Question

Consider the following waveform f(t) which is a one cycle of a sinusoid for 0 seconds while zero elsewhere (Aperiodic Signal) 2- t π in fo) 0 -10 a) Represent f(t) mathematically. b) Determine the Laplace transform using the integral expression c) Repeat that using the properties of the Laplace transform

Answer 1

Mathematical representation for sine wave is given by f(t) = A sin wt where, A - maximum amplitude = 10 w = I pi/T, T - fundamental period = pi w = 2 i/pi = 2 therefore, f(t) = {10 sin 2t: 0 < t < pi 0: otherwise Laplace transform of signal x(t) is given by x(s) = integral^infinity_-infinity x(t) e^-st dt here, x(t) = 10 sin(2t) \r\n we know, sin A = e^I theta - e^-j theta/2i, hence 10 sin 2t = 10(e^i2t - e^-i2t)/2i so x(s) = integral^pi_0 10(e^jlt - e^-j2t)/2i e^-st dt simplify x(s) = 10/2i integral^pi_0 [e^-(s -2) t - e^- (s 2j)t[]dt integrate, x(s) = 10/2j {[e^-(s -2i)t/-(s- 2j)]^pi_0 - [e^-(s + 2i)t/-(s + 2j)]^T_0] apply the limits, \r\n x(s) = 10/2i [1 - e^-(s - 2j) pi/(s - 2j) + e^-(s + 2j) pi - 1/s + 2j] x(s) = 10/2j [s + 2j - (s + 2j) e^-(s - 2j) pi + (s - 2j) e^-(s + 2j) pi - s + 2j/s^2 + 4] = 10/2j [4j - se^-3 pi(e^I 2 pi - e^-I 2 pi) - 2j e^-s pi (e^j 2 pi + e^-j 2 pi)/s^2 + 4]