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by: Emily Braun II


Emily Braun II
GPA 3.82

Christopher Neils

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Christopher Neils
Class Notes
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This 9 page Class Notes was uploaded by Emily Braun II on Wednesday September 9, 2015. The Class Notes belongs to BIOEN 302 at University of Washington taught by Christopher Neils in Fall. Since its upload, it has received 67 views. For similar materials see /class/191998/bioen-302-university-of-washington in Bioengineering at University of Washington.




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Date Created: 09/09/15
Fourier Series scaling property Fullwave recti ed sine BIOEN 302 Timedomain 4 2 E u 2 4 Lecture 13 WE Fourier Transforms October 24 2007 Frequency domain an an 710 1D nenumw rmsnc Upcoming Events Fourier Series scaling property Homework 1 m ave recti ed sine Assigned on Friday due next Wednesday 5 TFT Quiz 3 This Friday October 26 Time domain Fourier series some multiple choice one magnitude and phase plot quot 2 W58 2 quot Practice Problems 161 ad 1610 ae 1617 concepts only Ampl amp phase plots of On examples throughout chapter See 2006 quiz question 2 bc on line F qu CY dmaiquot Quiz 4 Friday November 9 LTI systems Van ran 71 n 1 u an an nenuenw fadsec Fourier Series scaling property Fourier Series scaling property 1 Fullwave rectified sine T2 T no 1 t0T f t Z OneW Cquot F If 061W quotn to Complex coefficients Magnitude and can vary inversely with T 4 2 2 Coef cients become smaller and more closely 1U spaced as period increases Time domain 4 lCnl Frequency domain Van ran an an 71 in 1D rrmuency rmsnc Fourier Series scaling property 1 Fullwave recti ed sine T2 4T Time domain 4 2 4 it me E El 5 Frequency domain 2 an an 4n 1U quotequmw KaiSE and so on Fourier Series scaling Things to note As T gt oo can gt 0 Le the series becomes continuous As T gt oo Cn gt 0 but the sum ofthe coefficients over an interval on1 to 032 remains finite Therefore we can calculate CnT which is a finite continuous function of to The resulting function is The Fourier Transform 6a jgte w dt Radianform 39m 1 m t i G cu elm dcu 90 Mi Gm Igte Z7Wdt Henzfmm 7 go jaw air The Fourier Transform m 7 wt 1 m mt F0 my I d ft IFme dm Things to note The FT is a weighting function for sinusoidal or complex exponential content in signal The FT transforms a continuous aperio i function in timeinto an aperiodic continuous function in frequency Because both the FT and lFT contain complex exponentials there are many cases of duality among FT pairs A few Fourier airs Foo ifte39l a t f t TF03ede ft 1 or any constant This function has zero frequency or in nite period The area under the FT curve must be finite for the amplitude of the timedomain signal to be nonzero Therefore FT1 50 A few Fourier pairs ft 008m0t This function has a speci c frequency Negative and positive frequencies are both present to cancel the imaginary part Therefore Fcosmut 27E2509mu 5aHnD Note that Fcosmut is real and even A few Fourier pairs Oerational transforms ft sinm0t 170 061de f0 This function has a specific frequency 7m A 90 phase shift in a complex exponential means multiplication by Therefore When one domain is stretched out the other domain is com ressed Fismimut 11 5Wmu 5mamul p Note that Fsinmut is imaginary and odd TFooede Scale change Example Tincreases m0 decreases V der in time means narrower and taller in frequency ft square pulse of height A and Width 2b oerationai transforms centered at t 0 Fm 2Ab Sinmbmb 2Ab sincmb The amplitude F0 is the area under the pulse Foa is real and even if the pulse is centered on 1 Fm Ifte 1 dz mp inmimd Modulation eg AM radio Foa is complex if the pulse is not centered Amplitude of highfrequency carrier is modifed The lFT of a pulse in frequency is a sinc 39 39n time by amplitude of lowfrequency Signa function i Fi tcosmut 12 Fmoa 12 Roman The original signal would have Fm centered around 030 the modulated signal would have Fm duplicated and shifted along the 0 axis Interesting What was Fcosmut 2 