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8.2: Basic Cooley-Tukey FFT

  • Page ID
    2008
  • The Cooley-Tukey FFT always uses the Type 2 index map from Multidimensional Index Mapping. This is necessary for the most popular forms that have \(N=R^M\), but is also used even when the factors are relatively prime and a Type 1 map could be used. The time and frequency maps from Multidimensional Index Mapping are

    \[n=((K_1n_1+K_2n_2))_N\]

    \[k=((K_3k_1+K_4k_2))_N\]

    Type-2 conditions in the 2.2: The Index Map become

    \[K_1=aN_2\; \; or\; \; K_2=bN_1\; \; but\; not\; both\]

    and

    \[K_3=cN_2\; \; or\; \; K_4=dN_1\; \; but\; not\; both\]

    The row and column calculations in 2.2: The Index Map are uncoupled by Type-two index map which for this case are

    \[((K_1K_4))_N=0\; \; or\; \; ((K_2K_3))_N=0\; \; but\; not\; both\]

    To make each short sum a DFT, the KiKi" role="presentation" style="position:relative;" tabindex="0">

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