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20 jan 2016 · 1 2 The IDFT matrix To recover N values of the function from its discrete Fourier transform we simply have to invert the DFT matrix to obtain

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[PDF] The Fourier Matrix 1 p - mathchalmersse

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Holger Broman? August 19? 1999TheFourierMatrixLetus?rstrecalltheDiscreteFourierTransform?DFT?Givenasequenceofnumb ers?real or complex?x0

?x1 ?????xN?1 ?theDFTof thesequenceisX k

N?1Xn?0

x n e ?j2? kn

N?k?0?1???? ?N?1?Thereis an inversetransform?x

n 1 N

N?1Xn?0

X k e j2? kn N?n?0?1???? ?N?1?So? in the general case? the DFT is a mapping of anN?dimensional complex vector?x0 ???? ?xN?1

T?toanN?dimensionalcomplexvector?X0

???? ?XN?1

T?Asthemapping is linear? there must exist something we can call a Fourier matrix?Hereit is ?almost??A?

2 6 6 6 64

111???11a

1?1a 2?1a ?N?1??11a 1?2a 2?2a ?N?1??21 1a

1?N?1?a

2?N?1?a

?N?1??N?1? 3 7 7 7 75
?where a?e ?j 2?

N?Exercise?Checkthat forx??x0

???? ?xN?1

T? thevectorXgivenbyX?AxisX??X0

?????XN?1

T?TheinverseDFTcanalsoberepresentedbyamatrix?Pleaseconstructit?Weimmediately notesome ?problems???ThecolumnsoftheFouriermatrixallhavethesamenorm?namely

p

N?Pleasecheck??The ?inverse Fourier matrix?is not the inverse of the Fourier matrix? thereis a scaling problem?1

To?xtheseproblems? theFourier matrix is re?de?nedasfollows?F? 1 p N 2 6

666664

1111a
1?11a ?N?1??N?1? 3 7

777775Nowthings b ecomesimpler?Verify thatF

HF?I?whereF

HistheHermitiantransp ose?i?e?transp oseandcomplexconjugateoftheelements?Hint??Showthatthecolumns ofFall haveunit length??Showthatthecolumns ofFare orthogonal?This result veri?es thatF

H?F ?1represents the inverse DFT? It also shows thatFis anorm?preservinglinear mapping fromC NtoC

N?andthatall eigenvaluesofFfallontheunitcircle?TofurtherstudytheeigenvaluesofF?thesimplestwayis to notetheinteresting prop ertyofF

2?please verify??

1 0 F 2 =0 0 0 0 11 0

Oops?F

2isavery sp ecial p ermutation matrix?The structure is that of a circulantHankelmatrix? to b edescrib edin another shortnote?Now?squareF?square??

F 2 2?F

4?I?Oops again?This means that all eigenvalues ofFare found in the setf?1??jg?Note?TheFouriermatrixwillreapp earinthenotesonTo eplitzandHankelstructures?Youshould runthem??le fourmat in Matlab whenyouhavestudiedthelea?etson Fourier? To eplitz andHankel matrices?2

?fourmat?m?theFouriermatrix?clearN?4?

F?dftmtx?N??sqrt?N???MatlabdoesnotmakeFnorm?preserving?unitary??constructF?FasaHankelmatrixdum?zeros??N?1??1??dum?2??1?F2?hankel?dum?2?N?1??dum?1 ?N?? ?sprintf??TestonF?F?0?3g??norm?F2?F?F??fro?? ?pause?constructIaIa?hankel?flipud?dum?2?N?1 ?????constuctageneralciculantToeplitzmatrixdum?randn??N?1??1??dum?N?1??dum?1??T?toeplitz?dum?1?N??flipud ?dum ?2?N ?1?? ???andtheHankelH?T?Ia??illustratehowtodiagonalizethesematrisessprintf??TestifFT?hermit?F?isdiagonal?0?3g??norm?F?T?F??diag?diag ?F? T?F? ???? f

r

sprintf??TestifFHFisdiagonal?0?3g??norm?F?H?F?diag?di ag?F ?H?F ???? fro ???pause?comparethediagonalofF?T?F?totheeigenvaluesofTsprintf??NowfollowseigenvalsofT??sort?diag?F?T?F????sort?eig?T???norm?sort?diag?F?T?F????so rt?e ig?T ???? fro? ??dangerousassortisimperfectpause?comparethediagonalofF?H?FtotheeigenvaluesofHsprintf??NowfollowseigenvalsofH??sort?diag?F?H?F???

sort?eig?H????Nowdoittheproperwaydum?diag?F?H?F??fork?2?ceil?N?2?dum1?sqrt?dum?k??dum?N?k?2? ?? dum?k??dum1?dum?N?k?2???dum1?end3

sprintf??NowfollowseigenvalsofHtheproperway??sort?real?dum????lackofnumericalprecisionleavesasmallimaginarypartnorm?sort?real?dum???sort? eig? H??? ?fro ???nowitworksasreal?valueddata4

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