ADV ASTRONOMICAL DATA ANALYSIS
ADV ASTRONOMICAL DATA ANALYSIS AST 5765
University of Central Florida
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UCF Physics AST 57654762 Advanced Astronomical Data Analysis Fall 2009 Lecture Notes 9 Probability Distributions and Error Analysis Check In 1230 1240 10 min 0 Did anyone notice errors in Bevington equations 1 HW problem extra credit for rst time each reported Bevington Eq 4 on p 48 log formula ll in Bevington Eq 43 on p 52 H should be 7139 Bevington Eq 610 on p 105 z should be 1 in rst line and should be in last line HW4 rst real HW exercises teach data analysis First quiz Thursday Probability Basics Reference Description of behavior of many identical independentlyprepared systems Discrete vs continuous probability coin vs time interval In the limit of large N everything appears continuous Described by probability density function PDF that integrates to l 0 D0 de nite integral over range to get probability 7 Px in range a 7 b pzd l L pltzgtdz 1 2 0 mean i 30 pzdx median Px lt median Pz gt median 12 0 mode maxpx o variance 02 z 2gt2pltzdz 3 mean is also the expected value of z o variance is the expected value of x 7 2 the square of the deviations from the mean 1 standard deviation 039 population all possible measurements of a system proportionally represented often in nite in number draw one measurement of system sample a set of draws estimate mean std deV etc from sample Estimates of Mean etc 1240 1245 5 min All estimates use discrete formulae can t take an in nite number of measurements Estimate the mean and standard deviation of the parent distribution from a sample mean 1 92 N 95 4 standard deviation 1 82 NimZmiz 5 Factor of Niim takes into account effect of degrees of freedom makes it work for low N Gaussian Distribution 1245 1250 5 min 1 H 2 m 6 o O 6 Limit of sum of binomial cointoss distribution and most others for large N Normal distribution 10 65 of measurements closer to mean than this 20 95 of measurements closer to mean than this 3039 99 of measurements closer to mean than this Wellbehaved errors follow it Memorize it Not analytically integrable Find function to calculate integral values 5 Poisson Distribution 1250 100 10 min 0 Timing of random discrete events in time or space diagram don t confuse z with space 0 Example Light 0 Say there are 71 events per unit time on average 0 Then 739 l n is the average interval between events 0 In time t we would expect an average of tT tn N events 0 The probability of getting z events in time t is trwiT Nmi Pzt et geN o z N o a xN o For large N Poisson approaches a Gaussian with i N and 039 N 0 Gaussian BUT M and 039 are locked together 0 i is signal and 039 is noise so N 7 m N 0 Quality of data improves as xnumber of counts SN Qli2 0 IF uncorrected systematics are well below 039 6 Con dence Intervals 100 105 5 min 0 Con dence interval probability that a draw lies inside a given range 0 For a Gaussian la 68 of measurements closer to mean than this 20 95 of measurements closer to mean than this 3U 99 of measurements closer to mean than this 0 The 3039 con dence interval for a Gaussian is 99 7 8 7 Uncertainty of a Measurement 105 1 10 5 min Estimate the measurement uncertainty by taking sample measurements use 039 Compare to a priori uncertainty estimate If rough agreement use sample value otherwise nd problem Goal of Error Analysis 110 115 5 min How do you plug an uncertainty into a formula Called error propagation Given variables 111 mean values 1117 calculated from samples with Nu Nu draws 11 11 standard errors Uu av function fu1 want If Calculate f1117 This is the best estimate of It is approximate if 11 17 calculated with small N approaches true for large N Plot a curve choose z and Um linearize at x trace to f axis show f and 0f Error Propagation Equation 115 130 15 min Consider the spread variance in values of f f 11 11 7 lim Zurf Naoo 9 We re ignoring the fact that it should be N 7 m in the denominator but 771 would be the number of variables in the equation By the way it is N in the denominator if the mean you are using is the real mean