Signals & systems Ch.3 Fourier Transform of Signals and LTI System

Signals & systemsCh.3 Fourier Transform of

Signals and LTI System

04/21/23

Signals and systems in the Frequency domain

04/21/23 KyungHee University 2

Time [sec]

Frequency [sec-1, Hz]

Fourier transform

3.1 Introduction

Orthogonal vector => orthonomal vector

What is meaning of magnitude of H?

)2/1,2/1()2/1,2/1( 1 vvo

1,0,][110 1101 jiforjivvvvvvvv jioo

),(),( 1010 HHHhhh Fourier

110 vHvHh o

110 vhHvhH o

Any vector in the 2- dimensional space can be represented by weighted sum of 2 orthonomal vectors

Fourier Transform(FT)

Inverse FT

3.1 Introduction cont’

1 1 1 1( , , , )2 2 2 2ov

Orthogonal

? 1 0 1 10 1 1o ov v v v v v

[ ] , 0,1,2,3i jv v i j for i j

Any vector in the 4- dimensional space can be represented by weighted sum of 4 orthonomal vectors

Orthonormal

function?][)()(

2)cos()( 00 jidttvtv

TwheretkAtv

][)()(1

)( **0

0 kmdttvtvT

whereetvT kmkm

T km dteT

dttvtvT

0)(* 1)()(

1 1 1 1( , , , )2 2 2 2

3.2 Complex Sinusoids and Frequency Response of LTI Systems

cf) impulse response

.][ ][][][ )(

ekhknxkhny

,)( ][][ njj

kjnj eeHekheny

kjj ekheH

How about for complex z? (3.1) nznx ][

,)( )( )()( )( tjjtjtj ejHdehedehty

dehjH j

How about for complex s? (3.3) stetx )(

.)()( )})(arg{( jHtjejHty

Magnitude to kill or not? Phase delay

Fourier transform

1[ ] ( )

2j j nx n X e e d

( ) [ ]j j n

X e x n e

( ) ( )2

j tx t X j e d

( ) ( ) j tX j x t e dt

discrete time

Continuous time

Time domain frequency domain

1( ) ( )

2stx t X s e ds

j ( ) ( ) stX s x t e dt

Laplace transform

11[ ] ( )

2nx n X z z dz

j ( ) [ ] n

X z x n z

z-transform

(periodic) -

(discrete)

(discrete) -

(periodic)

3.6 DTFT: Discrete-Time Fourier Transform

(discrete) (periodic)

deeXnx njnj )(

njj enxeX ][)(

)(][ jDTFT eXnx

(3.31)

(3.32)

(a-periodic) (continuous)

Example 3.17 DTFT of an Exponential Sequence

Find the DTFT of the sequence ][][ nunx n

Solution :

njnnjnj eenueX0

][)( 1,1

sincos1

2/122/1222 )cos21(

)sin)cos1((

sinarctan()}(arg{

x[n] = nu[n]. magnitude

= 0.5 = 0.9

3.6 DTFT Example 3.17

Example 3.18 DTFT of a Rectangular Pulse

,1][ ].[nxLet Find the DTFT of

Solution : (square) (sinc)

Figure 3.30 Example 3.18. (a) Rectangular pulse in the time domain. (b) DTFT in the frequency domain.

,4,2,0,12

,4,2,0,1

2/)12(2/)12(

)2/2/2/

2/)12(2/)12(2/)12(

MjMjMjMjj

eeeeeX

)2/sin(

)2/)12(sin()(

12)2/sin(

)2/)12(sin(lim

,,4,2,0

)2/sin(

)2/)12(sin()(

Example 3.19 Inverse DTFT of a Rectangular Spectrum

Find the inverse DTFT of

Solution : (sinc) (square)

0),sin(1

njdenx njW

)sin(1

lim]0[0

)sin(1

][ Wnn

nx )/(][

Wnncsi

Figure 3.31 (a) Rectangular pulse in the frequency domain. (b) Inverse DTFT in the time domain.

