Exam2ECE3084

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Georgia Institute Of Technology**We aren't endorsed by this school

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ECE 3084

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Electrical Engineering

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Jan 14, 2025

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Mid-term exam #2 Date: November 9, 2022 ECE 3084 Signals and Systems Prof. Stanislav Emelianov Code of Honor Statement I have neither given nor received aid on this examination. Student Name (Last, First) Student Signature Rules: 1. This is a closed book examination. 2. One page (8.5 x 11) of notes written on both sides is permitted. 3. No graphic calculators, laptops, phones, or other sophisticated electronic devices allowed. 4. You must write your answer in the space provided on the exam paper itself. Only these answers will be graded. Circle your answers or write them in the boxes. If space is needed for scratch work, use the backs of previous pages. 5. Show your work for every problem! No credit will be given for irrelevant information and no credit will be given for the correct answer without relevant work. Instructions: 1. Write your name (last, first) on the front page of the exam only. 2. Make sure that I can tell what is your final answer. 3. You can solve every problem given below! 4. Note that problems have different weighting – see table below. Plan your work accordingly. 5. There should be no questions regarding any problems. If you have to make assumptions – state them clearly and proceed. However, make sure your assumptions are valid and necessary. 6. Please keep an eye on the whiteboard once in a while – there may be something useful written there as we go through the exam. 7. Keep the pages stapled. 8. If you finish exam early (i.e., attempted to solve all problems), there is a few bonus questions. Good luck – you will do well! Problem 1ab 2ab 3ab 4 5 6 7 Total Bonus Points 15 15 16 9 20 20 5 100 15

2Overall Grade: PROBLEM 1a (7 points) A continuous-time periodic signal 𝑥ሺ𝑡ሻis real-valued and has a fundamental period 𝑇଴ൌ8. The nonzero Fourier series coefficients for 0( )jntnnx tD eare 1*1552DDjDDExpress 𝑥ሺ𝑡ሻin the form 01cos( )nnnnx taatPROBLEM 1b (8 points) For the continuous-time periodic signal 25( )2cos4sin33x ttt, determine the fundamental frequency 0and the Fourier series coefficients Dnsuch that 0( )jntnnx tD e

3PROBLEM 2a (7 points) Use the Fourier transform analysis (direct integration) to calculate the Fourier transform of 21( )tx te(note 𝑎𝑏𝑠ሺ𝑡 െ1ሻhere) Show your analysis/derivations. Hint: |𝑡 െ1|ൌ ൜െሺ𝑡 െ1ሻ𝑓𝑜𝑟𝑡 ൏1ሺ𝑡 െ1ሻ𝑓𝑜𝑟𝑡 ൒1PROBLEM 2b (8 points) Consider signal 𝑥ሺ𝑡ሻgiven by ( )a tx tefor which Fourier transform 𝑋ሺ𝑗𝜔ሻexist. a) Derive 𝑋ሺ𝑗𝜔ሻusing direct integration. If you need to make any assumptions, clearly state them here. b) What are the values of 𝑥ሺ0ሻand 𝑋ሺ0ሻ? 𝑥ሺ0ሻ ൌ𝑋ሺ0ሻൌc) Sketch both 𝑥ሺ𝑡ሻand |𝑋ሺ𝑗𝜔ሻ|. 𝑥ሺ𝑡ሻ|𝑋ሺ𝑗𝜔ሻ|

4PROBLEM 3a (8 points) The left-hand column below shows four (1-4) continuous-time signals. The Fourier transform of each signal appears in the right-hand column in mixed-up order. Match the signal to its Fourier transform and briefly justify/explain your answer. # Letter Explanation #1 #2 #3 #4

5PROBLEM 3b (8 points) The magnitude (M) and phase (angle, A) of the Fourier transform of a signal x(t) are Four signals, derived from 𝑥ሺ𝑡ሻ, are shown in a table below. Six magnitude plots (M1-M6) and six phase or angle plots (A1-A6) are also shown. Determine which of these plots is associated with each of the four derived signals and place the appropriate label (e.g., M1 or A3) in the following table. Explain/justify your answer. Note that more than one derived signal could have the same magnitude or angle. Signal Magnitude (M1-M6) Angle of phase (A1-A6) 𝑥ሺ𝑡ሻ ∗ 𝑥ሺ𝑡ሻ𝑥ሺ𝑡 െ𝜋2ൗሻ𝑥ሺ2𝑡ሻ𝑥ଶሺ𝑡ሻ

