Question

Topics: Filter Design by Pole Zero Placement PROBLEM Problem #2 . a) Design a simple FIR second order filter with real coefficients, causal, stable and with unity AC gain. Its steady state response is required to be zero when the input is: xIn]cos [(T/3)n] u[n] H(z) R.O.C: answer: b) Find the frequency response for the previous filter. H(0) c) Sketch the magnitude frequency response. T/3 t/3 d) Find the filter impulse response. h[n] e) Verify that the steady state step response for the input in a) is cero. Yss[n] - f) Sketch the Pole - Zero plot for H(z) g) Write the difference equation for the filter h) Find the steady state response to the input x[n]-cos[Tnlu[n] yss[n] -

Answer 1

"(a) Design a simple FIR second order filter with real coefficients, causal, stable and with ""unity AC gain"". Its steady state response is required to be zero when the input is, x(n) = cos [(pi(3)n] u(n) sketch theta possible pole 3090 plat for H(z) z (cos omega_0 sigma) (-)^z vert l z^2 - z middot cos omega_0/z^2 -z 2 (cos omega_0 + 1) vert z vert > vert z(sin omega_0 sigma)(-)^z vert l = z son omega_0/z^2 - z 2 cos omega_0 + vert z vert > 1 x(n) doubleheadarrow^z vert l x(z) cos[(pi (3)n] u(n) doubleheadarrow^z vert l z^2 - x cos(pi/3)/z^2- 2z cos (pi//3) + 1 Hence omega_0 = pi/3 x(2) = z^2 - 2(1/2)/z^2 - 2z(1/x)+1 x(2) = z^2 - z/2 \r\n x_1(n) times x_2(n) doubleheadarrow^z vert l x_1 (z_1 middot x_2 (z) z_1 (n)middot u_2(n) doubleheadarrow z vert l 1/2 pi j [x_1(2) middot x_2(z) FIR filter: - finite input response is defined a) filter whose impose response is finite can having finite number of samples. given using DC gain means if (z) = o/p/I/p = y(z)/x(z) = 1 y(z) = xz H(z) = z = 75 l_3"