# Faculty of Engineering and Science

The International Doctoral School of Technology and Science

#
PhD Course

## Fourier Analysis

### March 4, 2005

The lectures dealt with the discrete Fourier transform and its properties.
I also presented the Uncertainty Principle in the discrete case
The material presented and its relation to the book
BNB, and comments on what to read, follows below:
- Section 6.1
- Covered in abbreviated form. The book contains a lot of
detailed computations
- Section 6.2
- Covered in detail. I also went through exercise 6.2.5, using the
book by Strang and Nguyen. See below.
- Section 6.3
- Covered.
- Section 6.4
- This material was covered using the book by Strang and
Nguyen. See below.
- The discrete uncertainty principle
- Covered using articles by Donoho et al., see below.

### Additional Course Material

I have used the following additional course material:
- DFT and FFT: The pages 61-68 and 265-271 from Strang and Nguyen:
Wavelets and Filter Banks. Copies have been distributed.
- Uncertainty principles: I have used material from the following
two papers:
- D. Donoho, P. Stark: Uncertainty principles and signal
recovery.
SIAM J. Appl. Math. 49 (1989), no. 3, 906--931.
- D. Donoho, X. Huo: Uncertainty principles
and ideal atomic decomposition.
IEEE Trans. Inform. Theory 47 (2001), no. 7, 2845--2862.

These two papers contain a lot of material and are good starting
points, if you want to see how the uncertainty principles can be
used in signal recovery. They are somewhat more advanced than
the course.

#### Problem set 4

There will be no problem set 4 concerning the discrete Fourier transform.
#### Exercises

The following exercise is suggested:
- Maple exercise
- Look at the way one can deal with the discrete Fourier transform
in Maple.
- Matlab exercise
- Compare the above with the possibilities in matlab.

Updated March 8, 2005, by Arne Jensen.