and not one estimated from the data But that rarely if ever happens Also if you get signi cantly diiTerent results with N or N 7 m you re almost certainly up to no good and should get more data First we need the deviations of f which we can estimate with a Taylor series in the variables taken about their mean values 10 o Combining these we can get a in terms of 0395 03 which we measured 2 2 N A i J if 2 if 0f 7 u u6uv v6v 11 N 1 2 6f 2 2 6f 2 5f 3f 2u 7 7 13 First two terms in the sum can be expressed in terms of 0393 and 03 Eq 9 since etc are constants evaluated at f Last term is the covariance 1 v3 N113 N 2 Km 7 um 7 m 14 0 So the Error Propagation Equation is 6f 2 6f 2 6f 6f 2 N 2 7 2 7 2 if Ufiau u 039v lt61 20W6u a 15 o If u and 1 are uncorrelated one s not positive when the other is and vice versa then 7 a 1 7 17 will be negative half the time and positive half the time and the sum will tend to zero 0 So for uncorrelated variables we can drop the covariance terms 0 Equation is important for two reasons Evaluate 0f Determine which input variables have the greatest in uence on the output and therefore need the smallest errors Or on other words how should I spend my telescope time y x MN 5 UCF Physics AST 57654762 Advanced Astronomical Data Analysis Fall 2009 Lecture Notes 19 PSF Check In 1030 1035 5 min 0 Questions before we start HW6 1035 1055 20 min N Dimensional Gaussian 1055 1105 10 min 0 We often use a 2D Gaussian to model a stellar PSF 0 Usually symmetric needn t be 0 Consider 1 1 p 75 0 G 27m2 0 Integrates to 1 For 2D function y direction must integrate to pGz at each x 0 Since pG integrates to 1 just multiply pGpGy o This smears z Gaussian in y 0 Repeat for higher dimensions N N wrewr 2 1 7i 39 2 20 pGNH 7 6 234 gt 2 i1 27m2 0 Note exponent now looks like an ellipse hence name elliptical Gaussian 0 See gaussian py in class Python dir PointSpread Function PSF 1105 1125 20 min 0 Atmosphere has cells of air w differing temperatures 0 Cells refract light also move 0 Highspeed images show stellar point source bouncing around 0 Central limit theorem a Gaussian U G o Diffraction rD 7 3 o Telescope aberrations astigmatism coma etc o All these effects produce PSF o Convolve reality image with PSF a observed 0 Why are bright stars bigger Same PSF 0 Every image and sometimes dilTerent parts of images can have a di erent PSF Centering 1125 1130 5 min 0 Need to know exact subpixel center of PSF for photometry 0 Many methods 0 For most stars t a 2D Gaussian with free z and y parameters 0 For Spitzer IRAC data good to 002 pixels 0 See gaussian py in class directory 0 For very dim stars noncircular PSFs and nonstars centeroflight Howell p 105 may be better Aperture Photometry 1130 1140 10 min 0 Want to measure relative ux of stars 0 Could just count all the DN but some sky remains 0 Do want all of signal do not want to add noise 0 Draw grid on board put star in it 0 Establish center accurately 0 Place aperture in GREEN 0 Place sky annuli in RED 0 Table Light Area Bad Pixels DN PiXZ Star F1 Ni must x Sky Fsky Nsky don t count 0 Do not count bad pixels in sky region why must count for star 0 Aperture photometry equation Fsky N sky Facal Faraw 7 N 0 Aperture correction W PSF l Size of the Sky Annulus 1140 1200 20 min 0 Draw Gaussian pro le sky with noise 0 Mark aperture locations in color 0 Want all of starlight but minimal sky noise 0 Establish 039 using bright star why bright 0 Use gausszdfit or just do FWHM and convert 0 Count DN within 340 from center for star 0 Count DN in sky annulus further out o How far out to count radial pro le 0 SN vs 039 for bright star SN vs 039 for dim star 0 Must do same for all stars to avoid systematic error 8 Or Elsel 1200 1205 5 min 0 Saturn occultation using aperture photometry vs raw star counts gV iso t39newps amp gV iso t39 oldbps amp gV run39invert39jh39horizons39rgf39imlcps amp l gV run39invert39jh39horizons39rgfimlc39V2ps amp y x MN 5 UCF Physics AST 57654762 Advanced Astronomical Data Analysis Fall 2009 Lecture Notes 34 Data Presentation Check In 1030 1035 5 min 0 Questions Go Over HW11 1035 1055 20 min Presentation