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Discrete Fourier Transformation

Module by: Phil Schniter. E-mail the author

Summary: This module covers the fundamentals of Discrete-Fourier Transformations.

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N-point Discrete Fourier Transform (DFT)

Xk=n=0N1xne(i)2πnkn k ,k=0N1: k=0N1 X k n 0 N 1 x n 2 n k n k k 0 N 1
xn=1Nk=0N1Xkei2πnkn n,n=0N1: n=0N1 x n 1 N k 0 N 1 X k 2 n k n n n 0 N 1

Note that:

  • Xk X k is the DTFT evaluated at ω=2πNk k ,k=0N1: k=0N1 ω 2 N k k k 0 N 1
  • Zero-padding xn x n to MM samples prior to the DFT yields an MM-point uniform sampled version of the DTFT:
    Xei2πMk= n =0N1xne(i)2πMk X 2 M k n 0 N 1 x n 2 M k
    Xei2πMk=n=0N1 x zp ne(i)2πMk X 2 M k n 0 N 1 x zp n 2 M k Xei2πMk= X zp k k ,k=0M1: k=0M1 X 2 M k X zp k k k 0 M 1
  • The NN-pt DFT is sufficient to reconstruct the entire DTFT of an NN-pt sequence:
    Xeiω= n =0N1xne(i)ωn X ω n 0 N 1 x n ω n
    Xeiω=n=0N11Nk=0N1Xkei2πNkne(i)ωn X ω n 0 N 1 1 N k 0 N 1 X k 2 N k n ω n Xeiω=k=0N1Xk1Nk=0N1e(i)(ω2πNk)n X ω k 0 N 1 X k 1 N k 0 N 1 ω 2 N k n Xeiω=k=0N1Xk1N(sinωN2πk2sinωN2πk2Ne(i)(ω2πNk)N12) X ω k 0 N 1 X k 1 N ω N 2 k 2 ω N 2 k 2 N ω 2 N k N 1 2

Figure 1: Dirichlet sinc, 1NsinωN2sinω2 1 N ω N 2 ω 2
Figure 1 (dirichletsinc.png)

  • The DFT has a convenient matrix representation. Defining W N =e(i)2πN W N 2 N ,
    ( X0 X1 XN1 )=( W N 0 W N 0 W N 0 W N 0 W N 0 W N 1 W N 2 W N 3 W N 0 W N 2 W N 4 W N 6 )( x0 x1 xN1 ) X 0 X 1 X N 1 W N 0 W N 0 W N 0 W N 0 W N 0 W N 1 W N 2 W N 3 W N 0 W N 2 W N 4 W N 6 x 0 x 1 x N 1
    where X=Wx X W x respectively. WW has the following properties:
    • WW is Vandermonde: the nnth column of WW is a polynomial in W N n W N n
    • WW is symmetric: W=WT W W
    • 1NW 1 N W is unitary: (1NW)1NWH=1NWH(1NW)=I 1 N W 1 N W H 1 N W H 1 N W I
    • 1NW¯=W-1 1 N W W -1 , the IDFT matrix.
  • For NN a power of 2, the FFT can be used to compute the DFT using about N2log2N N 2 2 N rather than N2 N 2 operations.

Table 1
N N N2log2N N 2 2 N N2 N 2
16 32 256
64 192 4096
256 1024 65536
1024 5120 1048576

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