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Practice Final Exam for Signal Processing I: Key Concepts

School
University Of Connecticut**We aren't endorsed by this school
Course
ECE 3780
Subject
Electrical Engineering
Date
Dec 10, 2024
Pages
10
Uploaded by SuperThunder12978
1 DEPARTMENT: Electrical and Computer Engineering COURSE: Signal Processing I (Winter 2025) EXAM TYPE: Practice Final Exam TOTAL PAGE: 10 test pages + 1 table page DATE: 11/04/2025COURSE NO.: ECE3780 DURATION: 1:30PM –4:30PM (3 hours) ------------------------------------------------------------------------------------------------------------ STUDENT NUMBER ----------------------------------------------------------------------------------------------------------- LAST NAME FIRST NAME ------------------------------------------------------------------------------------------------------------ SIGNATURE Material Allowed:Non-programmable calculators. One aid sheet (US Letter size, double-sided). Material Not Allowed:Books, notes, or other aids, all other electronics (including smart watches). Answer All Questions in the Provided Space. You may want to first work things through on scratch paper and then transfer to this booklet. Only this booklet will be considered in the grading. Academic Integrity Contract Cheating is a serious offense. As members of the University Community, students have an obligation to act with academic integrity. Any Student who engages in Academic Misconduct in relation to a University Matter will be subject to discipline. Instructions: Wait for the β€œBEGIN” signalbefore opening this booklet. In the meantime, fill out the requested info above. Put your ID card (with photo) on the desk. Your answers to multiple-choice questions do NOT require justification. For calculation/proof questions, give enough details to justify your answer. You will get only a portion of the points if you just give the final value/result. When the end of the exam is announced, you must stop writing immediately. Anyone caught writing after the exam is over will get a penalty of 10 points deducted from the total grade. For examiner use: Question 1 2 3 4 5 6 7 8 9 Grade /5 /3 /3 /5 /5 /5 /5 /5 /4 TOTAL GRADE/40
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2 Question 1.Multiple choices. Choose only onecorrect answer from the provided options. (1 ptfor each sub-question. 5 ptsin total) 1.1Suppose a periodic signal has the following two-sided Fourier spectra, which of the following accurately describes its time domain expression? ---------------------------------------------------( C ) (A) ?(?) = 10 + 6cos(2?) + 3sin(4?)(B) ?