DFTandFFTimplementationsandproofsusingACL2

MertcanTemel

11/10/17

WhatisFourierTransform?

• 3maintypes:• ContinuousTimeFourierTransform(CTFT)

• Input:Continuous,Output:Continuous• DiscreteTimeFourierTransform(DTFT)

• Input:Discrete,Output:Continuous• DiscreteFourierTransform(DFT)

• Input:Discrete,Output:Discrete

• Decompositionofasignalintothefrequenciesthatmakeitup• Applicationsinclude:• Differential/DifferenceEqs,FilterDesign,SpeechRecognition,FastLargeIntegerMultiplication…

• Theoneweareinterestedin• FastFourierTransform(FFT):someefficientalgorithmstocomputeDFT

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WhatisDiscreteFourierTransform(DFT)?

𝑋" = $ 𝑥&𝑒()*+&"

,-,(.

&/0

DFT

𝑥& =1𝑁$ 𝑋"𝑒

)*+&",-

,(.

"/0

InverseDFT(IDFT)

x:inputvectoroffinitelengthNxm:mth elementofxX:outputvectorofthesamelengthNXk:kth elementofXj:squarerootof-1

DFTImplementation:• Calculatethesumoverm from0toN-1foreverykin[0N-1]• TimecomplexityofO(N2)

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WhatisFastFourierTransform(FFT)?• AnefficientimplementationofDFT• Twomostcommonlyknownways:• Decimation-in-time(DIT)• Decimation-in-frequency(DIF)

• Restriction:vectorlengthNshouldbepowerof2.Ifnot,fillwith0s.• Arecursivealgorithm

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WhatisFastFourierTransform(FFT)?

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N/2pointDFT(FFT)

N/2pointDFT(FFT)

x0x2x4...

x1x3x5...

Evens

Odds

G0

G1

G2

.

.

.

H0

H1

H2

.

.

.

𝑊,0

𝑊,.

𝑊,*

𝑊,…

𝑊,,/*

𝑊,,/*6.

𝑊,,/*6*

𝑊,…

X0 =G0 +H0*𝑊,0

X1 =G1 +H1*𝑊,.

X2 =…

X…

XN/2XN/2+1XN/2+2X…

• where𝑊,& =𝑒()*+&/,

• Decimation-in-time• Recursivedefinitionwith

baseN=2• N=2stepisalsocalled

butterflystep• Timecomplexityof

O(Nlog(N))

WorkDoneinACL2

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• DFTImplementation• Prooffor(idft (dft xN)N)=x

• FFTdecimation-in-timeimplementation• Prooffor(fft xN)=(dft xN)

DFTImplementationinACL2

𝑋" = $ 𝑥&𝑒()*+&"

,-,(.

&/0

DFT

𝑥& =1𝑁$ 𝑋"𝑒

)*+&",-

,(.

"/0

InverseDFT(IDFT)

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ProofforIDFTofDFTofXisX

Goal:ProvethatinverseDFTofDFTofavectorgivestheoriginalvector.i.e.(idft (dft x))=x?

ACL2theorem:

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ProofforIDFTofDFTofXisX- Steps

Step1:PlugDFTsumintoIDFTsum.3variables:p,q,andm

1𝑁$($ 𝑥9𝑒

()*+:9,-

,(.

9/0

)𝑒)*+&:

,-,(.

:/0

Step2:Mergeexponentials

1𝑁$ $ 𝑥9𝑒

()*+:96)*+&:,-

,(.

9/0

,(.

:/0

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ProofforIDFTofDFTofXisX- Steps

Step2:Mergeexponentials

1𝑁$ $ 𝑥9𝑒

()*+:96)*+&:,-

,(.

𝒑/0

,(.

𝒒/0

Step3:Changesummationorder

1𝑁$ $ 𝑥9𝑒

)*+:(&(9),-

,(.

𝒒/0

,(.

𝒑/0

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ProofforIDFTofDFTofXisX- Steps

Step3:Changesummationorder

1𝑁$ $ 𝑥9𝑒

)*+:(&(9),-

,(.

:/0

,(.

9/0

Step4:Takexp out

1𝑁$ 𝑥9 $ 𝑒

)*+:(&(9),-

,(.

:/0

,(.

9/0

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ProofforIDFTofDFTofXisX- Steps

Step4:Takexp out

1𝑁$ 𝑥9 $ 𝑒

)*+:(&(9),-

,(.

:/0

,(.

9/0

Step5:Take1/Nin

$𝑥9(1𝑁$ 1 ∗ 𝑒

)*+: &(9,- )

,(.

:/0

,(.

9/0

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ProofforIDFTofDFTofXisX- Steps

Step5:Take1/Nin

$𝑥9(1𝑁$ 1 ∗ 𝑒

)*+: &(9,- )

,(.

:/0

,(.

9/0

Step6:Rewriteimpulsefromidft ofones

$𝑥9𝛿[𝑚 − 𝑝],(.

