f (t), period T = ω2πo
P∞
jnωo t
n=−∞ Fn e
ao
2
+
P∞
n=1
co
2
+
Form
Exponential
an cos (nωo t) + bn sin (nωo t)
P∞
n=1 cn
cos (nωo t + θn )
Trigonometric
Compact for real f (t)
Coefficients
R
Fn = T1 T f (t)e−jnωo t dt
an = Fn + F−n
bn = j (Fn − F−n )
cn = 2 |Fn |
θn = ∠Fn
Table 1: Fourier series forms.
1
2
3
4
5
6
7
8
Name:
Condition:
Property:
Scaling
Addition
Time shift
Derivative
Hermitian
Even function
Odd function
Average power
Constant K
f (t) ↔ Fn , g(t) ↔ Gn , . . .
Delay to
Continuous f (t)
Real f (t)
f (−t) = f (t)
f (−t) = −f (t)
K f (t) ↔ K Fn
f (t) + g(t) + . . . ↔ Fn + Gn + . . .
f (t − to ) ↔ Fn e−jnωo to
df
dt ↔ jnωo Fn
F−n = Fn∗
P∞
ao
f (t) = 2 + n=1 an cos (nωo t)
P∞
f (t) = n=1 bn sin (nωo t)
R
P∞
2
2
P ≡ T1 T |f (t)| dt = n=−∞ |Fn |
Table 2: Fourier series properties
Name:
Condition:
Property:
1
2
3
4
5
6
7
Amplitude scaling
Addition
Hermitian
Even
Odd
Symmetry
Time scaling
f (t) ↔ F (ω), constant K
f (t) ↔ F (ω), g(t) ↔ G(ω), · · ·
Real f (t) ↔ F (ω)
Real and even f (t)
Real and odd f (t)
f (t) ↔ F (ω)
f (t) ↔ F (ω) , real s
Kf (t) ↔ KF (ω)
f (t) + g(t) + · · · ↔ F (ω) + G(ω) + · · ·
F (−ω) = F ∗ (ω)
Real and even F (ω)
Imaginary and odd F (ω)
F (t) ↔ 2πf (−ω)
1
f (st) ↔ |s|
F ( ωs )
8
9
10
11
12
13
14
15
16
Time shift
Frequency shift
Modulation
Time derivative
Freq derivative
Time convolution
Freq convolution
Compact form
Parseval, Energy W
f (t) ↔ F (ω)
f (t) ↔ F (ω)
f (t) ↔ F (ω)
Differentiable f (t) ↔ F (ω)
f (t) ↔ F (ω)
f (t) ↔ F (ω), g(t) ↔ G(ω)
f (t) ↔ F (ω), g(t) ↔ G(ω)
Real f (t)
f (t) ↔ F (ω)
f (t − to ) ↔ F (ω)e−jωto
f (t)ejωo t ↔ F (ω − ωo )
f (t) cos(ωo t) ↔ 12 F (ω − ωo ) + 12 F (ω + ωo )
df
dt ↔ jωF (ω)
d
F (ω)
−jtf (t) ↔ dω
f (t) ∗ g(t) ↔ F (ω)G(ω)
1
f (t)g(t) ↔ 2π
F (ω) ∗ G(ω)
R
∞
1
f (t) = 2π 0 2|F (ω)| cos(ωt + ∠F (ω))dω
R∞
R∞
1
W ≡ −∞ |f (t)|2 dt = 2π
|F (ω)|2 dω
−∞
Table 3: Important properties of the Fourier transform.
f (t) ↔ F (ω)
1
2
3
4
5
6
7
8
9
10
11
12
13
−at
1
a+jω , a > 0
1
,a>0
eat u(−t) ↔ a−jω
2a
−a|t|
e
↔ a2 +ω2 , a > 0
a2
−a|ω|
,a>0
a2 +t2 ↔ πae
1
−at
te u(t) ↔ (a+jω)
2, a > 0
n!
