Question

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4. (20 points) Consider the system y(n)(-1 x(n) +x(n-1) (1) Find the frequency response of this system. (2) Find the steady-state response when x(n) = 5 cos(n, π)

Answer 1

4. y(n) - 0.5 y(n - 1) = 0.5 x(n) + 0.5 x(n - 1) (a) Taking z-transform on both sides, y(z) - 0.5 z^- 1 y(z) = 0.5 x (s) + 0.5 z^- 1 x(z) y(z) (1 - 0.5 Z^- 1) = 0.5 (1 + z^- 1) x(z) system Transfer function (or) frequency Response of this system H(Z) = y(z)/x(z) = 0.5 (1 + z^- 1)/(1 - 0.5 z^- 1) since z = e^j w H(e^j w) = y(e^j w)/x^(e^j w) = 0.5 (1 + e^- j w)/(1 - 0.5 e^- j w) (1) (b) The steady state Response for I/p x[n] = 5 cos (n pi/2 + pi) y[n] = 5 Vertical-bar H (E^j w) Vertical-bar _w = pi/2 cos (n pi/2 + pi + angle H (e^j w) Vertical-bar _w = pi/2) (ii) From equation (1) H (e^j w) Vertical-bar _w = pi/2 = 0.5 (1 + e^- j pi/2)/(1 - 0.5 e^- j pi/2) = 0.5 (1 - j)/1 - (0.5 ((- j)) = 0.5 (1 - j)/(1 + j 0.5) = 0.5 squareroot 1^2 + (- 1)^2/squareroot 1^2 + 0.5^2