Math 262 / CME 372
Applied Fourier Analysis
and
Elements of Modern Signal Processing

Winter 2021

Instructor
Emmanuel Candes
144 Sequoia Hall

Office Hours: Th 1:30-2:30
or by appointment

   

Lectures
Tuesday and Thursday
2:30-3:50 p.m.
Live lectures on Zoom

First Meeting: January 12

 

Home

Handouts

Homework


Description: The main goal of this course is to expose students to the mathematical theory of Fourier analysis, and at the same time, to some of its many applications in the sciences and engineering. In particular, the Fourier transform arises naturally in a number of imaging problems as in the theory of diffraction, magnetic resonance imaging (MRI), computed tomography (CT) and we shall explain how this happens. This course covers a broad range of topics and might be of interest to mathematicians and engineers alike.


Prerequisite: This is an upper-division undergraduate and/or lower-division graduate course. Some prerequisites include linear algebra (Math 104), real analysis (Math 115) and probability theory (Stats 217). Assignments would typically involve a fair amount of scientific programming in any language you like (e.g. Python, Julia, R, Matlab) together with more classical exercises.


Syllabus:

    Fundamental concepts of Fourier analysis
    • Continuous-time Fourier transform, Parseval identity (Plancherel theorem), inverse Fourier transform
    • Fourier series, sampling of bandlimited functions, Shannon's sampling theorem, aliasing
    • The Fourier transform and time/space invariant operators, convolutions
    • Fast Fourier transform (FFT) and non-uniform FFTs
    • The Weyl-Heisenberg uncertainty principle
    Selected applications
    • Spectral representation of stationary stochastic processes, Wiener filtering
    • Fourier optics: theory of diffraction, X-ray crystallography and phase retrieval problems
    • The mathematics of computed tomography (CT) and magnetic resonance imaging (MRI)
    • Short introduction to compressive sensing


    Textbooks:

    1. A First Course in Fourier Analysis by D. Kammler, Cambridge University Press, revised edition, 2008. (Required)
    2. Introduction to Fourier Optics by J. W. Goodman, Roberts and Company Publishers, 2005. (Optional)
    3. Fourier Analysis by T. W. Korner, Cambridge University Press, 1989. (Optional)
    4. Principles of Magnetic Resonance Imaging by Dwight Nishimura, 2010. (Optional)
    5. Fourier Series and Integrals by H. Dym and H. P. McKean, Academic Press, 1972. (Optional)
    6. Introduction to the Mathematics of Medical Imaging, Second Edition, by C. L. Epstein, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. (Optional)

    Handouts: All handouts will be posted online.

    Course assistant and office hours:

    • Jorge Guijarro Ordonez () Office hours TBD

    Grading (tentative):

    • Homework assignments: 50%
      • Homework assignments will generally be distributed on Thursdays and are due in class the following Thursday.
      • Late homeworks will NOT be accepted for grading (medical emergencies excepted with proof).
      • There will be about 4 assignments; the lowest score will be dropped in the final grade.
      • It is encouraged to discuss the problem sets with others, but everyone needs to turn in a unique personal write-up.

    • Final project: 50%. Students will have the freedom to select a project from a list according to their own interest. Students can also define their own project after communicating with the instructor.