Sheet with Useful Formulas PDF

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Description

selected Laplace transform pairs

useful formulas name

formula

x(t)

X(s)

x(t)

Z

δ(t)

1

all s

u(t)

1 s

Re(s) > 0

e−a t u(t)

1 s+a

Re(s) > −a

cos(ω o t) u(t)

s s2 + ωo2

Re(s) > 0

formulas for continuous-time LTI signals and systems

sin(ω o t) u(t)

ωo s2 + ωo2

Re(s) > 0

name

formula

e−a t cos(ω o t) u(t)

s+a (s + a)2 + ωo2

Re(s) > −a

area under impulse

Z

e−a t sin(ω o t) u(t)

ωo (s + a)2 + ωo2

Re(s) > −a

multiplication by impulse

f (t) δ (t) = f (0) δ (t)

. . . by shifted impulse

f (t) δ (t − to ) = f (to ) δ (t − to ) Z f (t) ∗ g (t) = f (τ ) g (t − τ ) dτ

Euler’s formula . . . for cosine

e

jθ

= cos(θ) + j sin(θ)

cos(θ) =

e e

. . . for sine

sin(θ) =

sinc function

sinc(θ) :=

convolution

jθ

jθ

+e 2 −e 2j

−jθ

−jθ

sin(π θ) πθ

δ(t) dt = 1

. . . 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

Z

frequency response

H f (ω) =

. . . their connection

H f (ω) = H (jω ) provided jω-axis ⊂ ROC

h(t) e−jωt dt

x(t) e−st dt

Note: a is assumed real.

Laplace transform properties 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)

x(t − to )

e−s to X(s)

ROC (def.)

Fourier transform properties

selected Fourier transform pairs x(t)

X f (ω)

x(t)

Z

Z

1 2π

X f (ω) ejωt dω

x(t) e−jωt dt

(def.)

X f (ω )

δ(t)

1

1

2 π δ(ω )

u(t)

π δ(ω) +

ejωo t

2 π δ(ω − ω o )

cos(ω o t)

π δ (ω + ω o ) + π δ (ω − ω o )

sin(ω o t)

j π δ (ω + ω o ) − j π δ (ω − ω o )

“ω ” ωo o t sinc π π

ideal LPF

symmetric pulse

2 sin ω

1 jω

cut-off frequency ω o „

T ω 2

«

width T , height 1 impulse train period T , height 1

impulse train period, height ω o =

2π T

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 f X (ω) ∗ Gf (ω ) 2π

x(t − to )

e−jto ω X(ω)

x(t) ejωo t

X(ω − ω o )

x(t) cos(ω o t)

0.5 X(ω + ω o ) + 0.5 X(ω − ω o )

x(t) sin(ω o t)

j 0.5 X(ω + ω o ) − j 0.5 X(ω − ω o )

dx(t) dt

j ω X f (ω)...