(?) = 10 + 6cos(2?) + 3cos(4?)(C) ?(?) = 10 + 12cos(2?) + 6sin(4?)(D) ?(?) = 10 + 12cos(2?) + 6cos(4?)1.2The Fast Fourier Transform is a very efficient algorithm to numerically compute the Fourier spectra compared to the Discrete Fourier Transform. For a signal sequence that contains 1024 data points, the number of multiplications required for obtaining the Fourier Spectra using the Fast Fourier Transform is ---------------------------------------------------------------------------------( B ) (A) of the order of 1000 (B) of the order of 10,000 (C) of the order of 100,000 (B) of the order of 1,000,000 1.3For an LTI system with an input x(t) and an initial state, the natural response (zero-input response) yNand the forced response (zero-state response) yFare ?𝑁= ?βˆ’2??(?) βˆ’ ?βˆ’??(?),?𝐹= 2cos(?)?(?) + 3?(?) + 2?βˆ’??(?). Then the expression of the transient response yTand the steady-state response yScan be expressed by: -------------------------------------------------------------------------------------------------------------------( D ) (A)??= ?βˆ’2??(?) βˆ’ ?βˆ’??(?), ??= 2cos(?)?(?) + 3?(?) + 2?βˆ’??(?)(B) ??= ?βˆ’2??(?) + ?βˆ’??(?), ??= 3?(?)(C)??= ?βˆ’2??(?) βˆ’ ?βˆ’??(?), ??= 2cos(?)?(?) + 3?(?)(D)??= ?βˆ’2??(?) + ?βˆ’??(?), ??= 2cos(?)?(?) + 3?(?)1.4The Laplace transform of an output signal from an LTI system is ?(?) =?βˆ’1?2+2?. The initial value and the final value of the output signal are -----------------------------------------------------( B ) (A) initial value = -0.5, final value = 1 (B) initial value = 1, final value = -0.5 (C) initial value = -1, final value = 0.5 (D) initial value = 0.5, final value = -1 1.5For the Laplace transform of a continuous-time signal ?(?) = ?2??(?) + ??cos(?)?(?), which of the following is its region of convergence (Re[s] = Οƒ) -------------------------------------------------------------------------------------------------------------------( B ) (A) Re[s] > 1 (B) Re[s] > 2 (C) Re[s] < 1 (D) Re[s] < 2
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3 Question 2 (Fourier Series) Find the trigonometric form of the Fourier Series for the following signal (3 pts): ?0=1𝑇0∫ ?(?)?0?? =13∫ 210?? βˆ’13∫ 221?? = 0??=2𝑇0∫?(?)cos(π‘›πœ”0?)?0?? =23∫ 2cos (23π‘›πœ‹?)10?? βˆ’23∫ 2 cos (23π‘›πœ‹?)21??=4π‘›πœ‹sin (23π‘›πœ‹) βˆ’2π‘›πœ‹sin (43π‘›πœ‹)??=2𝑇0∫?(?)sin(π‘›πœ”0?)?0?? =23∫ 2sin (23π‘›πœ‹?)10?? βˆ’23∫ 2 sin (23π‘›πœ‹?)21??=2π‘›πœ‹βˆ’4π‘›πœ‹cos (23π‘›πœ‹) +2π‘›πœ‹cos (43π‘›πœ‹)Therefore, ?(?) = βˆ‘2π‘›πœ‹(2sin (23π‘›πœ‹) βˆ’ sin (43π‘›πœ‹)) cos (23π‘›πœ‹?)∞?=1+2π‘›πœ‹(1 βˆ’ 2cos (23π‘›πœ‹) + cos (43π‘›πœ‹)) sin (23π‘›πœ‹?)
