Fourier Methods jjinddakorea ac kr lockdown 99korea ac

Fourier Methods 신상민 jjindda@korea. ac. kr 윤상기 lockdown 99@korea. ac. kr 최현상 realchs@korea. ac. kr 1

Fourier Series q q For periodic data, it is more appropriate to use sine and cosine functions for the approximation or interpolation Fourier Series m An investigation into data approximation and interpolation using trigonometric polynomials m The formulas for the coefficients are found by using the appropriate orthogonality results for the sine and cosine function. m Derivation from ‘Euler formulas’ m Korea Univ. wireless data communication laboratory 2

Fourier Series q Example n=1 n=2 Korea Univ. wireless data communication laboratory n = 10000 3

contents 1. 2. 3. Fourier Approximation and Interpolation Fast Fourier Transforms for n=2 r Fast Fourier Transforms for General n Korea Univ. wireless data communication laboratory 4

Fourier Approximation and Interpolation q Discrete Fourier Approximation m Formulas Ø If number of samples period are odd, Ø If number of samples period are even, Derivation of the formulas for the coefficients? ? Korea Univ. wireless data communication laboratory 5

Fourier Approximation and Interpolation q Discrete Fourier Approximation m Korea Univ. wireless data communication laboratory 6

Fourier Approximation and Interpolation q Example 10. 1 Trigonometric Interpolation Korea Univ. wireless data communication laboratory 7

Fourier Approximation and Interpolation q Example 10. 2 Trigonometric Approximation Korea Univ. wireless data communication laboratory 8

Fourier Approximation and Interpolation q 10. 1. 1 Matlab function for Fourier Interpolation or Approximation Korea Univ. wireless data communication laboratory 9

Fourier Approximation and Interpolation q Example 10. 3 A step function z = [ 1 1 0 0] m=4 [a, b]=Trig_poly(z, m) m=4 a = [ 0. 5 0. 25 0] b = [0 0. 6036 0 0. 1036 0] m=3 Korea Univ. wireless data communication laboratory 10

Fourier Approximation and Interpolation q Example 10. 4 Geometric Figures Korea Univ. wireless data communication laboratory 11

Fast Fourier Transform 12

Discrete Fourier Transform q discrete-time Fourier transform m The discrete-time Fourier transform X(ejw) of a sequence x[n] is defined by q discrete Fourier transform m uniformly sampling X(ejω) on the ω-axis between 0 ≤ ω ≤ 2π at ωk=2πk/N, 0 ≤ k ≤ N-1 Korea Univ. wireless data communication laboratory 13

Discrete Fourier Transform q Commonly used notation m We can rewrite DFT equation as q Inverse discrete Fourier transform (IDFT) Korea Univ. wireless data communication laboratory 14

Discrete Fourier Transform q Matrix Relations m The DFT samples can be expressed in matrix form as q q DFT can be computed in O(N 2) operations. FFT can reduce the computational complexity to about O(Nlog 2 N) operations Korea Univ. wireless data communication laboratory 15

contents 1. 2. 3. Fourier Approximation and Interpolation Fast Fourier Transforms for n=2 r Fast Fourier Transforms for General n Korea Univ. wireless data communication laboratory 16

Fast Fourier Transforms for n = 2 r q begin by considering the FFT when n is power of 2, i. e. , n=2 r q Example of n = 4 m Each value of j can be written in binary form as j=2 r-1 jr+… 22 j 3+2 j 2+j 1. j j 2 j 1 0 0 0 1 2 1 0 3 1 1 m We can also write k in binary form, but as k = 2 k 1+k 2 j j 2 j 1 k 2 k 1 k 0 0 0 1 0 1 2 2 1 0 1 3 1 1 3 Korea Univ. wireless data communication laboratory 17

Fast Fourier Transforms for n = 2 r q begin by writing out the linear system of equations for the Fourier transform components for the case n=4: q w 4 = w 0 = 1, and interchanging the order of the second and third equations Korea Univ. wireless data communication laboratory 18

Fast Fourier Transforms for n = 2 r q We now factor the coefficient matrix q Substituting the factored form of the coefficient matrix into DFT eq. Korea Univ. wireless data communication laboratory 19

Fast Fourier Transforms for n = 2 r q First we find the product q Then we form the second product Korea Univ. wireless data communication laboratory 20

