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where the complex number w is de ned by w = e 2 i/n Let F be the matrix whose elements are the complex conjugates of the elements of the matrix F The matrix F has the nice property that F F = F F = nI, (D2)
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where I is the identity matrix (the column vectors of the symmetric matrix F form an orthogonal system) To verify this, let w = e 2 i/n denote the complex conjugate of w The inproduct of the rth row of F and the sth column of F is given by rs = w0 w0 + wr ws + w2r w2s + + w(n 1)r w(n 1)s For r = s each term equals e 0 = 1 and so the sum rs is n For r = s the sum rs can be written as 1 + + + n 1 = (1 n )/(1 ) with = wr ws (= 1) Since wn = e 2 i = 1 and wn = e 2 i = 1, we have n = 1 and so rs = 0 for r = s This gives (D2) By (D2), we have F 1 = (1/n)F It now follows that the vector c of Fourier coef cients is given by c = (1/n)F f Componentwise, we have n 1 1 f e 2 i k/n , k = 0, , n 1 (D3) ck = n
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This inversion formula parallels the formula ck = (2 ) 1 f (x) e ikx dx in continuous Fourier analysis Notice that (D3) inherits the structure of (D1) In many applications, however, we proceed in reverse order: we know the Fourier coef cients ck and wish to calculate the original coef cients fj By the formula (D1) we can transform c back into f The matrix multiplications in (D1) would normally require n2 multiplications However, the discrete FFT method performs the multiplications in an extremely fast and ingenious way that requires only n log2 (n) multiplications instead of n2 The key to the method is the simple observation that the discrete Fourier transform of length n (n even) can be written
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as the sum of two discrete Fourier transforms, each of length n/2 Suppose we know the ck and wish to compute the f from (D1) It holds that
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The discrete Fourier transform of length n can thus be written as the sum of two discrete Fourier transforms each of length n/2 This beautiful trick can be applied recursively For the implementation of the recursive discrete FFT procedure it is convenient to choose n = 2m for some positive m (if necessary, zeros can be added to the sequence f0 , , fn 1 in order to achieve that n = 2m for some m) The discrete FFT method is numerically very stable (it is a fast and accurate method even for values of n with an order of magnitude of a hundred thousand) The discrete FFT method that calculates the original coef cients fj from the Fourier coef cients ck is usually called the inverse discrete FFT method Ready-to-use codes for the discrete FFT method are widely available The discrete FFT method is a basic tool that should be part of the toolbox of any applied probabilist It is noted that the discrete FFT method can be extended to a complex function de ned over a multidimensional grid Numerical inversion of the generating function Suppose an explicit expression is available for the generating function
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