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4 Question 3 (Fourier Series) Consider a continuous-time, periodic signal that can be described by the following Fourier Series (complex form) ?(?) = 2 + (5 + 12𝑗)???+ (3 + 4𝑗)??2?+ (5 βˆ’ 12𝑗)?βˆ’??+ (3 βˆ’ 4𝑗)?βˆ’?2?If we know that the fundamental frequency of the signal πœ”0= 1 ???/?, Derive its trigonometric form (cosine and sine functions) and the compact form (only cosine functions). (3 pts) ?(?) = 2 + 13??67.4Β°???+ 5??53.1Β°??2?+ 13?βˆ’?67.4Β°?βˆ’??+ 5?βˆ’?53.1Β°?βˆ’?2?= 2 +26cos(? + 67.4Β°) +10cos(2? + 53.1Β°)(compact form) = 2 +26cos(?)513βˆ’26sin(?)1213+10cos(2?)35βˆ’10sin(2?)45= 2 +10cos(?) βˆ’24sin(?) +6cos(2?) βˆ’8sin(2?) (trigonometric form)
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5 Question 4. (Fourier Transform) For a causal LTI system, suppose we have the following impulse response function Using the following input signal, verify the convolution property of the Fourier Transform, i.e. β„±(?(?) βˆ— β„Ž(?)) = ?(πœ”) Γ— ?(πœ”)(5 pts) Time domain convolution of x(t)and h(t): ?(?) βˆ— β„Ž(?) = ∫?(? βˆ’ 𝜏)βˆžβˆ’βˆžβ„Ž(𝜏)?𝜏= ∫1 Γ— ?βˆ’πœ?+10?𝜏 Γ— ((?(? + 1) βˆ’ ?(? βˆ’ 1)) + ∫1 Γ— ?βˆ’πœ?+1?βˆ’1?𝜏 Γ— ?(? βˆ’ 1)= (1 βˆ’ ?βˆ’?βˆ’1)?(? + 1) βˆ’ (1 βˆ’ ?βˆ’?βˆ’1)?(? βˆ’ 1) + (?βˆ’?+1βˆ’ ?βˆ’?βˆ’1)?(? βˆ’ 1)= (1 βˆ’ ?βˆ’?βˆ’1)?(? + 1) βˆ’ (1 βˆ’ ?βˆ’?+1)?(? βˆ’ 1)= ?(? + 1) βˆ’ ?(? βˆ’ 1) βˆ’ ?βˆ’(?+1)?(? + 1) + ?βˆ’(?βˆ’1)?(? βˆ’ 1)Note that ?(? + 1) βˆ’ ?(? βˆ’ 1) = rect(?/2), and β„±(rect(?/2)) = 2sinc(πœ”)β„±(?(?) βˆ— β„Ž(?)) = 2sinc(πœ”) βˆ’11 + π‘—πœ”??πœ”+11 + π‘—πœ”?βˆ’?πœ”=2sin(πœ”)πœ”βˆ’11 + π‘—πœ”(??πœ”βˆ’ ?βˆ’?πœ”)=2sin(πœ”)πœ”βˆ’2𝑗1 + π‘—πœ”(??πœ”βˆ’ ?βˆ’?πœ”2𝑗) =2sin(πœ”)πœ”βˆ’2𝑗1 + π‘—πœ”sin(πœ”)=2 + 2π‘—πœ” βˆ’ 2π‘—πœ”πœ”(1 + π‘—πœ”)sin(πœ”) =2πœ”(1 + π‘—πœ”)sin(πœ”)Frequency representation of x(t)and h(t): ?(πœ”) = ∫?βˆ’??βˆ’?πœ”?∞0?? =11 + π‘—πœ”?(πœ”) = 2sinc(πœ”)Multiplication of those two: ?(πœ”) Γ— ?(πœ”) =11 + π‘—πœ”Γ— 2sin(πœ”)πœ”=2πœ”(1 + π‘—πœ”)sin(πœ”)Therefore, we have β„±(?(?) βˆ— β„Ž(?)) = ?(πœ”) Γ— ?(πœ”)The convolution property of the Fourier Transform is verified.
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6 Question 5 (Frequency Response) (a) Find the Frequency Response of the following circuit, where x(t)is the input signal and y(t)is the output signal. C= 0.5 F, L = 1 H, R = 3 Ξ©(2 pts) In the frequency domain, using the current to represent corresponding voltages, we have the following description of the impedance: 𝑅 β†’ 𝑅; 𝐿 β†’ π‘—πœ”πΏ; 𝐢 β†’ 1/π‘—πœ”πΆTherefore, using Ohm’s Law:?(πœ”) =1π‘—πœ”πΆ1π‘—πœ”πΆ+ π‘—πœ”πΏ + 𝑅?(πœ”) =11 βˆ’ πœ”2𝐿𝐢 + π‘—πœ”π‘…πΆ?(πœ”)Or: ?(πœ”) =?(πœ”)?