Example 3.20 DTFT of the Unit ImpulseFind the DTFT of ][][ nnx

Solution : (impulse) - (DC) 1][)(

j eneX

1][ DTFTn

Example 3.21 Find the inverse DTFT of a Unit Impulse Spectrum.

Solution : (impulse train) (impulse train)

denx nj)(

j keX )2()(

1 DTFT

3.6 DTFT Example 3.20-21

Example 3.23 Multipath Channel : Frequency Response

]1[][][ naxnxny

Solution : jj aeeH 1)( })arg{(1 ajea })arg{sin(})arg{cos(1 aajaa

2/122/1222 }))arg{cos(21(}))arg{(sin}))arg{cos(1(()( aaaaaaaeH j

( arg{ })

1 1( )

1 1 cos( arg{ }) sin( arg{ })inv j

j aH e

a e a a j a a

}))arg{cos(21(

eHaaaeH

|)(| jeH (a) a = 0.5ej2/3. (b) a = 0.9ej2/3.

|)(| jinv eH (a) a = 0.5ej2/3. (b) a = 0.9ej2/3.

3.7 CTFT

(continuous aperiodic) (continuous aperiodic)

Inverse CTFT (3.35)

dwejwXtx jwt)(

CTFT (3.26)

dtetxjwX jwt)()(

)()( jwXtx CTFT

Condition for existence of Fourier transform:

3.7 CTFT Example 3.24

Example 3.24 FT of a Real Decaying ExponentialFind the FT of )()( tuetx at

Solution : 0,0

adte at Therefore, FT not exists.

dtedtetuejwXa

tjwajwtat

)()(,0

)/arctan()}(arg{ awjwX

LPF or HPF? Cut-off from 3dB point?

)()( tuetx at

Example 3.25 FT of a Rectangular Pulse

TtTtx Find the FT of x(t).

Solution : (square) (sinc)

0),sin(21

)()( 0

dtedtetxjwXT

jwtjwt

2)sin(2

lim,0 TwTw

Example 3.25. (a) Rectangular pulse. (b) FT.

)sin(2

)( 0wTw

jwX )/(2)( 00 wTincsTjwX

)sin(2)( 0

0/)sin(,

0/)sin(,0)}(arg{

wwTjwX

Example 3.26 Inverse FT of an Ideal Low Pass Filter!!Fine the inverse FT of the rectangular spectrum depicted in Fig.3.42(a) and given by

WwWjwX

Solution : (sinc) -- (square)

)()()sin(1

,/)sin(1

0),sin(1

Wtncsi

WtxorWt

3.7 CTFT Example 3.27-28

Example 3.27 FT of the Unit Impulse

)()( ttx

Solution : (impulse) - (DC)

1)()( dtetjwX jwt 1)( FTt

Example 3.28 Inverse FT of an Impulse Spectrum

Find the inverse FT of )(2)( wjwX

Solution : (DC) (impulse)

dwewtx jwt

)(21 wFT

Example 3.29 Digital Communication SignalsRectangular (wideband)

Separation between KBS and SBS. Narrow band

2/)),/2cos(1)(2/()(

TtTtAtx

Figure 3.44 Pulse shapes used in BPSK communications. (a) Rectangular pulse. (b) Raised cosine pulse.

Figure 3.45 BPSK (a) rectangular pulse shapes

(b) raised-cosine pulse shapes.

Solution : 22/

T rr AdtAT

dtTtTtT

dtTtAT

)]/4cos(2/12/1/2cos(21[4

))/2cos(1)(4/(1

2/ 000

the same power constraints

wTjwX r

)sin(2)( 2/0

fTjfX r

)sin()( 0'

))/2cos(1(3

jwtc dteTtjwX

)/2(2/

jwtc dtedtedtejwX

)2/sin(

)2/)/2sin((

)2/)2sin((

2)2/sin(

wTjwX c

))/1(sin(

fTjfX c

rectangular pulse.

