EE 442 Fourier Transform 1

The Fourier Transform

EE 442 Analog & Digital Communication Systems Spring 2017

Lecture 4

Review: Fourier Trignometric Series (for Periodic Waveforms)

2 EE 442 Fourier Transform

1 1 0

0 0 01

0 0

0

0

is defined in time interval of

( ) cos( 2 ) sin( 2 )

1where ( ) is of period , and .

We can calculate the coefficients from th

Periodic function g(

e equations:

t

2

)

n nn

n

t t t T

g t a a n f t b n f t

g t T fT

a gT

1 0

1

1 0

1

0

0

0

( ) cos( 2 ) for n = 1, 2, 3, 4,

2( ) sin( 2 ) for n = 1, 2, 3, 4,

t T

t

t T

n

t

t n f t dt

b g t n f t dtT

t

g(t)

T0

Lathi & Ding; pp. 50-61

3 EE 442 Fourier Transform

Review: Exponential Fourier Series (for Periodic Functions)

1 1 0

0 0

0

02

Again, is defined in time interval

( ) for 0, 1, 2, 3,

1where again ( ) is of period , and .

We can calculate the coefficients fr

periodic function

om the

g(t

eq i

)

uat o

nn

n

jn f t

t t t T

g t D n

g t T fT

D

e

00

02

n:

1( ) for ranging from - to .n

T

jn f tD g t dt n

Te

But, periodicity is too limited for our needs, so we will need to proceed on to the Fourier Transform.

The reason is that signals used in communication systems are not periodic (it is said they are aperiodic).

Lathi & Ding; pp. 50-61

EE 442 Fourier Transform 4

Sinusoidal Waveforms are the Building Blocks in the Fourier Series

Simple Harmonic Motion Produces Sinusoidal Waveforms

Sheet of paper unrolls in this direction

Future Past

Time t

Mechanical Oscillation

Electrical LC Circuit

Oscillation

LC Tank Circuit

EE 442 Fourier Transform 5

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 domain to frequency domain:

We use Fourier Transform

EE 442 Fourier Transform 6

Example: Periodic Square Wave

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

Odd function

Question: What would make this an

even function?

EE 442 Fourier Transform 7

Example: Periodic Square Wave (continued)

Line Spectra

3f0

f0

5f0

7f0

Even or Odd?

EE 442 Fourier Transform 8

https://en.wikipedia.org/wiki/Fourier_series

1 1 13 5 7

4( ) sin( ) sin(3 ) sin(5 ) sin(7 )f t t t t t

t

Square Wave From Fundamental + 3rd + 5th & 7th Harmonics

EE 442 Fourier Transform 9

Another Example: Both Sine & Cosine Functions 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

Gibb’s Phenomena at Discontinuities in Waveforms

About 9% overshoot at discontinuities of waveform. This is an artifact of the Fourier series representation.

time t

Am

plit

ud

e

The Gibbs phenomenon is an overshoot (or "ringing") of Fourier series occurring at simple discontinuities.

EE 442 Fourier Transform 11

Fourier Series

Discrete Fourier

Transform

Continuous Fourier

Transform

Discrete Fourier

Transform

Continuous time Discrete time

Periodic

Aperiodic

Fourier series for continuous-time periodic signals → discrete spectra Fourier transform for continuous aperiodic signals → continuous spectra

Fourier Series versus Fourier Transform

EE 442 Fourier Transform 12

Definition of Fourier Transform

2( )

j ftg t dte

2( )

j ftG f dfe

( )g t ( )G f

Time-frequency duality: ( ) ( ) ( ) ( )g t G f and G t g f

We say “near symmetry” because the signs in the exponentials are different between the Fourier transform and the inverse Fourier transform.

Figure 3.15 Lathi & Ding Cyclic relation:

EE 442 Fourier Transform 13

Fourier Transform Produces a Continuous Spectrum

FT [g(t)] gives a spectra consisting of a continuous sum of exponentials with frequencies from - to + .

( ) ( ) exp ( )G f G f j f

where |G(f)| is the continuous amplitude spectrum of g(t) and (f) is the continuous phase spectrum of g(t).

Supplemental Reading: Lathi & Ding;

pp. 97-99

EE 442 Fourier Transform 14

Example: Rectangular Pulse

1

G(f)

f time t

2

1

2

3

1

2

3

2

2

0 0

( ) ( ) ( / )g t rect t t

( ) ( ) ( / )g t rect t t 1 for

2 2

0 for all 2

t

t

sinc(f )

Remember = 2f

Lathi & Ding; pp. 101-102 Example 3.2

EE 442 Fourier Transform 15

Two Definitions of the Sinc Function

sin( )sinc ( )

and

sin( )sinc ( )

xx

x

xx

x

Ref.: Lathi and Ding, page 100; uses the definition sinc (x) = sin( )x

x

Sometimes sinc (x)

is written as Sa (x).

(1)

(2)

x

EE 442 Fourier Transform 16

Definition of the Sinc Function

Unfortunately, there are two definitions of the sinc function in use.

