Download Presentation
Chapter 4

Loading in 2 Seconds...

1 / 31

# Chapter 4 - PowerPoint PPT Presentation

Chapter 4. Fourier Series &amp; Transforms. Basic Idea. notes. 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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
Download Presentation

## PowerPoint Slideshow about 'Chapter 4' - vevay

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Chapter 4

Fourier Series & Transforms

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
• http://upload.wikimedia.org/wikipedia/commons/6/62/Exp_series.gif
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

Square Wave

K=1,3,5

K=1,3,5, 7

Frequency Components of Square Wave

Fourier Expansion

K=1,3,5, 7, 9, …..

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:
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
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)

Euler’s Relationship
• Review  Euler formulas

notes

Examples
• Find Fourier Series Coefficients for
• Find Fourier Series Coefficients for
• Find Fourier Series Coefficients for
• Find Fourier Series Coefficients for

C1=1/2; C-1=1/2; No DC

C1=1/2j; C-1=-1/2j; No DC

notes

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

Examples

Find Fourier Series Coefficients for

Find Fourier Series Coefficients for

Remember:

Examples

Find the Complex Exponential Fourier Series Coefficients

notes

textbook

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

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!!

|4V/p|

2|Ck|

|4V/3p|

|4V/5p|

notes

w0

3w0

5w0

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

Filter @ [kwo]

Sinusoidal waveform

X(t)

1

To/2

@ [kwo]

To

X(t)

To/2

0.5

To

-0.5

kwo

B changes depending on k value

Demo

Ck corresponds to frequency components

In the signal.

Example
• 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.

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

|4V/p|

2|Ck|

|4V/3p|

|4V/5p|

w0

3w0

5w0

Fourier Series - Applet

http://www.falstad.com/fourier/

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

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

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

To

-Xo/2

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

To

-Xo/2

Only DC value changed!

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

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

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
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