Transforms and its Applications

MA25C03

Transforms and its Applications

Course Objective:

• To provide a strong foundation in Fourier Series, Laplace, Fourier and Z- Transforms.

• To develop the ability to analyze and solve engineering problems in continuous and discrete time domains using appropriate transform techniques.

Laplace Transforms: Existence conditions, Properties of Laplace transform, Laplace transform of standard functions, derivatives and integrals, Unit step function and Dirac delta function, Laplace transform of periodic functions; Inverse Laplace transform: Partial fraction technique, Convolution theorem.

Application: Solution of second order ordinary differential equations using Laplace transform.

Activities: Compute the Laplace transform of time-domain functions, Inverse Laplace transform, Solution of ordinary differential equations using Laplace transform.

Z-Transform: Z-transform of standard functions, properties; Inverse Z – transform: Standard functions, Partial fraction technique, Convolution theorem.

Application: Solution of difference equation using Z – transform.

Activities: Compute the Z-transform of a discrete-time signal, Solution of linear constant-coefficient difference equations using Z-transform.

Fourier Series: Dirichlet’s conditions, General Fourier series, Convergence of Fourier series, Odd and even functions; Half range sine series, Half range cosine series, Root mean square value, Parseval’s identity.

Application: Solution of one-dimensional wave and heat equation.

Activities: Compute Fourier coefficients, Reconstruct signal using Fourier series (Partial sum), Plot convergence of Fourier series.

Fourier Transform: Complex Fourier transform, Properties, Relation between Fourier and Laplace transform, Fourier sine and cosine transforms, Parseval’s identity, Convolution theorem.

Application: Simple applications to solve partial differential equations using Fourier transform.

Activities: Compute the Fourier and inverse Fourier transform, Parseval’s theorem validation.

Weightage: Continuous Assessment: 40%, End Semester Examinations: 60%.

Assessment Methodology: Assignment (20%), Software activity (20%), Quiz (10%), Internal Examinations (50%).

References:

1. Kreyszig, G. E. (2018). Advanced engineering mathematics. John Wiley & Sons Ltd.

2. Grewal, B. S. (2021). Higher engineering mathematics. Khanna Publications.

3. Zill, D. G. (2022). Advanced engineering mathematics. Jones & Bartlett India Ltd.

4. Wylie, C. R., & Barrett, L. C. (2019). Advanced engineering mathematics. Tata McGraw Hill.

5. Duffy, D. G. (2017). Advanced engineering mathematics with MATLAB. CRC Press.

E-resources:

1. Stanford Engineering Everywhere | EE261 - The Fourier Transform and its Applications

2. https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-003- signals-and-systems-fall-2011/

3. https://www.coursera.org/learn/mathematics-engineers-fourier-laplace-z- transforms

4. Transforms and Applications Handbook | Alexander D. Poularikas, Artyom