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I'm looking at the inverse fast Fourier transform as calculated by Matlab. In the question "What's the correct way to shift zero frequency to the center of a Fourier Transform?" the way to implement Fast Fourier Transform in Mathematica from the fft(x) function in Matlab is discussed.

I have a dataset obtained by:

n=80;    
data = Table[0.25 Sin[2 Pi/n i], {i, 1, n}];
 (*data={0.019614773931961236`,0.03910861626005772`,0.05836134096397635`,0.07725424859373686`,0.09567085809127245`,0.11349762493488669`,0.1306246411789872`,0.14694631307311828`,0.16236201208254591`,0.17677669529663687`,0.19010149140000773`,0.20225424859373686`,0.21316004108852307`,0.22275163104709197`,0.    23096988312782168`,0.23776412907378838`,0.24309248009941914`,0.24692208514878444`,0.249229333433282`,0.25`,0.249229333433282`,0.24692208514878444`,0.24309248009941914`,0.23776412907378838`,0.23096988312782168`,0.22275163104709197`,0.21316004108852307`,0.20225424859373686`,0.19010149140000773`,0.17677669529663687`,0.16236201208254591`,0.14694631307311828`,0.1306246411789872`,0.11349762493488669`,0.09567085809127245`,0.07725424859373686`,0.05836134096397635`,0.03910861626005772`,0.019614773931961236`,0.`,-0.019614773931961236`,-0.03910861626005772`,-0.05836134096397635`,-0.07725424859373686`,-0.09567085809127245`,-0.11349762493488669`,-0.1306246411789872`,-0.14694631307311828`,-0.16236201208254591`,-0.17677669529663687`,-0.19010149140000773`,-0.20225424859373686`,-0.21316004108852307`,-0.22275163104709197`,-0.23096988312782168`,-0.23776412907378838`,-0.24309248009941914`,-0.24692208514878444`,-0.249229333433282`,-0.25`,-0.249229333433282`,-0.24692208514878444`,-0.24309248009941914`,-0.23776412907378838`,-0.23096988312782168`,-0.22275163104709197`,-0.21316004108852307`,-0.20225424859373686`,-0.19010149140000773`,-0.17677669529663687`,-0.16236201208254591`,-0.14694631307311828`,-0.1306246411789872`,-0.11349762493488669`,-0.09567085809127245`,-0.07725424859373686`,-0.05836134096397635`,-0.03910861626005772`,-0.019614773931961236`,0.`}*)

Applying the Fast Fourier Transform, fft(data), as in Matlab, we have:

