fundamentals of engineering supplied-reference handbook - Ventech!

Fourier Transform Pairs
x(t) X(f)
1 δ ( f )
δ () t
1
u (t)
1 1
δ ( f ) +
2 j2π f
Π(/ t τ )
τfsinc( τ f )
sinc( Bt )
1
( f / B)
B Π
Λ(/ t τ )
2
τfsinc ( τ f )
−at
e u() t
te u(t)
at −
a t
e −
2
(at)
e −
cos(2 π ft+θ
)
0
0
sin(2 π ft+θ
)
n=+∞
∑
n=−∞
δ( t −nT
)
s
1
a+ j2πf 1
( a+ j2 πf)
2a
2
2 2
a + (2 πf)
π
e
a
a > 0
2
−π ( f / a)
a > 0
a > 0
1 jθ − jθ
[ e δ( f − f0) + e δ ( f + f0)]
2
1 jθ − jθ
[ e δ( f − f0) −e δ ( f + f0)]
2 j
k =+∞
1
fs ∑ δ( f − kfs) fs
=
T
k =−∞
s
176
ELECTRICAL AND COMPUTER ENGINEERING (continued)
Fourier Transform Theorems
Linearity ax() t + by() t aX ( f ) + bY ( f )
Scale change
x( at )
1 ⎛ f ⎞
X ⎜
a ⎝
⎟
a⎠
Time reversal x( − t)
X( − f)
Duality Xt ()
x( − f )
Time shift x( t t0)
Frequency shift 2
xte () 0 π
− − j2πft0 j ft
X( f) e
X( f − f )
Modulation x()cos2 t π f0t 1
X( f − f0)
2
1
+ X( f + f0)
2
Multiplication x() t y() t X( f) ∗ Y( f)
Convolution x(t) ∗ y() t X( f) Y( f )
n
d x(
t)
Differentiation n
dt
Integration
∫
t
∞
- x λ d
( ) λ
n
0
( j2 π f) X( f)
1
X( f)
j2πf 1
+ X(0) δ(
f)
2
Frequency Response and Impulse Response
The frequency response H(f) of a system with input x(t) and
output y(t) is given by
Y( f)
H( f)
=
X( f)
This gives
Y( f) = H( f) X( f)
The response h(t) of a linear time-invariant system to a unitimpulse
input δ(t) is called the impulse response of the
system. The response y(t) of the system to any input x(t) is
the convolution of the input x(t) with the impulse response
h(t):
∫
+∞
−∞
yt () = xt () ∗ ht () = x( λ) ht ( −λ) dλ
∫
+∞
−∞
= ht ( ) ∗ xt ( ) = h( λ) xt ( −λ) dλ