of Data 1055 11 20 25 min 0 Video by Hans Rosling o httpVideotedcomtalkspodcastHansRos1ing39200639480mp4 0 As he speaks note how he discusses errors doing better than chance 0 Also see how simple and yet datarich his plots are 0 Keep an eye out for interesting ways to present data catalog for your future use 0 Just reading the matplotlib manual gives new ideas of how to present Rules for Plots 1120 1130 10 min 0 Know what you want your Viewer to learn from your plot 0 Know how your Viewer will be seeing your plot screen print projected live on web 0 Know with what knowledge your Viewer will be seeing your plot after reading something before 0 Let this information drive all your decisions 0 Don t do anything without a purpose 0 KEEP IT SIMPLE AVOID CLUTTER 0 Have a title axis titles values on axes o In a paper plot title is a sentence fragment that starts the caption 0 Size is according to angular size as seen by Viewer 0 Plot is small for slideshow big for paper computer screen U 0 Choose font sizes styles line thicknesses styles accordingly 0 Data are usually points connect with lines only if that adds info 0 Models are usually lines if using points make distinct from data 0 Present multiple datasets only to compare them 0 Keep keys far out of the way or eliminate o No grid or very light if critical 0 Keep data separated from axes on sides top and bottom so reader can visually extrapolate 0 Avoid distracting backgrounds line styles gratuitous visual elTects etc 0 Use shape linestyle or color to deliver information only 0 If needed highlight important information with thicker lines brighter color etc 0 Do not adjoin plots with unrelated axes 0 Show it to someone without comment see if they get your point Project Help 1130 1145 15 min 0 maskboxlevpy o Other questions y x UCF Physics AST 57654762 Advanced Astronomical Data Analysis Fall 2009 Lecture Notes 7 Python Functions Check In 1030 1035 5 min 0 Do you have VNC up python running and the lecture demo ready to go 0 Who has done the matplotlib tutorial How much of pydatatut 0 Any problems with system 0 Examplestemplates in python directory 0 HW3 is out there is a handout called codingpdf 0 Questions before we start Go over HW2 1035 1055 20 min Functions in General 1055 1110 15 min 0 Computer programs can be very long and complicated 0 Manage complexity by boxing away detail in functions 0 Difference between black box and glass brick o A function is a program unit that can be called by name 0 Also called routine or procedure or subprogram w nuances 0 Contains program statements 0 Executes same way each time it is called 0 Programmer says name of function rather than inserting whole block of statements in the main routine 0 Example def triplex y 3 x return y o Ifyou now say triple 4 it prints 12 0 Variables outside function can have different names from those inside z 5 triplez 15 o It works even though we used z and not x 4 When and Why to Write Functions 1110 1125 15 min 0 Avoids repetition if those statements will be called more than once 0 Avoids bugs in that case 0 Sure it will run same way each time with same inputs 0 Generally functions have few inputs and outputs 0 Functions may have many intermediate variables 0 Often write functions that will be called only once 0 This simpli es code like an outline o If a function calculates something it may return it to the caller o If so we say the function returns a value 0 sin234 for example has a numerical value 0 You can put such a function anywhere that type of value can be used 0 For example you can say x sin cos x just as in math notation U Python Functions 1125 1140 15 min 0 Start with def o Arguments in parens commaseparated 0 Optional arguments have assignments o Statements are indented o Nonparen continued lines have 0 Have return values even if None 0 printmaxpy example 0 Docstring 0 Documentation with example py see pythonwebsources 6 Arguments in Python 1140 1145 5 min J Python has two types of arguments positional and keyword Consider def triordupx y None prtl False prt2 False z 3 x if prtl print x if prt2 print y if y Z 2 y return z