1. Fourier Series
September 18, 2000 EE 64, Section 1 ©Michael R. Gustafson II Pratt School of Engineering

2. Previous Chapters Signals and signal properties
System and system properties
Impulse responses of LTI systems
Convolution - determining system responses using the impulse response
Correlation

3. This Chapter
Representing periodic signals as a summation of cosines and sines
Using this representation in concert with convolution to represent LTI system output as a summation of cosines and sines

4. Historical Perspective
Read Section 3.1 in OW
Main idea - progression of the idea of breaking up a signal into individual normal modes

5. Motivation
Following our previous progression for linear systems, add to the list the complex exponential:

6. System Response
So, the response to a complex exponential is the same complex exponential multiplied by a function of the exponential s

7. Why is this useful?
If it can be shown that a signal can be represented as a combination of complex exponentials, then the response of an LTI system to an input can be represented as a combination of complex exponentials times some function of the impulse response and the exponential.

8. Why is this useful? (Math)

9. Fourier Series
The representation: can be written for real periodic signals as: where w0 is the fundamental frequency, T is the fundamental period, and k is the harmonic number.

10. Synthesis Examples
Given: what is x(t)?

11. Finding Fourier Coefficients
Finding the Fourier Coefficients employs the idea of orthogonality: that is, the integral over a period of a signal created by multiplying two periodic exponentials together is zero unless the two exponentials have the same period

13. Fourier Series
Synthesis equation:
Analysis equation:

14. Analysis Examples
Given: what are the ak?
Given: what are the ak?

15. Plot of x(t) T
W/2
W/2 T

16. Rules of Thumb
For cos() and sin(), determine the fundamental frequency first, figure out what multiple the terms have, and remember that:

17. Wrap-Up Assignment Next time Read Chapter 3 of OW
Convergence of the Fourier Series.
Properties of the Fourier Series.