signal processing, communications, computer networks, information processing, and control systems. Specifically, major topics - Laplace Transform, Linear Algebra, Complex Analysis, Probability, Applied Statistics, and Random Processes will be studied in context of their engineering applications. The course also provides ample opportunity to identify the inter relations between the topics proposed in a context of an engineering problem and motivates the students to formulate the problem.
Objectives/Justification of Proposal
· Need for this module
The area of Electrical and Computer Engineering demands a comprehensive knowledge of analytical techniques. This core course provides an excellent opportunity to acquire the required mathematical and analytical skills in the field of electrical and computer engineering such as, electrical circuit analysis, signals and systems, communications, computer networks, information processing, and control systems. Specifically, the topics include the following. Laplace transforms, linear algebra, complex analysis, probability, and random processes, with an emphasis on the applicability of these concepts to various electrical and computer engineering topics. It provides the required mathematical foundation to courses like circuits, control systems, and signals.
· Methodology
Students will be introduced to the topics through rigorous analytical procedures and applied numerical examples. Emphases will be made on the applications of these mathematical techniques to problems closely related to electrical and computer engineering topics. MATLAB software will be used to demonstrate the concepts wherever required and MATLAB programming assignments for tutorial sessions will be provided.
· Outcomes
(a) understand the basic theory related to the proposed topics in Laplace transform, linear algebra, complex analysis, probability and random processes;
(b) identify the relevance of the topics proposed to a given engineering problem context and must be able to formulate the problem formally; and
(c) apply these concepts to solve relevant engineering problems.
Detailed Syllabus
1. Analytical Methods in Circuits and Systems: Laplace Transform (5 Hours)
Laplace Transform of Integrable Functions. Inverse Laplace Transform of Functions with (i) Distinct Linear Factors (ii) Repeated Linear Factors and (iii) Quadratic Factors. Laplace Transform Rules. Transform of Periodic Functions. Initial and Final-Value Theorems. Convolution. Initial-Value Problems, application of Laplace transforms to solve problems in ciruits and systems – step respone and impulse response, Response as related to the s-plane roots.
2. Analytical Methods in Signals and Systems: Linear Algebra (7 Hours)
Quadratic forms, symmetric, skew-symmetric, orthogonal matrices, Hermitian, skew-Hermitian, System representation in canonical forms, Unitary matrices;.Linear Programming - Introduction to linear programming, simplex method, application to scheduling problems.
3. Analytical Methods in Systems and Control: Complex Analysis (5 Hours)
Use of complex functions, Analyticity, Complex Differentiation, Cauchy-Riemann Equations, introduction to Conformal Mapping, Singularities and Zeros, Laurent Series, Application to system stability analysis.
4. Analytical Methods in Computer Networks and Systems: Probability and Random Variables (8 Hours)
Set Definitions. Probability Laws. Conditional probability, independence, Bayes theorem. Discrete and Continuous random variables. probability functions, Expectation and moments. Multiple random variables, Functions of random variables, Binomial distribution. Poisson distribution. Normal distribution and applications, Standard normal random variable, Central Limit theorem. Normal approximation to the Binomial distribution, Exponential distribution.
5. Analytical Methods in Information processing: Applied Statistics (6 Hours)
Point Estimation: Sampling and Statistical Concepts. Parameter Estimate Bias. Efficiency and Consistency. Sample Mean. Sample Variance and its application. Maximum Likelihood Estimation. Interval Estimation: Chi-Squared and t-Distribution. Confidence Interval Concept. Sampling Distributions of Sample Mean and Sample Variance. Hypothesis Tests. Application to system reliability, MTBF, MTTF.
6. Analytical Methods in Signal Processing: Random Processes (5 Hours)
Concepts of Random Processes. Ensemble Mean, Autocorrelation, Time Averages. Cross-Correlation. Stationarity and Ergodicity. Power Spectral Density. Cross-Power Spectral Density. Application to random signal analysis, filters.