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Discrete Fourier Transformation

Module by: Phil Schniter

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

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N-point Discrete Fourier Transform (DFT)

Xk=∑n=0N−1xnⅇ-ⅈ2πnkn ∀k,k=0…N−1 X k n 0 N 1 x n 2 n k n k k 0 … N 1 (1)
xn=1N∑k=0N−1Xkⅇⅈ2πnkn ∀n,n=0…N−1 x n 1 N k 0 N 1 X k 2 n k n n n 0 … N 1 (2)

Note that:

  • Xk X k is the DTFT evaluated at ω=2πNk ∀k,k=0…N−1 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=0N−1xnⅇ-ⅈ2πMk X 2 M k n 0 N 1 x n 2 M k (3)
    Xⅇⅈ2πMk=∑n=0N−1 x zp nⅇ-ⅈ2πMk X 2 M k n 0 N 1 x zp n 2 M k Xⅇⅈ2πMk= X zp k ∀k,k=0…M−1 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=0N−1xnⅇ-ⅈωn X ω n 0 N 1 x n ω n (4)
    Xⅇⅈω=∑n=0N−11N∑k=0N−1Xkⅇⅈ2πNknⅇ-ⅈωn X ω n 0 N 1 1 N k 0 N 1 X k 2 N k n ω n Xⅇⅈω=∑k=0N−1Xk1N∑k=0N−1ⅇ-ⅈω−2πNkn X ω k 0 N 1 X k 1 N k 0 N 1 ω 2 N k n Xⅇⅈω=∑k=0N−1Xk1NsinωN−2πk2sinωN−2πk2Nⅇ-ⅈω−2πNkN−12 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-ⅈ2πN W N 2 N ,
    X0X1⋮XN−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 …⋮⋮⋮⋮⋮x0x1⋮xN−1 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 (5)
    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: 1NW1NWH=1NWH1NW=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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