Table of Fourier Series Properties:
1
Fourier Analysis : ck =
T0
Fourier Synthesis : x(t) =
Z
x(t)e−jkω0 t dt
T0
∞
X
ck ejkω0 t
k=−∞
(ω0 is the fundamental angular frequency of x(t) and T0 is the fundamental period of x(t))
F
For each property, assume x(t) ←→ ck
F
and y(t) ←→ dk
Property
Time domain
Fourier domain
Linearity
Ax(t) + By(t)
Ack + Bdk
Time Shifting
x(t − t0 )
ck e−jkω0 t0
Frequency Shifting
x(t)ejM ω0 t
ck−M
Conjugation
x∗ (t)
c∗−k
Time Reversal
x(−t)
c−k
Circular Conv.
x(t) ~ y(t)
T0 ck dk
Multiplication
x(t)y(t)
ck ∗ dk
Differentiation
d
dt x(t)
Integration
Rt
(jkω0 )ck
1
jkω0 ck
−∞ x(τ ) dτ
Conjugate Symmetry for
x(t) is real
ck = c∗−k
Real and Even Signals
x(t) is real and even
ck is real and even
Real and Odd Signals
x(t) is real and odd
ck is purely imaginary and odd
1
T0
P∞
Real Signals
Parseval’s Relation for
Cont. Periodic signals
R
T0
|x(t)|2 dt
2
k=−∞ |ck |
1
Table of Special Functions:
Function name
Expression
Sinc function
sinc(x) =
Rectangle function
rect(x) =
Notes
sin(x)
x
Note
that
there
exists
an alternative
definition for
sinc(x)



 0
if
x ≥ 1/2


 1
if
− 1/2 ≤ x < 1/2
,
x < −1/2
rect(x) = u(x + 21 ) − u(x − 12 )
Unit triangle function
∆(x) =