9/0

where

𝛿 𝑎 = F1, 𝑎 = 00, 𝑎 ≠ 0

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ProofforIDFTofDFTofXisX- Steps

Step6:Rewriteimpulsefromidft ofones

$𝑥9𝛿[𝑚 − 𝑝],(.

9/0

where

𝛿 𝑎 = F1, 𝑎 = 00, 𝑎 ≠ 0

Step7:Equatethetermto𝑥&.Thisconcludestheproof.

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FFTImplementationinACL2

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N/2pointDFT(FFT)

N/2pointDFT(FFT)

x0x2x4...

x1x3x5...

Evens

Odds

G0

G1

G2

.

.

.

H0

H1

H2

.

.

.

𝑊,0

𝑊,.

𝑊,*

𝑊,…

𝑊,,/*

𝑊,,/*6.

𝑊,,/*6*

𝑊,…

X0 =G0 +H0*𝑊,0

X1 =G1 +H1*𝑊,.

X2 =…

X…

XN/2XN/2+1XN/2+2X…

• where𝑊,& =

𝑒()*+&/,• Recursivelycalculate

N/2pointFFTsofevenandoddindices

• PerformG+H*Wovereachelement(GandHvaluesareusedtwice)

FFTImplementationinACL2

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ProofforFFTisDFT

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• BasicideaofFFT:• Getridofredundant/repeatedmultiplications• Rememberintermediateresults

• DifferentderivationsforDecimation-in-time(DIT)andDecimation-in-frequency(DIF).OnlyDITwillbediscussedhere.• ACL2Theorem:

ProofforFFTisDFT- Steps

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DFTformula:

𝑋" = $ 𝑥J𝑒()*+J"

,-,(.

J/0

Step1:Divideintotwosummations

𝑋" = $ 𝑥*J𝑒()*+(*J)"

,-,/*(.

J/0

+ $ 𝑥*J6.𝑒()*+(*J6.)"

,-,/*(.

J/0

ProofforFFTisDFT- Steps

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Step1:Divideintotwosummations

𝑋" = $ 𝑥*J𝑒()*+(*J)"

,-,/*(.

J/0

+ $ 𝑥*J6.𝑒()*+(*J6.)"

,-,/*(.

J/0

Step2:Distributeandcommuteconstants

𝑋" = $ 𝑥*J𝑒()*+"

(,/*)-,/*(.

J/0

+ $ 𝑥*J6.𝑒()*+"J

(,/*)- 𝑒()*+"

,-,/*(.

J/0

ProofforFFTisDFT- Steps

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Step2:Distributeandcommuteconstants

𝑋" = $ 𝑥*J𝑒()*+"

(,/*)-,/*(.

J/0

+ $ 𝑥*J6.𝑒()*+"J

(,/*)- 𝑒()*+"

,-,/*(.

J/0

Step3:Taketheconstantexponentialout

𝑋" = $ 𝑥*J𝑒()*+"

(,/*)-,/*(.

J/0

+ (𝑒()*+"

,- ) $ 𝑥*J6.𝑒()*+"J

(,/*)-,/*(.

J/0

ProofforFFTisDFT- Steps

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Step3:Taketheconstantexponentialout

𝑋" = $ 𝑥*J𝑒()*+"

(,/*)-,/*(.

J/0

+ (𝑒()*+"

,- ) $ 𝑥*J6.𝑒()*+"J

(,/*)-,/*(.

J/0

Observation1:𝑋" = (𝑁/2𝑝𝑜𝑖𝑛𝑡𝐷𝐹𝑇𝑜𝑓𝑒𝑣𝑒𝑛𝑠𝑥) + (𝑒

()*+",- ) ∗ (𝑁/2𝑝𝑜𝑖𝑛𝑡𝐷𝐹𝑇𝑜𝑓𝑜𝑑𝑑𝑠𝑥)

Observation2:NpointDFTisperiodicwithN(i.e.Xk =Xk+N)

ProofforFFTisDFT- Steps

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Observation1:𝑋" = (𝑁/2𝑝𝑜𝑖𝑛𝑡𝐷𝐹𝑇𝑜𝑓𝑒𝑣𝑒𝑛𝑠𝑥) + (𝑒

()*+",- ) ∗ (𝑁/2𝑝𝑜𝑖𝑛𝑡𝐷𝐹𝑇𝑜𝑓𝑜𝑑𝑑𝑠𝑥)

Observation2:NpointDFTisperiodicwithN(i.e.Xk =Xk+N)=>1st and2nd DFTsgivethesameresultforkandk+N/2

(e.g.evendft(x)k=evendft(x)k+N/2)=>Insteadofcalculatingseparatelyforallk∈ [0N-1],

calculate1st and2nd DFTsfork∈ [0N/2-1]rememberandusethosevaluestwiceforkandk+N/2

Applyingtheseconcludestheproof.

FutureWork

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ImplementthisFFTweb,- intheDEsysteminACL2- asaself-timed,asynchronouscircuit