8Continuous-TimeFourier Transform

In this lecture, we extend the Fourier series representation for continuous-time periodic signals to a representation of aperiodic signals. The basic ap-proach is to construct a periodic signal from the aperiodic one by periodicallyreplicating it, that is, by adding it to itself shifted by integer multiples of anassumed period To. As To is increased indefinitely, the periodic signal then ap-proaches or represents in some sense the aperiodic one. Correspondingly,since the periodic signal can be represented through a Fourier series, thisFourier series representation as To goes to infinity can be considered to be arepresentation as a linear combination of complex exponentials of the aperi-odic signal. The resulting synthesis equation and the corresponding analysisequation are referred to as the inverse Fourier transform and the Fouriertransform respectively.

In understanding how the Fourier series coefficients behave as the peri-od of a periodic signal is increased, it is particularly useful to express theFourier series coefficients as samples of an envelope. The form of this enve-lope is dependent on the shape of the signal over a period, but it is not depen-dent on the value of the period. The Fourier series coefficients can then beexpressed as samples of this envelope spaced in frequency by the fundamen-tal frequency. Consequently, as the period increases, the envelope remains thesame and the samples representing the Fourier series coefficients become in-creasingly dense, that is, their spacing in frequency becomes smaller. In thelimit as the period approaches infinity, the envelope itself represents the sig-nal. This envelope is defined as the Fourier transform of the aperiodic signalremaining when the period goes to infinity.

Although the Fourier transform is developed in this lecture beginningwith the Fourier series, the Fourier transform in fact becomes a frameworkthat can be used to encompass both aperiodic and periodic signals. Specifical-ly, for periodic signals we can define the Fourier transform as an impulse trainwith the impulses occurring at integer multiples of the fundamental frequencyand with amplitudes equal to 27r times the Fourier series coefficients. Withthis as the Fourier transform, the Fourier transform synthesis equation in factreduces to the Fourier series synthesis equation.

Signals and Systems8-2

As suggested by the above discussion, a number of relationships exist be-tween the Fourier series and the Fourier transform, all of which are importantto recognize. As stated in the last paragraph, the Fourier transform of a peri-odic signal is an impulse train with the areas of the impulses proportional tothe Fourier series coefficients. An additional relationship is that the Fourierseries coefficients of a periodic signal are samples of the Fourier transform ofone period. Thus the Fourier transform of a period describes the envelope ofthe samples. Finally, the Fourier series of a periodic signal approaches theFourier transform of the aperiodic signal represented by a single period as theperiod goes to infinity.

We now have a single framework, the Fourier transform, that incorpo-rates both periodic and aperiodic signals. In the next lecture, we continue thediscussion of the continuous-time Fourier transform in particular, focusingon some important and useful properties.

Suggested ReadingSection 4.4, Representation of Aperiodic Signals: The Continuous-Time Four-

ier Transform, pages 186-195

Section 4.5, Periodic Signals and the Continuous-Time Fourier Transform,pages 196-202

Continuous-Time Fourier Transform

FOURIER REPRESENTATION OF APERIODIC SIGNALS

x Mt

(""N

x(t) = x(t)

T1 TO2

It' < 20

As To -- o x(t) -- x(t)

- use Fourier series to represent x(t)

- let To---ooto represent x(t)

TRANSPARENCY8.1Representation of anaperiodic signal as aperiodic signal withthe period increasingto infinity.

MARKERBOARD8.1(a)

Cw%-nwnsA -TWAe

Ferio. SAsDg

Nce pevb~e.t.

T . 4,)

r..4

(00,-TO

Signals and Systems

TRANSPARENCY8.2Fourier seriesrepresentation of theaperiodic signal 2(t).

(N

x(t) = x(t)

+00~

X~t ak e jkwot

k=-coTo /2

ak=

+f0a To fX (t)

T,2

21r=0 -T

i(t) e -jkwot dt

e-jkwot

TRANSPARENCY8.3Representation ofthe Fourier seriescoefficients assamples of theenvelope X(w).

__1aTO

Define:

Then:

+ co

fX(t) e- jkco 0t d

+ 00

X(W) f x (t) e-i" dt-00

Toak X(W) I=kcoo

X(w) is the envelope of To ak

+00

k=-coe jkoot = X(kwO) ejkwot

k=-00

+00

x(t) = 27 X(kco) kk=-Oo

r-KT-TI T(0- . T,

Continuous-Time Fourier Transform

+00

X(o) = x(t) ei jWt dt00

x(t) = E X(kwo) ejkwo tWOk=-0O

Fourier transform- analysis

As To-- oo,

coo -* 0 1x t) - x (t), we doE -+a(

x(t) = 2-2o

X(o) e jcot dcoInverseFourier transform- synthesis

TRANSPARENCY8.4The analysis andsynthesis equationsassociated with theFourier transform.

x(t)

0 T,

X w)

-- e" -- % N

\ I \~ ,.

