Post on 30-Jan-2020
EE 442 Fourier Transform 1
The Fourier TransformEE 442 Analog & Digital Communication Systems
Lecture 4
Voice signal
time
frequency (Hz)
ES 442 Fourier Transform 2
Jean Joseph Baptiste Fourier
March 21, 1768 to May 16, 1830
ES 442 Fourier Transform 3
Review: Fourier Trignometric Series (for Periodic Waveforms)
Agbo & Sadiku;Section 2.5pp. 26-27
0 0 0
n 1
0 0
0
0
0
0
Equation (2.10) should read (time was missing in book):
( ) cos( ) sin( )
2 1where and
and (Equations 2.12a, b, & c)
1( ) (DC term)
2( )cos( ) for 1, 2, 3,
n n
T
T
n
t
e
t
f t a a n b n
fT T
a f t dtT
a f t n
t
n t dtT
0
0
.
2( )sin( ) for 1, 2, 3, .
T
n
tc
b f t n t dt n etcT
ES 442 Fourier Transform 4
Fourier Trigonometric Series in Amplitude-Phase Format
Agbo & Sadiku;Section 2.5
Page 27
0 0
n 1
0 0
2 2
1
Equations (2.13) and (2.14) should read:
( ) cos( )
tan
Also known as polar form of Fourier series.
n n
n n n
nn
n
f t A A n
a A
A a b and
b
a
t
EE 442 Fourier Transform 5
Example: Periodic Square Wave as Sum of Sinusoids
Line Spectra
3f0
f0
5f0
7f0
Even or Odd?
EE 442 Fourier Transform 6
Example: Periodic Square Wave (continued)
http://ceng.gazi.edu.tr/dsp/fourier_series/description.aspx
1 1 13 5 7
4( ) sin( ) sin(3 ) sin(5 ) sin(7 )f t t t t t
Fundamental only
Five terms
Eleven terms
Forty-nine terms
t
This is an odd function Question:What would make this an
even function?
EE 442 Fourier Transform 7
Sinusoidal Waveforms are the Building Blocks in the Fourier Series
Simple Harmonic Motion Produces Sinusoidal Waveforms
Sheet of paper unrolls in this direction
FuturePast
Time t
MechanicalOscillation
ElectricalLC Circuit
Oscillation
LC Tank Circuit
EE 442 Fourier Transform 8
Visualizing a Signal – Time Domain & Frequency Domain
Source: Agilent Technologies Application Note 150, “Spectrum Analyzer Basics”http://cp.literature.agilent.com/litweb/pdf/5952-0292.pdf
Amplitude
To go from time domainto frequency domain
we use Fourier Transform
EE 442 Fourier Transform 9
Example: Where Both Sine & Cosine Terms are Required
http://www.peterstone.name/Maplepgs/fourier.html#anchor2315207
Period T0
Note phase shift in the fundamental frequency sine waveform.
EE 442 Fourier Transform 10
Fourier Series
DiscreteFourier
Transform
Fourier Transform
DiscreteFourier
Transform
Continuous time Discrete time
Periodic
Aperiodic
Fourier series for continuous-time periodic signals → discrete spectraFourier transform for continuous aperiodic signals → continuous spectra
Fourier Series versus Fourier Transform
ES 442 Fourier Transform 11
Definition of Fourier Transform
1
The Fourier transform (spectrum) of is ( ):
( ) ( ) ( )
1( ) ( ) ( )
2
Therefore, ( ) ( ) is a Fourier Transform pair
j t
j t
f(t) F
F FT f t f t e dt
f t FT F F e d
f t F
Agbo & Sadiku;Section 2.7;
pp. 40-41
Note: Remember = 2 f
EE 442 Fourier Transform 12
Fourier Transform Produces a Continuous Spectrum
FT { f(t)} gives a spectra consisting of a continuous sum of exponentials with frequencies ranging from - to + .
where |F()| is the continuous amplitude spectrum of f(t)and
() is the continuous phase spectrum of f(t).
,)()( )( jeFF
Often only the magnitude of F() is displayed and the phaseis ignored.
ES 442 Fourier Transform 13
Example: Impulse Function (t)
1)()()( 0
0
j
t
tjtj eedtettFTF
0if0
0if
t
t)(t
1
)(tFT
)(21
1)(
t
Delta function has unity area.