Fast Fourier Transforms for n = 2 r q Pathways with powers of w on them indicate that the quantity on the left is multiplied by that amount. z 0 s 0 z 1 s 1 z 2 s 2 w 2 z 3 g 0 g 2 w s 3 Korea Univ. wireless data communication laboratory g 3 w 2 1 st stage g 1 w 2 2 nd stage 21

Algebraic Form of FFT q Example of n = 4 m to calculate the discrete Fourier transform of the data zk, i. e. , m using binary factorization of j and k, we have Korea Univ. wireless data communication laboratory 22

Algebraic Form of FFT q q We first compute the inner summation for each value of j Writing the digits so that j is in natural order, we have k = k 2+2 k 1 and j=j 1+2 j 2; the first stage produces the values of s(j 1+2 k 2) z(k 2+2 k 1) k 1 k 2 j 1 j 2 s(j 1+2 k 2) z 0 0 0 z 0 w(0)(2)(0)+z 2 w(0)(2)(1) =s 0 z 3 1 0 z 0 w(1)(2)(0)+z 2 w(1)(2)(1) =s 1 z 1 0 1 z 0 w(0)(2)(0)+z 2 w(0)(2)(1) =s 2 z 4 1 1 z 0 w(1)(2)(0)+z 2 w(1)(2)(1) =s 3 Korea Univ. wireless data communication laboratory 23

Algebraic Form of FFT q We now compute the outer summation s 0 k 1 k 2 j 1 j 2 g s 0 0 0 s 0 w(0+2(0))0+s 2 w(0+2(0))1=s 0 w 0+s 2 w 0 s 1 1 0 s 1 w(0+2(0))0+s 3 w(0+2(0))1=s 1 w 0+s 3 w 1 s 2 0 1 s 0 w(0+2(1))0+s 2 w(0+2(1))1=s 0 w 0+s 2 w 2 s 3 1 1 s 1 w(1+2(1))0+s 3 w(1+2(1))1=s 1 w 0+s 3 w 3 Korea Univ. wireless data communication laboratory 24

MATLAB Function for FFT with n = 4 Korea Univ. wireless data communication laboratory 25

Example – FFT with Four Points Korea Univ. wireless data communication laboratory 26

contents 1. 2. 3. Fourier Approximation and Interpolation Fast Fourier Transforms for n=2 r Fast Fourier Transforms for General n Korea Univ. wireless data communication laboratory 27

FFT for General n q q The general FFT does not require the factorization of n example n=6, r 1 = 2 and r 2 = 3 j j 2 j 1 k k 1 k 2 0 0 0 1 0 1 2 1 0 2 3 1 1 3 1 0 4 2 0 4 1 1 5 2 1 5 1 2 j=2 j 2+j 1 j 2 j 1 k 2 K=3 k 1+k 2 0 0 0 1 0 1 1 0 3 2 1 0 0 1 1 3 1 1 4 4 2 0 0 2 2 5 2 1 1 2 5 Korea Univ. wireless data communication laboratory 28

FFT for General n q Using preceding factorization of j and k, we have Korea Univ. wireless data communication laboratory 29

Example - FFT for Six Data Points q z=[012321] m compute the inner sum for each pair of values of j 1 and k 2: zc k j k 1 k 2 j 1 j 2 s(2 k 2+j 1) 0 0 0 0 w 0+3 w 0=3=s 0 3 3 1 1 0 0 w 0+3 w 3=-3=s 1 1 1 2 0 1 1 w 0+2 w 0=3=s 2 2 4 3 1 1 1 w 0+2 w 3=-1=s 3 2 2 4 0 2 2 w 0+1 w 0=3=s 4 1 5 5 1 2 2 w 0+1 w 3=1=s 5 Korea Univ. wireless data communication laboratory 30

Example - FFT for Six Data Points q compute the outer sum J k 1 k 2 j 1 j 2 g(2 j 2+j 1) g 0 0 0 s 0 w 0+s 2 w 0+s 4 w 0 g 0=9 1 1 0 s 1 w 0+s 3 w 1+s 5 w 2 2 0 1 s 0 w 0+s 2 w 2+s 4 w 4 3 1 1 s 1 w 0+s 3 w 3+s 5 w 6 4 0 2 s 0 w 0+s 2 w 4+s 4 w 8 5 1 2 s 1 w 0+s 3 w 5+s 5 w 10 Korea Univ. wireless data communication laboratory 31

MATLAB Function for FFT with n=rs Korea Univ. wireless data communication laboratory 32