erational transforms 0 F0 Ifte 1 dz mp inmimd erational transforms Translation in the time domain Fm Ifte 1 dz mp inmimd Convolution Equivalent to multiplication in the frequency The output yt from a system with unit impulse domain response I7t when the input is xt can be Translation in the frequency domain represented in M ways Equivalent to multiplication in the frequency 39 by convom on in the time domain domain by multiplication in the frequency domain Not a smart thing to do Y WW Fm is real and even ifthe pulse is centered on t0 Similar to VOUTs VWsHs in Laplace domain See Lecture 14 for definition of convolution Lalace vs Fourier no Moi dz f0 IFooede Laplace transforms are better For systems analysis convergence for wider variety of functions For control systems analysis Fourier transforms are better Easier to understandjoa axis than 5 plane Basis for FFT for discrete data V dely used in signal processing Fourier Transforms reading Book sections 172 You can skip the derivation starting with eqn 1714 but note the result in eqn 1718 173 ifyou really like Laplace transforms 174 although these functions are also in the tables 175 Properties 176 Operational transforms 136 Convolution see lecture 14 slide BIOEN 302 Lecture 9 Circuit analysis in the s domain October 15 2007 Unit impulse response Given a system transferfunction Hs Voms VWS Let ht L 1Hs 9 vomt vmt via mm vout ht when the input is the unit impulse 5t an ht ht Convergence and stability BIBO stability A system with transfer function Hs is BIBO stable if its unit imulse response meets the following criterion 1 hold lt oo Examples 100equot2ut is stable cos2nt is not 100e V2ut is not BIBO stability is guaranteed ifthe ROC contains thejm axis In other words Physical system poles in left halfplane Homework 3 Part1 Delete quotand only itquot from 1A Part 2 Circuit analysis in s domain coming up next Atemative problem using impulse response Convergence of LT The part of the sdomain where the Laplace transform integral converges is called the Region of Convergence Example LTe 3 does not converge when Reslt3 For a causal system ie one that we can build the ROC is a righthand plane where the poles define the left boundary ROC not essential except when two functions have the same LT Then the ROC decides which LT 1 is correct Circuit analysis in the frequency domain It is possible to write KVL and KCL directly in the Laplace domain but we need to knowthe relationship between Vs and ls for each component Sections 131 133 to p 595 Section 94 definition of Z Section 96 series and parallel Z Impedance For every R L and C component or system made of R L and C components Vs Zs where Z is impedance units Z is the complex equivalent to resistance includes effect of timevarying signal on magnitude and phase oftransfer function Circuit analysis with impedances Combine impedancesjust as you would resistances In series Z Z In parallel Z l Zfl 4 Use voltage and current divider relationships For example V2S VMS 22 Z1 22 A look ahead Convolution How is ht related to vuut and vnt 2 Convolution integral Section 136quot Memorize either the 1St sentence of section 136 or this Convolution is a mathematical operation that descrbes the way an input to a system combines t with the system s unit impulse response 0 produce an output 136 is also the speci c gravity of Hg and the energy in e of an electron in the lowest energy state in a H atom Impedance of R L C components Resistor LTvt Rit gt VsR ls ZR R lnductor LTvt L ditdt gt VsL s ls i0 Z sL Capacitor LTit C dvtdt gt ls C s Vs v0 ZC 1 le R is purely resistive component L amp C are purely reactive components ie they react when V orl changes Circuit analysis procedure Replace the circuit components with impedances Analyze the impedance networkjust as you would a resistive network Find Vows as a function ofVNs Vows can be across any components Vows VNs gives the transfer function Hs Multiply Hs by VNs to get Vows Use partial fractions to break Vows into pieces for which you know the inverse LT Take inverse LT to get vuut BIOEN 302 Lecture 12 Fourier Series October 22 2007 