EC3202 / MC3207Signals and Systems

Jonghyun Choijhc@gist.ac.kr

Slides adapted from Kuk-Jin Yoon

Continuous Time Fourier Transform

Part I

• Signals• Systems• LTI Systems• Con2nuous Time Fourier Series• Discrete Time Fourier Series

What We’ve LearnedBefore the Mid-term

Review: Continuous Time Fourier Series

Review: Continuous Time Fourier Series

Fourier Transform

! " = $%&'(

()% *+%,-.

Fourier Series →)% =

1123

! " *'+%,-.4"Fourier Transform →

)% =112'(

(! " *'+%,-.4"

5 67 where 7 = 879

Inverse Fourier Transform

! " = 12&'()

)* +, -./01,

Continuous Time Fourier Transform (CTFT)

• Aperiodic con+nuous-+me signals can be represented as an infinite-length discrete-frequency signals

• Aperiodic signal !(#) → periodic extension of !(#)• Assuming that !(#) is repeated every %& seconds

and deno+ng it as !'((#)• if we take the limit as %& → ∞, we obtain a precise

model of an aperiodic signal for which all rules that govern periodic signals can be applied

From CTFS to CTFT• By three steps1. Make aperiodic signal !(#) to a fake periodic

signal %!(#) with period &2. Represent %!(#) into a Fourier Series3. Let & go to ∞

Step 1: Fake Periodic Signal• We assume that an aperiodic signal x(t) has finite

dura7on, i.e., !(#) = 0 for |#| > )/2, for some ). Since !(#) is aperiodic, we first construct a periodic signal ,!(#):

,!(#) = !(#),

for −)/2 < # < )/2, and ,!(# + )) = ,!(#)

Step 2: !"($) to CTFS• Since x (̃t) is periodic, we may express x (̃t) using

Fourier Series:

&' ( = *+,-.

./+01+234

where /+ = 56 ∫6 &' ( 0-1+2348(

• We can represent /+ by

/+ =1:;-.

.' ( 0-1+2348(

where

' ( =&' ( , −:2 < ( < :

20, ( > :

2

Step 2: !"($) to CTFS

• We define & '( = ∫+,, - . /+012345., then

61 =18 & '9(:

Consequently, subs2tu2ng this into ;- . =∑1=+,, 61/01234

;- . = >1=+,

, 18 & '9(: /01234

= >1=+,

, 12@ & '9(: /01234(:

(Cont’d)

Step 3: Let ! go to ∞• Note that x ̃(t) is the periodic padded version of x(t). • When the period T → ∞, the periodic signal x ̃(t) approaches

x(t). Therefore #$ % → $(%)

as ! → ∞.• Moreover, when ! → ∞, or equivalently )* → 0, the limit

of the sum becomes an integral as

lim/0→*12345

5 128 9 :;)* <=2/0>)* = @

45

5 128 9 :;)* <=2/0A)*

RewriEng the last equaEon,

$ % = 128@45

59 :;)* <=2/0>A)*

CTFT, a formal version• The Fourier Transform X(jω) of a signal x(t) is given

by

! "# = %&'

'( ) *&+,-.)

and the inverse Fourier Transform is given by

( ) = 121%&'

'! "# *+,-.#

(analysis equa=on)

(synthesis equa=on)

Called Fourier Transform pair.

Next Class• Details about Con.nuous .me Fourier Transform