You can call this several ways triordup3 prints 9 triordupx3 prints 9 triordup3 2 prints 4 triordup3 prtlTrue prints 39 triordup3 2 prt2True prints 24 triordup3 2 prtlTrue prints 34 triordup3 2 prt2True prtlTrue prints 32 x is a positional parameter know which is 2 and which is 3 from the position of x and y in the de nition prtl and prt2 have defaults can appear in any order or be omitted Programming Style 1145 1200 15 min Programming done in text Text is communication follow the 3 rules of communication Know your subject Know your recipient Viewer listener etc Know what you want your recipient to receive How do these apply to code Knowing subject can be hard algorithms and data can be complex Recipients computer you sooner amplater colleagues students examples Must make all happy 0 Documentation with template py 0 look at goodcodepro badcodepro using IDL code because IDL lets you not indent check input values documentation no gotos no globals indent for structure and readability use horizontal and vertical spaces for clarity and ease of veri cation Try not to hardcode values except 0 and 1 Put at top in variables if necessary or pass into functions There are limits to this Query data for items like image sizes that vary from one analysis to another or pass them if necessary 0 Document it if you modify an input ok for arrays otherwise avoid UCF Physics AST 57654762 Advanced Astronomical Data Analysis Fall 2009 Lecture Notes 14 General Fitting CCD Systematics y x Check In 1030 1035 5 min 0 Questions before we start 0 Gone rest of today tomorrow 0 Return quiz Grads 66 Undergrads 75 average N General Linear Fitting 1035 1045 55 min 0 Summarize general linear tting with sketch 0 If model has just multiply parameter intercept is used as a nal background t 0 Can still do if data are in more than 2 axes o Eg model depends on x and y in an image o 3 Function Minimization Fitting 1045 1100 105 min 0 Linear tting doesn t work if there is more than a multiply and an add parameter 0 E g multiply add and shift 0 Could derive a new expression like linear least squares formulae 0 Or do it numerically De ne an Python function that has parameters of model as inputs Usually the parameters are in a vector Run scipy optimi ze leastsq on the data with the function It nds the optimal values and errors Can be slow depending on model and data space Can be fooled if data space is strange local minima in X2 space 0 What scipy optimize leastsq does is Evaluate the function Compute X2 vs data uh Vary the parameters and repeat 0 It explores parameter space looking for the minimal X2 o It uses a function minimization routine 0 Calculates gradients in X2 and walks down them 0 Parameter errors are related to slopes near X2 minimum CCD Characteristics and Systematics 1235 1250 15 min o 0 Quantum e iciency mainly in Visible cut off at 11 pm 0 Modern QE 80 0 QB varies from pixel to pixel by 10 or more o 0 Dark Current thermal Vibrations in Si can bump 6 into conduction band 0 Charge integrates even in dark Id exp fakT l o Strongly temperaturedependent so CCDs usually coolded by lN2 to 77 K o Onchip ampli er has a bias voltage to attract the charge from the shift register 0 It also has a gain Vout QNF Vbias 2 o Vout digitized into ADU AnalogtoDigital Units or DN Data Numbers 0 Ampli ers are not perfect pixel capacitance adds read noise 0 Good CCDs have lt5 6 read noise or lt25 6 charge added per pixel 5 Summary of CCD Systematics 1250 1255 5 min 0 QB lt100 0 QB varies 0 Id 0 Id varies pixel to pixel o I d varies in time with temp etc 0 Id has its own noise Vbias o Read noise 0 Ampli er not perfectly linear o Pixels not perfectly linear Too much charge cancels some bias voltage 0 Cosmic rays deposit many 6 in wells 0 Some pixels have defects bad pixels G Basic CCD Error Correction 1255 100 5 min 0 Do readout backwards 0 QB di erences were multiplied so divide them out 0 Dark current was added so subtract it o Bias voltage was added so subtract it 0 Basic formula C 3 7 Making the Calibration Frames 100 110 10 min 0 This is an art 0 Sensitive analyses and quirky instruments a more