Question

1. (a) Use the property of impulse function to evaluate the integral (2-at) (-4) dr (b) Find the power in the signal x(t)-Ae-u(t). Is this a power signal? (e) Find the 3dB bandwidth of the filter whose impulse response is h(t) - Ae-at?

Answer 1

Given solution: - integral^infinity_-infinity delta(2 - at) (t^2 - 4) dt = we know delta(t) = delta(-t) = integral^infinity_-infinity delta(at - 2) (t^2 - 4) dt = integral^infinity_-infinity delta (a(t - 2/a)) (t^2 - 4) dt = 1/a integral^infinity_-infinity (t - 2/a) (t^2 - 4) dt x(t_0) = integral^infinity_-infinity delta (t - t_0) x(t) dt 1/a (a/a^2 - 4) = 1/a ((2/a)^2 - 4) = 4 - a^2 - 4/a^3 \r\n power in the signal x(t) = Ae^-t a(t) The signal has no Time period so, power = lt_T_0 rightarrow infinity integral^infinity_-infinity 1/T_0 (Ae^-t)^2 dt = lt_T_0 rightarrow infinity 1/T_0 integral^infinity_0 A^2 e^-2t dt A^2 /T_0 middot e^-2t/-2 integral^infinity_0 = A^2 /-2 T_0 (0 - 1) = A^2 /-2 T_0 as = As T_0 rightarrow infinity power becomes rightarrow 0 power = 0 so the signal 'm' Not power signal \r\n Given data: h(t) = A e^-at^2 for '3' dB Band width, = -pi^2 f^2 /e a h(t) = A squareroot pi/a e we know, e^-at^2 fourier/Transform squareroot pi/a -pi^2 f^2/ea For 3 dB Band width e^-pi^2/a t^2 = 1/squareroot 2