Transcription of Analog Digital - UMIACS
1 Slides adapted from ME Angoletta, time, tk [ms] Voltage [V] time, tk [ms] Voltage [V]tsAnalog & Digital signalsAnalog & Digital signalsContinuous functionContinuous functionV of continuouscontinuousvariable t (time, space etc) : V(t).AnalogDiscrete functionDiscrete functionVkof discretediscretesampling variable tk, with k = integer: Vk= V(tk). [ms]Voltage [V]Uniform (periodic) sampling. Sampling frequency fS= 1/ tSSlides adapted from ME Angoletta, CERND igital vsanalog proc ingDigital vsanalog proc ingDigital signal processing (DSPing) More flexible. Often easier system upgrade. Data easily stored. Better control over accuracy requirements. A/D & signal processors speed: wide-band signals still difficult to treat (real-time systems). Finite word-length effect. Obsolescence ( Analog electronics has it, too!)
2 LimitationsLimitationsSlides adapted from ME Angoletta, CERND igital system exampleDigital system examplems V Analog Analog DOMAINDOMAINms V FilterAntialiasing k A Digital Digital DOMAINDOMAINA/Dk A Digital Processingms V Analog Analog DOMAINDOMAIND/Ams V FilterReconstructionSometimes steps missing- Filter + A/D(ex: economics);- D/A + filter(ex: Digital output wanted).General schemeSlides adapted from ME Angoletta, CERND igital system implementationDigital system implementation Sampling rate. Pass / stop DECISION POINTS:KEY DECISION POINTS:Analysis bandwidth, Dynamic range No. of bits. ProcessingA/DAntialiasingFilterANALOG INPUTANALOG INPUTDIGITAL OUTPUTDIGITAL OUTPUT Digital to use for processing ? See slide DSPing aim & tools Slides adapted from ME Angoletta, CERNS amplingSamplingHow fast must we sample a continuous signal to preserve its info content?
3 Ex: train wheels in a frames (=samples) per second. Frequency misidentification due to low sampling starts wheels go accelerates wheels go *Sampling: independent variable (ex: time) continuous : dependent variable (ex: voltage) continuous we ll talk about uniform sampling.**Slides adapted from ME Angoletta, CERNS ampling -2 Sampling -2__s(t) = sin(2 f0t) (t) @ fSf0= 1 Hz, fS= 3 (t) = sin(8 f0t) (t) = sin(14 f0t) (t) = sin( 2 (f0+ k fS) t ) , k s(t) @ fSrepresents exactly all sine-waves sk(t) defined by:1 Slides adapted from ME Angoletta, CERNThe sampling theoremThe sampling theoremA signal s(t) with maximum frequency fMAXcan be recovered if sampled at frequency fS> 2 on fS?fS> 300 Hzt)cos(100 t) sin(30010t) cos(503s(t) + =F1=25 Hz, F2= 150 Hz, F3= 50 HzF1F2F3fMAXE xample1 Theo**Multiple proposers: Whittaker(s), Nyquist, Shannon, Kotel frequency (rate) fN= 2 fMAXorfMAXor fS,MINor fS,MIN/2 Naming getsconfusing !
4 Slides adapted from ME Angoletta, CERNF requency domain (hints)Frequency domain (hints) Time & frequencyTime & frequency: two complementary signal descriptions. Signals seen as projected onto time or frequency BandwidthBandwidth: indicates rate of change of a signal . High bandwidth signal changes + brain act as frequency analyser: audio spectrum split into many narrow bands low-power sounds detected out of loud adapted from ME Angoletta, CERNS ampling low-pass signalsSampling low-pass signals -B 0 B fContinuous spectrum(a)Band-limited signal : frequencies in [-B, B] (fMAX= B).(a) -B 0 B fS/2 f Discrete spectrumNo aliasing (b)Time sampling frequency > 2 B no aliasing.(b)1 0 fS/2 f Discrete spectrum Aliasing & corruption (c)(c)fS2 B aliasing !
5 Aliasing !Aliasing: signal ambiguity Aliasing: signal ambiguity in frequency domainin frequency domainSlides adapted from ME Angoletta, CERNA ntialiasing filterAntialiasing filter -B 0 B fSignal of interest Out of band noise Out of band noise -B 0 B fS/2 f(a),(b)Out-of-bandnoise can alias into band of interest. Filter it before!Filter it before!(a)(b) -B 0 B fAntialiasing filter Passband frequency (c)Passband: depends on bandwidth of interest. Attenuation AMIN: depends on ADC resolution ( number of bits N). AMIN, dB~ N + Out-of-band noise parameters: ripple, (c)AntialiasingAntialiasingfilterfilter1 Slides adapted from ME Angoletta, CERNADC - Number of bits NADC - Number of bits NContinuous input signal digitized into 2 Nlevels.
