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EXAMPLE 6.1: We determine the Laplace transform of the signal x(t) (which may correspond to the voltage observed across a resistor in an electrical circuit) given by Eq. (6.13). x(t) = 15eat - 2)u(t), for a > 0 EXAMPLE 6.2: Given that the transform of a unit step function is L[u(t)] = we obtain the transform of cos(w t)u(t) using the multiplication/modulation property given by Eq. (6.15). EXAMPLE 6.3: Given that L[u(t)] =, what is the transform of the delayed unit step uſt - 2) ? EXAMPLE 6.4: From L[sin(t) u(t)=; we derive the Laplace transform of sin(at)u(t) for a > 0. EXAMPLE 6.6: Given that the Laplace transform of cos (at), t 2 0 is s2 + a2 we derive the transform of sin (at) fort 2 0. EXAMPLE 6.9: Given that sin (at)u(t) obtain the transform of te-bisin (at)u(t)
EXAMPLE 6.10: The impulse response of an RC lowpass circuit is given by h(t) = e-atu(t). What is the step response of the circuit? EXAMPLE 6.12: Determine the inverse Laplace transform (t) for X(5) given by 452 + 15s + 15 X($) = +65 + 115 +6 EXAMPLE 6.14: Find the inverse Laplace transform of 26 X(3) = s(82 + 45 +13) + EXAMPLE 6.19: Determine the transfer function of a system described by dy dy dx 4y(t) + 4 at de? = X(t) + āt What are the poles and zeros of the system? What is the impulse response of the system? 6.6 Determine the Laplace transform of each of the following signals. (a) sin (120nt)u(t) (b) e-2 cos (120xt)u(1) (c) 2[u(t) - ut -5)] (d) 5[1 --*]u(t) (e) e2u(t)-e-2(t-5)
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