1

Review of Continuous-Time Fourier Series

39.3.........)(1

38.3...............)(

2)()(

T periodic x(t)

SeriesFourier Time-Continuous

0

0

0

eqndtetxT

aAnalysis

eqneatxSynthesis

TTtxtx

tjk

T

k

k

tjkk

2

Example 3.5

T/2T1-T/2 -T1

This periodic signal x(t) repeats every T seconds.x(t)=1, for |t|<T1 , and x(t)=0, for T1 <|t|< T/2

Fundamental period= T,Fundamental frequency Choosing the period of integration to be between-T/2 and +T/2. Use eqn 3.39 to get at Fourier Series Coefficients.

3

Example 3.5 continued

T

Tdt

T

T

TT

12

T

10

a

0k i.e. first, period, aover

valueaverageor ermconstant t dc, get the usLet

dtt

T

2jk

x(t)eT

1dt

t0

jkx(t)e

T

1k

a

1

1

400

00

1010

1

1

01

1

)(1

|)sin(2)sin(2

ka

)(|)sin(2)sin(2

ka

)sin(2]

2[

2k

a

,|1

dtt

0jk

eT

1k

a

0.k have we

-:harmonicsorder higher andorder first lfundamentaFor

dtt

T

2jk

x(t)eT

1dt

t0

jkx(t)e

T

1k

a

1

0

10

1

0

10

0

10

0

0

kk

kk

TjkTjk

TT

tjkT

T

TT

XTT

T

Tk

Tk

XT

k

TkT

Tk

Tk

j

ee

Tk

eTjk

5

6

7

8

Continuous-Time Fourier Transform

)(txsignalAperiodic

)(~

txsignalPeriodic

1T1T 0

0 1T1T TT)()(.2||)()(

~~

txtxTTtfortxtx

2T

9

Continuous-Time Fourier Transform

2

2

~

~

0

~~

~

0

0

)(x1

)(x

2)(x)(x

T periodic (t)x

SeriesFourier Time-Continuous

T

T

tjkk

k

tjkk

dtetT

aAnalysis

eatSynthesis

TTtt

10

0

0

0

)(

)(5.4........)()(

:)(

.)(1

,2||0)(

.)(1

,2||)()(

2

2

~

kk

tj

k

tjkk

T

T

tjkk

jXTa

eqnanalysisorFTeqndtetxjX

asTaofjXenvelopetheDefining

dtetxT

aTtfortxSince

dtetxT

aequationAnalysisThe

TtfortxtxSince

Continuous-Time Fourier Transform

11 .8.4..)(2

1)(

,,,),()(,

)(2

1)(,2

,)(1

)(

)(

).(1

00

~

00

~

0

0

~

~

0

0

0

0

FTInverseoreqnsynthesiseqndejXtx

dktxtxTAs

ejkXtxTwith

ejkXT

tx

eatxeqnsynthesisfrombut

jkXT

ahaveWe

tj

k

tjk

k

tjk

k

tjkk

k

Continuous-Time Fourier Transform

12

9.4........)()(

)(

8.4..)(2

1)(

.

eqndtetxjX

eqnanalysisorFT

eqndejXtx

FTInverseoreqnsynthesis

tj

tj

Continuous-Time Fourier Transform Pair Equation

13

)(|)(|

)}(Im{)}(Re{)(

)()(

)()(

)(2

1)(

Xj

FT

t

t

tj

tj

eX

XjXX

Xtx

dtetxXAnalysis

deXtxSynthesis

TransformFourierTimeContinuous

14

jatue

eja

edtee

dtetxX

tuetx

Example

FTat

tja

jattjat

t

t

tj

at

1)(

|1

)()(

)()(

1.4

0)(

0

)(

0

15

).(tan)(,1

|)(|,01

)(

4.1 Example

1

22 ajX

ajXa

jatue FTat

An exponential time function and its Fourier Transform

16

Bode Plotrepresentationof the FourierTransform of

an exponentialtime function

17

18

19

20

21

Properties of Fourier Transform

• Linearity• Time Shifting• Conjugation• Differentiation in the time-domain• Integration in the time-domain• Time and Frequency Scaling.• Duality• Parseval’s Relation• Convolution• Multiplication

22

FT Property associated with Linearity

.)()()()(

)()(

)()(

2121

22

11

jbXjaXtbxtax

jXtx

jXtx

FT

FT

FT

23

FT Properties associated with Time shifting & Conjugation

).()(then

-:nConjugatio

).()(then

-:Shifting Time

),()(

**

00

jXtx

jXettx

jXtx

FT

tjFT

FT

24

FT Properties associated with Differentiation & Integration in Time Domian

).()0()(1

d)(then

-:domain Time in then Integratio

).()(d

then

-:Domain timein theation Differenti

).()(

t

-

XjXj

x

jXjdt

tx

jXtx

FT

FT

FT

25

FT Properties associated with Time & Frequency Scaling.

).()( have we-1a by taking

).(||

1)(then

-:ScalingFrequency and Time

).()(

jXtx

a

jX

aatx

jXtx

FT

FT

FT

26

FT Properties associated with Duality

.||,1)(sin

)(

.sin2

)(||,1)(

-:Duality

).()(

22

1111

WpulsejXt

Wttx

TjXTtpulsetx

jXtx

FT

FT

FT

27

FT Properties –Duality continued.

4.25) eqn. analysis ofation differenti from (Derive

.)(

)(

4.24) eqn. synthesis ofation differenti from (Derive

).()(

-:domainfrequency & in timeation differenti ofDuality

).()(

d

jXtjtx

jXjdt

tdx

jXtx

FT

FT

FT

28

FT Properties – Duality continued.

}.{))(()(

).()(}{

-:shiftingfrequency &shifting timeofDuality

).()(

0

0

0

0

shiftingFrequencyjXtxe

jXettxshiftingtime

jXtx

FTtj

tjFT

FT

29

FT Properties-Duality continued

domian}.frequency in on {integrati

.d)()()0()(1

. domain} in timeon {integrati

).()0()(1

d)(

-domainfrequency in n integratio &

domain in timen integratio ofDuality

).()(

-

t

-

xtxtxjt

XjXj

x

jXtx

FT

FT

FT

30

FT Properties- Duality continued

.d|)(|2

1d|)(|

-:Relation sParseval'

).()(

-

2

-

2

jXttx

jXtx FT

31

FT Property associated with Convolution in time domain

domain) (freqtion multiplicadomain) (timen convolutio

).()()(*)()(

)()(

)()(

FT

FT

FT

FT

jHjXthtxty

jHth

jXtx

32

FT Property associated with Multiplication in time domain

domain.equency in time/fr

cation n/multipliconvolutio ofDuality

.)(()(2

1

)(*)(2

1)().()(

)()(

)()(

djHjX

jHjXthtxty

jHth

jXtx

FT

FT

FT