Question

1. Express the real part of the following signal in the form Ae-atcosωt+θ where A, a, ω and θ are real and -π<θ≤π
   1. xt= e-5tsin⁡6t-π4 (4 points)

2. xt= 9je-6+j400t   (8 points)

Answer 1

-St ca) X(t) = è st sinlet - To(43) Re{z(t)} = Reşist sinkt - TT (4))} Splitting -40 "+(-9%) = Reş -st e sin (6t + 1/2 9%)} = Coso sin (/te) - Re { est, cos(6t - 97 )} 97 UTH / Refest.cos (6- 47 = ") } : Re{est cos(6+-"]} est cos (61-14) - Cos (n (271) + 0) Coso Cos(-0) = coso -T<OZT st X(t) = ca) if sin (6t - Tyy Re{a(t)} = . Reş -5t e sin (6t - T/4)} splitting - Tly as T/ - 3T 4 = Resest sin(6+ +14-37) 3 - Repest cos ( 64 -5%)] -st cos ( cos (6+ -37) .. Re {x4+1} e

2) x(+) = 9; et 6+ juoolt Re{x(t)} Re{aje 6t4 400t' } 30 Re { qj ebt juoot } = (40+ isino -bt - Re{qje [cos (400 t) + jsinyoot}} . Reqq. est { scoshoot + j' sin 400t]} {q. [jcos 400t -1 sinyoot] } Re{ -bt q.e imajinary Red part -6t =-9. sinuoot -9.elt sin (400t + 7) sin(TTO) - Sino -9.6!. sin(uoot +%+7) 9.86€. cos( 400t + c) (i sin(%+0): Cojo) Re[r(t)} = q. élt cos (400t +