BME3508_Fall2021_test1-solution

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BME3508 Biomedic a l Signals and Systems Fall 2021 September 17 th BME3508: Biosignals and Systems Test #1 Name: ___________________________________ UFID: ________________________ Q1 [30] Q2[30] Q3 [30] Q4 [10] Total [100] Instructions: 1. Write your name ON EVERY PAGE. 2. Number the pages. 3. Write your answers on the front pages only. DO NOT write on the back of any page. 4. Show all your work. 5. Make sure you scan every page. 6. Sign the academic honesty pledge. Academic Honesty: All students admitted to the University of Florida have signed a statement of academic honesty committing themselves to be honest in all academic work and understanding that failure to comply with this commitment will result in disciplinary action. This statement is a reminder to uphold your obligation as a UF student and to be honest in all work submitted and exams taken in this course and all others. I affirm that I have acted in accordance with the UF Academic Honesty Guidelines. Signature ____________________________________________

Biosignals & Systems Name:______________________________ BME3508-Fall 2021 Test #1 Page 2 of 7 Question 1: [30pts] You have joined the Engineering Research & Development team at Medtronic. The team is working on a revision of their patent, “ Programmable waveform pulses for an implantable medical device ”, aiming to change the old square wave-like stimulation waveform. The goal is to replace the sharp rise of the first pulse of the stimulation waveform (“hard - start”) , with a smoothly rising pulse (“soft - start” ), described by x 1 (t). The second pulse of the stimulation waveform is described by x 2 (t). It is essential for the overall stimulation waveform to be charge-balanced, which means that both pulses, first and second must have the same size (strength). Figure Q1.A Left : x 1 (t) depicts the “ soft start” puls e (first pulse) of the stimulation waveform. Right: x 2 (t) depicts the second pulse, of the stimulation waveform. A) [15pts] Choose a proper measure of size (Energy or Power) and calculate the size of each signal. What is the amplitude ratio ( ࠵? ࠵? ) for which ࠵? ࠵? (࠵?) and ࠵? ࠵? (࠵?) would be of equal size. B) [10pts] The team needs to mathematically describe the overall stimulation waveform ࠵?(࠵?) , where ࠵?(࠵?) = ࠵? ࠵? (࠵?) + ࠵? ࠵? (࠵?) . Sketch ࠵?(࠵?) and describe it mathematically using common signals and time domain operations. ࠵?(࠵?) = √࠵? √࠵? Xzlt )={-MB, ⾨ tE8 0,01W ✗ if )={ ☒ hostel ☒ , lets } 0,01W - binalsau aperiodic Ifinite • • Exifexttiidt µ× i÷÷÷ " " ¥ A=s ☐ =/ Bdt =§Atidt+ / Adt = , µ ; | € A # 1.313 =A ¥ /dtÉt| ? =A+2A=IA Ma - - B- rant cult ) - net -113 + Frutti ) - ult -33 ] - cult -5 ) - ult - 8) ]

Biosignals & Systems Name:______________________________ BME3508-Fall 2021 Test #1 Page 3 of 7 Question 1: [30pts] continued C) [2.5pts] The lead engineer assumed arbitrary values for the amplitude A and used MALTAB to plot the overall stimulation waveform, ࠵?(࠵?) , depicted in Figure Q1.C. The lead engineer referrs to x(t) as a zero-mean signal. Do you agree/disagree? Why? [1-2 sentences] Answer: D) [2.5pts] How would you change your ࠵?(࠵?) expression in part B) to match the waveform if part C? ࠵?(࠵?) = Figure Q1. C I disagree . The signal is a non - Zero mean signal as the waveform is centered around ~ 1 and not Zero . It seems that He waveform is shifted up by 1 . ✗ fixed ( t ) = ✗ ( t ) ¥ ⊥ me value by whid the entire waveform was shifted up .

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Biosignals & Systems Name:______________________________ BME3508-Fall 2021 Test #1 Page 4 of 7 Question 2: [30pts] The Engineers at Medtronic got bored and are using the programmable waveform generator to output a little house waveform, ࠵? ࠵? (࠵?) , shown in Figure Q2.1 A) [10pts] Give a piece-wise mathematical description of ࠵? ࠵? (࠵?). Answer: Figure Q2.1 B) [5pts] Calculate the signal mean ࠵? ࠵? ࠵? . C) [15pts] Sketch ࠵?(࠵?) = −࠵? ࠵? ( ࠵? ࠵? + ࠵?) − ࠵? in the blank axes below. Answer: 2 ; -2 ⾨ too ✗ • (f) = t ; cost EZ - flt - 8) ;2EtE5 ( o ; on IT " . . . . . , I 5 £ 2 M × °=÷+ , / x.lt/dt=s.,-.y.I2dttftattf-t-+g- at ti = ¥ [ ztl.it#1i+l-E+sft)/i1--EK4+2+(-Ei------E-D--0; ? ---- 1.5 ① shift left by 3 flip around x-axis & ② scale by { → expand shift down by 1 nsylt )

