Basic Measurement Issues. Sampling Theory and AnalogtoDigital Conversion


 Ashley Griffith
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1 Theory ad AalogtoDigital Coversio Itroductio/Defiitios Aalogtodigital coversio Rate Frequecy Aalysis Basic Measuremet Issues Reliability the extet to which a measuremet procedure yields the same results o repeated trials Validity the extet to which a measuremet procedure measures what it is supposed to measure Sesitivity resposiveess of a istrumet to a chage i the icomig sigal Precisio the degree of exactess with which a measuremet is made, similar meaig as accuracy Resolutio The fieess of detail that ca be distiguished i a measuremet Defiitios Sigal a fluctuatig electric quatity, such as a voltage or curret, whose variatios represet coded iformatio Aalog sigal represets data by a cotiuously variable quatity Digital sigal represets data i a discrete, umerical form Defiitios Digitizatio the process of covertig a aalog sigal ito digital form Aalogtodigital coverter (ADC) a device that digitizes a aalog sigal a aalog sigal is said to be sampled to produce the digital sigal Quatizatio to subdivide ito small but measurable icremets: digital sigals are limited by the umber of biary digits ( bits ) that ca be used to represet them Aalogtodigital coversio process aalog sigal ADC digital sigal computer memory Aalogtodigital coversio Parameters of the ADC process: Iput rage establishes max voltage rage to be detected bipolar: ± 1V, ± 5V, ± 10V uipolar: 01V, 05V, 010V usually selectable i hardware or software Iput cofiguratio sigleeded: oe iput relative to groud differetial: differece betwee 2 iputs is sampled System gai degree to which sigal is amplified before samplig 1
2 Aalogtodigital coversio Parameters of the ADC process: Amplitude resolutio how precisely amplitude of sigal ca be quatified determied by bit value of ADC commo: 12bit (2 12 = 4096 discreet levels) 16bit (2 16 = 65,536 discreet levels) Temporal resolutio how frequetly the ADC takes samples of the aalog sigal determied by samplig rate values from 30 to 2000 are commo Together, amplitude ad temporal resolutio greatly affect the quality of the sampled sigal ADC Amplitude Resolutio +5 V 0 V 5 V A 4bit A/D set up (2 4 = 16 discreet levels) biary values ADC Amplitude Resolutio A 4bit A/D set up closeup view V V V 1011 Aalogtodigital coversio How precisely does a 12bit A/D board detect amplitude chages i a sigal? 12 bit ADC = 2 12 = 4096 discreet levels, distributed over the whole iput rage For a ± 10 V iput, you have a rage of 20 V (20 V) / (4096 ADC uits) = V per ADC uit V V 1001 Is 4.9mV per ADC uit good or bad resolutio? Force Plate: if +10 V = 2500 N (for a give FP gai settig), the 1 ADC uit = 1.22 N I other words, you will be uable to detect chages i force that are less tha 1.22 N, this is probably OK ADC Temporal Resolutio Rate: how ofte is the sigal sampled? t 1 t 2 t t 3 period: time betwee adjacet samples e.g. τ = t 2 t 1 rate: frequecy with which samples are take e.g. f = 1 / τ ADC Temporal Resolutio What is a adequate samplig rate? Govered by Shao s samplig theorem: the sigal must be sampled at a frequecy at least twice as high as the highest frequecy preset i the sigal itself This miimally acceptable samplig frequecy is kow as the Nyquist frequecy But what exactly does sigal frequecy cotet mea? 2
3 Fourier (Harmoic) Aalysis The sie fuctio The frequecy cotet of a sigal ca be determied by performig a Fourier aalysis Fourier discovered that ay sigal, o matter how complex, ca be represeted by a ifiite sum of weighted sie ad cosie waves y( t) = A + [ B si( f t) + C cos( f t )] =1 I practice, oly a fiite umber of harmoics are eed to adequately represet a sigal A related techique, the Fourier trasform, allows the frequecy compoets of a sigal to be revealed y( t) = A + B si ( f t ) A the mea (D.C.) offset from zero B the amplitude of the sigal f the frequecy of the sigal A = 0 B = 1 f = 1 Fourier series approximatio costat = Frequecy Compoet These Aother sigals ca be discussed with represetatio respect to the time domai of the or Fourier the frequecy series domai. A approximatio sie (or cosie) waveform is a sigle frequecy; ay other waveform ca be the sum of a umber of sie ad cosie waves. Fourier series approximatio y(t) = A + B 1 si(f 1 t) + C 1 cos(f 1 t) + B 2 si(f 2 t) + C 2 cos(f 2 t) + Approximatio of the data improves as the umber of harmoics used icreases However, 94% of the total sigal power was cotaied i just the 1 st harmoic umber of harmoics used Fourier Trasform Coverts sigals from the time domai to the frequecy domai The Fourier trasform separates a sigal ito siusoids of differet frequecy which sum to the origial waveform: it distiguishes the differet frequecy siusoids ad their respective amplitudes Discrete Fourier Trasform (DFT) Fast Fourier Trasform (FFT) Typically, the square of the FFT is used istead, which is called the Power Desity Spectrum 3
