hw3 (1)

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Signals & Systems II Homework #3 158 points total ECE 316 – Winter 2024 Due Friday, March 15, 2024 at 10 p.m. 1. Consider the system y ′′ ( t ) − 2 y ′ ( t ) − 8 y ( t ) = x ′′ ( t ) + 5 x ( t ) , which is assumed to be causal. (35 points total) (a) Find the transfer/system function, including the ROC. (5 points) (b) Determine the poles and zeros of the system function. (5 points) (c) Draw the pole-zero diagram, including the ROC. (5 points) (d) Is the system stable? Why or why not? (5 points) (e) Assume that the system is invertible and that the inverse system is causal. Find the system function for the inverse system, including the ROC. (5 points) (f) Draw the pole-zero diagram for the inverse system, including the ROC. (5 points) (g) Is the inverse system stable? Why or why not? (5 points) 2. Consider the system y [ n ] + 19 15 y [ n − 1] + 6 15 y [ n − 2] = x [ n ] + 4 x [ n − 1] + 3 x [ n − 2] , which is assumed to be causal. (35 points total) 1

(a) Find the transfer/system function, including the ROC. (5 points) (b) Find the poles and zeros of the transfer function. (5 points) (c) Draw the pole-zero diagram, including the ROC. (5 points) (d) Is the system stable? Why or why not? (5 points) (e) Assume that the system is invertible and that the inverse system is causal. Find the system function for the inverse system, including the ROC. (5 points) (f) Draw the pole-zero diagram for the inverse system, including the ROC. (5 points) (g) Is the inverse system stable? Why or why not? (5 points) 3. (26 points total) (a) Use the MATLAB or SciPy function ellipap to find the poles, zeros, and gain for a third-order normalized elliptic lowpass filter with 2 dB ripple in the passband and stopband ripple that has an upper limit of -50 dB. Draw the pole-zero diagram for the filter, including the ROC. This can be done using MATLAB or python or by hand. (8 points) (b) Use MATLAB or SciPy to determine the coefficients { b k } and { a k } of the numerator and denominator of the transfer function. Write the transfer function and plot the frequency response on Bode plots using the MATLAB or SciPy function freqs . (5 points) (c) Use the transfer function found above to convert the original low- pass filter into the transfer function of a bandstop filter with edge frequencies ω L = 2 πf L , where f L = 200 Hz, and ω H = 2 πf H , where f H = 2000 Hz, by hand. Plot the frequency re- sponse on Bode plots using the MATLAB or SciPy function freqs . (8 points) (d) Use MATLAB or python to plot the magnitude response | H ( f ) | and the phase response ̸ H ( f ) of the bandstop filter on Bode plots for 0 Hz ≤ f ≤ 4000 Hz. (5 points) 2

4. (32 points total) (a) Use the Filter Builder in MATLAB or pyFDA, which is based on python, to create a DT lowpass filter based on the minimum- order Butterworth filter that matches the specification described below. This is the same specification used to create IIR filters using impulse invariance and the bilinear transformation in class. • The gain in the passband must be between 0 and -1 dB. • The stopband attenuation must be at least -15 dB. • The angular frequency range of the passband is 0 ≤ Ω ≤ 0 . 2 π . • The angular frequency range of the stopband is 0 . 3 π ≤ Ω ≤ π . If you are using Filter Builder, create the filter using the bilinear transformation. Include a screenshot of the settings you used to create the filter in your solutions. Determine the poles, zeros, and coefficients { b k } and { a k } for the filter created by the tool. (8 points) (b) Determine the coefficients { b k } and { a k } for the filter created using the bilinear transformation in class and compare the filter order N , cutoff frequency Ω c , and the coefficients to those for the filter created using the tool. Are they the same? If not, is there a way to make them match? (6 points) (c) Find the LCCDEs of the filter created using the tool and for the filter created using the bilinear transformation in class. (5 points) (d) Determine the poles and zeros of the filter created in class and draw the pole zero diagram, including the ROC, by hand or using MATLAB or python. Do the poles and zeros match those of the filter created using the tool? (8 points) (e) Are the filter created using the tool and the filter created using the bilinear transformation in class BIBO stable? Explain your answer. (5 points) 5. (30 points total) (a) Use the Filter Builder in MATLAB or pyFDA, which is based on python, to create a minimum-order lowpass FIR filter using a 3

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Hann window that matches the specification in Problem 4. Include a screenshot of the settings you used to create the filter in your solutions. Determine the poles, zeros, and coefficients { b k } and { a k } for the filter created by the tool. (8 points) (b) Determine the LCCDE of the filter. (5 points) (c) How many values of the input signal x [ n ] are needed to determine the value of the output signal y [ n ] at time n ? So, how long a delay would have to be added to the filter to make the overall system causal? (5 points) (d) Plot the frequency responses of the FIR filter and the IIR filter created using the bilinear transformation in class on the same Bode plots. (8 points) (e) How many multiplications and additions would be needed to cal- culate the value of the output signal y [ n ] at time n using the LCCDE for the FIR filter? How many multiplications and addi- tions would be needed to calculate the value of the output signal y [ n ] at time n using the LCCDE for the filter created in class using the bilinear transformation? (The LCCDE for this filter is deter- mined in Part (c) of Problem 4.) Which filter is computationally less expensive to apply? Explain your answer. (4 points) 4