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Chapter 4. Fourier Series & 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

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chapter 4

Chapter 4

Fourier Series & Transforms

taylor series
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
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 wave5
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
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
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 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
Euler’s Relationship
  • Review  Euler formulas

notes

examples
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
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

examples12
Examples

Find Fourier Series Coefficients for

Find Fourier Series Coefficients for

Remember:

examples13
Examples

Find the Complex Exponential Fourier Series Coefficients

notes

textbook

example
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

example15
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
Practical Application
  • Using a XTL oscillator which produces positive 1Vp-p how can you generate a sinusoidal waveforms with different frequencies?
practical application17
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

slide18
Demo

Ck corresponds to frequency components

In the signal.

example19
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
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
Fourier Series - Applet

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

using fourier series table
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 table23
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
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 transformation25
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 transformation26
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 transformation27
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
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 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