EE 442 Fourier Transform14
Example: Fourier Transform of Single Rectangular Pulse
1
time t2
2
2
0
Remember = 2f
1 for2 2
0 for all 2
t
t
)/(IIrect)( tttf
)/(IIrect)( tttf
2
2sin2sin2
)()(
2/2/2/
2/
2/
2/
j
j
j
ee
j
e
dtedtetfF
jjtj
tjtj
Pulse ofwidth
ES 442 Fourier Transform 15
fF sinc
2
2sin)(
Fourier Transform of Single Rectangular Pulse (continued)
F()
0
3
2
2
3
1
time t2
2
2
0
sinc function
Note the pulse is
time centered
EE 442 Fourier Transform 16
Properties of the Sinc Function
Definition of the sinc function:
Sinc Properties:1. sinc(x) is an even function of x2. sinc(x) = 0 at points where sin(x) = 0, that is,
sinc(x) = 0 when x = , 2, 3, ….3. Using L’Hôpital’s rule, it can be shown that sinc(0) = 14. sinc(x) oscillates as sin(x) oscillates and monotonically
decreases as 1/ x decreases with increasing | x |5. sinc(x) is the Fourier transform of a single rectangular pulse
sin( )sinc( )
xx
x
ES 442 Fourier Transform 17
Periodic Pulse Train Morphing Into a Single Pulse
https://www.quora.com/What-is-the-exact-difference-between-continuous-fourier-transform-discrete-Time-Fourier-Transform-DTFT-Discrete-Fourier-Transform-DFT-Fourier-series-and-Discrete-Fourier-Series-DFS-In-which-cases-is-which-one-used
EE 442 Fourier Transform 18
Sinc Function Tradeoff: Pulse Duration versus Bandwidth
1( )g t
t t
2 ( )g t
3( )g t
t
1( )G f
2 ( )G f
3( )G f
f
f
f
1
2
T1
2
T 2
2
T2
2
T
3
2
T3
2
T
2T
1T
3T
1
1
T1
1
T
2
1
T2
1
T
3
1
T3
1
T
T1 > T2 > T3
Also called the
Time ScalingProperty
(2.8.2)
EE 442 Fourier Transform 19
Properties of Fourier Transforms
2.8.1 Linearity (Superposition) Property2.8.2 Time-Scaling Property2.8.3 Time-Shifting Property2.8.4 Frequency-Shifting Property2.8.5 Time Differentiation Property2.8.6 Frequency Differentiation Property2.8.7 Time Integration Property2.8.8 Time-Frequency Duality Property2.8.9 Convolution Property
Agbo & SadikuSection 2.8;pp. 46 to 58
EE 442 Fourier Transform 20
2.8.1 Linearity (Superposition) Property
Given f(t) F() and g(t) G() ;
Then f(t) + g(t) F() + G() (additivity)
also kf(t) kF() and mg(t) mG() (homogeneity)
Combining these we have,
kf(t) + mg(t) kF () + mG ()
Hence, the Fourier Transform is a linear transformation.
This is the same definition for linearity as used in your circuits and systems EE400 course.
ES 442 Fourier Transform 21
2.8.2 Time Scaling Property
aF
aatfFT
1)(
)()()( Hence,
1)()(
,&Let
)()(
*
FF-tfFT
aF
aa
defatfFT
adtdat
dteatfatfFT
tj
tj
EE 442 Fourier Transform 22
Time-Scaling Property (continued)
Time compression of a signal results in spectral expansion andtime expansion of a signal results in spectral compression.
aF
aatfFT
1)(
EE 442 Fourier Transform 23
2.8.3 Time Shifting Property
FettfFT
tj 0)( 0
Fedefe
defttfFT
ttdtdtt
dtettfttfFT
tjjtj
tj
tj
09
0
)(
)()(
&,Let
)()(
)(
0
00
00
EE 442 Fourier Transform 24
Time-Shifting Property (continued)
Delaying a signal by t0 seconds does not change its amplitudespectrum, but the phase spectrum is changed by -2ft0.
Note that the phase spectrum shift changes linearly with frequency f.