Format 1 (Lathi and Ding, 4th edition – See pp. 100 – 102) Format 2 (as used in many other textbooks)

Sinc Properties: 1. sinc(x) is an even function of x 2. 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) = 1 4. 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

sin( )sinc( )

xx

x

We will use this form

EE 442 Fourier Transform 17

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

Lathi & Ding; pp. 110-111

EE 442 Fourier Transform 18

Sinc Function Appears in Both Pulse Train & a Single Pulse

Pulse Train – Fourier Series Single pulse – Fourier Transform

Continuous Spectrum

Discrete Spectrum

What does this imply for a digital binary communication signal?

sin( )

( )

ftF f

ft

2 21( ) ( ) ( )f t u t u t

EE 442 Fourier Transform 19

Some Insight into the Sinc Function

Lathi & Ding; pp. 100-101

sin( )sinc( )

xx

x 1

x

1

x

sinc( )x

x

sin( )x

EE 442 Fourier Transform 20

Properties of Fourier Transforms

1. Linearity (Superposition) Property 2. Time-Frequency Duality Property 3. Transform Duality Property 4. Time-Scaling Property 5. Time-Shifting Property 6. Frequency-Shifting Property 7. Time Differentiation & Time Integration Property 8. Area Under g(t) Property 9. Area Under G(f) Property 10.Conjugate Functions Property

EE 442 Fourier Transform 21

Linearity (Superposition) Property

Given g1(t) G1(f) and g2(t) G2(f) ; Then g1(t) + g2(t) G1(f) + G2(f) (additivity) also kg1(t) kG1(f) and kg2(t) kG2(f) (homogeneity) Combining these we have,

kg1(t) + mg2(t) kG1(f) + mG2(f) Hence, the Fourier Transform is a linear transformation.

This is the same definition for linearity as used in your circuits and systems EE400 course.

EE 442 Fourier Transform 22

Time-Frequency Duality Property

Lathi & Ding; pp. 106-109

0

0

0

0

2

2

Given ( ) ( ), then

( - ) ( )

and

g(t) ( )

j ft

j ft

g t G f

g t t G f

G f f

e

e

This leads directly to the transform Duality Property

2( )

j ftg t dte

2( )

j ftG f dfe

( )g t ( )G f

EE 442 Fourier Transform 23

Transform Duality Property

Lathi & Ding pp. 108-109

Given ( ) ( ), then

( ) ( )

and

( ) ( )

g t G f

g t G f

G t g f

See illustration on next page for anexample!

EE 442 Fourier Transform 24

Illustration of Fourier Transform Duality

1( )G f

2( )G f

1( )g t

2( )g t

1

2

T1

2

T t

1

1

T1

1

T

f

ft

1T

1

1

T1

1

T

1T

1

2

T1

2

T

1

2

T1

2

T

1

1

1f

T

Lathi & Ding; Page 109

EE 442 Fourier Transform 25

Time-Scaling Property

Given ( ) ( ),

then for a real constant a,

1( ) ( )

g t G f

fg at G

a a

Time compression of a signal results in spectral expansion and time expansion of a signal results in spectral compression.

Lathi & Ding; pp. 110-112

EE 442 Fourier Transform 26

Time-Shifting Property

002

Given ( ) ( ),

then ( ) ( )j ft

g t G f

g t t G f e

Delaying a signal by t0 seconds does not change its amplitude spectrum, but the phase spectrum is changed by -2ft0.

Note that the phase spectrum shift changes linearly with frequency f.

Frequency f

Frequency 2f

=

= 2

Lathi & Ding; pp. 112-113

27

( )g t

( ) ( )g t G f

( )G f

fBB

fCf

( )g t ( )cos(2 )Cg t f

Cf

2A

A

t

t

( )g t2B

Lathi & Ding; pp. 114-119

Frequency-Shifting Property

0

0

2

Given ( ) ( ),

then ( ) ( )j f t

g t G f

g t G f fe

aka Modulation Property

Multiplication of a signal g(t) by the factor places G(f) centered at f = fC.

cos(2 )Cf

EE 442 Fourier Transform

EE 442 Fourier Transform 28

Frequency-Shifting Property (continued)

Note phase shift

0( )cos(2 )g t f

0( )sin(2 )g t f

( )g t( )G f

( )G f

( )g f

2

2

a b

d c

e f

Lathi & Ding; Figure 3.21 Page 115

Multiplication of a signal g(t) by the factor places G(f) centered at f = fC.

0cos(2 )f

EE 442 Fourier Transform 29

Time Differentiation & Time Integration Property

Lathi & Ding; pp. 121-123

12

Given ( ) ( )

For time differentiation:

( )2 ( )

( )and 2 ( )

For time integration:

( )( ) (0) ( )

2

nn

n

t

g t G f

dg tj f G f

dt

d g tj f G f

dt

G fg d G f

j f

EE 442 Fourier Transform 30

Area Under g(t) & Area Under G(f) Properties

Given ( ) ( ),

Then ( ) (0)

g t G f

g t dt G

That is, the area under a function g(t) is equal to the value of its Fourier transform G(f) at f = 0.

