HW3 ECE2100 Spring 2014
Problem 1: Determine the System Function H(z) corresponding to the following difference equation:
HW3 ECE2100 Autumn2014Solutions
Problems encircled in red were graded
HW3 ECE2100 Page 1
HW3_sol Page 1
Problem 2: Find the poles and zeros of the System Function found in Problem 1. Plot the poles and zeros on the z plane. Which poles and zeros are trivial?
HW3 ECE2100 Page 2
HW3_sol Page 2
Problem 3: Analytically, determine the magnitude Frequency response corresponding to the difference equation of Problem 1.
Problem 4: Use MATLAB to plot the Magnitude Frequency Response evaluated in Problem 3 vs. the normalized frequency (units: cycles/sample). Is this system a Low Pass or High Pass filter, and why? Staple your MATLAB .code and the plot to this HW right after this page.
HW3 ECE2100 Page 3
HW3_sol Page 3
w=[0:pi/50:pi];fr_n = sqrt( (cos(w)+0.5).^2 + sin(w).^2);fr_d=sqrt( ( cos(2*w)‐cos(w)+0.5 ).^2 + ( sin(2*w)‐sin(w) ).^2 );fr = fr_n ./ fr_d;f = w/(2*pi);plot(f, fr)
HW3 ECE2100 Page 3‐1 Attachment
HW3_sol Page 4
Problem 5: Use MATLAB to plot the Magnitude Frequency Response of the system described by the difference equation in Problem 1 by using the freqz function of MATLAB (see Page 15 of Lecture39). Compare your result with the plot obtained in Problem 4. Staple your MATLAB code and plot to this page of the HW.
num = [ 1 0.5 0];den = [1 ‐1 0.5];[H, w] = freqz(num, den, 50);f = w/(2*pi);plot(f, abs(H))
HW3 ECE2100 Page 4
HW3_sol Page 5
Problem 6: Determine the System Function H(z) corresponding to the following poles and zeros:
HW3 ECE2100 Page 5
HW3_sol Page 6
Problem 7: Determine the Difference Equation corresponding to the poles ad zeros of Problem 6:
HW3 ECE2100 Page 6
HW3_sol Page 7
Problem 8: Analytically determine the Magnitude Frequency Response of the system corresponding to the following difference equation (your answer will depend on p and b0):
HW3 ECE2100 Page 7
HW3_sol Page 8
Problem 9: Determine the bandwidth of the filter given in Problem 8 in terms of p.
HW3 ECE2100 Page 8
HW3_sol Page 9