formulas for discrete-time LTI signals and systems
useful formulas
name
formula
name
formula
area under impulse
X
Euler’s formula
ej θ = cos(θ) + j sin(θ)
. . . for cosine
cos(θ) =
ej θ + e−j θ
2
. . . for sine
sin(θ) =
ej θ − e−j θ
2j
Sinc function
sinc(θ) :=
δ(n) = 1
n
multiplication by impulse
f (n) δ(n) = f (0) δ(n)
. . . by shifted impulse
f (n) δ(n − no ) = f (no ) δ(n − no )
convolution
f (n) ∗ g(n) =
X
f (k) g(n − k)
k
. . . with an impulse
f (n) ∗ δ(n) = f (n)
. . . with a shifted impulse
f (n) ∗ δ(n − no ) = f (n − no )
transfer function
H(z) =
X
Z-transform transform pairs
h(n) z −n
n
frequency response
. . . their connection
f
H (ω) =
X
f
sin(π θ)
πθ
x(n)
X(z)
x(n)
X
ROC
−jωn
h(n) e
jω
H (ω) = H(e )
provided unit circle ⊂ ROC
x(n) z −n
(def.)
n
δ(n)
1
all z
u(n)
z
z−1
|z| > 1
formulas for continuous-time LTI signals and systems
an u(n)
z
z−a
|z| > |a|
name
−an u(−n − 1)
z
z−a
|z| < |a|
formula
Z
area under impulse
δ(t) dt = 1
z 2 − cos(ωo ) z
− 2 cos(ωo ) z + 1
|z| > 1
sin(ωo n) u(n)
sin(ωo ) z
z 2 − 2 cos(ωo ) z + 1
|z| > 1
an cos(ωo n) u(n)
z 2 − a cos(ωo ) z
z 2 − 2 a cos(ωo ) z + a2
|z| > |a|
an sin(ωo n) u(n)
a sin(ωo ) z
z 2 − 2 a cos(ωo ) z + a2
|z| > |a|
cos(ωo n) u(n)
multiplication by impulse
f (t) δ(t) = f (0) δ(t)
. . . by shifted impulse
convolution
f (t) δ(t − to ) = f (to ) δ(t − to )
Z
f (t) ∗ g(t) = f (τ ) g(t − τ ) dτ
. . . with an impulse
f (t) ∗ δ(t) = f (t)
. . . with a shifted impulse
f (t) ∗ δ(t − to ) = f (t − to )
Z
H(s) = h(t) e−st dt
transfer function
z2
Z-transform transform properties
frequency response
. . . their connection
f
H (ω) =
Z
−jωt
h(t) e
dt
x(n)
X(z)
a x(n) + b g(n)
a X(z) + b G(z)
x(n − no )
z −no X(z)
x(n) ∗ f (n)
X(z) F (z)
f
H (ω) = H(jω)
provided jω-axis ⊂ ROC
Page 8 of 169
selected laplace transform pairs
x(t)
selected Fourier transform pairs
X(s)
Z
x(t)
x(t) e−st dt
ROC
X f (ω)
x(t)
Z
(def.)
x(t)
δ(t)
1
all s
u(t)
1
s
Re(s) > 0
e−a t u(t)
1
s+a
cos(ωo t) u(t)
Z
1
2π
X f (ω) ejωt dω
x(t) e−jωt dt
(def.)
X f (ω)
δ(t)
1
Re(s) > −a
1
2 π δ(ω)
s
s2 + ωo2
Re(s) > 0
u(t)
π δ(ω) +
sin(ωo t) u(t)
ωo
s2 + ωo2
Re(s) > 0
ejωo t
2 π δ(ω − ωo )
s+a
(s + a)2 + ωo2
cos(ωo t)
π δ(ω + ωo ) + π δ(ω − ωo )
e−a t cos(ωo t) u(t)
Re(s) > −a
sin(ωo t)
j π δ(ω + ωo ) − j π δ(ω − ωo )
e−a t sin(ωo t) u(t)
ωo
(s + a)2 + ωo2
“ω ”
ωo
o
sinc
t
π
π
ideal LPF
symmetric pulse
2
sin
ω
Re(s) > −a
Note: a is assumed real.
1
jω
cut-off frequency ωo
„
T
ω
2
«
width T , height 1
Laplace transform properties
impulse train
impulse train
period T , height 1
period, height ωo =
2π
T
x(t)
X(s)
a x(t) + b g(t)
a X(s) + b G(s)
x(t) ∗ g(t)
X(s) G(s)
dx(t)
dt
s X(s)
Fourier transform properties
x(t − to )
e−s to X(s)
x(t)
X f (ω)
a x(t) + b g(t)
a X f (ω) + b Gf (ω)
x(a t)
“ω”
1
X
|a|
a
x(t) ∗ g(t)
X f (ω) Gf (ω)
x(t) g(t)
1
X f (ω) ∗ Gf (ω)
2π
x(t − to )
e−jto ω X(ω)
x(t) ejωo t
X(ω − ωo )
x(t) cos(ωo t)
0.5 X(ω + ωo ) + 0.5 X(ω − ωo )
x(t) sin(ωo t)
j 0.5 X(ω + ωo ) − j 0.5 X(ω − ωo )
dx(t)
dt
j ω X f (ω)
Fourier series
If x(t) is periodic with period T then
X
x(t) =
c(k) ej k ωo t
where
2π
ωo =
T
and
c(k) =
1
T
Z
x(t) e−jkωo t dt
hT i
Page 9 of 169