One sinc

Raised cosine pulse 3 sinc’s

The narrower main lobe, the narrower bandwidth. But, the more error rate as shown in the time domain

Figure 3.47 sum of three frequency-shifted sinc functions.

Fourier transform

1[ ] ( )

2j j nx n X e e d

( ) [ ]j j n

X e x n e

1( ) ( )

2j tx t X j e d

( ) ( ) j tX j x t e dt

Discrete time

Continuous time

Time domain frequency domain

3.9.1 Linearity Property

)()()(][][][

)()()()()()(

jjjDTFT

ebYeaXeZnbynaxnz

jwbYjwaXjwZtbytaxtz

2)( tytxtz

)2/sin(2

1)4/sin(

)2/sin())/(1(][)(

)4/sin())/(1(][)(

kkkYty

kkkXtx

3.9.1 Symmetry Properties

Real and Imaginary Signals

dtetxjwX

)()((3.37)

(real x(t)=x*(t)) (conjugate symmetric)

dtetxjwX twj )()()( )()( jwXjwX (3.38)

3.9.2 Symmetry Properties of FT

EVEN/ODD SIGNALS

(even) (real)(odd) (pure

imaginary)

)(txe )}(Re{ jX

)(txo )}(Im{ jXj

For even x(t), .* jwXdexdtetxjwX jwtjw

3.10 Convolution Property

(convolution) (multiplication)

dtxhtxthty *

dwejwXtx tjw

dwejwXtx jwt

But given

ddweejwXhty jwjwt

change the order of integration

1dwejwXdeh jwtjw

1dwejwXjwHty jwt

40.3.* jwHjwXjwYtxthty FT

3.10 Convolution Property Example 3.31

Example 3.31 Convolution problem in the frequency domain

tttx sin1Input to a system with impulse response .2sin1 ttth

Find the output .* thtxty

Solution:

wjwXtx FT

wjwHth FT

,* jwHjwXjwYthtxty FT

wjwY .sin1 ttty

Example 3.32 Find inverse FT’S by the convolution property

Use the convolution property to find x(t), where

.sin4 2

wjwXtx FT

jwZ ,1,0

1,1jwZ

ttz FT

Ex 3.32 (p. 261). (a) Rectangular z(t). (b) txtztz )(*)(

3.10.2 Filtering

Continuous time Discrete time(periodic with 2π

Figure 3.53 (p. 263)

Frequency dependent gain (power spectrum) .log20log20 jeHorjwH

jwXjwHjwY kill or not (magnitude)

Example 3.34 Identifying h(t) from x(t) and y(t)

The output of an LTI system in response to an input is .Find frequency response and the impulse response of this system.

tuetx t2 tuety t

Solution:

jwjwX .

jwjwY But

jwYjwH

jwjwjw

jwjwH .tuetth t

sY But note sX

ssH .tuetth t

tue Lat

EXAMPLE 3.35 Equalization(inverse) of multipath channel

jwYjwHjwX inv , jjinvj eYeHeXor

Consider again the problem addressed in Example 2.13. In this problem, a distorted received signal y[n] is expressed in terms of a transmitted signal x[n] as .1,1 anaxnxny .

otherwise

.* nnhnh inv ,1 jjinv eHeH

Then .1

.1 jjDTFT aeeHnh .1

.nuanh ninv

3.11 Differentiation and Integration Properties

1dwejwXtx jwt

dwjwejwXtx

21 .jwjwXFT

EXAMPLE 3.37 The differentiation property implies that

d FTat

. tuaetetuaetuedt

d atatatat

d FTat

dssesXtx

21 .ssXL

3.11 Differentiation and Integration Properties

예제 한 두개

3.11.2 DIFFERENTIATION IN FREQUENCY

dtetxjwX jwt Differentiate w.r.t. ω, ,

dtetjtxjwXdw

Then, .jwXdw

dtjtx FT

Example 3.40 FT of a Gaussian pulse

Use the differentiation-in-time and differentiation-in-frequency properties for the FT of the Gaussian pulse, defined by and depicted in Fig. 3.60.