FettfFT
tj 0)( 0
22ImRe)( FFF
t
t
This time shifted pulseis both even and odd.
Even function. Both must be
identical.
)(F)(tf
A time shiftproduces a phase
shift in itsspectrum.
ES 442 Fourier Transform 25
2.8.4 Frequency Shifting Property
00)(
FetfFTtj
021
021
0
21
0
0
)(
()(cos
;cos Apply to
:napplicatio Special
)(
)()(
00
0
00
FFtFT
eet
Fdetf
dteetfetfFT
tjtj
tj
tjtjtj
ES 442 Fourier Transform 26
Important Formula to Remember
exp cos( ) sin( )j j
1sin( ) exp exp
2
1cos( ) exp exp
2
j jj
j j
22 1
/2 1 for
1 for
, tan
j jn
i
n evenj and
n odd
ba jb r a b
a
e e
e where r
jrejba
Euler's formula
( )g t
( ) ( )g t G f
( )G f
fBB
fCf
( )g t ( )cos(2 )Cg t f t
Cf
2A
A
t
t
( )g t2B
ES 442 Fourier Transform 27
Frequency Shifting Property is Very Useful in Communications
Multiplication of a signal g(t) by the factor [cos(2fct)]places G(f) centered at f = fC.
Carrier frequency is fc &g(t) is the message signal
EE 442 Fourier Transform 28
Frequency-Shifting Property (continued)
Note phase shift
0cos() 2 )(g t f t
0sin() 2 )(g t f t
( )g t( )G f
( )G f
( )g f
2
2
a b
d c
e f
Multiplication of a signal g(t) by the factor places G(f) centered at f = fC.
0cos(2 )f t
Message signal
29EE 442 Fourier Transform
Modulation Comes From Frequency Shifting Property
00)(then,
)(:pair FTGiven
Fetf
Ftf
tj
Amplitude Modulation Example:
Audio tone: sin(t)
Sinusoidal carrier signal:
ES 442 Fourier Transform 30
Fourier Transform of AM Tone Modulated Signal
Only positive frequencies shown;Must include negative frequencies.
f(t)
G()Carrier signal
Modulated AM sidebands
ES 442 Fourier Transform 31
Modulation of Baseband and Carrier Signals
EE 442 Fourier Transform 32
Transform Duality Property
Given ( ) ( ), then
( ) ( )
and
( ) ( )
g t G f
g t G f
G t g f
Note the minus sign!
Because of the minus sign they are not perfectly symmetrical – See the illustration on next slide.
EE 442 Fourier Transform 33
Illustration of Fourier Transform Duality
2
2
1( )G f
2 ( )G f
1( )g t
2 ( )g t
t f
ft
1T
1
1
T
What doesthis imply?
2
2
1
1
2
2
3
1
1
2
4
4
EE 442 Fourier Transform 34
Fourier Transform of Complex Exponentials
1
1
1
1
2
2 2
2
2
2 2
2
( ) ( )
Evaluate for
( )
( )
( ) ( )
Evaluate for
( )
( )
c
c
c C
c
c
f f
c
c C
c
c
f f
c
c c
c
c c
j f t
j f t j f t
j f t
j f t
j f t j f t
j f
F f f f f df
f f
F f f df
f f and
F f f f f df
f f
F f f df
f f
e
e e
e
e
e e
e
ct
EE 442 Fourier Transform 35
Fourier Transform of Sinusoidal Functions
2 2
2 212
!Taking ( ) and ( )
! !
We use these results to find of cos(2 ) and sin(2 )
Using the identities for cos(2 ) and sin(2 ),
cos(2 ) & cos(2 )
c cc c
c c
j f t j f t
j f t j f t
nf f f f
r n r
FT ft ft
ft ft
ft ft
e e
e e
2 212
1212
Therefore,
cos(2 ) ( ) ( ) , and
sin(2 ) ( ) ( )
c c
c c
c cj f t j f t
j
j
ft f f f f
ft f f f f
e e
ff fcfc
-fc
-fc
sin(2fct)cos(2fct)
*
ImRe
sin
EE 442 Fourier Transform 36
Summary of Several Fourier
Transform Pairs See
Agbo & Sidiku;Table 2.5,Page 54;------------
See also theFourier
TransformPair Handout
http://media.cheggcdn.com/media/db0/db0fffe9-45f5-40a3-a05b-12a179139400/phpsu63he.png
EE 442 Fourier Transform 37
Spectrum Analyzer Shows Frequency Domain
A spectrum analyzer measures the magnitude of an input signal versus frequency within the full frequency range of the instrument. It measures frequency, power, harmonics, distortion, noise, spurious signals and bandwidth.
