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.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 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 $\lg N$ stages of DIT (where $\lg N$ denotes the log-base-2 of ), we see that the total number of multiplies for a length DFT is reduced from ${\cal O}(N^2)$ to ${\cal O}(N\lg N)$ , where ${\cal O}(x)$ means ``on the order of ''. More precisely, a complexity of ${\cal O}(N\lg N)$ means that given any implementation of a length- radix-2 FFT, there exist a constant and integer such that the computational complexity ${\cal C}(N)$ satisfies