21 tetg

48.3.2 22

ttgettgdt

Figure 3.60 (p. 275) Gaussian pulse g(t).

,)( jwjwGttgtgdt

d FT jwGdw

dtjtg FTand

jttg FT 1 .jwwGjwG

.22wcejwG Then (But, c=?) .1210 22

dtejG t

.21 22 22 wFTt ee

Laplace transform and z transform

tue Lat

astute Lat

znu zn

znun zn

3.11.3 Integration

dxty 52.3.1

53.3.01

wjXjwXjw

dx FTt

dtu .1w

jwjwUtu FT Ex) Prove

54.3.sgn2

1ttu Note where a=0)()( tuetu at

t We know .21 wFT

.2sgn ttdt

d .2jwjwSgn

Fig. a step fn. as the sum of a constant and a signum fn.

jtu FT 1

since linear

Differentiation and Integration Properties

Common Differentiation and Integration Properties.

jXjtxdt

3.12.1 Time-Shift Property

dtettxetzjZ tjtj 0

jXedexedex tjjtjtj 000

)](arg[]arg[ 0

0 jXtjXe tj

Table 3.7 Time-Shift Properties of Fourier Representations

jnjDTFT

eXennx

jXettx

Fourier transform of time-shifted z(t) = x(t-t0)

Note that x(t-t0) = x(t) * δ(-t0) and 0)( 0tjFT ett

3.12 Time-and Frequency-Shift Properties

Example)

Figure 3.62 1Ttxtz jXejZ Tj 1

TejZ Tj

3.12.2 Frequency-Shift Property

dejZtz

txedejXe

dejXtz

tjtjtj

jXettx tjFT 00

Recall

Table 3.8 Frequency-Shift Properties

FSjk t

jDTFTj n

DTFSjk n

e x t X j

e x t X k k

e x n X e

e x n X k k

3.12.2 Frequency-Shift Property

Example 3.42 FT by Using the Frequency-Shift Property

Solution: We may express as the product of a complex sinusoid and a rectangular pulse

tz tje 10

,1 )(10 txetz tj

sin2 jXtx FT

10sin10

210)(10

jXtxetz FTtj

3.12 Shift Properties Ex. 3.43

Example 3.43 Using Multiple Properties to Find an FT

23 tuetuedt

dtx tt

Sol) Let and tuetw t3 2 tuetv t

Then we may write tvtwdt

By the convolution and differentiation properties

jVjWjjX

The transform pair

tue FTat

222 tueetv t

3.12 Shift Properties Ex. 3.43

Example 3.43 Using Multiple Properties to Find an FT

23 tuetuedt

dtx tt

Sol) Let and tuetw t3 2 tuetv t

Then we may write tvtwdt

By the convolution and differentiation properties

X s s W s V s

The transform pair

1Laplaceate u ta s

222 tueetv t

sseX s e

seV j e

3.13 Inverse FT: Partial-Fraction Expansions

3.13.1 Inverse FT by using

1FTate u ta s

B sb s b s bX s

s a s a s a A s

B sX s f s

Nkfordk ,,1N roots,

b sX j

partial fraction

10sdte u t for d

N Nd t s k

kk k k

Cx t C e u t X s

3.13 Inverse FT: Partial-Fraction Expansions

3.13.1 Inverse FT by using

tue FTat

ajajaj

bjbjbjX N

jBjfjX

Let then ,jv 0011

1 avavav N

Nkfordk ,,1N roots,

partial fraction

dfordj

tue FTdt

kFTtdk dj

CjXtueCtx k

Inverse FT: Partial-Fraction Expansions

1Laplaceate u ts a

'( ) ( )Laplacex t sX s 2 ( ) 3 '( ) ''( ) 2 ( ) 3 '( )y t y t y t x t x t 22 ( ) 3 ( ) ( ) 2 ( ) 3 ( )Laplace Y s sY s s Y s X s sX s

