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Descripción: .
07.04.2010
Table of Fourier Transform Pairs of Energy Signals Function name
Time Domain x(t)
Frequency Domain X( )
x t e
X
x t
FT
j t
dt F x t
x t
IFT
1
X e 2
j t
d F 1 X
X
Rectangle Pulse
t t 1 rect T T 0
Triangle Pulse
t t 1 W W 0
Sinc Pulse
sinc Wt
Exponential Pulse
e
Gaussian Pulse Decaying Exponential Sinc2 Pulse
t T
T 2
2
T sinc
elsewhen
W 2
t W
W sinc 2
elsewhen
sin( Wt )
1
Wt
W
a t
2 W
rect
2a
a 0
exp(
t 2 2 2
exp(at )u (t )
a 2 2
)
2 exp(
2
2
2
)
1
Re a 0
a j 1
sinc 2 Bt
B
Rect Pulse
2 B
Gaussian Gaussian Pulse
1.5
1.5
T=1
2=1
1
1
0.5
0.5
) a ( t c e r
0
0
-3
-2.5
-2
-1.5
-1
-0.5
0
0. 5
1
1. 5
2
2. 5
3
-3
-2.5
-2
-1. 5
-1
-0. 5
a
0
0. 5
1
1. 5
2
Sinc Pulse
3
Triangle Pulse
1.5
1.5 W=1
W=1 1 ) a ( c n i S
2.5
a
1
0.5 0.5 0 -3
-2. 5
-2
-1. 5
-1
- 0. 5
0
0.5
1
1. 5
2
2. 5
3
0 -3
-0.5
a
-2. 5
-2
-1. 5
-1
-0.5
0
a
0. 5
1
1. 5
2
2. 5
3
Table of Fourier Transform Pairs of Power Signals Function name
Time Domain x(t)
Frequency Domain X( )
xt e
X
x t
FT
j t
dt F x t
IFT
x t
1
X e 2
j t
d F 1 X
X
Impulse
(t )
1
DC
1
2 ( )
Cosine
cos 0t
Sine
sin 0t
j e j ( 0 ) e j ( 0 )
Complex Exponential
exp j 0t
2 ( 0 )
e j ( 0 ) e j ( 0 )
1 t 0 0 t 0
Unit step
u t
Signum
sgn(t )
Linear Decay
1
( )
1 t 0 1 t 0
j
2
j
j sgn( )
t
Impulse Train
1
2
t nT s
T s
n
k
2
k T
s
Fourier Series
ae
k
ak
jk 0t
k
1
, where
x (t )e T 0 T 0
2 jk 0t
dt
a k k
k
0
Table of Fourier Transforms of Operations Operation
FT Property Given g t G
Linearity
af t bg t aF bG
Time Shifting
g t t0 e
Time Scaling
g (at )
Modulation (1)
g t cos 0t
Modulation (2)
g t e
If f t
Differentiation
dt
1
a
G
a
G
1
G 0 G 0 2
j 0 t
d g t
j t 0
G 0
, then F ( ) j G
t
Integration
If f t
g d , then F ( )
1
j
G G 0
g t f t G F , where
Convolution
g t f t
g f t d
Multiplication Duality Hermitian Symmetry
f t g t
2
F G
If g t z , then z t 2 g If g(t) is real valued then G - G* ( G - G and G - G ) g * t G *
Conjugation
Parseval’s Theorem
1
Pavg
2
g t dt
1
G 2
2
d
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 x sin(x) ; sinc x Sa (x ) x x 2. The impulse function, aka delta function, is defined by the following three relationships: a. Singularity: t t 0 0 for all t t0
(t )dt 1
b. Unity area:
t b
c. Sifting property:
f (t ) (t t )dt f (t ) for ta < t0 < tb. 0
0
t a
3. Many basic functions do not change under a reversal operation. Other change signs. Use this to help simplify your results. 1 a. t t (in general, at t ) a b. rect t rect t c. t t d. sinc t sinc t e. sgn t sgn t 4. The duality property is quite useful but sometimes a bit hard to understand. Suppose a known FT pair g t z 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 z t 2 g , 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(t ) 2 j , what is the Fourier transform of x(t)=1/t? First, modify the given pair to j 2 sgn(t ) 1 by multiplying both sides by j/2. Then, use the duality function to show that 1 t 2 j 2sgn j sgn j sgn .