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Fast Fourier Transform

Fast Fourier Transform. Definition. All Periodic Waves Can be Generated by Combining Sin and Cos Waves of Different Frequencies Number of Frequencies may not be finite Fourier Transform Decomposes a Periodic Wave into its Component Frequencies. DFT Definition.

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Fast Fourier Transform

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  1. Fast Fourier Transform

  2. Definition • All Periodic Waves Can be Generated by Combining Sin and Cos Waves of Different Frequencies • Number of Frequencies may not be finite • Fourier Transform Decomposes a Periodic Wave into its Component Frequencies

  3. DFT Definition • Sample consists of n points, wave amplitude at fixed intervals of time:(p0,p1,p2, ..., pn-1) (n is a power of 2) • Result is a set of complex numbers giving frequency amplitudes for sin and cos components • Points are computed by polynomial:P(x)=p0+p1x+p2x2+ ... +pn-1xn-1

  4. DFT Definition, continued • The complete DFT is given byP(1), P(w), P(w2), ... ,P(wn-1) • w Must be a Primitive nth Root of Unity • wn=1, if 0<i<n then wi¹ 1

  5. Primitive Roots of Unity • wi is an nth root of unity (not primitive) • wn/2 = -1 • if 0£j£n/2-1 then w(n/2)+j = -wj • if n is even and w is a primitive nth root of unity, then w2 is a primitive n/2 root of unity • Example: w=cos(2p/n) + isin(2p/n)

  6. Divide and Conquer • Compute an n-point DFT using one or more n/2-point DFTs • Need to find Terms involving w2 in following polynomial • P(w)=p0+p1w+p2w2+p3w3+p4w4+ ... +pn-1wn-1 Here They Are

  7. Even/Odd Separation • P(w)= P1(w)+P2(w) • P1(w)=p0+p2w2+p4w4+ ... +pn-2wn-2 • P1(w)=Pe (w2)=p0+p2w+p4w1+...+pn-2w(n-2)/2 • P2(w)=p1w+p3w3+p5w5+ ... +pn-1wn-1 • P2(w)= w P3(w)=p1+p3w2+... +pn-1wn-2 • P3(w)=Po(w2)= p1+p3w+... +pn-1w(n-2)/2 • P(w)= Pe(w2)+ wPo(w2) • Pe & Po come from n/2 point FFTs

  8. The Algorithm DFFT(P:Array;k,m:Integer):Array; begin If k=0 Then DFFT[0]=P[0];DFFT[1]=P[0]; Else Evens = DFFT(EvenElemOf(P),k-1,2m); Odds = DFFT(OddElemOf(P),k-1,2m); For i := 0 to 2k-1-1 Do x := Odds[j]*wmj DFFT[j] := Evens[j] + x DFFT[2k-1+j] := Evens[j] - x End For End If end

  9. Iterative Algorithm For i := 0 To n-2 By 2 Do T[i] = p[f(i)] + p[f(t+1)]; T[i+1] := p[f(i)] - p[f(t+1)]; End For m := n/2; n := 2; For k := lg n - 2 To 0 By -1 Do m := m/2; n := 2*n; For i := 0 To (2k-1)*n By n Do For j := 0 To (n/2)-1 Do x := wmj * T[i+n/2+j]; T[i+n/2+j] := T[i+j] - x; T[i+j] := T[i+j] + x; End For End For End For

  10. What is f(i)? i f(i) 000000 - 000 000 - 000 000 001 010 -010 100 - 100 100 010 100 - 100 010 - 010 010 011 110 - 110 110- 110 110 100 001 - 001 001 - 001 001 101 011 - 011 101 - 101 101 110 101 - 101 011 - 011 011 111 111 - 111 111 - 111 111

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