n −at
t e u(t) ↔ (a+jω)n+1 , a > 0
rect( τt ) ↔ τ sinc( ωτ
2 )
π
ω
sinc(W t) ↔ W rect( 2W
)
2 ωτ
t
τ
4( τ ) ↔ 2 sinc ( 4 )
ω
sinc2 ( W2 t ) ↔ 2π
W 4( 2W )
ωo
e−at sin(ωo t)u(t) ↔ (a+jω)
2 +ω 2 , a >
o
a+jω
−at
e
cos(ωo t)u(t) ↔ (a+jω)2 +ω2 , a >
o
2 2
√
t2
− 2σ
− σ 2ω
2
e
e
u(t) ↔
14
δ(t) ↔ 1
15
1 ↔ 2πδ(ω)
16
17
18
δ(t − to ) ↔ e−jωto
ejωo t ↔ 2πδ(ω − ωo )
cos(ωo t) ↔ π[δ(ω − ωo ) + δ(ω + ωo )]
19
sin(ωo t) ↔ jπ[δ(ω + ωo ) − δ(ω − ωo )]
20
jω
π
2 [δ(ω − ωo ) + δ(ω + ωo )] + ωo2 −ω 2
o
sin(ωo t)u(t) ↔ j π2 [δ(ω + ωo ) − δ(ω − ωo )] + ω2ω−ω
2
o
2
sgn(t) ↔ jω
1
u(t) ↔ πδ(ω) + jω
P∞
P
∞
2π
2π
n=−∞ δ(t − nT ) ↔ T
n=−∞ δ(ω − n T )
P∞
P∞
1
2π
n=−∞ f (t)δ(t − nT ) ↔
n=−∞ T F (ω − n T )
21
22
23
0
24
0
25
cos(ωo t)u(t) ↔
↔ σ 2πe
Table 4: Important Fourier transform pairs. The left-hand column includes only “energy signals” f (t) ,
while the right-hand column includes “power signals” and distributions.
Name
Commutative
Distributive
Associative
Shift
Derivative
Reversal
Start-point
Property
h(t) ∗ f (t) = f (t) ∗ h(t)
f (t) ∗ (g(t) + h(t)) = f (t) ∗ g(t) + f (t) ∗ h(t)
f (t) ∗ (g(t) ∗ h(t)) = (f (t) ∗ g(t)) ∗ h(t)
h(t) ∗ f (t) = y(t) ⇒ h(t − t0 ) ∗ f (t) = h(t) ∗ f (t − t0 ) = y(t − t0 )
d
d
d
h(t) ∗ f (t) = y(t) ⇒ ( dt
h(t)) ∗ f (t) = h(t) ∗ ( dt
f (t)) = dt
y(t)
f (t) ∗ h(t) = y(t) ⇒ h(−t) ∗ f (−t) = y(−t)
If h(t) = 0 for t < tsh and f (t) = 0 for t < tsf
then y(t) = h(t) ∗ f (t) = 0 for t < tsy = tsh + tsf .
If h(t) = 0 for t > teh and f (t) = 0 for t > tef
then y(t) = h(t) ∗ f (t) = 0 for t > tey = teh + tef .
h(t) ∗ f (t) = y(t) ⇒ Ty = Th + Tf
where Th , Tf , and Ty denote the widths of h(t), f (t), and y(t).
End-point
Width
Table 5: Convolution properties.
Name
Convolution
Sifting
Sampling
Symmetry
Scaling
Area
Definite integral
Unit-step derivative
Derivative
Fourier transform
Impulse properties
δ(t) ∗ f (t) = f (t)
δ(t)f (t)dt = f (0) and
−∞
(
Rb
f (0) if a < 0 < b
δ(t)f (t)dt =
a
0,
otherwise
R∞
Shifted-impulse properties
δ(t − t0 ) ∗ f (t) = f (t − t0 )
δ(t − t0 )f (t)dt = f (t0 ) and
−∞
(
Rb
f (t0 ) if a < t0 < b
δ(t − t0 )f (t)dt =
a
0,
otherwise
R∞
f (t)δ(t − t0 ) = f (t0 )δ(t − t0 )
δ(t0 − t) = δ(t − t0 )
1
δ(t − t0 ), a 6= 0
δ(a(t − t0 )) = |a|
f (t)δ(t) = f (0)δ(t)
δ(−t) = δ(t)
1
δ(t), a 6= 0
δ(at) = |a|
R∞
δ(t)dt = 1 and
(
Rb
1 if a < 0 < b
δ(t)dt =
a
0, otherwise
Rt
δ(τ )dτ = u(t)
−∞
d
dt u(t) = δ(t)
d
d
( dt
δ(t)) ∗ f (t) = dt
f (t)
δ(t) ↔ 1
−∞
R∞
δ(t − t0 )dt = 1 and
(
Rb
1 if a < t0 < b
δ(t − t0 )dt =
a
0, otherwise
Rt
δ(τ − t0 )dτ = u(t − t0 )
−∞
d
dt u(t − t0 ) = δ(t − t0 )
d
d
( dt
δ(t − t0 )) ∗ f (t) = dt
f (t − t0 )
−jωt0
δ(t − t0 ) ↔ e
−∞
Graphical symbol
Table 6: Properties of the impulse and shifted impulse.
