HO12_Midterm

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Apr 3, 2024

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1 ELEN234: Winter 2015 Midterm This is a closed book exam. As discussed in class, you are allowed one 8.5”x11” page of notes (both sides). You will have the full 110 minutes to work on the exam. Please budget your time accordingly. Do not rush so quickly that you make silly mistakes. Many of the problems on this exam require a bit of thought and a minimal amount of ‘grinding’ – so, think before you write. You should realize that you will get the most credit for showing that you understand the concepts involved in the problem. This does not mean listing everything that you know about a particular topic or relaying things that could be copied from an equation sheet – instead, I’m looking for a careful analysis of the problem at hand. If there is a stumbling block between you and a final answer, explain what that stumbling block is, make an assumption and move on. If you’re really stuck, come talk to me and I may give you a hint at the cost of a certain number of points. I’ll tell you how many points before I give you the hint. In general, the scale of your assumptions and their impact on the solution will determine how much partial credit is possible. There are no purposeful ambiguities on this test. If you have any questions, ask. Please show all work. State any assumptions where relevant. Make your final answer clear. When you are finished, please turn in this exam and your equation sheet (with your name on it). Good luck!!! Name:____________________________________________ Section 1: __________/ 28 Section 2: __________/ 10 Section 3: __________/ 18 Section 4: __________/ 8 Section 5: __________/ 20 Section 6: __________/ 16 Total: __________/ 100 On my honor as a Santa Clara University student, I have neither received nor given unauthorized assistance on this examination. Furthermore, I have not witnessed any such inappropriate activity. SIGNATURE: _____________________________

Name: 2 1. (28 points) The impulse response of a filter is: ݄ ଵ ሾ݊ሿ ൌ ߜሾ݊ሿ ൅ ߜሾ݊ െ 4ሿ a) (4 points) Write an expression for the frequency response, ܪሺ݁ ௝ఠ ሻ , for this filter. Reduce this expression as much as makes sense - we have done this many times and you should know what “makes sense.” b) (8 points) Plot the magnitude and phase of the frequency response found in part (a). c) (2 points) What is the delay of this system? This is the only delay we’ve talked about in this course – that is, I am looking for the delay that has made this a causal system.

Name: 3 d) (6 points) A new filter is created by adding a zero at ݖ ൌ ൅1 to create ݄ ଶ ሾ݊ሿ . Add an appropriate pole (or poles) to keep the system causal. What are the new coefficients? e) (2 points) What symmetry (if any) does ݄ ଶ ሾ݊ሿ have? Justify your answer. f) (6 points) Draw a pole-zero plot for the new filter, ݄ ଶ ሾ݊ሿ .

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Name: 4 2. (10 points) You are given a low pass filter (cutoff of ߱ ௖ ൌ 5ߨ/8 ) of order 8 with coefficients: [a b c d e d c b a]. a) (7 points) In terms of the coefficients above (a, b, c, d and e), what are the coefficients of a low pass filter with a cutoff of ߱ ௖ ൌ 3ߨ/8 . (Hint: we’ve learned two different ways to manipulate the frequency domain through turning an LPF into an HPF. Both could be useful here.) b) (3 points) Using your answer from (a), what are the coefficients, in terms of a, b, c, d and e, of an order 8 filter that produces a bandpass filter with a range from 3ߨ/8 to 5 ߨ/8 ?

Name: 5 3. (18 points) You are given the magnitude (top) and phase (bottom) plots of a frequency response for two systems. The ߱ -axis in each case is written in multiples of ߨ . So, the far left value of -1 represents – ߨ and the far right value of +1 represents ߨ . If you have any trouble reading anything, let me know. System 1: System 2: a) (2 points) Which system(s) has/have an impulse response with real coefficients? Briefly justify your answer. -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -4 -2 0 2 4 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -4 -2 0 2 4

Name: 6 b) (12 points) The signal ݔሺݐሻ ൌ cosሺ2ߨሺ10ሻݐሻ ൅ cosሺ2ߨሺ32ሻݐሻ is sampled at 100Hz to produce x[n]. i. Determine the angular frequency, ߱ , that each frequency component of x[n] resides at after x(t) is sampled. ii. x[n] is put through System 1. Provide your best guess for the equation of the output y[n] – if you can determine delay, make sure to include it in your equation. If it is not predictable, state that. Give a justification for your response. ii. x[n] is put through System 2. Provide your best guess for the equation of the output y[n] – if you can determine delay, make sure to include it in your equation. If it is not predictable, state that. Give a justification for your response.

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Name: 7 c) (4 points) The signal ݔሺݐሻ ൌ cosሺ2ߨሺ32ሻݐሻ is sampled at 100Hz and then put through System 2. Provide your best guess for the equation of the output y[n] – if you can determine delay, make sure to include it in your equation. If it is not predictable, state that. Give a justification for your response. Keep in mind exactly what non-linear phase means and how it impacts a signal. In other words, does the fact that this signal only has one frequency affect how a system with non-linear phase impacts it?

Name: 8 4. (8 points) An ideal Hilbert Transformer has an impulse response given by         odd for 2 even for 0 n n n n h  . You wish to implement an approximate realizable filter of length 7 using a rectangular window. a.) (4 points) Briefly list the steps of the procedure needed to create the approximate realizable filter from the ideal impulse response. b.) (4 points) Find the values of the filter coefficients for the length 7 filter.

Name: 9 5. (20 points) Block diagrams for two systems are shown below. System 1: System 2: a) (4 points) Write the difference equation for ݕ ଵ ሾ݊ሿ , the output of System 1, if its input is ݔ ଵ ሾ݊ሿ . b) (6 points) For System 2, write the transfer function ܪ ଶ ሺݖሻ .

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Name: 10 c) (6 points) If the two systems are cascaded, what is the order of the new system? Write four inequalities that the four parameters must satisfy to maintain stability? d) (4 points) If a = -1/2, b = -1/2, c = +1/2, d = -2, what is the new transfer function and the new difference equation?

Name: 11 6) (16 points) a) (3 points) What type of FIR filter (1, 2, 3 or 4) is represented by the pole-zero plot below? b) (3 points) What type of FIR filter (1, 2, 3 or 4) is represented by the pole-zero plot below? -1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 6 Real Part Imaginary Part -1 -0.5 0 0.5 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 3 2 7 Real Part Imaginary Part

Name: 12 c) (4 points) Sketch (and label) the magnitude of the frequency response for the pole-zero plot in part (a). d) (6 points) You are given the lattice structure below and the corresponding transfer function. ܪሺݖሻ ൌ ܻሺݖሻ ܺሺݖሻ ൌ ߚ ൅ ሺߚߜ ൅ ߛߝሻݖ ିଵ ൅ ߛݖ ିଶ 1 ൅ ሺߜ ൅ ߙߝሻݖ ିଵ ൅ ߙݖ ିଶ What are ߙ, ߚ, ߛ, ߜ ܽ݊݀ ߝ to realize the system: ݕሾ݊ሿ ൌ ݔሾ݊ሿ ൅ 2ݔሾ݊ െ 1ሿ െ 0.5ݕሾ݊ െ 1ሿ െ 0.25ݕሾ݊ െ 2ሿ ?

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