Linear Systems and Signal Processing

Notes will be updated here periodically. 

This page is last updated on 29 Oct 2026.

Pre-requisites: 

MA-I (MA001), MA-II (MA002), Numerical Linear Algebra (MA307), MATLAB.

Course Syllabus: 

[Unit-1] Introduction and Mathematical Modeling of Linear Systems: Definition of system, control system; Classifications of system; Mathematical modeling of physical systems -  electrical, mechanical and electronic systems; General differential equations representation of a physical system; Initial and boundary value problems; Existence, uniqueness and Well-posedness of the model equation; Solution of differential equations, Dependence of initial and boundary conditions on solution; Recall Laplace and inverse Laplace transformation; Laplace transformation based method to solve the linear Differential equation.  

[Unit-2] Block Diagram Representation to Correlate Input and Output of the System: Transfer function and its derivation from linear differential equations; transfer functions of electrical, mechanical and electronic systems; system order and type; poles and zeros; block diagram representation of systems; series, parallel and feedback interconnections; signal flow graphs and Mason's gain formula; impulse response, convolution, and responses to impulse, step and ramp inputs; transient and steady-state responses; time-domain analysis of first- and second-order systems (rise time, peak time, overshoot, settling time, natural frequency, damping coefficient and damping ratio); effect of poles and zeros on system response; limitations of linear models and effect of common nonlinearities; Stability, Routh Hurwitz criterion; Steady-state error, sensitivity analysis.

[Unit-3] State Space Analysis: State variables; state space; state space representation of the physical system from transfer function and from differential equation; Convert transfer function to state space model; Solution of state space model; Similarity transformation; canonical realizations of transfer functions (phase-variable, cascade and parallel forms); Various matrix representation and their correlation with canonical realizations of the system; Steady state error using state space; Stability using state space; Stability in the sense of Lyapunov for linear systems, Lyapunov function and equation; Controllability and Kalman rank criterion; Observability and rank criterion; State estimation. 

[Unit-4] Frequency Domain Analysis and Signal Processing: Signals, type of signals; Fourier analysis – Fourier series representation, Sine and Cosine forms; Fourier transform – continuous time; Fast Fourier transform; Frequency response, convolution, filters.  

Course Outcomes: 

The present course will provide a mathematical and engineering perspective of the linear system theory and signal processing. By attending the course, students will be able to understand  

1. Mathematical modeling of the physical system using differential equation. 

2. Block diagram representation of the system and its transfer function.

3. State space model representation of the system and its stability. 

4. Effect of various signals on the system performance.

Recommended Books:

Text Books:

1. I. J. Nagrath, M. Gopal, Control System Engineering, New Age International, 5th Edition, (2011).

2. Norman S. Nise, Control System Engineering, John Wiley \& Sons, 6th Edition, (2011). 

3. H. L. Trentelman, A. A. Stoorvogel, M. Hautus,  Control Theory for Linear Systems, Springer Nature, (2001).

4. B. P. Lathi, Signal Processing and Linear Systems, Berkeley Cambridge, (1998).

5. J. G. Proakis, D. G. Manolakis, Digital Signal Processing, Pearson Prentice Hall, (2025)

Reference Books:

1. M. Gopal, Control Systems: Principles and Design, McGraw-Hill, 4th Edition, (2012). 

2. Katsuhiko Ogata, Modern Control Engineering, Prentice Hall, 5th Edition, (2010).

3. Jerzy Zabczyk, Mathematical Control Theory An Introduction, Modern Birkhauser Classics, (2008).

4. A. V. Oppenheim, A. S. Willsky, S. H. Nawab, Signals and Systems, (1997).

Evaluation Scheme:

Students will be evaluated on a scale of 160 = {MTE, ETE, component-1, component-2} where

MTE - Open Book Mid Term Exam - 30 Marks for 1.5 Hr duration

ETE - Open Book End Term Exam - 50 Marks for 2 Hr duration

There are 4 compnents. A student can choose any two (or best will be counted, in case more than 2 components opted). 

(Componet) Project - A two page report + 15 min presentation - 40 Marks 

(Component) Practical - MATLAB Coding of 40 Marks

(Component) Self-Learning - (a) Discrete System Analysis or (b) Wavelet Transformation - two page creative writing - 40 Marks

(Component) Quiz - 40 Marks    

Note: A minimum of M = min{0.5*Class Average, 56} out of 160 marks is required to pass the course. The grading will be relative-scaled.

Exam/Test Schedule: 

1. Mid Term Exam will be held during Mid Term Exams (1-9 Oct 2026). 

2. Self-Learning written exam (open book, creative writing) will be held on 23 Oct 2026. 

3. Project presentations will be held in 10-20 Nov 2026. No further date extension request will be accepted. 

4. MATLAB Exam will held on 13 Nov 2026. 

5. Quiz exam will held on 26 Nov 2026. 

6. End Term Exam will be held during End Term Exams (9-18 Dec 2026).