6 -4-3-2-10123-4-3-2-101234000001111010 VVFSRU niform, bipolar transfer function (N=3)Uniform, bipolar transfer function (N=3)Quantization stepQuantization stepq =V FSR2 NEx: VFSR= 1V , N = 12 q = VLSBLSBV oltage ( = q)Scale factor (= 1 / 2N )Percentage (= 100 / 2N ) q / 2q / 2 Quantisation errorQuantisation error2 Slides adapted from ME Angoletta, CERNADC - Quantisation errorADC - Quantisation error2 Quantisation Error eqin[ q, + q]. eqlimits ability to resolve small signal . Higher resolution means lower [ms]Voltage [V]Slides adapted from ME Angoletta, CERNF requency analysis: why?Frequency analysis: why? Fast & efficient insight on signal s building blocks. Simplifies original problem - ex.: solving Part. Diff. Eqns. (PDE). Powerful & complementary to time domain analysis techniques. The brain does it?
7 Time, tfrequency, fFs(t)S(f) = F[s(t)]analysisanalysissynthesissynthesi ss(t), S(f) : Transform PairGeneral Transform as General Transform as problemproblem--solving toolsolving toolSlides adapted from ME Angoletta, CERNF ourier analysis - tools Fourier analysis - tools Input Time SignalFrequency spectrum = =1N0nNnk 2jes[n]N1kc~DiscreteDiscreteDFSDFSP eriodic (period T)ContinuousDTFTA periodicDiscreteDFTDFTnf 2jens[n]S(f) + == 4681012time, , tk = =1N0nNnk 2jes[n]N1kc~**Calculated via FFT**dttf j2es(t)S(f) + = dtT0t kjes(t)T1kc =Periodic (period T) , 81012time, tNote: j = -1, = 2 /T, s[n]=s(tn), N = No. of samples Slides adapted from ME Angoletta, CERNA little historyA little history Astronomic predictions by Babylonians/Egyptians likely via trigonometric sums. 16691669: Newton stumbles upon light spectra (specter= ghost) but fails to recognise frequency concept (corpuscularcorpusculartheory of light, & no waves).
8 1818ththcenturycentury: two outstanding problemstwo outstanding problems celestial bodies orbits: Lagrange, Euler & Clairaut approximate observation data with linear combination of periodic functions; Clairaut,1754(!) first DFT formula. vibrating strings: Euler describes vibrating string motion by sinusoids (wave equation). 18071807: Fourier presents his work on heat conduction Fourier presents his work on heat conduction Fourier analysis analysis born. Diffusion equation series (infinite) of sines & cosines. Strong criticism by peers blocks publication. Work published, 1822 ( Theorie Analytique de la chaleur ). Slides adapted from ME Angoletta, CERNA little history -2A little history -2 1919thth/ 20/ 20ththcenturycentury: two paths for Fourier analysis two paths for Fourier analysis --Continuous & & Fourier extends the analysis to arbitrary function (Fourier Transform).
9 Dirichlet, Poisson, Riemann, Lebesgue address FS convergence. Other FT variants born from varied needs (ex.: Short Time FT - speech analysis).DISCRETE: Fast calculation methods (FFT)DISCRETE: Fast calculation methods (FFT) 18051805- Gauss, first usage of FFT (manuscript in Latin went unnoticed!!! Published 1866). 19651965- IBM s Cooley & Tukey rediscover FFT algorithm ( An algorithm for the machine calculation of complex Fourier series ). Other DFT variants for different applications (ex.: Warped DFT - filter design & signal compression). FFT algorithm refined & modified for most computer adapted from ME Angoletta, CERNF ourier Series (FS)Fourier Series (FS)**see next slidesee next slideA A periodicperiodicfunction s(t) satisfying function s(t) satisfying DirichletDirichlet ssconditions conditions **can be expressed can be expressed as a Fourier series, with harmonically related sine/cosine termsas a Fourier series, with harmonically related sine/cosine [] += +=1kt) (ksinkbt) (kcoska0as(t)a0, ak, bk: Fourier coefficients.
10 K: harmonic number,T: period, = 2 /TFor all t but discontinuitiesFor all t but discontinuitiesNote: {cos(k t), sin(k t) }kform orthogonal base of function space. =T0s(t)dtT10a =T0dtt) sin(ks(t)T2kb- =T0dtt) cos(ks(t)T2ka( signal average over a period, DC term & zero-frequency component.)analysisanalysissynthesissynt hesisSlides adapted from ME Angoletta, CERNFS convergence FS convergence s(t) piecewise-continuous;s(t) piecewise-monotonic; s(t) absolutely integrable , < T0dts(t)(a)(b)(c)Dirichlet conditionsIn any period:Example: square wave T (a) (b) Ts(t) (c)if s(t) discontinuous then |ak|<M/k for large k (M>0)Rate of convergenceRate of convergenceSlides adapted from ME Angoletta, CERNFS analysis - 1FS analysis - 1* Even & Odd functionsOdd :s(-x) = -s(x)xs(x)s(x)xEven :s(-x) = s(x)FS of odd*function: square square signal , sw(t)2 0 02 1)dt(dt2 10a= + = 0 02 dtktcosdtktcos 1ka= = {}= == = k cos1 02 dtktsindtktsin 1kb- =evenk,0oddk, k41 2 T= =(zero average)(zero average)(odd function)(odd function).
