Lectures

Group project presentation

Friday, Dec 07, 2012

Test 6

Wednesday, Dec 05, 2012

Group project

Monday, Dec 03, 2012

Group project

Friday, Nov 30, 2012

Group project

Wednesday, Nov 28, 2012

Group project

Monday, Nov 26, 2012

Haar wavelet reconstruction

Wednesday, Nov 21, 2012

Haar wavelet decomposition, II

Monday, Nov 19, 2012

Haar wavelet decomposition

Friday, Nov 16, 2012
We look at the process of decomposing a function using the Haar wavelet. In order to do this we need to use the 2-scale relations proved earlier. Next time we will derive the decomposition and see that in some ways it resembles the FFT

Problems

4.1, 4.2

Reminders

  • read chapter 4.2.3, 4.3.1

Haar wavelets, III

Wednesday, Nov 14, 2012
We discuss translates and dilates of psi

Haar wavelets, II

Wednesday, Nov 07, 2012

Haar wavelets, I

Monday, Nov 05, 2012
Discussion of the process of approximating a function on the unit interval using averages over subintervals. We learn, that this process can be described in terms of an orthonormal expansion.

Wavelets

Friday, Nov 02, 2012

Guest lecture on Frames

Wednesday, Oct 31, 2012

Test 4

Friday, Oct 26, 2012

Linear time-invariant operators, II

Wednesday, Oct 24, 2012
We will work out some examples of LTI operators and discuss causal systems

Reminders

  • Finish reading 3.2

Linear time-invariant operators, I

Monday, Oct 22, 2012
We will show that every linear time-invariant operator (LTI) is given by convolution with a fixed sequence.

Problems

Assigned in class

Reminders

  • Finish reading 3.2

Discrete Filters

Friday, Oct 19, 2012
Today we will consider the problem of filtering signals. Some applications include removing low and high frequency components. We will begin to show that every linear filter is given by a convolution.

Problems

Assigned in class

Reminders

  • Finish reading 3.2

Fast Fourier Transform

Wednesday, Oct 17, 2012
We complete our discussion of the fast Fourier transform and compute the running time of the algorithm

Problems

3.7, 3.8, 3.9

Reminders

  • Read 3.2 page 147, 148

Fast Fourier Transform

Monday, Oct 15, 2012
By splitting the Fourier transform of a vector into its even and odd parts we obtain a way to speed up the computation of a Fourier Transform. On Wednesday we will finish this and work on problems

Test 3

Friday, Oct 12, 2012

Convolution of sequences and the Fourier matrix

Wednesday, Oct 10, 2012

The Discrete Fourier Transform

Friday, Oct 05, 2012

The Fourier Basis

Wednesday, Oct 03, 2012

Periodic sequences

Monday, Oct 01, 2012

Reminders

  • Read chapter 3.1, first two pages
  • Do problems assigned in class

Wrapping up convergence results

Wednesday, Sep 26, 2012

Reminders

  • Read chapter 3.1 first two pages

Convolution

Monday, Sep 24, 2012

Test 2

Friday, Sep 21, 2012

Computations with the Dirichlet kernel and convolution

Wednesday, Sep 19, 2012

The Dirichlet kernel

Monday, Sep 17, 2012

The partial sums of a Fourier series and Dirichlet kernel

Friday, Sep 14, 2012
We will begin to show that the Fourier series for a differentiable function converges to the function uniformly.

First convergence results

Wednesday, Sep 12, 2012
We proved, under smoothness assumptions, that the fourier coefficients converge to zero and that the Fourier series converges absolutely (and uniformly).

The partial sums of a Fourier series and convergence theorems

Monday, Sep 10, 2012
We will state some theorems about the convergence of Fourier series and look talk about uniform convergence.

Test 1

Friday, Sep 07, 2012

Problem session

Wednesday, Sep 05, 2012
Work practice problems for the first test.

Inner products, part II.

Tuesday, Sep 04, 2012
Orthonormal bases.

Problems

0.6, 0.7, 0.11

Inner products.

