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Radix 2 FFT

When $ N$ is a power of $ 2$, say $ N=2^K$ where $ K>1$ is an integer, then the above DIT decomposition can be performed $ K-1$ times, until each DFT is length $ 2$. A length $ 2$ 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.3On 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 $ N$ DFT from the $ N/2$ length-$ 2$ 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 $ N$ (complex) multiplies needed for each stage of the DIT decomposition, and only $ \lg N$ stages of DIT (where $ \lg N$ denotes the log-base-2 of $ N$), we see that the total number of multiplies for a length $ N$ DFT is reduced from $ {\cal O}(N^2)$ to $ {\cal O}(N\lg N)$, where $ {\cal O}(x)$ means ``on the order of $ x$''. More precisely, a complexity of $ {\cal O}(N\lg N)$ means that given any implementation of a length-$ N$ radix-2 FFT, there exist a constant $ C$ and integer $ M$ such that the computational complexity $ {\cal C}(N)$ satisfies

$\displaystyle {\cal C}(N) \leq C N \lg N $

for all $ N>M$. In summary, the complexity of the radix-2 FFT is said to be ``N log N'', or $ {\cal O}(N\lg N)$.
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