Reading list for the Breadth Examinations – Engineering Mathematics

Ph.D. Qualifying Examination

 

Question 1: Engineering Mathematics

The student needs to answer any one from the following problems. The topics and recommended reading lists for the six problems are:

 

Problem No.

Topics

Reading list

1

Fourier analysis of continuous-time signals: Fourier series and Fourier transform representations of continuous time signals, properties

 

Sample problem

Signals and Systems, by Oppenheim, Willsky, and Nawab, 2nd edition, sections 3.0 – 3.5, AND 4.0 – 4.5.

2

The z-transform – region of convergence, properties, frequency response, poles and zeros.

 

Sample problem

Discrete Time Signal Processing, by Oppenheim, Schafer, and Buck, 2nd edition, chapter 3, and section 5.3.

 

3

Probability theory and random variables: concepts of probability and random variables, functions of one and two random variables, joint statistics.

 

Sample problem

Probability, Random Variables, and Random Signal Principles, 4th edition, by Peyton Z. Peebles, Jr., chapters 1 – 4, and sections 5.0 – 5.3

4

Laplace transforms: pole-zero plots, properties, transfer function, steady state response characteristics due to standard inputs like step, ramp, exponential, sinusoidal, and their composite inputs. Bode plots.

 

Signals and Systems, by Oppenheim, Willsky, and Nawab, 2nd edition, sections 9.1 – 9.7

 

 

 

5

Counting techniques, Recurrence Relations, Graphs.  Boolean Algebra. Order analysis (Big-Oh notation).

 

Sample problem

Discrete Mathematics and Its Applications, 3rd Ed. by Kenneth H. Rosen (ISBN 0-07-053965-0).  Chapters 5, 6, 7, and 9.

 

6

Linear algebra: Matrix inversion, solutions to fully determined systems of linear equations, method of elimination, determinants and Cramer’s rule.

 

Sample problem

Linear Algebra and Its Applications, 3rd Edition, by David C. Lay.  Sections 1.1-1.9, 2.1-2.3, 3.1-3.3 and  Elementary Differential Equations and Boundary Value Problems, by Edwards and Penney, 3rd edition, sections 4.1-4.2, 5.1

7

Differential equations: Solutions to first and second order linear ODEs with constant coefficients, solutions to linear systems of differential equations.

 

Sample problem

Elementary Differential Equations and Boundary Value Problems, by Edwards and Penney, 3rd edition, sections 1.1-1.5, 3.1-3.3, 4.1-4.2, 5.1-5.2, 7.1-7.6