6PROBLEM 4 (9 points) Given the following spectrum, ()X, sketch the spectrum of the sampled function, ( )x t, for the following sampling frequencies and comment if the original signal can be reconstructed using ideal low-pass filter. Note circular frequency 𝜔above and linear frequencies 𝑓௦below! (a) fs= 2 Hz The original signal can be / cannot be reconstructed using ideal low-pass filter. (b) fs= 1 Hz The original signal can be / cannot be reconstructed using ideal low-pass filter. (c) fs= 0.5 Hz The original signal can be / cannot be reconstructed using ideal low-pass filter -230-2-3X-230-2-3X-230-2-3X-230-2-3X

7PROBLEM 5 (20 points) The real-valued signal 𝑥ሺ𝑡ሻhas a Nyquist rate (or frequency) 𝜔ேൌ2𝜔஻. Determine the Nyquist rate for the following signals. Justify your answer. (a)ௗమ௫ሺ௧ሻௗ௧మ𝜔ேൌ_____ Explanation:(b) 𝑥ሺ𝑡ሻ ∗ 𝑥ሺ𝑡ሻ𝜔ேൌExplanation: (c) 𝑥ଶሺ𝑡ሻ𝜔ேൌ_____ Explanation: (d) ௫ሺ௧ሻଶఠಳെ௫ሺି௧ሻଶగఠಳ𝜔ேൌ_____ Explanation: (e) 𝑥ሺ2𝜔஻𝑡ሻ𝜔ேൌ_____ Explanation: PROBLEM 6 (20 points) The Nyquist sampling periods for 1-D band-limited signals 𝑓ሺ𝑡ሻand 𝑔ሺ𝑡ሻare 𝑇௙and 𝑇௚, respectively. Find the Nyquist sampling periods 𝑇ேfor the following signals 𝑥ሺ𝑡ሻ(i.e., if possible, state 𝑇ேexplicitly in terms of 𝑇௙and/or 𝑇௚). Explain. A) ( )ox tftt𝑇ேൌB)  x t= ft+ g t𝑇ேൌC)  ( )*( )x tftf t𝑇ேൌD)  ( )( )x tftg t𝑇ேൌE) ( )f t𝑇ேൌPROBLEM 7 (5 points) The signal 𝑦ሺ𝑡ሻis generated by convolving a band-limited signal 𝑥ଵሺ𝑡ሻwith another band-limited signal 𝑥ଶሺ𝑡ሻ, that is 12( )( )*( )y tx txtwhere 12()0 1,000()0 2,000XforXforThe signal 𝑦ሺ𝑡ሻis sampled using impulse-train with sampling period 𝑇௦. Determine the maximumsampling interval 𝑇௦such that 𝑦ሺ𝑡ሻis recoverable from the sampled signal using ideal lowpass filter.

8PROBLEM BONUS 1 (3 points) Determine whether the following signal is periodic. If so, determine the fundamental frequency. a) 𝑥ሺ𝑡ሻ ൌ2cos൫√2𝑡൯ ൅sinሺ𝑡ሻ𝜔଴ൌ____. Explain: b) 𝑥ሺ𝑡ሻ ൌ7cosሺ7𝑡ሻ െ3 cosሺ3𝑡ሻ𝜔଴ൌ____. Explain: c) 𝑥ሺ𝑡ሻ ൌ2𝑒ି௝ቀ௧ାഏరቁ𝜔଴ൌ____. Explain: PROBLEM BONUS 2 (9 points) Suppose you are given the following information about a signal 𝑥ሺ𝑡ሻ: 1) 𝑥ሺ𝑡ሻis a real signal 2) 𝑥ሺ𝑡ሻis periodic with period 𝑇଴ൌ4and has Fourier coefficients 𝐷௡3) 𝐷௡ൌ0for |𝑛|൒24) The signal with Fourier series coefficients 𝐵௡ൌ 𝑒ି௜ഏమ௡𝐷ି௡is odd 5) ଵସ׬|𝑥ሺ𝑡ሻ|ଶ𝑑𝑡ସൌ0.56) D1is a positive real number Determine the signal 𝑥ሺ𝑡ሻ. Justify your answer. PROBLEM BONUS 3 (3 points) The band-limited signals 𝑥ଵሺ𝑡ሻand 𝑥ଶሺ𝑡ሻare multiplied together and the product 𝑦ሺ𝑡ሻ ൌ 𝑥ଵሺ𝑡ሻ ∙ 𝑥ଶሺ𝑡ሻis sampled by a periodic impulse train. Given that 1122()0, ()0, XX, determine the maximumsampling interval 𝑇௦such that 𝑦ሺ𝑡ሻis recoverable from the sampled signal through the use of ideal lowpass filter (LPF).