fftdata = Fourier[data, FourierParameters -> {1, -1}];
(*dataf={0.` +0.` I ,0.7845909572784495`-9.969173337331279`I ,0.` +0.`I ,-1.0132180413547489`*^-16+4.185075451078514`*^-17 I ,0.` +0.`I ,1.3040059557750693`*^-16+2.6410775711049507`*^-16 I ,0.` +0.`I ,1.0308349246799976`*^-16-8.596848547326573`*^-17 I ,0.` +0.`I ,6.195506605672999`*^-17-1.6170461347458928`*^-16 I ,0.` +0.`I ,-1.9168912264360062`*^-16+2.0378880969845271`*^-16 I ,0.` +0.`I ,8.455403687140775`*^-17-6.729919522439383`*^-17 I ,0.` +0.`I ,-2.7755575615628914`*^-17+0.` I ,0.` +0.` I ,-3.893734216480464`*^-16-4.990258457628567`*^-16 I ,0.` +0.` I ,1.2592784991282853`*^-16-9.763574303309503`*^-17 I ,0.` +0.`I ,1.212504750441666`*^-16+1.0253813603191403`*^-16 I ,0.` +0.` I ,-4.253376814093551`*^-17+1.8137312595174412`*^-16 I ,0.` +0.` I ,-7.937975984179655`*^-17-3.8528542273131027`*^-16 I ,0.` +0.` I ,-2.5700391915483936`*^-17-8.898211079852906`*^-17 I ,0.` +0.` I ,-1.9723049703512393`*^-17-4.864219214722241`*^-18 I ,0.` +0.` I ,2.220446049250313`*^-16-8.881784197001252`*^-16 I ,0.` +0.` I ,-2.849755553531622`*^-17-7.58912956781088`*^-16 I ,0.` +0.` I ,-2.0048093906176696`*^-17-1.8985077351448689`*^-16 I ,0.` +0.` I ,3.512738297648335`*^-17+1.2979997047241213`*^-16 I ,0.` +0.` I ,9.565284753614687`*^-17-8.728316437179962`*^-17 I ,0.` +0.` I ,9.565284753614687`*^-17+8.72831643717996`*^-17 I ,0.` +0.` I ,3.5127382976483314`*^-17-1.2979997047241205`*^-16 I ,0.` +0.` I ,-2.0048093906176696`*^-17+1.8985077351448686`*^-16 I ,0.` +0.` I ,-2.3791346432658713`*^-17-9.169375983050538`*^-17 I ,0.` +0.` I ,2.325315327167679`*^-16-9.49924319129199`*^-16 I ,0.` +0.` I ,-1.972304970351243`*^-17+4.864219214722272`*^-18 I ,0.` +0.` I ,-2.5700391915483936`*^-17+8.898211079852904`*^-17 I ,0.` +0.` I ,-7.937975984179656`*^-17+3.852854227313102`*^-16 I ,0.` +0.` I ,-4.2533768140935505`*^-17-1.8137312595174415`*^-16 I ,0.` +0.` I ,1.2125047504416658`*^-16-1.0253813603191397`*^-16 I ,0.` +0.` I ,1.2592784991282855`*^-16+9.763574303309504`*^-17 I ,0.` +0.` I ,-4.56351734695374`*^-16+6.196258398420469`*^-16 I ,0.` +0.` I ,3.2980515196221534`*^-17-4.539341797738586`*^-16 I ,0.` +0.` I ,8.455403687140775`*^-17+6.729919522439382`*^-17 I ,0.` +0.` I ,-1.9168912264360062`*^-16-2.037888096984527`*^-16 I ,0.` +0.` I ,6.195506605673002`*^-17+1.6170461347458923`*^-16 I ,0.` +0.` I ,1.0308349246799979`*^-16+8.596848547326578`*^-17 I ,0.` +0.` I ,1.304005955775069`*^-16-2.6410775711049507`*^-16 I ,0.` +0.` I ,-1.0132180413547492`*^-16-4.185075451078515`*^-17 I ,0.` +0.` I ,0.7845909572784494` +9.96917333733128` I }*)

I would expect to get data when applying InverseFourierTransform to fftdata. I also tried using FourierParameters for that.

ifftdata1 = InverseFourier[fftdata];
ifftdata2 = InverseFourier[fftdata, FourierParameters -> {1, -1}];

However it doesn't work. Please, can anybody help me?

SAC
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1 Answers1

1

Works with v12.0 on my Mac

Clear["Global`*"]

$Version


(* "12.0.0 for Mac OS X x86 (64-bit) (April 7, 2019)" *)


n = 80;
data = Table[0.25 Sin[2 Pi/n i], {i, 1, n}];


fftdata = Fourier[data, FourierParameters -> {1, -1}];


data - InverseFourier[fftdata, FourierParameters -> {1, -1}] // Chop


(* {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0} *)
Bob Hanlon
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  • but the expected result is {0.019614773931961236,0.03910861626005772,0.05836134096397635,0.07725424859373686,0.09567085809127245,...}, ie, the same as date. – SAC Nov 24 '21 at 18:27
  • InverseFourier[fftdata, FourierParameters -> {1, -1}] // Chop is equal to data as shown by the fact that each term differs from its corresponding term in data by 0 – Bob Hanlon Nov 24 '21 at 18:32