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2 Fundamentals of Information, Signal and System Theory

The transfer function of the series resonant circuit H(p) can be given as the quotient

of transformed output quantity I(p) and input quantity S(p):

H(p) = ^I(p)

S(p) ^{= 1}

L

p

p² + ^R

L ^{p + 1}

LC

.

(2.65)

Excitation of the system by a unit step function,s(t) = u(t), leads to the Laplace trans-

form T{u(t)} = S(p) = ¹

p^{. Substitution into Equation 2.65 leads to:}

I(p) = ¹

L

1

p² + ^R

L ^{p + 1}

LC

.

(2.66)

Converting the polynomial in Equation 2.66 to zero form yields

I(p) = ¹

L

1

(p −p1) (p −p2) ^,

(2.67)

with

p1,2 = −^R

2L ^{± √R2}

4L^{2 −1}

LC ^.

(2.68)

Using partial fraction decomposition, Equation 2.67 can be written as

I(p) = ¹

L ^[

A

p −p1

+

B

p −p2

] ,

(2.69)

with the coefficients A and B still to be determined. The coefficient comparison of

Equation 2.69 with Equation 2.67 yields

A =

1

p1 −p2

(2.70)

and

B =

1

p2 −p1

.

(2.71)

The partial fraction form in Equation 2.69 can be easily transformed back by using

Table 2.5 and the linearity of the Laplace transform thus

i(t) = ¹

L ^{[Aep1t + Bep2t]u(t)}

(2.72)

with the previously calculated values for A, B, p1 and p2. Note that the excitation by

the unit step function s(t) = u(t) can cause oscillations if the root term in Equation 2.68

is negative, thus leading to complex zeros.

2.4.3 Continuous Short-Time Fourier-Transform and Wavelet Transform

The Fourier transform according to Equation 2.50 performs a integration out over all

times. of the signal Therefore, the spectrum obtained in this way does not contain any