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Table of Fourier Transform Pairs of Energy Signals

Function

name

Time Domain x(t) Frequency Domain X()

FT xt e jt

Xxtdtxt

F IFT 1 1 e 2 jt xtXdX FX

Rectangle

Pulse 1 2 0 T t tt rect TT elsewhen sinc 2 T T

Triangle

Pulse 1 0 t ttW W W elsewhen 2 sinc 2 W W Sinc Pulse sin() sinc Wt Wt Wt 1 2 rect WW

Exponen-

tial Pulse 0 at ea 22
2a a

Gaussian

Pulse 2 2 exp() 2 t 22

2exp()

2

Decaying

Exponen-

tial exp()()Re0atuta 1 aj Sinc 2 Pulse 2 sincBt 2 1 BB

Rect Pulse

0 0.5 1 1.5 -3-2.5-2-1.5-1-0.500.511.522.53 a r ect a T=1

Sinc Pu

lse -0.5 0 0.5 1 1.5 -3-2.5-2-1.5-1-0.500.511.522.53 a Si n c a W=1

Gaussian Pulse

0 0.5 1 1.5 -3-2 .5-2-1.5-1-0.500.511.522.53 a 2 =1

Triangle Pulse

0 0.5 1 1.5 -3-2.5-2-1.5-1-0.500.511.522.53 a W=1

BAF Fall 2002 Page 1 of 4

Table of Fourier Transform Pairs of Power Signals

Function

name

Time Domain x(t) Frequency Domain X()

FT xt e jt

Xxtdtxt

F IFT 1 1 e 2 jt xtXdX FX

Impulse

()t 1 DC 1 2()

Cosine

0 cost 00 jj ee Sine 0 sint 00 jj jee

Complex

Exponential

0 expjt 0 2()

Unit step

10 00 t ut t 1 j

Signum

10 sgn() 10 t t t 2 j

Linear

Decay 1 t sgn()j

Impulse

Train s n tnT 22
k ss k TT

Fourier

Series

0 jkt k k ae , where 0 jkt te 0 0 1 k T ax T dt 0 2 k k ak

BAF Fall 2002 Page 2 of 4

Table of Fourier Transforms of Operations

Operation

FT Property

Given gtG

Linearity aftbgtaFbG

Time Shifting

0 0 e jt gttG

Time Scaling

1 ()gatG aa

Modulation (1)

00 1 cos 2 gttGG 0

Modulation (2)

0 0 e jt gtG

Differentiation If

dgt ft dt , then ()FjG

Integration If

t ftgd , then 1 ()0FGG j

Convolution

gtftGF gtftgft , where d

Multiplication

1 2 ftgtFG

Duality

If gtz, then 2ztg

Hermitian Symmetry

If g(t) is real valued then

GG- (G-Gand G-G)

Conjugation

gtG

Parseval's Theorem

221
2 avg

PgtdtGd

BAF Fall 2002 Page 3 of 4

Some Notes:

1. There are two similar functions used to describe the functional form

sin(x)/x. One is the sinc() function, and the other is the Sa() function. We will only use the sinc() notation in class. Note the role of in the sinc() definition: sin sin() x x sincxSax xx

2. The impulse function, aka delta function, is defined by the following three

relationships: a. Singularity: 0 0 tt for all t t 0 b. Unity area:

1)(dtt

c. Sifting property: for t b a t t tfdttttf)()()( 00 a < t 0 < t b

3. Many basic functions do not change under a reversal operation. Other

change signs. Use this to help simplify your results. a. (in general, tt 1 att a b. recttrectt c. tt d. sincsinctt e. sgnsgntt

4. The duality property is quite useful but sometimes a bit hard to

understand. Suppose a known FT pair gtz is available in a table. Suppose a new time function z(t) is formed with the same shape as the spectrum z() (i.e. the function z(t) in the time domain is the same as z() in the frequency domain). Then the FT of z(t) will be found to be

2ztg , which says that the F.T. of z(t) is the same shape as

g(t), with a multiplier of 2 and with - substituted for t. An example is helpful. Given the F.T. pair sgn()2tj, what is the Fourier transform of x(t)=1/t? First, modify the given pair to

2sgn()1jt by multiplying both sides by j/2. Then, use the duality

function to show that 22sgnsgnsgntjjj1.

BAF Fall 2002 Page 4 of 4

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