robust methods 0 Here are the basic principles 0 Bias frame average many 0exposure images B 2 B 00 Dark frame D i E D 7 B ttot LZD139 NB 7 ttot average many unlit images with your exposure time or take long exposures subtract mean bias sum divide by total itime or t line to image values for each pixel position nding bias and dark rate Can then reconstruct dark current for range of exposure times Beware of big extrapolations best to avoid if possible Dark current is usually very small in the visible 0 Flat eld E Fi B i D F ltEF139 B Dgtall 4 Take many images of uniformlylit defocussed subject halfwell Subtract dark and bias Average Normalize so mean value l o Remaining instrument systematics are consistency problems 0 For consistency problems take atsbiases through night and use only those nearest in time to exposure 0 Need ats in each lter Infrared Arrays 110 130 20 min 0 Can dope Si so it is sensitive in visible and still act as CCD 0 Need other materials to be sensitive at long wavelengths InSb a A lt 5 pm HngTe a A lt 24 pm HngTe a A lt10 um SiAs a A lt 28 um SiSb a A lt 40 um HngTe QE curve Olirqepng 0 Most IR materials can t work as semiconductor like Si CCDs 0 Also the background can be high in the IR 0 Air telescope dome and camera can emit in IR brighter than astronomical sources Background limited sources ll wells fast so need to read out up to 1 kHz CCD takes several seconds to read Solution IR absorbing layer connected to Si multiplexer Indium bump bonds connect them Each unit cell read out directly in turn to ampli er no charge motion across chip Often have many ampli ers each reading quadrants or every 4th row etc Huge read noise up to 2000 6 Can measure voltage on each unit cell diode more than once nondestructive reads NDR Correlated multiple sampling reduces read noise by m Tradeoff between read speed and noise CCD slow and accurate multiplexer fast and noisy Major problems w thermal contraction mismatch can crack y x N UCF Physics AST 57654762 Advanced Astronomical Data Analysis Fall 2009 Lecture Notes 13 General Fitting Check In 1230 1235 5 min 0 Questions before we start 0 Level check General Linear Fitting 105 min 0 First example extends the line t to linear scaling of arbitrary models 0 This is a quickanddirty method useful because everyone has a lin t routine 0 Take each prediction of a nominal model 0 Multiply by a constant 0 Add another constant 0 2 free parameters like a line t 0 Can use linear tting routine to nd parameters Calculate a nominal model Treat x as the parameter in a parametric equation Calculate model for x values in data Pair those with y values in data Plot d ydata VS m ymodel No error in m but error in d So d goes on vertical axis Fit a line Slope is how much you multiply model by Intercept is how much you add to model If model has just multiply parameter intercept is used as a nal background t 0 Can still do if data are in more than 2 axes o Eg model depends on x and y in an image 3 Function Minimization Fitting 105 min 0 Linear tting doesn t work if there is more than a multiply and an add parameter E g multiply add and shift 0 Could derive a new expression like linear least squares formulae Or do it numerically De ne an Python function that has parameters of model as inputs Usually the parameters are in a vector Run scipy optimi ze leastsq on the data with the function It nds the optimal values and errors Can be slow depending on model and data space Can be fooled if data space is strange local minima in X2 space 0 What scipy optimize leastsq does is Evaluate the function Compute X2 vs data Vary the parameters and repeat It explores parameter space looking for the minimal X2 It uses a function minimization routine Calculates gradients in X2 and walks down them Parameter errors are related to slopes near X2 minimum UCF Physics AST 57654762 Advanced Astronomical Data Analysis Fall 2009 Lecture Notes 27 Spectroscopic Calibration and Applications 1 Check In 1230 1235 5 min 0 Questions 2 Calibration Sources 1235 1245 10 min 0 Common calibration sources Wavelength Resolution Doppler Slit Shifts Extra Time Object