Question

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Question 2 (25 Points) Consider a system represented by the following differential equation: 3 de o + 13 "%) + 4y(e) = 5 y) +9«ce) a) Determine the system's frequency response, H(jw) b) Determine the system's unit impulse response, h(t) c) Define the difference equation describing the system whose frequency response is Hlem"= (1 - že-jo)(1-Le-jw) (1 – e-fu)

Answer 1

Given differential equation 3a?ye) +12 dy@) +ugt) = 5 deg til +9 2003 S at at (a) frequency response Hfjw 30124t) + 8 dyct) +4 y(t) =920 atz at Apply Fourier transform 3 (jw) ²ycjw, to jw ygw) +4% GW) =qXjw) ycjw), Hjw) - xjw, . 3 (jw) ² +8 jwty (b) impulse response het) HW) = (w +2) (jw +243 ) Hlw)= -6.75 16.15 - jwtzt jw+23 Apply inverie Fourie'transform h(t)=(-6.75 ezt +6+7522/34 gult) a(t)= 6.750 13t_ ezt just (c) theow) T . (1-12 w )(1-2 ejw ) e-edw ] Hleda) - det senjaw) (-:w)

§ 1 Bleiw) - 3w 3-jw , -jzw -ju ette - e + 2 / 4 e yleiw] Xeous Zowy z jew =i3w yeowo) - zdarmow) + 3 pune do te meteow) xrediw) - Apply" inverse DIFT to get differenke equation yon] -7 4 4 4h-17 + 7 yn -2] - Lyfn-3] = refo]