TRANSPARENCY8.5Transparencies 8.5-8.7 illustrate that theFourier seriescoefficients of 2(t) aresamples of the Fouriertransform of oneperiod. As the periodTo increases, thesamples become moreclosely spaced. Here isshown x(t) (i.e., oneperiod of :(t)) and itsFourier transform.

Signals and Systems

TRANSPARENCY8.6z(t) and its Fourierseries coefficientswith To = 4T1. [As thefigure is drawn, To andT1 are not to scale.]

see

0 T, TO

X(w)

T, - 4T1

TRANSPARENCY8.7±(t) and its Fourierseries coefficientswith To = 8T1. [As thefigure is drawn, To andT, are not to scale.]

JT

x(t)1-T1 0 T, To

/

'it/I

90

F7-1 *..

T,1X(W)

t V T - S T 1t. e I

Continuous-Time Fourier Transform

OCAvhawA&-Tt-&

foe~ar Tpiwd . 4S

SIV4

3is

X..Xvt~*-b IS

ExCapkt (Td' 4.'1)

e.e

~-~-

I I~

~ c*w-+ a4-30&

=i )

-i

- e.

1Example 4.7: eat u(t) +-+ a > o

a+)j>

TRANSPARENCY8.8An exponential timefunction and itsFourier transform.[Example 4.7 from thetext.]

V~V

MARKERBOARD8.1(b)

IX(MOI

1/aV2

-a a

-a

vr/2

1/4

- r/2

Signals and Systems

TRANSPARENCY8.9Bode plotrepresentation of theFourier transform ofan exponential timefunction.

B20 log10 iX()I =' decibels (dB)Bode Plot4X(W)

0 dB

-20

-40

-601 I I 1

-r/4

-~ /2

-37r/4

0.1a

0.1a

10a 100a

10a 100a

frequency (CO )

TRANSPARENCY8.10Relationship betweenthe Fourier seriescoefficients of aperiodic signal and theFourier transform ofone period.

Fourier Series coefficients equal

1 times samples of FourierTotransform of one period

7(t)

27>T /2

x(t) = one period of x(t)

x(t) + k

x(t) -) X((I)

- _ ~ _ ~ I I

.111 1

. .r __

Continuous-Time Fourier Transform

x(t)

see

0

X(w)

T, - 4T,

TRANSPARENCY8.11The Fourier transformof x(t) as the envelopeof the Fourier seriescoefficients of 2(t).As the period Toincreases, the samplesbecome more closelyspaced. Thistransparency showsx(t) and its Fouriertransform.[Transparency 8.5repeated]

TRANSPARENCY8.12±(t) and its Fourierseries coefficientswith To = 4T1.[Transparency 8.6repeated]

0 T1

SX( )

-1 - -. I

N. I- -..~

\ ' .'/

W-K

-.01

Signals and Systems8-10

TRANSPARENCY8.132t(t) and its Fourierseries coefficientswith To = 8T 1.[Transparency 8.7repeated]

69 1 1

I(t)1- TO -T, 0 T, T,

-r-IN.

/ii'

Xlw)t - 1Tok T

TRANSPARENCY8.14Representation of theFourier series in termsof the Fouriertransform.

FOURIER TRANSFORM OF A PERIODIC SIGNAL R(t)

(t) + ak

N(t) X

Fourier seriescoefficients

Fourier transform

+00

i(t) - X(o) eict dw21r _ .,

+00

2r k=- 00

+00

27r ak f (w-kco0 ) e- jet dw

I71 19,1

Continuous-Time Fourier Transform8-11

1. x(t) APERIODIC

- construct periodic signal x(t) for

which one period is x(t)

- x(t) has a Fourier series

- as period of 'x(t) increases,

xM(t)-.x(t) and Fourier series of

x(t)-- Fourier Transform of x(t)

TRANSPARENCY8.15Illustration of theFourier seriescoefficients and theFourier transform fora symmetric squarewave.

TRANSPARENCY8.16Summary of thedevelopment of theFourier transformfrom the Fourierseries. [The periodicsignal has beencorrected here to read

o(t), not x(t).]

Signals and Systems

8-12

TRANSPARENCY8.17Summary of therelationship betweenthe Fourier transformand Fourier series fora periodic signal.

2. x(t) PERIODIC, x(t) REPRESENTS ONE PERIOD

- Fourier series coefficients of x(t)

= (1/To) times samples of Fourier

transform of x(t)

3. S(t) PERIODIC

-Fourier transform of x(t) defined as

impulse train:+oo

27rak 6(W - kwo)

TRANSPARENCY8.18Summary exampleillustrating some ofthe relationshipsbetween the Fourierseries and Fouriertransform.

X(W) =

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