any depends on object incl incl no OH39 sky lines near IR Rgt600 to resolve blends no no no Lamp He Ne Ar Xe UVinear IR all watch blends no no yes Planetary nebulae H He visibleim id IR low only few lines extra maybe yes Star lines UV near IR depends on line extra extra yes Telluric lines in starlight near IR depends on line no extra yes 3 Calibration Observations 1245 1250 5 min How often Often enough to make sure the solution t has not changed significantly 0 Depends on Instrument mechanics exure Instrument temperature changes Resolution How accurate a calibration the program requires ie what s signi can o 17spec ex 4 Gravitational Redshift 1250 1255 5 min 18specaps Photons leaving a massive body increase their potential energy 0 This decreases their kinetic energy Since E hy ifE drops so does V So they shift to longer wavelength 5 7 Doppler Shift 1255 100 5 min 0 If wave source moves toward observer wave is perceived at higher frequency AA 1 Z T 2 1 0 At high 1 1 2 1 1 2 i E211 21 m o Wavelengths shift shorter for approaching source blue shift 0 Wavelengths shift longer for receding source red shift 0 Ifa lum line appears at 7 pm 2 6 1 0 Hubble ow for distant galaxies students from exams etc Spectral Line Pro les 100 105 5 min 0 At high R spectral lines have a shape the line pro le Several effects combine some intrinsic and instrumental Instrumental mainly the nite slit width Intrinsic nature has no 6 functions Line broadening comes from Don t go into detail that s next Temperature Doppler broadening Lifetimecollisional broadening Zeeman magnetic eld and elecric eld broadening Final pro le is convolution of all independent effects Electrical and Zeeman Broadening 105 110 5 min Effects that shift or split quantum levels also shift line wavelengths o EXl High densitypressure distorts orbital shape and energy levels 0 EXZ Zeeman effect high magnetic elds e g near surfaces of neutron stars or in sunspots split and shift quantum states Can measure amount of split and thereby directly measure eld strength in a remote object 8 Temperature Broadening 110 115 5 min D o Molecules in a gas move E x kT o 771122 x kT v x W o Molecules move in all directions with a distribution of speeds around 1 0 Each molecule s lines Dopplershifted according to its thermal motion 0 Lines broaden according to T 0 Thermal linebroadening pro le is a Gaussian V0 2kT CYD 7 3 my 7 1 exp 7 my 4 a1237r O D LifetimeCollisional Broadening 1 15 125 10 min 0 Molecules excited to high energy state spontaneously decay to low state at some rate 0 OR deexcited in a collision emitting a photon o 2 effects combine average lifetime of state is At 0 Line gets broad when At is small compared to lV o Heisenberg Uncertainty Principle 0 Apr 2 g AEAt 2 g 0 But E hV We observe V so we must know E 0 But we can t know E precisely so there must be a spread of V o A solution to the Schrodinger equation gives the Lorentzian pro le width is 04L y 7 V0 7 2 5 7139 V 7 V0 04 l 04L m 6 m 7 A0 7 C 7 W 6 7 efw l o 04L related to Einstein absorption coe icient Am 1 At y x MN UCF Physics AST 57654762 Advanced Astronomical Data Analysis Fall 2009 Lecture Notes 28 Project Check In 1030 1035 5 min 0 Questions HW9 1035 1050 15 min The Project 1050 1105 15 min o 0 Final assignment is a project paper due in nals week 0 Worth half the grade 0 Each project has you reduce and analyze some raw data and write a paper 0 Do the analysis exactly as a homework Main le with commands and comments on analysis Generic analysis routines written for general use Create gures and tables as needed Include any tests We will be running them with a from projsol import command Make sure they work Project description says what to do step by step You may run into memory problems You must nd workarounds For example there may be too much data for a median lter But you can diVide the data cube into quarters do each and sew the results together 0 Graduate project weighs the dark matter in a galaxy o Undergraduate project is exoplanetary photometry 0 Project analyses are harder than homework 0 You must do all the work on the project yourself 0 