Friday, Aug 31, 2012
The definition of an inner product, its relationship to distance. Focus on the L^2 inner priduct and the fact that the cosine, sine functions are orthonormal. A little bit about complex Fourier series.

Reminders

  • Try reading 1.3 for Tuesday.

Convergence of Fourier series, Part II

Wednesday, Aug 29, 2012
Definition of distance. The uniform and L^2 norms.

Convergence of Fourier series, Part I

Monday, Aug 27, 2012
Even and odd functions and Fourier series. Review of convergence for sequences and series.

Reminders

  • Read chapter 1.3, for Wednesday

Computing Fourier Series

Friday, Aug 24, 2012
None

Problems

0.1, 0.2, 0.3, 0.4 (optional)

Reminders

  • Read chapter 1.3, for Monday

Computing Fourier Series

Wednesday, Aug 22, 2012
Computing some examples of Fourier series and trying to draw a picture.

Problems

0.1, 0.2, 0.3, 0.4 (optional)

Reminders

  • Read chapter 0.3, for Friday

Introduction to Fourier Series

Monday, Aug 20, 2012
What is signal processing? Some background on where the subject arose. The practical matter of computing Fourier series.

Problems

(page 82) 1.1, 1.2, 1.7, 1.9, 1.10

Reminders

  • Read chapters 0.2, 1.2 for Wednesday.

Course Overview

Instructor

Mrinal Raghupathi

About this course

This class is an introduction to the mathematical theory of signal processing. We will cover Fourier series, Fourier transform, and Wavelets. Time permitting, we will study Multiresolution analysis.

Course Policy Statement

Textbook

A First Course in Wavelets with Fourier Analysis, by Albert Boggess and Francis Narcovich. Second edition, Wiley.

Links

Piazza forum

Syllabus

DateTopicChapter
Mon, Oct 15, 2012 Fast Fourier Transform 3.1.3
Wed, Oct 17, 2012 Time-invariant Discrete Filter 3.2.1
Fri, Oct 19, 2012 Time-invariant Discrete Filter 3.2.1
Mon, Oct 22, 2012 Z-transform and Transfer function 3.2.2
Wed, Oct 24, 2012 Z-transform and Transfer function 3.2.2
Fri, Oct 26, 2012 Test 4 None
Mon, Oct 29, 2012 The Haar scaling function 4.2.1
Wed, Oct 31, 2012 The Haar scaling function 4.2.1
Fri, Nov 02, 2012 Basic Properties of the Haar Scaling function 4.2.2
Mon, Nov 05, 2012 The Haar wavelet 4.2.3
Wed, Nov 07, 2012 The Haar wavelet 4.2.3
Fri, Nov 09, 2012 Test 5 None
Wed, Nov 14, 2012 Decomposition 4.3.1
Fri, Nov 16, 2012 Decomposition 4.3.1
Mon, Nov 19, 2012 Reconstruction 4.3.2
Wed, Nov 21, 2012 Reconstruction 4.3.2
Mon, Nov 26, 2012 Filters and diagrams 4.3.3
Wed, Nov 28, 2012 Filters and diagrams 4.3.3
Fri, Nov 30, 2012 Test 6 None
Mon, Dec 03, 2012 Final group project None
Wed, Dec 05, 2012 Final group project None
Fri, Dec 07, 2012 Final group project None

Exams

  • Test 1
  • Friday, Sep 07, 2012
  • Computing Fourier series (1.2), inner products (0.2, 0.3), uniform and L^2 distance.
  • Test 2
  • Friday, Sep 21, 2012
  • Complex Fourier series, convergence of a Fourier series.
  • Test 3
  • Friday, Oct 12, 2012
  • Discrete Fourier transform, n-periodic sequences, convolution.
  • Test 4
  • Friday, Oct 26, 2012
  • Fast fourier transform, Convolution, Linear time invariant operators
  • Test 5
  • Friday, Nov 09, 2012
  • Wavelet basics
  • Test 6
  • Wednesday, Dec 05, 2012
  • Wavelet decomposition, wavelet reconstruction