Given ( ) ( ),

Then (0) ( )

g t G f

g G f df

That is, the value of a function g(t) at t = 0 is equal to the area under its Fourier transform G(f).

EE 442 Fourier Transform 31

Conjugate Functions Property

Given ( ) ( ),

Then for a complex-valued time function g(t),

we have

* ( ) * ( )

where the star symbol (*) denotes the complex

conjugate operation.

The corollary to this property is

* ( ) * ( )

g t G f

g t G f

g t G f

--- End of Fourier Transform Property Slides ---

EE 442 Fourier Transform 32

Some Important Fourier Transform Pairs

Impulse Function Impulse train Unit Rectangle Pulse Unit Triangle Pulse Exponential Pulse Signum Function

Refer to Handout

on FT Pairs

EE 442 Fourier Transform 33

The Impulse Function (t) (Impulse Function)

( ) 0 for all t 0t

( ) 1t dt

EE 442 Fourier Transform 34

Fourier Transform of the Impulse Function (t)

2 2 (0)( ) ( ) 1

because the exponential function to zero power is unity.

Therefor, we can write (t) 1

j ft j fF t t dte e

t f

g(t) =(t) G(f) =1 1

(t) is often called the Dirac delta function

EE 442 Fourier Transform 35

Inverse Fourier Transform of the Impulse Function (t)

1 2 (0)( ) ( ) 1

because the exponential function to zero power is unity.

Therefor, we can write 1 (t)

j ft j tF f f dfe e

This is just a DC signal. A DC signal is zero frequency.

t f

g(t) G(f) = (f) 1

0

EE 442 Fourier Transform 36

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 37

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 38

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

EE 442 Fourier Transform 39

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

Lathi & Ding; pp. 104

EE 442 Fourier Transform 40

Fourier Transform of Sinusoidal Functions

2 2

2 2 21 12 2

Taking ( ) and ( )

We use these results to find FT 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 jj

f f f f

ft ft

ft ft

ft ft

e e

e e e

2

1212

Therefore,

cos(2 ) ( ) ( ) , and

sin(2 ) ( ) ( )

c c

c c

c cf t j f t

j

ft f f f f

ft f f f f

e

f f fc fc

-fc

-fc

sin(2fct) cos(2fct)

Lathi & Ding; pp. 105

*

EE 442 Fourier Transform 41

Fourier Transform of Signum (Sign) Function

1, 0

sgn( ) 0, 0

1, 0

for t

t for t

for t

t

+1

-1

Sgn(t)

We approximate the signum function using

exp( ), 0

( ) 0, 0

exp( ), 0

at for t

g t for t

at for t

t

+1

-1

2 2

2 20

4 1( ) , because

(2 )

4 1lim

(2 )a

j fG f

a f j f

j f j

a f f j f

Lathi & Ding; pp. 105-106

EE 442 Fourier Transform 42

Summary of Several Fourier Transform Pairs

Lathi & Ding; Table 3.1 Page 107;

For a more complete Table of Fourier

Transforms ------------

See also the Fourier

Transform Pair Handout

EE 442 Fourier Transform 43

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 44

Re

f

fc

-fc

2

A

2

A

cos(2 )A f t

Blue arrows indicate positive phase directions

Fourier Transform of Cosine Signal

cos(2 ) ( ) ( )2

c c c

AA f t f f f f

3D View Not in

Lathi & Ding

ES 442 Fourier Transform 45

Fourier Transform of Cosine Signal (as usually shown in textbooks)

t

0f

FT

f

Real axis

0( )f cos( )t

0f

0T 0T

0

0

1f

T

EE 442 Fourier Transform 46

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 90 from cos(x) to get sin(x)

EE 442 Fourier Transform 47

Fourier Transform of Sine Signal (as usually shown in textbooks)

sin( )t

t

0( )f

0f0f

2

Bj

2

Bj

FT

Imaginary axis

f

B

EE 442 Fourier Transform 48

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 49

Even, Odd Relationships for Fourier Transform (1)

Re Re

Re Re

g(t)

g(t)

G(f)

G(f)

Im Im

Im Im

Real; Even Real; Even

Real; Odd Imaginary; Odd

f

f

t

t

From: Frederic J. Harris, Trignometric Transforms, Spectral Dynamics Corporation, 1976.

EE 442 Fourier Transform 50

Re Re

Re Re

g(t)

g(t)

G(f)

G(f)

Im Im

Im Im

Imaginary; Even Imaginary; Even

Imaginary; Odd Real; Odd

f

f

t

t

Even, Odd Relationships for Fourier Transform (2)

From: Frederic J. Harris, Trignometric Transforms, Spectral Dynamics Corporation, 1976.

EE 442 Fourier Transform 51

Effect of the Position of a Pulse on Fourier Transform

2 2

( ) Re( ( )) Im( ( ))G f G f G f

t

t

f

f

This time shifted pulse is both even and odd.

Even function.

Both must be

identical.

( )G f

EE 442 Fourier Transform 52

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 interesting.

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 text).

ES 442 Fourier Transform 53

Jean Joseph Baptiste Fourier

March 21, 1768 to May 16, 1830

ES 442 Fourier Transform 54