( ) 2 3( )

( ) 2 3 1 2

Y s s A BH s

X s s s s s

( 1)( )( 1)

B sH s s A A

2( ) ( ) ( )t th t Ae Be u t

( )x t( )h t

( ) ( )* ( )y t x t h t( ) ( ) ( )Y s X s H s( )X s

3.13.2 Inverse DTFT

3.13.2

NjN edeee

NNNNN vvvv

DTFTnk ed

1where

nkk nudCnx

Inverse FT: Partial-Fraction Expansions

1Zna u n

1[ ] ( )Z

ox n n z X z 2 [ ] 3 [ 1] [ 2] 2 [ ] 3 [ 1]y n y n y n x n x n 1 2 12 ( ) 3 ( ) ( ) 2 ( ) 3 ( )Z transform Y z z Y z z Y z X z z X z

1 2 1 1

( ) 2 3( )

( ) 2 3 1 1 2

Y z z A BH z

X z z z z z

(1 )( )(1 )

B zH z z A A

[ ] ( 2 ) [ ]nh n A B u n

[ ]x n[ ]h n

[ ] [ ]* [ ]y n x n h n( ) ( ) ( )Y z X z H z( )X z

3.13.2 Inverse DTFT by z-transform

3.13.2

1 0( 1) 1

N NN N

z zX z

1 1 11 1

NNN N k

z z z d z

d u nd z

nkk nudCnx

3.13 Inverse FT Example 3.45

Example 3.45 Inversion by Partial-Fraction Expansion

Solution:

Using the method of residues described in Appendix B, We obtain

eeCAnd

Hence, nununx nn 3/12/14 2011.5.4

3.13 Inverse FT Example 3.45

Example 3.45 Inversion by Partial-Fraction Expansion

Solution:1

1 2 1 1

61 1 1 1

1 1 16 6 2 3

z z z z

Using the method of residues described in Appendix B, We obtain

5 55 51 6 61 4

1 1 12 1 16 6 3z z

z zC z

5 55 51 6 61 1

1 1 13 1 16 6 3z z

z zC z

Hence, nununx nn 3/12/14 2011.5.4

3.14 Multiplication (modulation) Property

Given and

dvejvXtx jvt

dejZtz tj

dvdejZjvXtztxty tvj

Change of variable to obtain

dedvvjZjvXty tj

jZjXjYtztxty FT 2

dvvjZjvXjZjX Where

jjjDTFT eZeXeYnznxny21

(3.56)

(3.57)

denotes periodic convolution.

Here, and are -periodic. jeX jeZ 2

deZeXeZeX jjjj

Modulation property

23年 4月 21日 MediaLab , Kyunghee University

jZjXjYtztxty

nconvolutiotionmultiplica

jZjXjYtztxty

ionmultilicatnconvolutio

3.14 Modulation Property Ex 3.46

Example 3.46 Truncating the sinc function

Sol) truncated by

otherwise

Mnnhnht

DTFTt eHnh

tH e H e W e d

2/,1jeH

2/12sin

Figure 3.66 The effect of Truncating the impulse response of a discrete-time system. (a) Frequency response of ideal system. (b) for near zero. (c) for slightly greater than (d) Frequency response of system with truncated impulse response.

3.15 Scaling Properties

dteatxetzjZ tjtj

adexajZ

dexajZ aj //1

ajXaatxtz FT //1 (3.60)

3.15 Scaling Properties Example 3.48

Example 3.48 SCALING A RECTANGULAR PULSE

Let the rectangular pulse

Solution :

