CT LTI system
The system is:
- $ y(t)=10x(t)+x(t-1) $
unit impulse response
Obtain the unit impulse response h(t) and the system function H(s) of your system. :
- $ d (t) => System =>10 d (t) + d(t-1)\, $
- $ h(t)=10d(t) +d(t-1)\, $
- $ H(s)=\int_{-\infty}^{\infty} h(t)e^{-s t}dt $
- $ H(s)=\int_{-\infty}^{\infty} (10d(t) +d(t-1))e^{-s t}dt $
Using the shifting property,
- $ H(s)=10 e^{0 s} + e^{-1 s} \, $
- $ H(s)=10 + e^{- s} \, $, where s =jw
Part B
Compute the response of your system to the signal you defined in Question 1 using H(s) and the Fourier series coefficients of your signal.
- $ x(t)=e^{.5 j t \pi}-e^{-.5 j t \pi}+3 \, $
- $ Response = H(s)*x(t) \, $
- $ x(t)=(e^{.5 j t \pi}-e^{-.5 j t \pi}+3)(10+e^{-j.5\pi}) \, $
- $ x(t)=10e^{.5 j t \pi}-10e^{-.5 j t \pi}+30 +e^{.5 j t \pi}e^{-j.5\pi}-e^{-.5\pi j t }e^{-j.5\pi}+3e^{-j.5\pi} \, $
- $ x(t)=10e^{.5 j t \pi}-10e^{-.5 j t \pi}+30 +e^{.5 j (t-1) \pi}-e^{-.5 j (t-1) \pi}+3e^{-j.5\pi} \, $
