2.4 Signal Processing Transformations

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39

Tab. 2.5: Laplace transformations of various causal signals.

Signal

Time domain s(t) for t ≥0

Frequency domain S(p)

Dirac pulse

δ(t)

1

Constant

1

1

p

Power-function

t^k

k!

p^k+1

Exponential-function

e^−at

1

p+a

Exponential-function

1 −e^−at

a

p(p+a)

Cosine-function

cos(ω0t)

p

p²+ω²

0

Sine-function

sin(ω0t)

ω0

p²+ω²

0

effect of Equation 2.59 to outweigh the exponential increase e^at, σ > a must hold. The

Laplace transform of selected causal signals are listed in Table 2.5.

The interpretation of the Laplace transform is less descriptive than that of the

Fourier transform because the image domain has a two-dimensional variable space

due to Equation 2.56. For σ = σ0 = const. within the convergence region, the Laplace

transform S(σ0, jω) is equivalent to the Fourier transform for a signal damped with

e^−σ0t. The theorems of the Laplace transform are identical to those of the Fourier trans-

form according to Table 2.3 when jω is replaced by p.

Analogous to the Fourier transform, the inverse Laplace transform can be written

as follows:

T⁻¹ {S(p)} = s(t) =

1

2πj ^{∫S(p)eptdp .}

(2.61)

The integration limits in Equation 2.61 are given by [σ0 −j∞, σ0 + j∞], where σ0 must

again lie in the convergence region. The back transformation, is usually done by cor-

respondence tables (Table 2.5). This will be presented by means of an example. Let a

series resonant circuit with the given electrical components R, L and C (resistance, in-

ductance and capacitance respectively), be excited by the signal s(t) The output quant-

ity sought is the current i(t) of the series resonant circuit. The voltage equation of the

mesh through the series resonant circuit yields

s(t) = Ri(t) + L ^∂i

∂t ^{+ 1}

C ^{∫i(t)dt .}

(2.62)

Laplace transforming the differential equation 2.62 using the theorems for differenti-

ation and integration, leads to:

S(p) = RI(p) + LpI(p) + ¹

pC ^I(p)

(2.63)

⇐⇒pS(p) = (pR + p²L + ¹

C ^{) I(p) .}

(2.64)