Computational Methods (PHYS 2030)

York UniversityWinter 2018Lecture 25

Instructors: Prof. Christopher Bergevin (cberge@yorku.ca)

Schedule: Lecture: MWF 11:30-12:30 (CLH M)

Website: http://www.yorku.ca/cberge/2030W2018.html

à Consideranemptybulletinboard….

A ”computer vision” problem…

“thestapleproblem”

Ø Howmanystaplearethere?

Ø Distributionofangles?

Ø Howtowriteacodetoobjectivelymeasuresuch?

A ”computer vision” problem…

EXquantizeF.m

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Ø Asurprisinglylargeeffectisapparenthere![NotethattheverticalaxisisindB,whichislogarithmic]

DC vs AC

Ø DC (or‘directcurrent’)typicallyreferstostaticconditions(i.e.,|w|=0)

Ø AC (or‘alternatingcurrent’)typicallyreferstosinusoidalconditions(i.e.,|w|>0)

Ø Thissimpledistinctionhasawidevarietyifimplicationthroughoutphysicsandengineering

Ex.Ohm’slaw

V = IR DCversion

V = IZ ACversion(allvariablesarecomplexFouriercoefficients)

à Here,Z isknownastheimpedance (we’llcomebacktothis)

EXspectrogram.mSpectrograms

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‘time waveform’ recorded from mic

Spectrogram=ShortTimeFourierTransform(STFT)

STFT

Kutz (2012)

Ø BasicideaistocomputeDFTovershortintervals(i.e.,shortsegmentsfromalongerwaveform)andtherebyseehowfrequencycontentischangingwithtime

Ø Clearlythereisatradeoffbetweentimeandfrequencyresolution

P.windowL= 2048; % length of window segment for FFT {2048}P.overlap= 0.8; % fractional overlap between window, from 0 to 1 {0.8}pts= round(P.windowL*P.overlap); % convert fractional overlap to # of pointsspectrogram(A,blackman(P.windowL),pts,P.windowL,P.SR,'yaxis');

‘overlapping’andwindowingcanhelp

EXspectrogram.m

Summary

Uncertainty– Localizationinonedomainmeansbroadeningintheother

Informationcanrepresentedindifferentwaysbydifferent‘basisfunctions’

Practically,therewillbevarioustrade-offsThreesidesofthesamecoin

Fourierseries

Fouriertransform

DiscreteFouriertransform

à Consideranemptybulletinboard….

A ”computer vision” problem…

“thestapleproblem”

Ø Howmanystaplearethere?

Ø Distributionofangles?

Ø Howtowriteacodetoobjectivelymeasuresuch?

A ”computer vision” problem…

Ex. Photoshop & ‘sharpening’

à Howdoesthe‘sharpen’tool(orSharpen‘filter’)workinPhotoshop?

Ex. Photoshop & ‘sharpening’

Devries(1994)

Ex. Electric circuit’s ‘impulse response’

Ø Considerarelativelysimpleelectriccircuit:

Vin(t) Vout(t)Ø Wewillattempttocharacterizethis

(linear)‘system’intermsofthe‘signals’Vin(t) andVout(t)

Ø Consideriftheinputsignal[Vin(t)]isanimpulse.....

Ø Thecircuithaselementsthatwilldynamicallyshapetheinput(e.g.,energywillchargeupthecapacitor,theresistorwilllaterdissipatestoredenergysuchthatavoltagepersistsaftertheinputsignalpasses,....)

Ø ...Sotheoutput[Vout(t)]ofthesystemdiffersretheinput

Devries(1994)

Ex. Electric circuit’s ‘impulse response’

Ø Itappearsthattheproperties/characteristicsofthesystemarethusmanifestsomehowintherelationshipbetweentheinputandtheoutput

Ø WecanuseKirchoff’s lawtoexplicitlydeterminesucharelationship

§ Assumecurrentflowsinaclock-wise(CW)direction.Thenthecurrentflowingintopointb is:

§ Thecurrentthroughthecapacitorisproportionaltotherateofchangeofthepotentialdifferenceacrossit:

§ ButKirchoff’s currentlawrequires:

§ Forsimplicity,letVc=0.ThenVa=Vin andVb=Vout .Then:

Devries(1994)

Ex. Electric circuit’s ‘impulse response’

Ø Wethusendupwithasingle,linear,non-autonomousfirst-orderODE.Thegeneralsolutionis:

Specificsolution:

à Keepthisexampleinmind.We’llcomebacktothisshortly....

Ø Nowlet’sconsidertheinputasanimpulse:

Vin(t) Vout(t)

Convolution

Ø Adeceptivelycomplexquantity.Considertwofunctionsp(t)andq(t).Thentheconvolutionisdefinedas:

Notethatthisquantitywillbeafunctionoft (t isthe‘sliding’variablehere)

wikipedia (convolution)Devries(1994)

à Fromacomputationalpointofview,arelativelysimpleoperation

EXconvolution1.m

% ### EXconvolution1.m ### 11.07.14% Example code to perform convolution between a two user-defined binary sequencesclear% --------------------------------wf1=[0 0 0 1 1 1 0 0];wf2= [0 0 0 0 0 0 0 0 1 1 1 0 0 0 0];% --------------------------------% Use custom code (convolve1.m) or Matlab's built-in function? % [should return identical answers]if 1==1

