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Hello and welcome to Week 2 of Digital Signal Processing. It looks like we are off to a good start!

Last week we had to digest a bit of prep work but this week we're ready to discover the beauty of Fourier analysis, both in theory and in practical applications. Fourier analysis is the cornerstone of most scientific explorations and digital signal processing is no exception. The topic will take us two weeks to cover properly; this week we will concentrate on the basic tools and next week we will see more advanced issues related to usage and algorithms.

Day 3

Video lectures:

• 4.0 - Introduction
• 4.1 - Exploration via a change of basis
• 4.2 - The Discrete Fourier Transform
• 4.3 - The DFT in practice

Signal of the day:

• The First Man-Made Signal From Outer Space

Practice homework:

• Practice Homework 3 (Solutions)

Homework (due Monday November 9, 5:00pm CET):

• Homework 3
• Feedback 3

Day 4

Video lectures:

• 4.4 - DFT, DFS and DTFT
• 4.5 - DTFT: intuition and properties

Numerical example:

• Fourier Transforms (Python), (access on GitHub)

Practice homework:

• Practice Homework 4 (Solutions)

Homework (due Monday November 9, 5:00pm CET):

• Homework 4
• Feedback 4

Notes and external resources

This week is based on Chapter 4 of the textbook of the class. If you like the history of science, we can highly recommend James Gleick’s biography of Isaac Newton, where his work in "Opticks" is discussed in historical context (Isaac Newton, Pantheon Books, 2003).

Fourier series, Fourier analysis, and applications of the Fourier transform are the topic of many books in both mathematics and engineering. Below we just mention a selection.

• For a historical perspective, the original book by Fourier, ‘’Théorie Analytique de la Chaleur’’, is interesting. One can see how difficult it was, in the early 19th century, to prove properties that now follow easily from Hilbert space geometry (e.g. orthogonality of harmonic sines and cosines).
• From an engineering perspective, we recommend the ‘’The Fourier Transform and its Applications,’’ by R.Bracewell, McGraw Hill, 2000. http://www.amazon.com/The-Fourier-Transform-Its-Applications/dp/0073039381
• A beautiful book about Fourier analysis is by Thomas Körner, Fourier Analysis, Cambridge University Press, 1988.