3. Consider the first order linear system y(t) = 2y(t) + 10u(t). Suppose the system is...

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3. Consider the first order linear system y(t) = 2y(t) + 10u(t). Suppose the system is initially at rest at t and u(t) = sin wot, t (-, ). (a) (1pt) Compute the Fourier series coefficient of the input for k = 0, denoted co, when u(t) is represented by the Fourier series as 1 u(t) - k=- (b) (1pt) Compute the Fourier series coefficient of the input u(t) for k0, denoted Ck. (c) (1pt) With co and ck from (a) and (b), verify 8 W 3 1 k=- ckejkwot k=- Ckej kwot (d) (1pt) Compute the frequency response function of the system, denoted H(jw). (e) (1pt) Compute the Fourier series coefficient of the output for k = 0, denoted o, when y(t) is represented by the Fourier series as = sin wot. (f) (1pt) Compute the Fourier series coefficient of the output y(t) for k0, denoted . (g) (1pt) Write the Fourier series representation of the output y(t) and show that 1 y(t) = = [kwot k=- kejkwot = |H(jwo) | sin (wot + ZH (jwo)). 3. Consider the first order linear system y(t) = 2y(t) + 10u(t). Suppose the system is initially at rest at t and u(t) = sin wot, t (-, ). (a) (1pt) Compute the Fourier series coefficient of the input for k = 0, denoted co, when u(t) is represented by the Fourier series as 1 u(t) - k=- (b) (1pt) Compute the Fourier series coefficient of the input u(t) for k0, denoted Ck. (c) (1pt) With co and ck from (a) and (b), verify 8 W 3 1 k=- ckejkwot k=- Ckej kwot (d) (1pt) Compute the frequency response function of the system, denoted H(jw). (e) (1pt) Compute the Fourier series coefficient of the output for k = 0, denoted o, when y(t) is represented by the Fourier series as = sin wot. (f) (1pt) Compute the Fourier series coefficient of the output y(t) for k0, denoted . (g) (1pt) Write the Fourier series representation of the output y(t) and show that 1 y(t) = = [kwot k=- kejkwot = |H(jwo) | sin (wot + ZH (jwo)).