Fourier Methods

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Fourier Methods

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• ??? realchs_at_korea.ac.kr

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Fourier Series

• For periodic data, it is more appropriate to use
  sine and cosine functions for the approximation
  or interpolation
• Fourier Series
• An investigation into data approximation and
  interpolation using trigonometric polynomials
• The formulas for the coefficients are found by
  using the appropriate orthogonality results for
  the sine and cosine function.
• Derivation from Euler formulas

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Fourier Series

• Example

n 1
n 2
n 10
n 10000
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contents

• Fourier Approximation and Interpolation
• Fast Fourier Transforms for n2r
• Fast Fourier Transforms for General n

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Fourier Approximation and Interpolation

• Discrete Fourier Approximation
• Formulas
• If number of samples per period are odd,
• If number of samples per period are even,
• ? Derivation of the formulas for the
  coefficients??

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Fourier Approximation and Interpolation

• Discrete Fourier Approximation

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Fourier Approximation and Interpolation

• Example 10.1 Trigonometric Interpolation

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Fourier Approximation and Interpolation

• Example 10.2 Trigonometric Approximation

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Fourier Approximation and Interpolation

• 10.1.1 Matlab function for Fourier Interpolation
  or Approximation

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Fourier Approximation and Interpolation

• Example 10.3 A step function
• z 1 1 1 1 0 0 0 0
• m4
• a,bTrig_poly(z,m)
• a 0.5 0.25 0 0.25 0
• b 0 0.6036 0 0.1036 0

m4
m3
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Fourier Approximation and Interpolation

• Example 10.4 Geometric Figures

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Fast Fourier Transform
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Discrete Fourier Transform

• discrete-time Fourier transform
• The discrete-time Fourier transform X(ejw) of a
  sequence xn is defined by
• discrete Fourier transform
• uniformly sampling X(ej?) on the ?-axis between 0
  ? 2p at ?k2pk/N, 0 k N-1

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Discrete Fourier Transform

• Commonly used notation
• We can rewrite DFT equation as
• Inverse discrete Fourier transform (IDFT)

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Discrete Fourier Transform

• Matrix Relations
• The DFT samples can be expressed in matrix form
  as
• DFT can be computed in O(N2) operations.
• FFT can reduce the computational complexity to
  about O(Nlog2N) operations

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contents

• Fourier Approximation and Interpolation
• Fast Fourier Transforms for n2r
• Fast Fourier Transforms for General n

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Fast Fourier Transforms for n 2r

• begin by considering the FFT when n is power of
  2, i.e., n2r
• Example of n 4
• Each value of j can be written in binary form as
  j2r-1jr22j32j2j1.
• We can also write k in binary form, but as k
  2k1k2

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Fast Fourier Transforms for n 2r

• begin by writing out the linear system of
  equations for the Fourier transform components
  for the case n4
• w4 w0 1, and interchanging the order of the
  second and third equations

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Fast Fourier Transforms for n 2r

• We now factor the coefficient matrix
• Substituting the factored form of the coefficient
  matrix into DFT eq.

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Fast Fourier Transforms for n 2r

• First we find the product
• Then we form the second product

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Fast Fourier Transforms for n 2r

• Pathways with powers of w on them indicate that
  the quantity on the left is multiplied by that
  amount.

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Algebraic Form of FFT

• Example of n 4
• to calculate the discrete Fourier transform of
  the data zk, i.e.,
• using binary factorization of j and k, we have

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Algebraic Form of FFT

• We first compute the inner summation for each
  value of j
• Writing the digits so that j is in natural order,
  we have k k22k1 and jj12j2 the first stage
  produces the values of s(j12k2)

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Algebraic Form of FFT

• We now compute the outer summation

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MATLAB Function for FFT with n 4
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Example FFT with Four Points
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contents

• Fourier Approximation and Interpolation
• Fast Fourier Transforms for n2r
• Fast Fourier Transforms for General n

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FFT for General n

• The general FFT does not require the
  factorization of n
• example n6, r1 2 and r2 3

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FFT for General n

• Using preceding factorization of j and k, we have

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Example - FFT for Six Data Points

• z 0 1 2 3 2 1
• compute the inner sum for each pair of values of
  j1 and k2

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Example - FFT for Six Data Points

• compute the outer sum

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MATLAB Function for FFT with nrs