Signals and Systems: Chapter 4 Fourier transform

FOURIER TRANSFORM• Inverse Fourier Transform • Fourier Transform – given xt, we can find its Fourier transform – given , we can find the time domain signal xt – signal is decomposed in

ELEG 3124 SYSTEMS AND SIGNALS

Ch 4 Fourier Transform

Dr Jingxian Wu

wuj@uark.edu

(These slides are taken from Dr Jingxian Wu, University of Arkansas, 2020.)

EE 2000 SIGNALS AND SYSTEMS

• Introduction

• Fourier Transform

• Properties of Fourier Transform

• Applications of Fourier Transform

INTRODUCTION: MOTIVATION

• Motivation:

– Fourier series: periodic signals can be decomposed as the

summation of orthogonal complex exponential signals

jn t

c t

x

n

n exp 0 )

Time domain Frequency domain

INTRODUCTION: TRANSFER FUNCTION

• System transfer function

• System with periodic inputs

t j

ejn t

t jn n

• Introduction

• Fourier Transform

• Properties of Fourier Transform

• Applications of Fourier Transform

FOURIER TRANSFORM

• Inverse Fourier Transform

• Fourier Transform

– given x(t), we can find its Fourier transform

– given , we can find the time domain signal x(t)

– signal is decomposed into the “weighted summation” of complex

exponential functions (integration is the extreme case of summation)

t

2

1)

(

) ( 

X

) ( 

FOURIER TRANSFORM

• Example

– Find the Fourier transform of x(t) = exp(−a|t |) a  0

FOURIER TRANSFORM

• Example

– Find the Fourier transform of x(t) = exp(−at)u(t) a  0

FOURIER TRANSFORM

• Example

– Find the Fourier transform of x(t) =(t −a)

FOURIER TRANSFORM: TABLE

FOURIER TRANSFORM

−+| x(t)| dt  

)()exp(

)(t t u t

• Example

–

• The existence of Fourier transform

– Not all signals have Fourier transform

– If a signal have Fourier transform, it must satisfy the following two

• Introduction

• Fourier Transform

• Properties of Fourier Transform

• Applications of Fourier Transform

1 t X 

x  x2(t)  X2()

)()

()

()

(t X 

]exp[

)()

c j c

e c

cos

|

| c

2 2

|

| c = a +b  = atan(b/a)

phase shift

time shift in time domain ➔ frequency shift in frequency domain

– Phase shift of a complex number c by : 0 cexp( j0) =|c|expj( +0)

PROPERTY: TIME SHIFT

• Example:

– Find the Fourier transform of x(t) = rectt − 2

PROPERTY: TIME SCALING

(

 1 / 2)

( = rect  −

(t X 

)()

(  X* 

(t X 

)()

dx )(

)(t t

)()

(t X 

)()()

()

(t h t

)(

H

)()( H 

X

PROPERTY: CONVOLUTION

• Example

– An LTI system has impulse response

If the input isFind the output

exp)

exp)(

)

)0,

0,

0(a  b  c 

(t X 

 ( ) ( )2

1)

()

t m t

(t G 

)(

2)

(t  g −

G

/ (

rect t

PROPERTY: DUALITY

• Example

– Find the Fourier transform of x(t) =1

t j

e t

x( ) = 0

– Find the Fourier transform of

PROPERTY: SUMMARY

PROPERTY: EXAMPLES

• Examples

– 1 Find the Fourier transform of x(t) = cos(0t)

– 2 Find the Fourier transform of x(t) = u(t)

sgn( ) 1

2

1 )

PROPERTY: EXAMPLES

• Examples

– 3 A LTI system with impulse response

Find the output when input is

  ( )exp

)

)()

()

(t X 

)()(t m t x

– 6 If , find x(t)





j a

X

+

= 1)

(

PROPERTY: DIFFERENTIATION IN FREQ DOMAIN

• Differentiation in frequency domain

– If:

– Then:

)()

(t X 

n

n n

d

X d t

x jt



)

()

()

PROPERTY: DIFFERENTIATION IN FREQ DOMAIN

),()exp( at u t

• Example

– Find the Fourier transform of

PROPERTY: FREQUENCY SHIFT

• Frequency shift

– If:

– Then:

)()

(t X 

)(

)exp(

)(t j0t  X −0

x

• Example

– If , find the Fourier transform X() = rect (  −1)/2 x(t)exp(− j2t)

PROPERTY: PARSAVAL’S THEOREM

• Review: signal energy

2

1

|)(

|

PROPERTY: PARSAVAL’S THEOREM

• Example:

– Find the energy of the signal x(t) = exp(−2t)u(t)

PROPERTY: PERIODIC SIGNAL

• Fourier transform of periodic signal

– Periodic signal can be written as Fourier series

jn t

c t

x

n

n exp 0 )

2 ) (  c   n0

• Introduction

• Fourier Transform

• Properties of Fourier Transform

• Applications of Fourier Transform

APPLICATIONS: FILTERING

• Filtering

– Filtering is the process by which the essential and useful part of a

signal is separated from undesirable components

• Passing a signal through a filter (system)

• At the output of the filter, some undesired part of the signal (e.g noise) is removed

– Based on the convolution property, we can design filter that only

allow signal within a certain frequency range to pass through

(t h t

)(

H

)()( H 

X

APPLICATIONS: FILTERING

• Classifications of filters

Low pass filter

Passband Stop

band High pass filter

Passband Stop

band

Stop band

Stop band Passband Passband

APPLICATION: FILTERING

• A filtering example

– A demo of a notch filter

)(

X

)(

H

)()( H 

X

APPLICATIONS: FILTERING

• Example

– Find out the frequency response of the RC circuit

– What kind of filters it is?

RC circuit

APPLICATION: SAMPLING THEOREM

• Sampling theorem: time domain

– Sampling: convert the continuous-time signal to discrete-time signal

t

p( )  ( )

sampling period

) ( ) ( )

APPLICATION: SAMPLING THEOREM

• Sampling theorem: frequency domain

– Fourier transform of the impulse train

• impulse train is periodic

n

s

s

e T

nT t

n T

• Time domain multiplication ➔ Frequency domain convolution

 ( ) ( )2

1)

()

t p t

n

X T

t p t

APPLICATION: SAMPLING THEOREM

• Sampling theorem: frequency domain

– Sampling in time domain ➔ Repetition in frequency domain

APPLICATION: SAMPLING THEOREM

• Sampling theorem

– If the sampling rate is twice of the bandwidth, then the original

signal can be perfectly reconstructed from the samples

APPLICATION: AMPLITUDE MODULATION

• What is modulation?

– The process by which some characteristic of a carrier wave is

varied in accordance with an information-bearing signal

– Information bearing signal (modulating signal)

• Usually at low frequency (baseband)

• E.g speech signal: 20Hz – 20KHz– Carrier wave

• Usually a high frequency sinusoidal (passband)

• E.g AM radio station (1050KHz) FM radio station (100.1MHz), 2.4GHz, etc

– Modulated signal: passband signal

APPLICATION: AMPLITUDE MODULATION

• Amplitude Modulation (AM)

)2cos(

)()

)

(t

m

)2

APPLICATION: AMPLITUDE MODULATION

• Amplitude Modulation (AM)

2

) ( f A c M f f c M f f c

Amplitude modulation