The (Dirac) delta function δ(x) is defined such that δ(x) = 0 for all x = 0, The sequence an is the inverse discrete Fourier transform of the sequence Ak The for -
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[PDF] Lecture 7 - The Discrete Fourier Transform
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The (Dirac) delta function δ(x) is defined such that δ(x) = 0 for all x = 0, The sequence an is the inverse discrete Fourier transform of the sequence Ak The for -
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Notes 3, Computer Graphics 2, 15-463
Fourier Transforms and the
Fast Fourier Transform (FFT) Algorithm
Paul Heckbert
Feb. 1995
Revised 27 Jan. 1998
We start in the continuous world; then we get discrete.De®nition of the Fourier Transform
The Fourier transform (FT) of the functionf.x/is the functionF.!/, where:F.!/DZ
1 -1 f.x/e -i!x dx and the inverse Fourier transform is f.x/D1 2Z 1 -1 F.!/e i!x d!Recall thatiDp
1ande iDcosCisin.
Think of it as a transformation into a different set of basis functions. The Fourier trans- form uses complex exponentials (sinusoids) of various frequencies as its basis functions. (Other transforms, such as Z, Laplace, Cosine, Wavelet, and Hartley, use different basis functions). A Fourier transform pair is often writtenf.x/$F.!/,orF.f.x//DF.!/whereF
is the Fourier transform operator. Iff.x/isthoughtofasasignal(i.e. inputdata)thenwecallF.!/thesignal'sspectrum. Iffisthoughtofas theimpulseresponseofa®lter(whichoperates on inputdatatoproduce output data) then we callFthe ®lter'sfrequency response. (Occasionally the line between what's signal and what's ®lter becomes blurry). 1Example of a Fourier Transform
Suppose we want to create a ®lter that eliminates high frequencies but retains low frequen- cies (this is very useful in antialiasing). In signal processing terminology, this is called an ideal low pass ®lter. So we'll specify a box-shaped frequency response with cutoff fre- quency! cF.!/D1j!j!
c0j!j>!
cWhat is its impulse response?
We know that the impulse response is the inverse Fourier transform of the frequency response, so taking off our signal processing hat and puttingon our mathematics hat, all we need to do is evaluate: f.x/D1 2Z 1 -1 F.!/e i!x d! for this particularF.!/: f.x/D1 2Z c c e i!x d! D 1 2e i!x ix c !D-! c D1 xe i! c x -e -i! c x 2i D sin! c x xsince sinDe i -e -i 2i D! c sinc.! c x/ where sinc.x/Dsin.x/=.x/. For antialiasing with unit-spaced samples, you want the cutoff frequency to equal the Nyquist frequency, so! c D.Fourier Transform Properties
Rather than write ªthe Fourier transform of anXfunction is aYfunctionº, we write the shorthand:X$Y.Ifzis a complex number andzDxCiywherexandyare its real and imaginary parts, then the complex conjugate ofziszDx-iy. A functionf.u/isevenif
f.u/Df.-u/,itisoddiff.u/D-f.-u/, it is conjugate symmetric iff.u/Df .-u/, and it is conjugate antisymmetric iff.u/D-f .-u/. 2 discrete$periodic periodic$discrete discrete, periodic$discrete, periodic real$conjugate symmetric imaginary$conjugate antisymmetric box$sinc sinc$boxGaussian$Gaussian
impulse$constant impulse train$impulse train (can you prove the above?) When a signal is scaled up spatially, its spectrum is scaled down in frequency, and vice versa:f.ax/$F.!=a/for any real, nonzeroa.Convolution Theorem
The Fourier transform of a convolution of two signals is the product of their Fourier trans- forms:fg$FG. The convolution of two continuous signalsfandgis .fg/.x/DZ C1 -1 f.t/g.x-t/dt So R C1 -1 f.t/g.x-t/dt$F.!/G.!/. The Fourier transform of a product of two signals is the convolution of their Fourier transforms:fg$FG=2.Delta Functions
The (Dirac) delta function.x/is de®ned such that.x/D0 for allx6D0,R C1 -1 .t/dtD1, and for anyf.x/: .f/.x/DZ C1 -1 f.t/.x-t/dtDf.x/ The latter is called thesifting propertyof delta functions. Because convolution with a delta is linear shift-invariant ®ltering, translating the delta byawill translate the output bya: f.x/.x-a/ .x/Df.x-a/ 3Discrete Fourier Transform (DFT)
When a signal is discrete and periodic, we don't need the continuous Fourier transform. Instead we use the discrete Fourier transform, or DFT. Suppose our signal isa n fornD0:::N-1, anda
n Da nCjN for allnandj. The discrete Fourier transform ofa, also known as the spectrum ofa,is: A k D N-1 X nD0 e -i 2 N kn a nThis is more commonly written:
A k D N-1 X nD0 W knN a n (1) where W N De -i 2 N andW kN forkD0:::N-1arecalledtheNth roots of unity. They're called this because, in complex arithmetic,.W kN N D1 for allk. They're vertices of a regular polygon inscribed in the unit circle of the complex plane, with one vertex at.1;0/. Below are roots of unity forND2,ND4, andND8, graphed in the complex plane. W 42ReIm N=2W 20 W 21
N=4W 40
W 43
W 41
1-1 -1
1i -i W 84N=8W 80
W 86
W 82
-1 1i -i W 87
W 85
W83 W 81
Powers of roots of unity are periodic with periodN, since the Nth roots of unity are N isequiv- alent to rotation clockwise by this angle. Multiplication byW NN is rotation by 2radians, that is, no rotation at all. In general,W kN DW kCjN N for all integerj. Thus, when raisingW N to a power, the exponent can be taken moduloN.