Biosignals & Systems Name:______________________________ BME3508-Fall 2021 Test #1 Page 5 of 7 Question 3: [30pts] Consider the sinusoid ࠵?(࠵?) = −࠵? ࠵? ࠵? ࠵?(࠵?.࠵?࠵?࠵? − ࠵?.࠵?࠵?) + ࠵? A) [7.5pts] Determine the imaginary part of the sinusoid, ࠵?࠵?{࠵?(࠵?)} , which we will refer to as ࠵? ࠵? (࠵?) from now on. ࠵? ࠵? (࠵?) = ࠵?࠵?{࠵?(࠵?)} = ____________________________________________________ B) [5pts] Consider ࠵? 2 (࠵?) from par A). Find the following parameters: Answer: A= ____________ ࠵? = _____________ ࠵? = ____________ T = _____________ f =_______________ C) [2.5pts] Determine the start time, ࠵? ࠵?࠵?࠵?࠵? , of ࠵? 2 (࠵?) . D) [2.5pts] If ࠵? ࠵?࠵?࠵?࠵? was equal to -3s. What would the new phase shift, ࠵? ࠵?࠵?࠵? , be? E) [7.5pts] Without using integrals. Calculate the Power of ࠵? 3 (࠵?) , where ࠵? 3 (࠵?) = ࠵? 2 (࠵?) + 4 cos(0.3 ࠵? ࠵? + 1.8 ࠵? ) F) [5pts ] Consider the general sinusoid ࠵?(࠵?) = ࠵? cos( ࠵?࠵?). Hand-sketch the signal y(t) = 0.5 ࠵? (࠵?)[࠵?(࠵?) − ࠵?(࠵? − 2࠵?)] . ✗ ( t ) = -3C - j ) [ cos 10.4Mt -1.317 ) tjs.info -4Mt -1-317 ) ] + j = 3J cos C- . ) + 3J ' sin C- . . ) c- j = 3J cos C. . ) -3 Sint - . . ) + j = -3 sin C- . . ) + j( scosc . . ) -11 ) 3 cos( 0.4Mt -1.317 ) -11 w= a. 417=21^1 3 1.317rad → f=° " # 0.417 racks 5s 0 - ZHZ T=5s 1- peak happens when tfearswmantot 0.617 tpak -1.317=0 HE cos is = Zero . 0.4 tpeak = 1. 3 tpeek =3 -25s s 4 = - Wotpeek new = - 0.4M C- 5) chew = 217 Rolls F- → sum of 2 sinusoids Pxs = Pxztpx , = Iz + ¥ =/ 2.5 monogamous E) Mxz = 1 since it hes an offset 47 .

Biosignals & Systems Name:______________________________ BME3508-Fall 2021 Test #1 Page 6 of 7 Question 4: [10pts] A. A nerdy way to ask someone to change the playback rate of a video to 0.5x is to say: Can you please time-expand the signal by 0.5?. [ True | False ] B. The mean of a signal is a good measure of its fluctuations. [ True | False ] C. Time scaling a signal does not change its energy. [ True | False ] D. The product of an odd and an odd function is an odd function. [ True | False ] E. Time-shifting a signal changes its power. [ True | False ] 0 ☐ the variance or He standard den . ate ① It changes He energy since He waveform becomes compressed / expanded & this will change He ate under the entire . ① it's even time - shifting doesn't change Energy & hence ① won't change poor either , as power is the time - avenge of the Energy

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Biosignals & Systems Name:______________________________ BME3508-Fall 2021 Test #1 Page 7 of 7 List of Formulas ࠵?(࠵?) = ࠵? cos( ࠵?࠵? + ࠵?) ࠵? = − ࠵? 0 ࠵? ࠵?࠵?࠵?࠵? Euler’s Formula ࠵?࠵? ࠵?࠵? = A [cos( ࠵?) + ࠵? sin( ࠵?) ] ࠵?࠵? −࠵?࠵? = A [cos( ࠵?) − ࠵? sin( ࠵?)] ࠵?(࠵?) = ℜ{࠵?࠵? ࠵?࠵? } ࠵? = ࠵? + ࠵?࠵? = ࠵? ࠵?࠵?࠵?(࠵?) + ࠵? ࠵? ࠵?࠵?࠵?(࠵?) = ࠵?࠵? ࠵?࠵? ࠵? 2 = ࠵? 2 + ࠵? 2 ࠵? = t an −1 ( ࠵? ࠵? ) ࠵? ࠵? = ∫ ࠵?(࠵?) 2 ࠵?࠵? ࠵?2 ࠵?1 ࠵? ࠵? = 1 ࠵? 2 − ࠵? 1 ∫ ࠵?(࠵?) 2 ࠵?࠵? ࠵? 2 ࠵? 1 ࠵? ࠵? = ࠵? ࠵? ࠵? 2 − ࠵? 1 ࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? = ࠵? 2 2 ࠵? ࠵?࠵?࠵? = √࠵? ࠵? ࠵? ࠵? = 1 ࠵? 2 − ࠵? 1 ∫ ࠵?(࠵?) ࠵?࠵? ࠵?2 ࠵?1