You may discuss techniques but not quantitative results 0 You may not share code look at each others screens etc 1 0 Everything counts the right answer a clean analysis Right answer 25 Quality of analysis coding style 15 Presentation and explanation in paper 10 0 Paper should look like a research paper e g ApJ uh Title usually generic serious not ashy Author a iliation Abstract Introduction look on ADS andor Simbad for at least 1 fact Data Description table good Data Correction Measurement name after actual measurement Results Discussion Conclusions References at least 1 on topic 1 for methods Can use Latex or own SW There are latex templates online No longer than 5 journal pages please Shorter is better but everything must be there No gaps This is a big deal It will take 40 hours START NOW Dark Matter 1105 1120 15 min 0 151628jpg We measure rotation by Doppler shift 0 galslitjpg galrotspecpng Place slit on galaxy axis get spectrum vs 7 151629jpg 151582jpg 151630jpg Rotation curve o Enclosed mass formula 2 47rzr3 GM 1 47rzr3 M G102 2 v 2 3 4 1 Substitute and cancel M 7 2 5 T G 0220 kmsec r28000 ly Milky way MT 2gtlt1041 kg 1011 MG 1011 MG win Sun s orbit But we measure constant 1 with increasing 7 This means increasing M even though decreasing luminosity dark matter Other applications of Doppler shift include nding extrasolar planets measuring the veloci ties of stellar winds and measuring planetary zonal winds Use crosscorrelation to nd shift 05 Lfltz sgtgltzgtdz 6 Evaluate for reasonable range of shifts maX is best alignment Here x A and s AA Generally want to use f 7 f and g 7 g IDL help for ccorrelate gives discrete versions of formula y x N UCF Physics AST 57654762 Advanced Astronomical Data Analysis Fall 2009 Lecture Notes 32 Discrete Fourier Transfrom Check In 1030 1040 10 min 0 Questions 0 HWlO due 9 November HWll due 16 November 0 Correction Basis Functions 739 Orthonormal Basis Functions 0 Who read Press Amplitude Phase Power Spectrum in Python 1040 1130 50 min o 0 Steps for FFT 0 Variables are t gt f 0 Times and frequencies of the elements of t and f are implicit even spacings o Read data into array It can be complex 0 fnp fft fft t o tnpfftifftf 0 Power Pnp abs f 2 0 Power Spectral Density PSDPP O o Chance that Poisson process eg photons produces PSD F by chance is pF e F o A 30 detection thus requires PSD 65 o Amplitude anp abs f 0 Phase phnp arctanz np imag f np real f o Frequencies nu np arange N2 Ndt 0 Plot pltplot nu P O N2 l o Signals and noise 3 FFT Algorithm 1130 1140 10 min 0 A DFT of length N is the sum of2 DFTs oflength N2 0 One of these is from the evennumbered n the other from the odd N71 1 H Z 627n16nNhk 160 N21 I N21 I Z 627Tlnlt21 Nh2k Z 627Tl lt21 1Nh2k1 160 160 N21 I N21 I Z 627Tlnlt21 Nh2kWn Z 627Tlnlt21 Nh2k1 10 k0 H Wan W 62wiN712N o W is a constant complex unit vector with 6 27139 N Raise that to n 0N 7 1 and it rotates once in N samples H is the DFT of length N2 from the even hks 0 H3 is from the odd ones 71 still runs 07 7 N 7 1 so Hf and H3 span the interval but take half the data 0 But you can compute Hf and H3 using the same method giving H35 and Hff Each of these now spans the interval but samples 14 of the data and again until there is H355550500quot39550 hn 1point transforms are identities plug in N 1 above The splits make a binary tree of sums Which value ofn corresponds to 066660600660 39 J39 39 39 ofthe data We got Pm by The rst 60 decision was like the 1 bit in a binary number The second was like the 2 bit and so on Replace e a 0 and 0 a 1 Reverse the bit order This is the binary representation of n 1 2 3 4 5 k 0 Reorder hn according to new 71 it s just swaps 0 Do each set of sums in binary tree to reconstruct Hn in steps 0 If you don t have 2 quot elements pad with 0 to next 2 quot boundary FFT Speed 1140 1145 5 min 0 Compute time each hn gets summed once but there are only log N binary splits 0 As long as the binary reordering is quick there are N Fs to calculate each taking logN time o FFT is an N logz N algorithm 0 On a l MHZ CPU the time to calculate a 106point DFT is 1 week 0 With FFT it s 30 sec