Embed Size (px)
Citation preview
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
• 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(2ft)
S(t)=1/3[sin(2f)t)]
S(t)= 4/{sin(2ft) +1/3[sin(2f)t)]}
oddK k
kftAts
/1
)2sin(4)(
Fourier Expansion
Square Wave
oddK k
kftAts
/1
)2sin(4)(
Frequency Components of Square Wave
K=1,3,5 K=1,3,5, 7
K=1,3,5, 7, 9, ….. Fourier Expansion
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: o• Example:
oo
tj
o
T
AetxComplex
ttxal
txnTtxPeriodic
o
/2
)(
)cos()(Re
)()(
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 0
– |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
k
oo
jkk
tjkk
tjkk
kk
eCC
eCeC
CC
&*
*
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
)cos()( ttx o
)sin()( ttx o
)4/2cos()( ttx
notes
C1=1/2; C-1=1/2; No DC
C1=1/2j; C-1=-1/2j; No DC
)(sin)( 2 ttx o
Different Forms of Fourier Series
• Fourier Series Representation has three different forms
Also: Harmonic
Also: Complex
Exp.
oddK k
kftAts
/1
)2sin(4)(
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
ttttx oo
oo 3sin4)302cos(5cos310)(
notes
textbook
Example• Find the average power of x(t)
using Complex Exponential Fourier Series – assuming x(t) is periodic
dttxPoT 2
)(
2
**
**
2
)()(
)(
)(
)(
kk
kk
k
T
tkj
kk
tkj
kk
T
C
CCdttxtxP
eCtx
eCtx
dttxP
o
o
o
o
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
V To/2
-V
To
notes
X(t)
)sin(4
)90cos(4
2)(
/1
/1
2/
/
tkk
V
tkk
V
eek
Vtx
ooddk
ooddk
tkjj
oddk
o
2|Ck| |4V/|
|4V/5|
|4V/3|
0 0 0
Use aLow Pass Filter to pick any tone you want!!
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?
)sin()( tkBty o
Square Signal@ wo
Filter @ [ko]Level ShifterSinusoidal waveform
1 To/2
To
X(t)
0.5To/2
To
X(t)
-0.5
@ [kwo]
kwo
)sin(4
2)(
/1
2/
/
tkk
V
eek
Vtx
ooddk
tkjj
oddk
o
B changes depending on k value
Demo
Ck corresponds to frequency componentsIn the signal.
Example
• Given the following periodic square wave, find the Fourier Series representations and plot Ck as a function of k.
Sinc Function
Only a functionof freq.
1
Note: sinc (infinity) 1 & Max value of sinc(x)1/x
Note: First zero occurs at Sinc (+/-pi)
Use the Fourier Series Table (Table 4.3)
• Consider the following periodic square wave
• Find the Exponential Fourier Series of representations of x(t)
• X0V
)sin(4
)90cos(4
2)(
/1
/1
2/
/
tkk
V
tkk
V
eek
Vtx
ooddk
ooddk
tkjj
oddk
o
V To/2
-V
To
X(t)
2|Ck| |4V/|
|4V/5|
|4V/3|
0 0 0
tkj
oddk
oek
Vjtx
/
20)(
Fourier Series - Applet
tkj
oddk
oek
Vjtx
/
20)(
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)
X01
C0=T/To
T/2=T1T=2T1
Ck=T/T0 sinc (Tkw0/2)
tkj
ko
o
oo
o
ok
oekTcT
T
kTcT
TTkc
T
TC
)(sin2
)(sin2
2sin
11
11
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.
To
Xo
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.
To
Xo
To
Xo/2
-Xo/2
From the table:
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.
tkj
oddk
oo oek
XXtx
/2
2
2)(
To
Xo
tkj
oddk
o
tkj
oddk
ooo
o
o
o
ek
X
ek
XXX
txX
ty
/2
/2
2
2
22
)(2
)(
To
Xo/2
-Xo/2
Only DC value changed!
From the table:
Fourier Series Transformation• Express the Fourier Series for a
sawtooth waveform?
• Express the Fourier Series for this sawtooth waveform?
To
Xo
To
Xo
From the table:
-3
1
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
tkj
k
joo oeek
XXtx
0/
2/
22)(
To
Xo
To
Xo
1)(4
)(
txX
tyo
From the table:
-3
1
Example
Example
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 k
• 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
