Question

Let a be a positive real number. Consider a discrete-time echo system (called system 1) given by the difference equation y[n] = v[n] + av[n – 4). Here v is the input signal and y is the output signal. A. (1 mark) Determine the systems' transfer function H1 (2). B. (1 mark) What are the pole(s) of this system? Plot the pole(s) in the complex plane. C. (2 marks) Is this system stable? Explain your answer. D. (2 marks) Determine the system's step response. E. Next, consider the echo removal system (called system 2) given by the difference equation w[n] + au[n – 4] = y[n]. 1. (1 mark) Compute the transfer function H2 (2) of system 2. II. (2 marks) What are the pole(s) of system 2? Plot the pole(s) in the complex plane for a = III. (3 marks) For which values of a is system 2 unstable? Explain your answer and give a physical interpretation for this situation in terms of the echo. F. (2 marks) By using the transfer functions that you computed in parts A and E, show that system 2 is the inverse of system 1. 16

Answer 1

0 Solo Given that Yn) = V(h) +2.V (1-4] Apply 2-transform on both sides YCZ) = V(z) ta.z7. Vz] Y(2) = [Itazt]. V [z] Y(2) V(2) Cleaza Hi (2) et ta Zt ® H1 (2) has 4-poks at Z=0' im (2) Z=0, Four poles ke(z] [ii AI poles of H, (2) are lies inside unit circle. Therefore the system is stable For all bounded inputs [des, werd constant oc], the difput ylhy bounded. Therefore system is stable is

Y(Z) = v(2) +2.zt. 12 ) NOTE Z-Transform of win) 2 IZIDI Z-1 Therefore Step response, "V(n)= UG) Y (2) = 2 2-1 ži+d. + d. 27. 2 - V(2)=2 2-1 Z-1 Apply Inverse Z-Transform on both sides Y(n) = um t2. U(n-1)