4. Find the convolution of the following functions a. f(t)=t g(t) = sin 3t b. f(t)=é…
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Question “4. Find the convolution of the following functions a. f(t)=t g(t) = sin 3t b. f(t)=é…”
Answer
The convolution theorem states that t (fxg), t = flus gCt–u) du la), f(t), = t and Glt, and glt), = sin(3t-zu) du = u 3 t u cos (37-34) Sin (3t-zu + o 3 Sin (3) cos = Cos = Cos + 4 sun 6) -0 – wilt sin (3t) – ) y ) ) ) ANSER 9 Sin C3t-3u) du Seos I du f16) + 34) = (f*g)t = 5 – [.econoy ) menali Cuce cos(3t-zu) du = u 3 t u cos (37-34) Sin (3t-zu + o 3 Sin (3) cos = cos Co) + 4 sun 6) -0 – wilt sin (3t) o Therefore, sin (3t) (f*9) t- wit ANSWER 9
(b) F(t.=et and G(t. = cos(at. t. flt. *g(t.) = (f*8)t = J). eu.cos(at-su) da t t =[e T. cos Cat-w) e”. 2 sin (2t-24) du – t t du – [ecos( 2t-su) – Sze “sin Cat-su) 0 First find, 2e4 sin(at-9u)du = 2 sivC2t-2u)eu (-2005 (2t-zu) du => [e”sin (2t-zu) -SC-22 *cos(24-zu) du 4 Jetcoscat-su) du 2e^sin (21-24) + Integral Jeuco cos (21-24) was already calculated above. t t Therefore, from O (*g) t = (e cose t-su) – geosin (2t -zu)) cos (22-24) du D -Le e cos (21-24) -e a sin(at-zu)], 2e” sin (21-24) 4 teu cos (21-24) 5 S e “cas (2t-24)
Therefore, *-}. [ ** (v3), emington t -2 Sin (at-24) + cos (2t – 2u o 5 et – 2 sin (at) cos (26) = 5 t a sin (at) cos(at) = elh + 5 5 Therefore, t asin (2) (fxg) t = cos (at) + elb 5 5 ANSWER
Conclusion
Above is the solution for “4. Find the convolution of the following functions a. f(t)=t g(t) = sin 3t b. f(t)=é…“. We hope that you find a good answer and gain the knowledge about this topic of math.