4 Fourier Trasform origial sigal 20 oise added Fourier Trasform Raw EMG sigal Power Spectral Desity time (s) FFT time (s) FFT samples frequecy () frequecy () frequecy () Frequecy Cotet Frequecy cotet of some commo sigals ecoutered i biomechaics: Activity Stadig posture Walkig Ruig Heelstrike trasiet EMG (surface) EMG (idwellig) Max Freq of Iterest ADC Temporal Resolutio Now, back to samplig rate: What happes if you sample at a rate lower tha the Nyquist frequecy? A error, kow as aliasig, will occur, corruptig your data (aliased sigals show up at a freq of f samp f sig, so 40 sigal sampled at 50HZ will show up as a 10 sigal) Whe aliasig occurs, frequecies greater tha ½ the samplig rate fold back ito the lower frequecy compoets, distortig the sigal High frequecy iformatio is ot simply lost, it actually reappears as false low frequecy cotet Oe commo form of aliasig, which most people have observed, is the 'wago wheels' effect i films. Sice the rate of the film is usually much lower tha the frequecy at which the spokes of a wheel pass ay oe poit, aliasig takes place ad the wheels appear to tur either much more slowly tha they really are, or eve seem to go backwards sometimes." (D.W. Grieve et. al., Techiques for the Aalysis of Huma Movemet. Priceto Book Compay, 1975, p. 41.) 4
5 Frequecy aalog 4 sie wave <2hz This is a 4 sigal, so the Nyquist freq is hz 12.5 Aliasig occurs at samplig rate of Frequecy Rate Take home message(s): at the Nyquist frequecy esures that frequecy cotet of sigal is preserved I practice, samplig at 45 times the highest frequecy preset i the sigal esures that sigal amplitude characteristics are preserved as well Aother example of aliasig: 1 sigal appears to be 0.33 Whe possible, a atialiasig filter should be used to elimiate oise that exists above ½ the samplig rate Rate Typical samplig rates used to collect various sigals i biomechaics: Sigal Motio Force EMG (surface) EMG (idwellig) Typical SR How do we determie the legth of samplig period? I repetitive motio the time eeds to be more tha the time for 1 cycle so if gait is 120 steps (60 strides) per miute the the rate is 1 ad you eed to sample loger tha 1s to obtai the full data set. You may icrease time to collect multiple repetitios. 5
6 I o repetitive activities such as liftig the you eed to go from start to fiish. I cases where frequecy of movemet is ot obvious pilot data eeded to eable appropriate decisio. Witer reported that i quiet stadig sigificat data foud at.002 which would require a 8 miute trial. Measuremet Errors No measuremet process is perfect, there will be some degree of error i the data Measuremet errors fall ito two categories Radom Errors Systematic Errors Both ca be miimized, but either ca be completely elimiated I particular, radom errors may be reduced through data processig techiques, systematic errors geerally ca ot Radom Error  Example 3 people measure the distace betwee two lies that are 1.5 cm apart Systematic Error  Example 1 perso measures the distace betwee two lies that are 1.5 cm apart, but places the ruler dow icorrectly cm 1.4 cm 1.6 cm They do t all get the same value, but this radom fluctuatio ca be largely elimiated by averagig the three values ( ) / 3 = 1.5 cm 1.2 cm 1.2 cm 1.2 cm This idividual was very cosistet i his measuremets, but he has itroduced a systematic error ito the data No amout of processig ca remove this error Measuremet Errors i Biomechaics Systematic Errors: cover frequecy spectrum Poor placemet of aatomical markers Movemet of markers/ski relative to boes Calibratio oliearities Digitizatio errors (some) Radom Errors: typically high frequecy Digitizatio errors (some) Marker jigglig/vibratio Electrical iterferece Miimizig Measuremet Errors Use of cosistet, proper techiques for data collectio Use of 3D techiques for motio capture Proper traiig of persoel Use of high quality equipmet 6
7 Up Next Topic Kiematics Readigs Robertso et al. (2004) Chapter 1, pp Witer (1990) Chapter 2, pp
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