 0
if
|x| ≥ 1/2


 1 − 2|x|
if
|x| < 1/2
2
Note
that
there
are
alternative definitions with
different values
for rect(±1/2)
Table of Fourier Transform Pairs:
Z
∞
Fourier Transform : X(ω) =
Inverse Fourier Transform : x(t) =
1
2π
x(t)e−jωt dt
−∞
Z ∞
X(ω)ejωt dω .
−∞
x(t)
X(ω)
condition
e−at u(t)
1
a+jω
a>0
eat u(−t)
1
a−jω
a>0
e−a|t|
2a
a2 +ω 2
a>0
te−at u(t)
1
(a+jω)2
a>0
tn e−at u(t)
n!
(a+jω)n+1
a>0
δ(t)
1
1
2πδ(ω)
ejω0 t
2πδ(ω − ω0 )
cos(ω0 t)
π (δ(ω − ω0 ) + δ(ω + ω0 ))
sin(ω0 t)
jπ (δ(ω + ω0 ) − δ(ω − ω0 ))
u(t)
πδ(ω) +
sgn(t)
2
jω
cos(ω0 t)u(t)
π
2
sin(ω0 t)u(t)
π
2j
e−at sin(ω0 t)u(t)
ω0
(a+jω)2 +ω02
a>0
e−at cos(ω0 t)u(t)
rect Tt
a+jω
(a+jω)2 +ω02
a>0
T sinc
W
π sinc(W t)
rect
∆
t
T
W
2
2π sinc
∞
X
WT
2
δ(t − nT )
n=−∞
−t2 /(2σ 2 )
e
jω
ω02 −ω 2
(δ(ω − ω0 ) + δ(ω + ω0 )) +
(δ(ω − ω0 ) − δ(ω + ω0 )) +
ωT
2
ω
2W
T
2
2 sinc
1
jω
∆
ω0
ω0
ω02 −ω 2
ωT
4
ω
2W
∞
X
δ(ω − nω0 )
ω0 =
n=−∞
√
2 2
σ 2πe−σ ω /2
3
2π
T0
Table of Fourier Transform Properties: For each property, assume
F
x(t) ←→ X(ω)
F
and y(t) ←→ Y (ω)
Property
Time domain
Fourier domain
Linearity
Ax(t) + By(t)
AX(ω) + BY (ω)
Time Shifting
x(t − t0 )
X(ω)e−jωt0
Time Scaling
x(αt)
1
|α| X
Frequency Shifting
x(t)ejω0 t
X(ω − ω0 )
Conjugation
x∗ (t)
X ∗ (−ω)
Time Reversal
x(−t)
X(−ω)
Convolution
x(t) ∗ y(t)
X(ω)Y (ω)
Multiplication
x(t)y(t)
1
2π X(ω)
Differentiation
d
dt x(t)
Integration
Rt
(jω)X(ω)
1
jω X(ω)
−∞ x(τ ) dτ
ω
α
∗ Y (ω)
Conjugate Symmetry for
x(t) is real
X(ω) = X ∗ (−ω)
Real and Even Signals
x(t) is real and even
X(ω) is real and even
Real and Odd Signals
x(t) is real and odd
X(ω) is purely imaginary and odd
R∞
1
2π
Real Signals
Parseval’s Relation for
Aperiodic signals
2
−∞ |x(t)| dt
4
R∞
2
−∞ |X(ω)| dω
Table of Laplace Transform Pairs:
Z
∞
Bilateral Laplace Transform : X(s) =
x(t)e−st dt
Z−∞
∞
x(t)e−st dt
Z c+j∞
1
Inverse Laplace Transform : x(t) =
X(s)est ds .
2πj c−j∞
Unilateral Laplace Transform : X(s) =
0
x(t)
X(s)
δ(t)
1
1
s
1
s2
n!
n+1
s
1
s−a
1
(s − a)2
n!
(s − a)n+1
s
2
s + b2
b
s2 + b2
s+a
(s + a)2 + b2
b
(s + a)2 + b2
(r cos(θ))s + (ar cos(θ) − br sin(θ))
s2 + 2as + (a2 + b2 )
0.5rejθ
0.5re−jθ
+
s + a − jb s + a + jb
As + B
2
s + 2as + c
u(t)
tu(t)
tn u(t)
eat u(t)
teat u(t)
tn eat u(t)
cos(bt)u(t)
sin(bt)u(t)
e−at cos(bt)u(t)
e−at sin(bt)u(t)
re−at cos(bt + θ)u(t)
re−at cos(bt + θ)u(t)
q
A2 c+B 2 −2ABa
c−a2
re−at cos(bt + θ)u(t)
√
θ = tan−1 AAa−B
c−a2
,
r=
,
b=
e−at Acos(bt) +
sin(bt) u(t)
B−Aa
b
√
c − a2
,
b=
√
c − a2
5
s2
As + B
+ 2as + c
Table of Laplace Transform Properties: For each property, assume
L
x(t) ←→ X(s)
L
and y(t) ←→ Y (s)
Property
Time domain
Laplace domain
Linearity
Ax(t) + By(t)
AX(s) + BY (s)
Time Shifting
x(t − t0 )
Time Scaling
x(αt) ,
X(s)e−st0
1 s
X
α
α
Frequency Shifting
x(t)es0 t
X(s − s0 )
Time Reversal
x(−t)
X(−s)
Convolution
x(t) ∗ y(t)
X(s)Y (s)
Multiplication
x(t)y(t)
1
X(s) ∗ Y (s)
2πj
Differentiation
α≥0
d
x(t)
dt
d2
x(t)
dt2
d3
x(t)
3
Zdt
t
Integration
x(τ ) dτ
0−
Z t
x(τ ) dτ
−∞
Frequency Differentiation
−tf (t)
Frequency Integration
f (t)
t
Initial value
f (0+ )
Final value
f (∞)
sX(s) − f (0− )
s2 X(s) − sf (0− ) − f˙(0− )
s3 X(s) − s2 f (0− ) − sf˙(0− ) − f¨(0− )
1
X(s)
s
Z −
1 0
1
X(s) +
f (t)dt
s
s −∞
dF (s)
Z ds
∞
F (z)dz
s
lim sF (s)
s→∞
lim sF (s)
s→0
6
(if poles of sF (s) are in left-hand plane)