ASSIGNMENT 2 1. Find the Fourier Transform of the signal (𝑡) = 𝑡𝑒−𝑎𝑡𝑢(𝑡) . 2. State the conditions for the convergence of Fourier Series. 3. State and prove Parseval’s theorem for DTFS. CO2 4. A signal x[n] has DTFT (𝑒𝑗𝜔 ) = 1 1−𝑎𝑒−𝑗𝜔 , |𝑎| ≤ 1. Determine the DTFT of [𝑛 + 2]𝑒𝑗𝜋2𝑛 . 5. Consider the discrete LTI system [𝑛] − 12 𝑦[𝑛 − 1] = 𝑥[𝑛] + 12 𝑥[𝑛 − 1]. Determine the frequency response of the system. 6. Find continuous time Fourier series for a half wave rectifier system. 7. Compute the Fourier transform of the signal x(t) and plot the magnitude and phase response. CO2 (𝑡)={1+𝑐𝑜𝑠𝜋𝑡 |𝑡|≤00 |𝑡| >0 8. The input and the output of a stable and causal LTI system are related by the differential equation: 𝑑2𝑦(𝑡)𝑑𝑡2+6𝑑𝑦(𝑡)𝑑𝑡+8𝑦(𝑡)=2𝑥(𝑡) (a) Find the impulse response of this system. (b) What is the response of this system if (𝑡)=𝑡𝑒−2𝑡𝑢(𝑡)?