➢ It is an electronic receiver➢Measure magnitude of signals➢ Does not measure phase of signals➢ Complements time domain
Bluetooth spectrum
EE 442 Fourier Transform 38
Re
f
fc
-fc
2
A
2
A
cos(2 )A f t
Blue arrows indicatepositive phase directions
Fourier Transform of Cosine Signal
cos(2 ) ( ) ( )2
c c c
AA f t f f f f
3D ViewNot in
Lathi & Ding
ES 442 Fourier Transform 39
Fourier Transform of Cosine Signal (as shown in textbooks)
t
0f
FT
f
Real axis
0( )f cos( )t
0f
0T 0T
0
0
1f
T
EE 442 Fourier Transform 40
Re
f
fc
-fc
2
B
2
B
sin(2 )B f t
Fourier Transform of Sine Signal
sin(2 ) ( ) ( )2
c c c
BB f t j f f f f
We must subtract 90from cos(x) to get sin(x)
EE 442 Fourier Transform 41
Fourier Transform of Sine Signal (as usually shown in textbooks)
t
0( )f
0f0f
2
Bj
2
Bj
FT
Imaginary axis
f
B
Bsin(0t)
ES 442 Fourier Transform 42
Visualizing Fourier Spectrum of Sinusoidal Signals
Multiplying byj is a phase
shift
EE 442 Fourier Transform 43
Re
f
fc
-fc
2
B
2
B
2
A
2e j ftR
2
A
2e j ftR
-
2 2e e e ej j ft j j ftR R
2 2
1tan2 2
A B AR and
B
Fourier Transform of a Phase Shifted Sinusoidal Signal(with phase information shown)
EE 442 Fourier Transform 44
Selected References
1. Paul J. Nahin, The Science of Radio, 2nd edition, Springer, New York, 2001. A novel presentation of radio and the engineering behind it; it has some selected historical discussions that are very insightful.
2. Keysight Technologies, Application Note 243, The Fundamentals of Signal Analysis; http://literature.cdn.keysight.com/litweb/pdf/5952-8898E.pdf?id=1000000205:epsg:apn
3. Agilent Technologies, Application Note 150, Spectrum Analyzer Basics; http://cp.literature.agilent.com/litweb/pdf/5952-0292.pdf
4. Ronald Bracewell, The Fourier Transform and Its Applications, 3rd ed., McGraw-Hill Book Company, New York, 1999. I think this is the best book covering the Fourier Transform (Bracewell gives many insightful views and discussions on the FT and it is considered a classic textbook).
ES 442 Fourier Transform 45
Auxiliary Slides For Introducing Sampling
EE 442 Fourier Transform 46
Fourier Transform of Impulse Train (t) (Shah Function)
0f 02 f0f02 f 0 f0T 02T0T02T 0 t
0Period T0
1Period
T
Ш(t) = 0 0( ) ( )n n
t nT t nT
12
12
( ) 1
n
n
t dt
Ш
Shah function (Ш(t)):
aka “Dirac Comb Function,” Shah Function & “Sampling Function”
Ш(t)
EE 442 Fourier Transform 47
Shah Function (Impulse Train) Applications
0( ) ( ) ( )n
f t f n t nT
Ш(t)
0( ) ( )n
f t f t nT
Ш(t)
The sampling property is given by
The “replicating property” is given by the convolution operation:
Convolution
Convolution theorem:
1 2 1 2( ) ( ) ( ) ( )g t g t G f G f
1 2 1 2( ) ( ) ( ) ( )g t g t G f G f
and
EE 442 Fourier Transform 48
Sampling Function in Operation
0( ) ( ) ( )n
f t f n t nT
Ш(t)
0( )t nT
( )f t
(0)f0( ) (1)f T f
0(2 ) (2)f T f
0T
t
t
t
0T