Ph.D. student — University of British Columbia — Mathematics

• New updates and Notes from (selected) meetings

Find new updates and notes from past meetings here!

• Project Outline

The philosophy of this mini-course, as well as what it hopes to achieve. Included is a regularly-updated syllabus for what has been covered so far, and what will be covered in the future.

• Interactive Guide to Fourier Transform

(external link) Gives preliminary intuition on how the Fourier transform works. Suggests intuition of mapping between different spaces/different bases. Nice interactive animations.

• The Heat Equation

(external link) An engineer's derivation of the heat equation. For a purely mathematical derivation, see almost any introductory book on PDEs.

### Matlab scripts

• Example script to compute Fourier transform of Dirac Comb

Contains code to compute the Fourier transform of a Dirac Comb. This script is easily modified to see how modifying the height of the Comb's tines affects the output of the Fourier transform.

• Gabor Transform (function)

A working (but unfinished) function for approximating the Gabor transform of vector data. Note that this function only accepts data of length `3N+1` and `2N+1`.

• Example script to compute Gabor transforms

Script file that depends on the Gabor transform function, above. For two signals, `x` and `x2`, the script plots the Gabor transform over a set of timeshifts and frequencies. Default output should resemble these figures.

• Importing songs

A script whose purpose is to elucidate how to import .wav files into Matlab, and showcase basic manipulations using Fourier series — showcases companion material used in our initial introduction to Matlab/coding/visualization.

• Example script to make test images

To explore what it means to take the Fourier transform of an image, this script file contains some skeleton code showing the FFT of a line, of an image, and a circle. Change the number `egNumber` to change which example is run when the script file is run. Note that `fur.png` is expected to be in the same directory as the script file. Feel free to change the file name to any other image. Some sample images (including `fur.png`) are included in the images directory.

• Fourier partial sums example (visualization)

This script file was created to visualize what it means to approximate a function with Fourier (sine) series. Note that while the first half of the code is "fancier" (i.e., more efficient, but much harder to read), it only computes coefficients for the function f(x)=x. In contrast, the second half of the code can be easily modified to compute Fourier partial sums for other functions.

To run the code, open the file, run the script, and then hit Enter to move through each graph (beware: there's a lot of graphs [100 by default]). Note that because this file only uses sine series, only odd functions can "actually" be approximated (otherwise things get weird!). What happens if you try to approximate a function that isn't odd?