Formulas para engenharia

EE3160 FALL 2010

USEFUL FORMULAS rect(t) Height = 1 Area = 1 Width = 2 −1 1 t 1 tri(t)

Definitions: sinc(x) =

sin(πx) πx

Height = 1 Area = 1 Width = 1

1

t −0.5 0.5

Signal Energy

E=

x2 (t) dt for three pulse shapes x(t) shown below
Half-Cycle Sinusoid E = A 2b / 2 b
∞

A b

Rectangular Pulse E = A2 b
∞

A

A b

Triangular Pulse E = A2 b / 3
∞

Sifting Integral: δ(t) x(t) = x(t)

x(t)δ(t−α) dt = x(α)
−∞

Convolution: y(t) = rect(t) rect(t) = tri(t)
∞

x(τ )h(t−τ ) dτ =
−∞

x(t−τ )h(τ ) dτ
−∞

u(t) u(t) = r(t)

e−αt u(t) e−αt u(t) = te−αt u(t)
∞

FOURIER TRANSFORM: x(t) ⇐ ft ⇒ X(f ) δ (t) (1) t -0.5 0.5 1 t rect(t) 1

X(f ) =
−∞

x(t)e−j2πf t dt

x(t) =
−∞

X(f )ej2πf t df

sinc (f)

e −α t t

1 α + j 2π f

sinc2 (t) ⇐ ft ⇒ tri(f )

tri(t) ⇐ ft ⇒ sinc2 (f ) 1 X(f /α) |α|

te−αt ⇐ ft ⇒

1 (α + j2πf )2

FT Properties Scale: x(αt) ⇐ ft ⇒

Shift: x(t − α) ⇐ ft ⇒ X(f )e−j2πf α X(f + f0 ) + X(f − f0 ) 2

Derivative: x (t) ⇐ ft ⇒ j2πf X(f )

Modulation: x(t) cos(2πf0 t) ⇐ ft ⇒

Convolution: x(t) h(t) ⇐ ft ⇒ X(f )H(f )
∞

Multiplication: x1 (t)x2 (t) ⇐ ft ⇒ X1 (f ) X2 (f )
∞

Central Ordinates: x(t)

t=0 ∞

= x(0) =
−∞ ∞ −∞

X(f ) df

X(f )

f =0

= X(0) =
−∞

x(t) dt

Signal Energy =
−∞

x2 (t) dt =

|X(f )|2 df

(|X(f )| is magnitude spectrum)

∞

∞

FOURIER SERIES: xp (t) = k=−∞ Dk ej2πkf0 t = D0 + k=1 2|Dk | cos(2πkf0 t+θk )

(where Dk = |Dk |ejθk )

FS Coefficients: X[k] = Dk =

1 X1 (f ) f =kf0 (where X1 (f ) is the Fourier transform of one period) T The k-th harmonic of xp (t) is 2|Dk | cos(2πkf0 t + θk ) or 2|X[k]| cos(2πkf0 t + θk ). 1 T
T 0 ∞ ∞

Signal Power =

x2 (t) dt = p k=−∞ |Dk |2 = k=−∞ |X[k]|2

(|..|2 means magnitude squared)

c Ashok Ambardar, Fall 2010

1

LAPLACE TRANSFORMS and PROPERTIES δ(t) ⇐ lt ⇒ 1 te−αt u(t) ⇐ lt ⇒ u(t) ⇐ lt ⇒ 1 (s + α)2 1 s tu(t) ⇐ lt ⇒ 1 s2 e−αt u(t) ⇐ lt ⇒ 1 s+α ω