I Consider the causal discrete-time LTI system described by the following constant coefficient difference equation: 5 3 y(n) = y(n − 1) + y(n − 2) + x(n) (a) Determine the total response y(n), n ≥ 0 when the input signal is x(n) = ()u(n) and y(-1) = C₁, y(-2) = c2 are the given initial conditions. Is x(n) a bounded signal? Is y(n) a bounded signal? (b) Based on your answer in (a) determine the zero-input system response and the zero-state system response. (c) Determine the impulse response h(n) of this system. Is the system BIBO-stable? Is the system FIR or IIR? Is the system an-all pole system?

Problem 4. Given the Linear Difference Equation y[n] = 0.5y[n–1]+2x[n–1] Determine the output sequence for each of the following inputs. In each case the final answer y[n] has to be a real signal (nothing complex): Q1: x[n] = 2 cos(0.5zn)u[n] ; Q2. x[n] = 2 cos (0.5zn) Q3: x[n] =2 cos (0.5zn)u[-n-1]

Consider the system shown in Figure (1). This is a PID control of a second-order plant G(s). For the disturbance input in the absence of the reference input, the closed-loop transfer function is given as below: Ca(s) S D (s) (s + 10) (s² + 4s + 16) Rewrite the transfer function of PID controller in following form: 1 Ge(s)= K₂ (1+: + Tas) Tis R(3) K(as+1)(bs+1) PID controller D(s) Figure (1) +3.68+9 Plant G(s) C(s)

Q2: Draw the output waveforms (S, X, and Y) for the below sequential circuit by assuming the propagation delays of all the components that are shown in below figure equal to zero second. cik Cik DQ5

For the digital system shown in below: 1) Write down the difference equation that represents the system. 2) Find the transfer function of the system, i.e. H(Z) x(n) y(n) 0.75 0. 125 Delay one sample Delay one sample

6 - Forward path transfer function is applied to the input of a negative unit feedback system given as below, and the overrun rate at the output of the system is given as% OS = 5.4%. K G(s) = s(s +1) In this case, what is the damping rate of the system? A) 0.63 B) 0.57 C) 0.61 D) 0.68 E) 0.55

1. Consider an input x[n] and a unit impulse response h[n] of a discrete-time LTI system given by: h[n] = 38[n + 3] + 28[n+ 1] + 8[n] – 8[n – 1] + 28[n – 2] 101 2 3 Determine and plot the output signal y[n] = x[n] + h[n].