C= convolve1(wf1,wf2); % custom codeelse

C= conv(wf1,wf2); % Matlab's built-in functionend% +++figure(1); clf;subplot(211)stem(wf1,'b'); hold on; grid on;stem(wf2,'diamondr','filled'); xlabel('Sample #'); ylabel('Amplitude');subplot(212)stem(C,'k'); hold on;

convolve1.m

function y= convolve1(wf1,wf2); % convolve two 1-D row vectors (should work similar to Matlab's conv.m) % +++error(nargchk(2, 2, nargin)), error(nargoutchk(0, 1, nargout))if ~isvector(wf1) || ~isvector(wf2)

error('Parameters must be vectors.')end% ensure they are row vectorsif (~isrow(wf1)), wf1= wf1'; endif (~isrow(wf2)), wf2= wf2'; endm = length(wf1); n = length(wf2); % extract relevant dimensions% create new arrays as needed for operationg= fliplr(wf2); % flipped wf2f= [zeros(1,n) wf1 zeros(1,n)];NN= m+n-1;for k=1:NN

% Note: It took me awhile to get this code right!y(k)= sum(f.*[zeros(1,k) g zeros(1,m-k+n)]); % shifted wf2

endreturn

à PresumablyasimplifiedversionofMatlab’s built-inconv.m

EXconvolution1.m

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wf1=[0 0 0 1 1 1 0 0];wf2= [0 0 0 0 0 0 0 0 1 1 1 0 0 0 0];C= convolve1(wf1,wf2);

EXconvolution1.m

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EXconvolution1.mwf1=[0 0 1 1 1 0 0];% y= exp(-5*x) for x=[0,1]wf2= [0 0 0 0 0 0 1.0000 0.6065 0.3679 0.2231 0.1353 0.0821 0.0498 0.0302 0.0183 0.0111 0.0067];wf2= [0 0 0 0 0 0 0 0 1 1 1 0 0 0 0];C= convolve1(wf1,wf2);

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% ### EXconvolution2.m ###% Example code to perform convolution between a sinusoid and narrow digital pulseclear% --------------------------------SR= 44100; % sample rate [Hz]Npoints= 8192; % length of fft window (# of points)f= 2580.0; % wf1: Frequency (for waveforms w/ tones) [Hz]CLKbnd= [1000 1001]; % wf2: indicies at which pulse turns 'on' and then 'off'% --------------------------------% +++t=[0:1/SR:(Npoints-1)/SR]; % create an array of time points% +++% create two waveforms (same dimensions)wf1= cos(2*pi*f*t);clktemp1= zeros(1,Npoints);clktemp2= ones(1,CLKbnd(2)-CLKbnd(1));wf2= [clktemp1(1:CLKbnd(1)-1) clktemp2 clktemp1(CLKbnd(2):end)];% +++% Use custom code (convolve1.m) or Matlab's built-in function? [should return identicalanswers]if 1==1

C= convolve1(wf1,wf2); % custom codeelse

C= conv(wf1,wf2); % Matlab's built-in functionend% +++figure(1); clf;subplot(211)plot(t,wf1,'b'); hold on; grid on;plot(t,wf2,'r'); xlabel('Time [s]'); ylabel('Amplitude');subplot(212)plot(C,'k'); hold on;

EXconvolution2.m

EXconvolution2.m

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Zoomed-inversion

à Interestingly,youseemtogetbacksomethingthatlookslike(aslightlymodifiedversionof)theoriginal.....[thisrelatesbacktowhywesamplewithpulses!]

Connecting back to the electric circuit

Ø Whatistheconnectionbetweenthecircuitandtheconvolution?

Ø Imaginethattheinputisastringofimpulses.Sincethesystemislinear,theoutputwilljustbeasumofallthe‘individual’responses

Singleimpulse Twosuccessiveimpulses

Ø Wecanimaginealimitwithmoreandmoreimpulsesarrivingatshorterandshorterinterval,therebypassingtoacontinuouscase

Thereisa‘history’!

Herewehavejustwrittena(arbitrary)continuousfunctionasanintegraloverdeltafunctions

Connecting back to the electric circuit

Ø Letr(t)representthe‘impulseresponse’ofthesystem(i.e.,theresponsetoadeltafunctioninput)

Ø Writingourinputvoltageas...

Ø ...leadstoanexpressionfortheoutputas:

à Thatis,theoutputofthesystemistheconvolutionoftheinputsignalandthesystem’simpulseresponse

Putmoregenerally,theimpulseresponsetotallycharacterizesthe(linear)system!

Ø Remarkably,allthistiesverynicely/importantlybacktoFouriertransforms....[we’llcomebacktothatinabit]

Ø Intuitivelydefinedintwodifferent(butequivalent)ways:

1. Timeresponseof‘system’whensubjectedtoanimpulse(e.g.,strikingabellw/ahammer)

2. Fouriertransformofresultingresponse(e.g.,spectrumofbellringing)

ex.Harmonicoscillator Temporalimpulseresponse

Spectralimpulseresponse

Impulse response

(Important)Note:TheFouriertransformoftheimpulseresponseiscalledthetransferfunction

Devries(1994)

Ex. Electric circuit’s ‘impulse response’

Ø Howdoesthisalltieback(intuitively)toouroriginalcircuitproblem?

Vin(t) Vout(t)Ø Fouriertransformoftheimpulseresponse

‘attenuates’athighfrequencies

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à Low-passfilter

à Thisintroducesanimportanttopicknownas‘filtering’

à Howdoesthe‘sharpen’tool(orSharpen‘filter’)workinPhotoshop?

Ex. Photoshop & ‘sharpening’