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2.22.2  Discrete Fourier Transform

Let N be an integer. The Discrete Fourier Transform (DFT) is a transformation FN defined on the set of periodic sequences of period N, it depends on a choice of a primitive N-th root of unity ωN. If the DFT is defined on sequences with complex coefficients, we take:

ωN=e
i π
N
 

If x is a periodic sequence of period N, defined by the vector x=[x0,x1,...xN−1] then FN(x)=y is a periodic sequence of period N, defined by:

(FNN(x))k=yk=
N−1
j=0
xjωNk· jk=0..N−1 

where ωN is a primitive N-th root of unity. The discrete Fourier transform may be computed faster than by computing each yk individually, by the Fast Fourier Transform (FFT). Xcas implements the FFT algorithm to compute the discrete Fourier transform only if N is a power of 2.

The properties of the Discrete Fourier Transform

The Discrete Fourier Transform FN is a bijective transformation on periodic sequences such that

 FNN−1=
1
N
 FNN−1 
 =
1
N
 
FN
     on  ℂ

i.e. :

(FN−1(x))k=
1
N
N−1
j=0
xjωNk· j 

Inside Xcas the discrete Fourier transform and it’s inverse are denote by fft and ifft:

fft(x)= FN(x), ifft(x)= FN−1(x)

Definitions
Let x and y be two periodic sequences of period N.

Properties :

N*FN(x · y)=FN(x) * FN(y)
FN(x * y)=FN(x) · FN(y)

Applications

  1. Value of a polynomial
    Define a polynomial P(x)=∑j=0N−1cjxj by the vector of its coefficients c:=[c0,c1,..cN−1], where zeroes may be added so that N is a power of 2.
  2. Trigonometric interpolation
    Let f be periodic function of period 2π, assume that f(2kπ/N)=fk for k=0..(N−1). Find a trigonometric polynomial p that interpolates f at xk=2kπ/N, that is find pj, j=0..N−1 such that
    p(x)= 
    N−1
    j=0
     pj exp(ijx),    p(xk)=fk
    Replacing xk by it’s value in p(x) we get:
    N−1
    j=0
     pj exp(i
    j2kπ
    N
    ) = fk
    In other words, (fk) is the inverse DFT of (pk), hence
    (pk)= 
    1
    N
     FN(  (fk)  ) 
    If the function f is real, pk=pk, hence depending whether N is even or odd:
    p(x)=
    p0+ 2 ℜ(
    N
    2
    −1
    k=0
    pkexp(ikx))+ℜ(p
     
    N
    2
     exp(i
    Nx
    2
    )) 
    p(x)=
    p0+ 2 ℜ(
    N−1
    2
    k=0
    pkexp(ikx))
  3. Fourier series
    Let f be a periodic function of period 2π, such that
    f(xk)=yk,    xk=
    2kπ
    N
    k=0..N−1 
    Suppose that the Fourier serie of f converges to f (this will be the case if for example f is continuous). If N is large, a good approximation of f will be given by:
     
    N
    2
     ≤ n<
    N
    2
     cn exp(inx
    Hence we want a numeric approximation of
    cn=
    1
     


    0
    f(t)exp(−int)dt 
    The numeric value of the integral ∫0f(t)exp(−int)dt may be computed by the trapezoidal rule (note that the Romberg algorithm would not work here, because the Euler Mac Laurin developpement has it’s coefficients equal to zero, since the integrated function is periodic, hence all it’s derivatives have the same value at 0 and at 2π). If c_n is the numeric value of cn obtained by the trapezoidal rule, then
    c_n=
    1
    N
    N−1
    k=0
    ykexp(−2i
    nkπ
    N
    ),     −
    N
    2
     ≤ n<
    N
    2
     
    Indeed, since xk=2kπ/N and f(xk)=yk:
    f(xk)exp(−inxk)=
    ykexp(−2i
    nkπ
    N
    ), 
    f(0)exp(0)=f(2π)exp(−2i
    nNπ
    N
    )
    =y0=yN 
    Hence :
    [c0,..c
     
    N
    2
    −1
    ,c
     
    N
    2
    ,..c−1]=
    1
    N
    FN([y0,y1...y(N−1)]) 
    since

    Properties

    Example, input

    f(t):=cos(t)+cos(2*t)
    x:=f(2*k*pi/8)$(k=0..7)

    Output :

    x={2,(√2)/2,-1,-((√2)/2),0,-((√2)/2),-1,(√2)/2}
    fft(x)=[0.0,4.0,4.0,0.0,0.0,0.0,4.0,4.0]

    After a division by N=8, we get

    c0=0,c1=4.0/8,c2=4.0/2,c3=0.0,
    c−4=0,c−3=0,c−2=4.0/8,=c−1=4.0/8

    Hence bk=0 and ak=ck+ck is equal to 1 if k=1,2 and 0 otherwise.

  4. Convolution Product
    If P(x)=∑j=0n−1ajxj and Q(x)=∑j=0m−1bjxj are given by the vector of their coefficients a=[a0,a1,..an−1] and b=[b0,b1,..bm−1], we may compute the product of these two polynomials using the DFT. The product of polynomials is the convolution product of the periodic sequence of their coefficients if the period is greater or equal to (n+m). Therefore we complete a (resp b) with m+p (resp n+p) zeros, where p is choosen such that N=n+m+p is a power of 2. If a=[a0,a1,..an−1,0..0] and b=[b0,b1,..bm−1,0..0], then:
    P(x)Q(x)=
    n+m−1
    j=0
    (a*b)jxj 
    We compute FN(a), FN(b), then ab=FN−1(FN(aFN(b)) using the properties
    NFN(x · y)=FN(x) * FN(y),    FN(x * y)=FN(x) · FN(y

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