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The system function of a causal LTI system is given as H₁ (s) = 2s+5 s²+55+6 a) (2) Write down the differential equation relating the input x(t) and the output y(t). b) (2) Determine the output y(t) when the input is x(t) = e¯¹ u(t) is applied to H₁(s). Another causal LTI system has the system function H₂ (s) = 20 (s+1) s²+4s+2504 c) (2) Sketch the pole-zero plot. d) (2) Specify the ROC. Explain your answer. e) (2) Is the system stable? Explain your answer. f) (2) What is the value of the natural frequency of this system? What is the value of its zeta parameter ? g) (2) Is the system oscillatory? Explain your answer. h) (2) Is the system over-damped, under-damped or critically damped ? Explain your answer. i) (2) Specify the maximum gain, the half-power gain and the half-power frequency / frequencies. j) (2) Roughly sketch the magnitude response. Show important values. If an input x (t) = 1 + 4 sin(52t) + 2 sin(1000t) is applied to this stable LTI system, k) (2) Estimate the frequency response (in exponential form) at w = 0, w = 52 rad/s and w = 1000 rad/s. 1) (2) Represent the output y(t) as the sum of real sine signals.