### Radix 2 FFT

When is a power of , say where is an integer,
then the above DIT decomposition can be performed times, until
each DFT is length . A length DFT requires no multiplies. The
overall result is called a *radix 2 FFT*. A different radix 2
FFT is derived by performing decimation in frequency.

A *split radix* FFT is theoretically more efficient than a pure
radix 2 algorithm [73,31] because it
minimizes real arithmetic operations. The term ``split radix'' refers
to a DIT decomposition that combines portions of one radix 2 and two
radix 4 FFTs [22].^{A.3}On modern general-purpose
processors, however, computation time is often not minimized by
minimizing the arithmetic operation count (see §A.7 below).

#### Radix 2 FFT Complexity is N Log N

Putting together the length DFT from the length- DFTs in a radix-2 FFT, the only multiplies needed are those used to combine two small DFTs to make a DFT twice as long, as in Eq.(A.1). Since there are approximately (complex) multiplies needed for each stage of the DIT decomposition, and only stages of DIT (where denotes the log-base-2 of ), we see that the total number of multiplies for a length DFT is reduced from to , where means ``on the order of ''. More precisely, a complexity of means that given any implementation of a length- radix-2 FFT, there exist a constant and integer such that the computational complexity satisfies

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Fixed-Point FFTs and NFFTs

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Decimation in Time