MinimumMinimum PhasePhase RevisitedRevisited JeffJeff Grossman,Grossman, GaryGary Margrave,Margrave, andand MichaelMichael LamoureuxLamoureux IntroductionIntroduction

•• MinimumMinimum phasephase forfor digitaldigital systemssystems •• FourierFourier synthesis:synthesis: anan exampleexample •• TemperedTempered distributionsdistributions •• CausalityCausality andand thethe HilbertHilbert transformtransform •• MinimumMinimum phasephase forfor analoganalog systemssystems •• ExamplesExamples •• ConclusionsConclusions MinimumMinimum phasephase forfor DigitalDigital systemssystems MinimumMinimum phasephase forfor DigitalDigital systemssystems

A minimum phase digital signal… • has all the poles and zeros of its Z-transform inside the unit circle of the complex plane • is causal, stable, and always has a minimum phase convolutional inverse • has its energy concentrated toward time 0 more than any other causal signal having the same magnitude spectrum MinimumMinimum phasephase forfor DigitalDigital systemssystems

• Every causal, stable, all-pole digital filter is minimum phase • Fermat’s principle of least traveltime implies seismic wavelets are minimum phase • Wiener and Gabor deconvolution assume that a constant-Q-attenuated seismic wavelet is minimum phase FourierFourier synthesissynthesis

anan exampleexample

FourierFourier synthesissynthesis forfor aa RandomRandom spikespike seriesseries FourierFourier synthesizedsynthesized signalssignals CrosscorrelationsCrosscorrelations withwith originaloriginal signalsignal FourierFourier SynthesisSynthesis

•• So,So, itit seemsseems phasephase accuracyaccuracy isis moremore importantimportant thanthan amplitudeamplitude spectrumspectrum accuracyaccuracy……

•• But,But, inin Gabor/WienerGabor/Wiener deconvolution,deconvolution, aa phasephase spectrumspectrum isis computedcomputed fromfrom anan estimateestimate ofof thethe amplitudeamplitude spectrumspectrum TemperedTempered distributionsdistributions

aa classclass ofof generalizedgeneralized functionsfunctions TemperedTempered distributionsdistributions

• A way to make sense of divergent integrals • A way to make sense of Dirac’s delta “function” • A tempered distribution is a continuous linear mapping T : S(R) Æ C •S(R) is the Schwartz space of smooth, rapidly decreasing functions, e.g., Gaussians TemperedTempered distributions,distributions, somesome examplesexamples

Given a nice enough function f that doesn’t grow too quickly,

∞ Tfxxdxf ()ϕϕ= ()() ∫−∞ defines a tempered distribution. TemperedTempered distributions,distributions, somesome examplesexamples

Any function f for which

defines a tempered distribution, Tf . TemperedTempered distributions,distributions, somesome examplesexamples

Dirac’s delta function, is a tempered distribution that doesn’t correspond to any function. CausalityCausality andand thethe HilbertHilbert transformtransform CausalityCausality andand thethe HilbertHilbert transformtransform

Given a tempered distribution, T, write it as a sum of its causal and anti-causal parts: CausalityCausality andand thethe HilbertHilbert transformtransform

It turns out that the Hilbert transform is given simply as: MinimumMinimum phasephase forfor AnalogAnalog systemssystems MinimumMinimum phasephase forfor AnalogAnalog systemssystems ExamplesExamples ExamplesExamples Why not?

Summary • In the analog examples, fˆ was invertible, but 1/ fˆ was not “stable” (not integrable, or it had infinite energy). • In each case, a convolution could reduce the original distribution to a spike, by division in the Fourier domain: 1 fˆ =1, fˆ which is the Fourier transform of δ . Summary •• SometimesSometimes thethe HilbertHilbert transformtransform relationrelation failsfails forfor analoganalog minimumminimum phasephase signals,signals, butbut itit worksworks veryvery wellwell onon bandlimitedbandlimited versions.versions. •• WeWe computedcomputed aa simplersimpler minmin phasephase spectrumspectrum thanthan thethe originaloriginal ringyringy one,one, yetyet reconstructedreconstructed thethe minimumminimum phasephase signalsignal withwith veryvery highhigh fidelity.fidelity. Acknowledgements

•• TheThe CREWESCREWES ProjectProject andand itsits sponsorssponsors •• TheThe POTSIPOTSI ProjectProject andand MITACSMITACS •• MyMy coco--authors/supervisorsauthors/supervisors •• MyMy energeticenergetic son,son, forfor wakingwaking meme upup atat nightnight soso II couldcould thinkthink aboutabout wavelets.wavelets.