δ 0 (t) ↔ s
u(t) ↔ 1s
tu(t) ↔ s12
1
2
3
δ(t) ↔ 1
1
↔ s−p
1
tept u(t) ↔ (s−p)
2
7
8
9
4
n!
(s−p)n+1
s
cos(ωo t)u(t) ↔ s2 +ω
2
o
s+α
−αt
e
cos(ωd t)u(t) ↔ (s+α)2 +ω2
d
10
tn u(t) ↔
11
sin(ωo t)u(t) ↔
5
6
ept u(t)
tn ept u(t) ↔
12
e−αt sin(ωd t)u(t)
n!
sn+1
ωo
s2 +ωo2
ωd
↔ (s+α)
2 +ω 2
d
Table 7: Laplace transforms pairs h(t) ↔ Ĥ(s) involving frequently encountered causal signals —
α, ωo , and ωd stand for arbitrary real constants, n for non-negative integers, and p denotes for an
arbitrary complex constant.
Name:
Condition:
Property:
1
2
3
4*
5
Multiplication
Addition
Time scaling
Time delay
Frequency shift
f (t) → F̂ (s), constant K
f (t) → F̂ (s), g(t) → Ĝ(s)· · ·
f (t) → F̂ (s) , real a > 0
f (t) ↔ F̂ (s), to ≥ 0
f (t) → F̂ (s)
6
Time derivative
Differentiable
f (t) → F̂ (s)
7
8
9*
10
11
Time integration
Freq. derivative
Time convolution
Freq. convolution
Poles
f (t) → F̂ (s)
f (t) → F̂ (s)
h(t) ↔ Ĥ(s), f (t) ↔ F̂ (s)
f (t) → F̂ (s), g(t) → Ĝ(s)
f (t) → F̂ (s)
12
ROC
f (t) → F̂ (s)
13*
Fourier transform
f (t) ↔ F̂ (s)
14
15
Final value
Initial value
Poles of sF̂ (s) in LHP
Existence of the limit
Kf (t) → K F̂ (s)
f (t) + g(t) + · · · → F̂ (s) + Ĝ(s) + · · ·
f (at) → a1 F̂ ( as )
f (t − to ) ↔ F̂ (s)e−sto
f (t)eso t → F̂ (s − so )
f 0 (t) → sF̂ (s) − f (0− )
00
f (t) → s2 F̂ (s) − sf (0− ) − f 0 (0− )
···
f (n) (t) → sn F̂ (s) − · · · − f (n−1) (0− )
Rt
1
0− f (τ )dτ → s F̂ (s)
d
−tf (t) → ds
F̂ (s)
h(t) ∗ f (t) ↔ Ĥ(s)F̂ (s)
1
f (t)g(t) → 2πj
F̂ (s) ∗ Ĝ(s)
Values of s such that |F̂ (s)| = ∞
Portion of s − plane to the right
of rightmost pole 6= ∞
F (ω) = F̂ (jω) if and only if
ROC includes s = jω
f (∞) = lims→0 sF̂ (s)
f (0+ ) = lims→∞ sF̂ (s)
Table 8: Important properties of the one-sided Laplace transform. Properties marked by * in the
first column hold only for causal signals.