).sin(2

jwX 2/1).2/()( atxty

).2sin(2

)2sin()2

)2(2)(

wiXjwY

Example 3.49 Multiple FT Properties for x(t) when

}.)3/(1

Solution)jw

jwStuetsFT

1)()()(

)}.3/({)( 2 jwSedw

djjwX wj

we define )3/()( jwSjwY ).(3)3(3)3(3)( 33 tuetuetsty tt

Now we define )()( 2 jwYejwW wj

).2(3)2()( )2(3 tuetytw t

Finally, since ),.()( jwWdw

).2(3)()( )2(3 tutettwtx t

Example 3.49 Multiple FT Properties for x(t) when

Solution)s

sStuetsL

1)()()(

)}.3/({)( 2 sSedw

dssX s

we define )3/()( sSsY ).(3)3(3)3(3)( 33 tuetuetsty tt

Now we define )()( 2 sYesW s

).2(3)2()( )2(3 tuetytw t

Finally, since ).()( sWds

).2(3)()( )2(3 tutettwtx t

3.16 Parseval’s Relationships

? |)(|?|)(| 22 에너지쓴주자가바이올린꽝언제 jwXtx

.|)(|2

).()(|)(|

dwjwXdttx

thatconcludeSo

dwjwXjwXW

dtdwejwXtxW

dwejwXtx

txtxtxthatNote

Representation

Parseval Relation

dwjwXdwjwX

Table 3.10 Parseval Relationships for the Four Fourier Representations

3.16 Parseval’s Relationships Example 3.50

Example 3.50 Calculate the energy in a signal

WnnxLet

sin ( )E = | [ ]| .

Use the Parseval’s theorem

Solution)

1 2 deXE j

||,1{)(][

WeXnx jDTFT

Signals & systems Ch.3 Fourier Transform of Signals and LTI System

Documents

Signals & systems Ch.3 Fourier Transform of Signals and LTI System

Fourier Transform of Signals

BYST SigSys - WS2003: Fourier Rep. 120 CPE200 Signals and Systems Chapter 3: Fourier Representations for Signals (Part I)

6/10/2015 Signals & systems Ch.3 Fourier Transform of Signals and LTI System

· Fourier Representations for Signals: ... Hwei Hsu, “Signals and ... Lecture 40- Transform Analysis of LTI Systems

Fourier Analysis of Video Signals

Fourier Series for CT & DT Signals

Signals and Systems - Lecture 3: Discrete-Time LTI Systems: … · 2018. 10. 10. · Complex Exponential Sequences Periodic Signals Eigenfunctions of LTI Systems The z-Transform Deﬁnition

REPRESENTATIONOFSIGNALSASSERIESOF …nal projection of functions or signals. Several kinds of decomposition are explored: Fourier, Fourier-Legendre, Fourier-Bessel series for 1D signals,

Chapter 3: Fourier Representation of Signals and LTI Systemstwins.ee.nctu.edu.tw/courses/ss_19/class_note.files/S14...Fourier Representation of Signals and LTI Systems Chih-Wei Liu

Extended Fourier Analysis of Signals

SIGNALS & SYSTEMS. Contents of the Lecture Signal & System? Time-domain representation of LTI system Fourier transform and its application Z transform

03 Signals Fourier-VP

1 Chapter 3 Fourier Series Representations of Periodic Signals Chapter 3 Fourier Series Representations of Periodic Signals

10.1 fourier analysis of signals using the DFT 10.2 DFT analysis of sinusoidal signals 10.3 the time-dependent fourier transform Chapter 10 fourier analysis

Chapter 3: Fourier Representation of Signals and LTI Systems

Fourier Transform (for Non-Periodic Signals)

Signals and Systems Fall 2003 Lecture #7 25 September 2003 1.Fourier Series and LTI Systems 2.Frequency Response and Filtering 3.Examples and Demos

Chapter 3: Fourier Representation of Signals and LTI Systemstwins.ee.nctu.edu.tw/courses/ss_17/S14-03-3-Fourier.pdf · Fourier Representation of Signals and LTI Systems ... Uppercase

Signals and Systems Lecture (S4) Fourier Series and …web.mit.edu/16.unified/www/archives 2007-2008/signals...Signals and Systems Lecture (S4) Fourier Series and LTI Systems March