ECE 351 Practice Final Fall 20

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California State University, Northridge *

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351

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Mechanical Engineering

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Oct 30, 2023

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pdf

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ECE 351 Name ___________________________ Fall’16, Exam 3, form 1 1/5 Part 1: This section is a closed-book and closed-notes. However, you may use your calculator and an 8.5” x 11” sheet of notes, with writing on both sides. Show all work in the space provided, and simplify and circle your answers to receive full credit. 1. The graph below shows the magnitude spectrum, |X[n]|, for a sinusoidal signal, obtained from a 64- point DFT. The original time-domain signal is of the form: x[k] = A cos(  1 k), where A is the amplitude and  1 is the discrete-time frequency. a. (10 pts) Find the amplitude A of the sinusoid. b. (10 pts) Find the discrete-time frequency,  1 , of the sinusoid. n

ECE 351 Name_____________________________________ Exam 3 – Fall ‘18 2/5 2. (Short-answer questions) a. (10 pts) The equations for finding the output, y[n] or Y(z), of a discrete-time LTI system with impulse response h[n] and transfer function H(z) are: y[n] = ____________________ (given the impulse response and the input x[n]) Y(z) = ____________________ (given the transfer function and the input X(z)) b. (5 pts) Find the transfer function, H(z), for the system with difference equation y[n+1] - .7 y[n] = x[n + 1] H(z) = ____________________________ c. (5 pts) Find the eigenvalues,  i , of the matrix: A = [ 2 0 1 3 ] . _________________________

ECE 351 Name_____________________________________ Exam 3 – Fall ‘18 3/5 3. (20 pts) Find the location of the poles for an analog second-order Butterworth filter with 3-dB cut-off frequency  c = 100 rad/sec. (You can express the locations in the complex s-plane in the form: r   , re j  , or a + bj.) You may use facts from the table of denominators 1 for H(s) for Butterworth filters, or you may find the poles analytically, by placement around the circle in the s-plane. Be sure to simplify your answer. Pole 1: ______________________________ Pole 2: ______________________________ 1 Recall for Butterworth filters, H (s) = 1/Q(s); for second-order Butterworth filters, Q(s) = 1 2 2 + + s s Re(s) Re(s)

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ECE 351 Name_____________________________________ Exam 3 – Fall ‘18 4/5 Part 2: This section of the exam is closed-book and closed-notes , but you may use a single 8.5” x 11” sheet of notes (with writing on both sides) and MATLAB. You also may consult MATLAB’s help or doc functions if you are unsure about a particular syntax. Once you have solved a problem, clean up your code before you submit it. 1. Use MATLAB ’s freqz function to plot the magnitude of the frequency response, |H(f)|, in decibels, for frequency values ranging from f = 0 to 400 Hz, for the FIR filter with transfer function: H(z) = .21𝑧 3 +.47𝑧 2 +.47𝑧+.21 𝑧 3 Assume a sample rate of f s = 800 samples/sec. If the printers work, print out your code and the plot, and hand them in when you’re done with Part 2. Back- up plan, only if the printers don’t work: email your code and plot to: dvanalphen@csun.edu.

ECE 351 Name_____________________________________ Exam 3 – Fall ‘18 5/5 2. Using MATLAB, generate the complex signal, x[k] = 2 + exp(j*2  k/3), for k values going from 0 to 63. Then find and plot |X[n]|, the magnitude of the 64-point DFT of x[k]. Don’t be surprised if your DFT is showing leakage. That will happen because  1 is not an integer multiple of the spacing  0 . If the printers work, print out your code and the plot, and hand them in when you’re done with Part 2. Back- up plan, only if the printers don’t work: email your code and plot to: dvanalphen@csun.edu