ECE 351 – Linear Systems II Homework #3 1. For a system described by the linear difference equation: yk−12yk −1= xk a. Find an expression for H(ejθ) [Answer: 11−12e− jθ ] b. Find an expression for |H| [Answer: 1(1−12cosθ)2+12sinθ"#$%&'2 ] c. Find an expression for arg H. [Answer: −tan−112sinθ1−12cosθ ] 2. For a system described by the linear difference equation: yk+ yk −1−12yk − 2= xk a. Find an expression for H(ejθ) [Answer: 11+ e− jθ−12e− j 2θ ] b. Find an expression for |H| [Answer: 1(1+ cosθ−12cos2θ)2+ −sinθ+12sin 2θ"#$%&'2 ] c. Find an expression for arg H. [Answer: −tan−1−sinθ+12sin 2θ1+ cosθ−12cos 2θ ] 3. For a system described by the linear difference equation: yk− 0.9yk −1+ 0.5yk − 2= 0.25xk− 0.25xk − 2 a. Find an expression for H(ejθ) [Answer: 0.25 − 0.25e− j 2θ1− 0.9e− jθ+ 0.5e− j 2θ ]b. Find an expression for |H| [Answer: (0.25 − 0.25cos2θ)2+ 0.25sin 2θ( )2(1− 0.9cosθ+ 0.5cos 2θ)2+ 0.9sinθ− 0.5sin 2θ( )2 ] c. Find an expression for arg H. [Answer: tan−10.25sin 2θ0.25 − 0.25cos 2θ− tan−10.9 sinθ− 0.5sin 2θ1− 0.9 cosθ+ 0.5cos 2θ ] 4. A low pass filter is described by the linear difference equation of the form: yk− 0.7254 yk −1= 0.1373xk+ 0.1373xk − 2 Find an expression for H(ejθ) [Answer: 0.1373 + 0.1373e− j 2θ1− 0.7254e− jθ ] Read and try the examples given in MATLAB Tutorial #3 before trying the problems below. 5. For the system described in problem 1: a. Use MATLAB to plot the expression obtained in problem 1, part a for |H| versus θ, the normalized frequency in radians/sample. b. Repeat part a using the freqz command. c. Use MATLAB to plot the expression obtained in problem 1, part a for arg H versus θ, the normalized frequency in radians/sample. d. What type of filter is this? e. What is the normalized frequency θ at the 3 dB point? 6. For the system described in problem 2: a. Use MATLAB to plot the expression obtained in problem 2, part a for |H| versus θ, the normalized frequency in radians/sample. b. Use MATLAB to plot the expression obtained in problem 2, part a for arg H versus θ, the normalized frequency in radians/sample. c. What type of filter is this? d. What is the normalized frequency θ at the 3 dB point?e. If the maximum frequency to be passed through this filter is 10 KHz, what sampling interval, T, should be used? f. At the sampling interval found in part e, what actual frequency does the 3 dB point correspond to? g. Use the freqz command to plot |H| versus frequency (Hz) for the sampling interval found in part e. 7. For the system described in problem 3: a. Use the freqz command in MATLAB to plot |H| versus normalized frequency, θ. b. Use the freqz command in MATLAB to plot arg H versus normalized frequency, θ. c. What type of filter is this? d. What are the normalized frequency(s) θ at the 3 dB point(s)? e. What is the normalized center frequency? f. Plot the sequence: xk = 1 + cosθ0k + cos πk for 0 ≤ k ≤ 20 where θ0 is the normalized center frequency of part e. g. Apply xk to the filter using the filter command and plot the output. h. What does this output correspond to? i. If a sampling interval T of 0.1 mS is used, what is the center frequency of this filter (in Hz)? j. If a sampling interval T of 0.1 mS is used, what are the 3 dB points (in Hz)? k. What is the maximum frequency that can be applied to the filter? l. Use the freqz command in MATLAB to plot |H| versus actual frequency, f. 8. For the system described in problem 4: a. Use MATLAB to plot |H| versus normalized frequency, θ. b. A sampled pulse, x1 = [ones(1,75), -1* ones(1,75), ones(1,75), -1* ones(1,75)] and a high frequency noise signal, x2k = 2cos(0.7πk) + 3cos(0.9πk) are input to this filter. Use MATLAB to plot the input x1, and the input with noise x1 + x2. c. Apply the input with noise to the filter and plot the filter output. Does the filter eliminate some of the