The Fourier Transform...Properties of Fourier Transforms . 2.8.1 Linearity (Superposition) Property...

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


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


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


Example: Periodic Square Wave as Sum of Sinusoids

Line Spectra

3f0

f0

5f0

7f0

Even or Odd?

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiRoOKRyrLKAhUK6WMKHXIOCYMQjRwIBw&url=http://opticalengineering.spiedigitallibrary.org/article.aspx?articleid=1891707&psig=AFQjCNFc9fLjzYXNu5WSVTnARVvhnmtmuQ&ust=1453178842432929


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?

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiD8pHh74vKAhUJ82MKHWu1C1QQjRwIBw&url=http://ceng.gazi.edu.tr/dsp/fourier_series/description.aspx&psig=AFQjCNEI8lPeKqVZhxEcdjM-Y8YrGVMBYQ&ust=1451849642678935

http://ceng.gazi.edu.tr/dsp/fourier_series/description.aspx


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

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiXoLKPpq_KAhVB3mMKHUlNAxIQjRwIBw&url=http://electronics.stackexchange.com/questions/152600/why-is-sine-wave-preferred-over-other-waveforms&psig=AFQjCNGYnmTGUQWybjSTajHNgAoblcixnQ&ust=1453066870437166


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

http://cp.literature.agilent.com/litweb/pdf/5952-0292.pdf


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.

http://www.peterstone.name/Maplepgs/fourier.html


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


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


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.


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


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

https://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiezdvb66LKAhUFxGMKHRmcB20QjRwIBw&url=https://www.linkedin.com/pulse/20141101222119-3577215-frequency-spectrum-digital-signals&psig=AFQjCNGEskK1UCqwwHHj28PNHgnsDy_z3g&ust=1452638575939142


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


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

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


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)


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


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.


2.8.2 Time Scaling Property

aF

aatfFT

1)(

)()()( Hence,

1)()(

,&Let

)()(

*

FF-tfFT

aF

aa

defatfFT

adtdat

dteatfatfFT

tj

tj


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)(

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwifxMCctqrKAhVJ_WMKHbt9BgIQjRwIBw&url=http://www.slideshare.net/moeenafridi18/chapter-3-31730980&psig=AFQjCNHQuF2K3xaVfJnWIeh-5IoptE0hcg&ust=1452899257086213


2.8.3 Time Shifting Property

FettfFT

tj 0)( 0

Fedefe

defttfFT

ttdtdtt

dtettfttfFT

tjjtj

tj

tj

09

0

)(

)()(

&,Let

)()(

)(

0

00

00


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.

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwitmrOx-6LKAhUS0mMKHVDrBFQQjRwIBw&url=http://www.wikiwand.com/en/Fourier_transform&psig=AFQjCNG7RRzPlVUXuI5QLMxc5RzRezsRqQ&ust=1452643076173768


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


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


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

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwjV6sPb6arKAhUOzWMKHVnGB7oQjRwIBw&url=http://technoengineers.blogspot.com/2011/03/communication-systems.html&psig=AFQjCNGQNBZVj8obTzoMRD0r58UjYbP6Qg&ust=1452913069658120


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:


Fourier Transform of AM Tone Modulated Signal

Only positive frequencies shown;Must include negative frequencies.

f(t)

G()Carrier signal

Modulated AM sidebands


Modulation of Baseband and Carrier Signals


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.


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


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


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


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

http://media.cheggcdn.com/media/db0/db0fffe9-45f5-40a3-a05b-12a179139400/phpsu63he.png


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

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwi43vSA_MLKAhVH-mMKHUfbCWgQjRwIBw&url=http://www.directindustry.com/prod/keysight-technologies/product-8981-531406.html&psig=AFQjCNFbcOMHZD4R171V6GpamJxwBUXXjg&ust=1453742383327003

https://en.wikipedia.org/wiki/File:Bluetooth_signal_behind_wireless_lan_signal.png


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

https://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiwrIr-jrrKAhUL-mMKHcJoDzgQjRwIBw&url=https://commons.wikimedia.org/wiki/File:Sine_cosine_one_period.svg&psig=AFQjCNEDFfYizm-Gpwjhrr-IEoNpN-LtnQ&ust=1453438610150373


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


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)

https://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiwrIr-jrrKAhUL-mMKHcJoDzgQjRwIBw&url=https://commons.wikimedia.org/wiki/File:Sine_cosine_one_period.svg&psig=AFQjCNEDFfYizm-Gpwjhrr-IEoNpN-LtnQ&ust=1453438610150373


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)

https://www.chem.purdue.edu/courses/chm621/text/ft/basiset/sine/sine.htm


Visualizing Fourier Spectrum of Sinusoidal Signals

Multiplying byj is a phase

shift


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)


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).

http://literature.cdn.keysight.com/litweb/pdf/5952-8898E.pdf?id=1000000205:epsg:apn

http://cp.literature.agilent.com/litweb/pdf/5952-0292.pdf


Auxiliary Slides For Introducing Sampling


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)

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiQ9KuV-rbKAhVK52MKHYd1Ai8QjRwIBw&url=http://opticalengineering.spiedigitallibrary.org/article.aspx?articleid=1096259&psig=AFQjCNGJo1e8yqzi8ym2oAvPzYXpwsJjIA&ust=1453329936371428


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


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

The Fourier Transform...Properties of Fourier Transforms . 2.8.1 Linearity (Superposition) Property...

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