(πœ”)=11 βˆ’ πœ”2𝐿𝐢 + π‘—πœ”π‘…πΆ=22 + (π‘—πœ”)2+ π‘—πœ”3(b) Suppose there is no initial states with the capacitor and the inductor, determine the output signal associated with the input signal ?(?) = 4cos(?) + 2sin(2?)(3 pts) Method 1: ?(πœ”) = 4(πœ‹π›Ώ(πœ” βˆ’ 1) + πœ‹π›Ώ(πœ” + 1)) + 2(π‘—πœ‹π›Ώ(πœ” + 2) βˆ’ π‘—πœ‹π›Ώ(πœ” βˆ’ 2))?(πœ”) = ?(πœ”)?(πœ”)= (4πœ‹π›Ώ(πœ” βˆ’ 1) + 4πœ‹π›Ώ(πœ” + 1) + 2π‘—πœ‹π›Ώ(πœ” + 2) βˆ’ 2π‘—πœ‹π›Ώ(πœ” βˆ’ 2))22 + (π‘—πœ”)2+ π‘—πœ”3= 8πœ‹π›Ώ(πœ” βˆ’ 1)1 + 𝑗3+ 8πœ‹π›Ώ(πœ” + 1)1 βˆ’ 𝑗3+ 4π‘—πœ‹π›Ώ(πœ” + 2)βˆ’2 βˆ’ 6π‘—βˆ’ 4π‘—πœ‹π›Ώ(πœ” βˆ’ 2)βˆ’2 + 6𝑗=45πœ‹(1 βˆ’ 𝑗3)𝛿(πœ” βˆ’ 1) +45πœ‹(1 + 𝑗3)𝛿(πœ” βˆ’ 1) +110π‘—πœ‹(βˆ’2 + 6𝑗)𝛿(πœ” + 2)βˆ’110π‘—πœ‹(βˆ’2 βˆ’ 6𝑗)𝛿(πœ” βˆ’ 2)=45πœ‹(𝛿(πœ” βˆ’ 1) + 𝛿(πœ” + 1)) +125π‘—πœ‹(𝛿(πœ” + 1) βˆ’ 𝛿(πœ” + 1))βˆ’15π‘—πœ‹(𝛿(πœ” + 2) βˆ’ 𝛿(πœ” βˆ’ 2)) βˆ’35πœ‹(𝛿(πœ” + 2) + 𝛿(πœ” βˆ’ 2))?(?) = β„±βˆ’1(?(πœ”)) =45cos(?) +125sin(?) βˆ’15sin(2?) βˆ’35cos(2?)Method 2: ?(πœ”) =2(1 + π‘—πœ”)(2 + π‘—πœ”)=21 + π‘—πœ”βˆ’22 + π‘—πœ”?(1) =21 + 3𝑗=15(1 βˆ’ 3𝑗) =√105??arctan(βˆ’3)?(2) =1βˆ’1 + 3𝑗=110(βˆ’1 βˆ’ 3𝑗) =√1010?j(arctan(3)βˆ’πœ‹)Therefore, ?(?) = 4|?(1)|cos(? + ∠?(1)) + 2|?(2)| cos(2? + ∠?(2) βˆ’ πœ‹/2)=4√105cos(? + arctan(βˆ’3)) +√105cos(2? + arctan(3) βˆ’ 3πœ‹/2)It can be proven that the two results are mathematically the same.
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7 Question 6. (Laplace Transform) For each of the following differential equations describing the input-output association of an LTI system, determine its transfer function and BIBO stability. (5 pts) (a) ?2?(?)??2+ 4??(?)??+ 4?(?) = ?(?)?(?) =1?2+ 4? + 4=1(? + 2)2BIBO stable as poles = -2, -2 (b) 5?2?(?)??2βˆ’ 4??(?)??βˆ’ ?(?) = 7?(?)?(?) =75?2βˆ’ 4? βˆ’ 1=7(? βˆ’ 1)(5? + 1)BIBO unstable as poles = 1, -0.2 (c) 5?2?(?)??2+ 4??(?)??βˆ’ ?(?) = 2??(?)??+ 7?(?)?(?) =2? + 75?2+ 4? βˆ’ 1=2? + 7(? + 1)(5? βˆ’ 1)BIBO unstable as poles = -1, 0.2 (d) ?2?(?)??2+ 2??(?)??+ ?(?) =?2?(?)??2+??(?)??+ ?(?)?(?) =?2+ ? + 1?2+ 2? + 1=?2+ ? + 1(? + 1)2BIBO stable as poles = -1, -1 (e) ??(?)??+ 2?(?) =?2?(?)??2+ 2??(?)??+ ?(?)?(?) =?2+ 2? + 1? + 2BIBO unstable as this is an improper transfer function
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8 Question 7. (Frequency Response) The frequency response ?(πœ”)of an ideal band pass filter can be depicted as follows: The Fourier Transform of an input signal is: (a) Craft the Fourier spectra of the filtered signal ?(?)(2 pts) (b) Determine the signal energy of the filtered signal ?(?)(2 pt) Using the Parseval’s energy theorem𝐸?=1πœ‹βˆ« 4232?πœ” =16πœ‹(c) Find the 95% Energy bandwidth of the filtered signal ?(?)(1 pts) 𝐸95%=1πœ‹βˆ« 42Ξ©2?πœ” =16πœ‹(Ξ© βˆ’ 2) =16πœ‹Γ— 0.95Ξ© = 2.95 (???/?)