Upcoming Events Homework None this week Quiz 3 This Friday October 26 Fourier series Some multiple choice One magnitude and phase plot Practice Problems 161 a d 1610 a e 1617 concepts only Ampl amp phase plots of On examples throughout chapter Fourier Series main points Infinite sum of sines cosines or both no a0 Z a1 00501030 bn sinnm0t 1 All frequencies are integer multiples of a fundamental frequency 030 FS can represent any periodic function that we can physically produce Fourier Series main points Book sections 162 163 164 as leadin for 168 168 169 No RMS or power calculations for now Underlying principle superposition m Fourier coefficients trig form p 760 f t a0 in a1 cosnm0t b1 sinnm0t a all tOJtTftdt 0 7 T 0 2 t0T a F jfzcoskmordz 0 2 t0T bk fawnmend Source of the Fourier coefficients p 761 Symmetry 0f functions IOT Even symmetry ft f t Icosm030t sin nwotdt 0 all m and n MW 395an 0 j ZOT 2 0 Icosm030t cosn00tdt 0 m 72 n 2 2 31 o l 2 0 Even Aperiodic Even Periodic tOT Odd symmetry ft f t i sinm030t Sinn00tdt 0 m n 10 Xquot5 5xquot3x O E 0395 Z0T ZOT T 3 5 j 3 2 1 5 06 q I coszmc00tdt Isin2mm0tdt 3 19 I O J 0 Odd Aperiodic Odd Periodic Symmetry 01 functions Fourier coefficients Complex exp form 0 Half wave Z Z Cnejnwot When the period limits are properly chosen 0 each halfperiod is a mirror image of the where complementary halfperiod 1 tOT ft ft T2 On J fte anot Each period is even but the overall function to may be even Odd 0r Either See example 166 on p 786 for periodic pulse Quarter wave train V T 00 Sinmw TZ In addition to halfwave symmetry each half Cn g 2 mo TOZ period has odd symmetry about the T4 and w 0 3T4 points Smx sincx Magnitude and phase plots Magnitude and phase plot example Magnitude plot shows Cn0 lCnl Phase shows tan 1lmCnReCn if V quot06 Plot eXIsts at no0 only 04 l mm Oil m 10 86 4 2 2 4 6 8 1 10 7 04l 6 180 l l 90 i llFit g i i i l 513 11 9 7 5 3 39139i393 5 7 14 12 10 8 6 4 2 The phase angle of On Measurement of ow velocity using Doppler ultrasound The doppler effect is the change in frequency of propagating waves due to relative motion between the transmitting7 receiving7 or re ecting objects Because the speed of sound in a given medium is constant7 the product of wavelength and frequency is also coanstant Any changes in wavelength caused by the motion of an object will result in a corresponding change in the frequency of the acoustic waves Our doppler ow meter uses two xed transducers one a transmitter and one a receiver to bounce ultrasound waves off a moving liquid Most of the high frequency carrier wave 1 MHZ in our case is re ected unchanged7 due to the quantity of material in the system that is not moving A small portion of the signal7 however7 is shifted in frequency The frequency shift can be stated in terms of the following variables fD Doppler frequency shift 1 f0 source or carrier frequency 2 i liquid velocity 3 c velocity of sound in water 4 t9 angle between ow direction and acoustic axis 5 Note that this de nition of 6 is different from the one presented in class speci cally7 it is the complement of the angle shown on the board Nonetheless7 if you choose the angle to be at or near 45 then there is no difference in the value of cos0 If we assume that the speed ofthe water is much smaller than the speed of sound in water7 then we can approximate fDfo as 2110 The resulting relationship between the frequency shift and the carrier frequency is then 2foi cost9 fD c This formula provides the magnitude of y but does not indicate its direction The 77lmaginary Quadrature7 or lQ processing that was performed by the MATLAB function is meant to provide direction information about the liquid ow Multiplying the received signal by the carrier wave in one signal processing channel and by a wave of the same frequency but with a 790 phase shift in a second7 parallel channel provides two output signals that can be compared to determine the direction of ow Sources J Webster7 Medical Instrumentation Application and Design7 3d Edition John Wiley and Sons7 1998 pp 347 352


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