1. Fourier Series & Transforms
Chapter 4
Fourier Series & Transforms

2. Basic Idea
notes

3. Taylor Series Complex signals are often broken into simple pieces
Signal requirements
Can be expressed into simpler problems
The first few terms can approximate the signal
Example: The Taylor series of a real or complex function ƒ(x) is the power series

4. Square Wave S(t)=sin(2pft) S(t)=1/3[sin(2p(3f)t)]
S(t)= 4/p{sin(2pft) +1/3[sin(2p(3f)t)]}
Fourier Expansion

5. Square Wave K=1,3,5 K=1,3,5, 7 Frequency Components of Square Wave
Fourier Expansion
K=1,3,5, 7, 9, …..

6. Periodic Signals
A Periodic signal/function can be approximated by a sum (possibly infinite) sinusoidal signals.
Consider a periodic signal with period T
A periodic signal can be Real or Complex
The fundamental frequency: wo
Example:

7. Fourier Series
We can represent all periodic signals as harmonic series of the form
Ck are the Fourier Series Coefficients; k is real
k=0 gives the DC signal
k=+/-1 indicates the fundamental frequency or the first harmonic w0
|k|>=2 harmonics

8. Fourier Series Coefficients
Fourier Series Pair
We have
For k=0, we can obtain the DC value which is the average value of x(t) over one period
Series of complex numbers
Defined over a period of x(t)

9. Euler’s Relationship
Review  Euler formulas
notes

10. Examples Find Fourier Series Coefficients for C1=1/2; C-1=1/2; No DC
C1=1/2j; C-1=-1/2j; No DC
notes

11. Different Forms of Fourier Series
Fourier Series Representation has three different forms
Also:
Complex Exp.
Also:
Harmonic
Which one is this?
What is the DC component?
What is the expression for Fourier Series Coefficients

12. Examples
Find Fourier Series Coefficients for
Remember:

13. Examples notes textbook
Find the Complex Exponential Fourier Series Coefficients
notes
textbook

14. Example
Find the average power of x(t) using Complex Exponential Fourier Series – assuming x(t) is periodic
This is called the Parseval’s Identity

15. Example Consider the following periodic square wave
Express x(t) as a piecewise function
Find the Exponential Fourier Series of representations of x(t)
Find the Combined Trigonometric Fourier Series of representations of x(t)
Plot Ck as a function of k
X(t) V
To/2 To -V
Use a
Low Pass Filter to
pick any tone
you want!!
2|Ck|
|4V/p|
|4V/3p|
|4V/5p|
notes w0
3w0
5w0

16. Practical Application
Using a XTL oscillator which produces positive 1Vp-p how can you generate a sinusoidal waveforms with different frequencies?

17. Practical Application
Using a XTL oscillator which produces positive 1Vp-p how can you generate a sinusoidal waveforms with different frequencies?
Square Signal
@ wo
Level Shifter
[kwo]
Sinusoidal waveform
X(t) 1
To/2
@ [kwo] To
X(t)
0.5
To/2 To
-0.5
kwo
B changes depending on k value

18. Demo
Ck corresponds to frequency components
In the signal.

19. Example Only a function of freq.
Given the following periodic square wave, find the Fourier Series representations and plot Ck as a function of k. 1
Note: sinc (infinity)  1 &
Max value of sinc(x)1/x
Sinc Function
Note: First zero
occurs at Sinc (+/-pi)
Only a function
of freq.

20. Use the Fourier Series Table (Table 4.3)
Consider the following periodic square wave
Find the Exponential Fourier Series of representations of x(t)
X0V
X(t) V
To/2 To -V
2|Ck|
|4V/p|
|4V/3p|
|4V/5p| w0
3w0
5w0

21. Fourier Series - Applet

22. Using Fourier Series Table
Given the following periodic square wave, find the Fourier Series representations and plot Ck as a function of k. (Rectangular wave)
X01
C0=T/To
T/2=T1T=2T1
Ck=T/T0 sinc (Tkw0/2)
Same as before
Note: sinc (infinity)  1 &
Max value of sinc(x)1/x

23. Using Fourier Series Table
Express the Fourier Series for a triangular waveform?
Express the Fourier Series for a triangular waveform that is amplitude shifted down by –X0/2 ? Plot the signal. Xo To

24. Fourier Series Transformation
Express the Fourier Series for a triangular waveform?
Express the Fourier Series for a triangular waveform that is amplitude shifted down by –X0/2 ? Plot the signal. Xo To
From the table:
Xo/2
-Xo/2 To

25. Fourier Series Transformation
Express the Fourier Series for a triangular waveform?
Express the Fourier Series for a triangular waveform that is amplitude shifted down by –X0/2 ? Plot the signal. Xo To
From the table:
Xo/2
-Xo/2 To
Only DC value changed!

26. Fourier Series Transformation
Express the Fourier Series for a sawtooth waveform?
Express the Fourier Series for this sawtooth waveform? Xo To
From the table: Xo 1 To -3

27. Fourier Series Transformation
Express the Fourier Series for a sawtooth waveform?
Express the Fourier Series for this sawtooth waveform?
We are using amplitude transfer
Remember Ax(t) + B
Amplitude reversal A<0
Amplitude scaling |A|=4/Xo
Amplitude shifting B=1 Xo To
From the table: Xo 1 To -3

28. Example

29. Example

30. Fourier Series and Frequency Spectra
We can plot the frequency spectrum or line spectrum of a signal
In Fourier Series k represent harmonics
Frequency spectrum is a graph that shows the amplitudes and/or phases of the Fourier Series coefficients Ck.
Amplitude spectrum |Ck|
Phase spectrum fk
The lines |Ck| are called line spectra because we indicate the values by lines

31. Schaum’s Outline Problems
Schaum’s Outline Chapter 5 Problems:
4,5 6, 7, 8, 9, 10
Do all the problems in chapter 4 of the textbook
Skip the following Sections in the text:
4.5
Read the following Sections in the textbook on your own
4.4