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9 Question 8 (Laplace Transform for System Analysis). Consider the following circuit (a) Determine the transfer function of the system. (2 pts) 𝑉?(?) = 𝑉?(?) Γ—?𝐿 + 𝑅2?𝐿 + 𝑅1+ 𝑅2?(?) =?𝐿 + 𝑅2?𝐿 + 𝑅1+ 𝑅2(b) when i(0-) = 2 A, vi(t) = u(t), R1= 1 Ξ©, R2= 2 Ξ©, L= 1 H. Find the zero-input response and the zero-state response. (3 pts) Using KVL, 𝑉?(?) = ? Γ— 𝑅1+ 𝑉?(?)Also 𝑉?(?) = ? Γ— (?𝐿) βˆ’ 𝐿?(0 βˆ’) + ? Γ— 𝑅2Therefore, 𝑉?(?) =𝑉?(?) βˆ’ 𝑉?(?)𝑅1?𝐿 βˆ’ 𝐿?(0 βˆ’) +𝑉?(?) βˆ’ 𝑉?(?)𝑅1𝑅2That is (𝑅1+ ?𝐿 + 𝑅2)𝑉?(?) = (?𝐿 + 𝑅2)𝑉?(?) βˆ’ 𝐿𝑅1?(0βˆ’)𝑉?(?) =? + 2? + 3𝑉?(?) βˆ’1? + 3?(0βˆ’)Zero-input response -> Vi(s)=0 𝑉?βˆ’?(?) = βˆ’2? + 3??βˆ’?(?) = βˆ’2?βˆ’3??(?)Zero-state response -> I(0-)=0 𝑉?βˆ’?(?) =? + 2? + 3Γ—1?=1/3? + 3+2/3?=13? + 9+23???βˆ’?(?) =13?βˆ’3??(?)+23?(?)
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10 Question 9 (Sampling Theory) Suppose that a sound recorder samples continuous-time sound waves at a constant sampling rate of fs= 5Hz (or Ο‰s= 10Ο€rad/s). (a) What is the Nyquist frequency of the recorder? (2 pts) The Nyquist frequency = 2.5 Hz. (b) If Sam uses the recorder to record a sound wave that can be expressed as ?(?) = cos (2πœ‹? +πœ‹3) + 2cos (4πœ‹? +23πœ‹)Will he be able to reconstruct the original continuous-time signal later by applying a low-pass filter to the sampled discrete-time sequence? If yes, please provide him with suggestions on the range of the cutoff frequency that he should be using in his low-pass filter design. If no, please tell him what range of sampling rate he should be looking for to get a new sound recorder that would allow him to perfectly reconstruct his sound signal afterwards. (2 pts) The bandwidth of the signal = 2Hz < the Nyquist frequency. Sam is able to reconstruct the signal. The Fourier Spectra of the sampled signal is: Therefore, the range of the cutoff frequency of the low-pass filter should be 2?? < ?𝑐???𝑓𝑓< 3??
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