226 Gray Hall

michaelr at american {dot} edu

Office hours:

- Tuesdays: 10am-noon,
- Wednesdays: 10am-noon, 1-2pm,
- Thursdays: 10am-noon, 4-5pm, or
- by appointment (please contact me 24 hours in advance to make arrangements)

Feel free to contact me with

Homework assignments

Course schedule

Information about exams

Some useful links

Course policies

The overall objectives of this course are to

- Introduce students to transform-based methods for
*analysis*and*synthesis*of functions on specific geometric domains - Teach students to compute transforms of elementary functions
- Introduce students to the practical application of these concepts in modern signal processing systems

*Principles of Applied Mathematics: Transformation and Approximation*by James Keener*Signals and Systems*by Alan Oppenheim, Alan Willsky, and S. Hamid Nawab

The programming projects will consist of some preliminary and background theory questions along with a set of guided programming exercises. **Instructions and submission of programming projects will be elecronic via the course BlackBoard page! Check there often for updates.**

Homework 1: due January 28

Homework 2: due February 6

Project 1: due February 14 (start early!)

Homework 3: due March 18

Homework 4: due March 27

Project 2: due April 8 (start early!)

Homework 5: due April 15

Homework 6: due April 24

Project 3: due April 25 (start early!)

- Prove orthogonality of trigonometric functions (over appropriate intervals) using complex exponentials
- Derive the Fourier series decomposition of an arbitrary function
- Decompose any function as a sum of an even function and an odd function
- Construct Fourier series in general Hilbert spaces using Gram-Schmidt orthonormalization (for instance, spherical harmonic decomposition)

January 14: Vector spaces, norms, and inner product spaces (Keener 1.1)

January 16: Spectrum of a matrix (Keener 1.2-1.3)

January 21: Least squares solutions (Keener 1.4-1.5)

January 23: Complete vector spaces (Keener 2.1)

January 28: General Fourier series (Keener 2.2)

January 30: Specific Fourier series (Keener 2.2)

February 4: Review for Exam 1

February 6: Exam 1

- Evaluate Fourier transforms of elementary functions
- Prove convergence of the Fourier transform under appropriate decay constraints
- Explain the construction of the fast Fourier transform for power-of-two length sequences
- Explain the Shannon-Nyquist sampling theorem

February 11: Complex calculus (Keener 6.1-6.2)

February 13: Contour integration (Keener 6.4)

February 18: The Fourier transform (Keener 7.1-7.2)

February 20: Computing Fourier transforms

February 25: Fourier transforms solve differential equations

February 27: Convergence of the Fourier transform (Keener 7.2)

March 4: Dr. Robinson on travel

March 6: Dr. Robinson on travel

March 18: Inverting the Fourier transform (Keener 7.2.1)

March 20: The Discrete (Fast) Fourier transform

March 25: Review for Exam 2

March 27: Exam 2

- Compute Laplace transforms of elementary functions
- Use Fourier and Laplace transforms to solve ordinary differential equations on unbounded domains
- Explain how the poles and zeros of Laplace transforms of operators determine stability
- Describe how to fit a specialized transform to a particular domain by diagonalizing the Laplace operator

April 1: The Laplace transform (Keener 7.3, Oppenheim 9.1)

April 3: Computing Laplace transforms (Oppenheim 9.6)

April 8: Convergence of the Laplace transform (Oppenheim 9.2)

April 10: Inverting the Laplace transform (Oppenheim 9.3)

April 15: Pole-zero plots (Oppenheim 9.4)

April 17: The

April 22: Transforms in context: group representations

April 24: Review for the final exam

**Final exam: May 1, 5:30-8:00pm**

20% Homeworks

20% Projects

20% Exam 1

20% Exam 2

20% Final exam