When the desired DFT length can be expressed as a product of smaller integers, the Cooley-Tukey decomposition provides what is called a mixed radix Cooley-Tukey FFT algorithm.A.2

Two basic varieties of Cooley-Tukey FFT are decimation in time (DIT) and its Fourier dual, decimation in frequency (DIF). The next section illustrates decimation in time.

### Decimation in Time

The DFT is defined by

where is the input signal amplitude at time , and

Note that . When is even, the DFT summation can be split into sums over the odd and even indexes of the input signal:
 (A.1)

where and denote the even- and odd-indexed samples from . Thus, the length DFT is computable using two length DFTs. The complex factors are called twiddle factors. The splitting into sums over even and odd time indexes is called decimation in time. (For decimation in frequency, the inverse DFT of the spectrum is split into sums over even and odd bin numbers .)

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 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

for all . In summary, the complexity of the radix-2 FFT is said to be N log N'', or .

### Fixed-PointFFTs and NFFTs

Recall (e.g., from Eq.(6.1)) that the inverse DFT requires a division by that the forward DFT does not. In fixed-point arithmetic (Appendix G), and when is a power of 2, dividing by may be carried out by right-shifting places in the binary word. Fixed-point implementations of the inverse Fast Fourier Transforms (FFT) (Appendix A) typically right-shift one place after each Butterfly stage. However, superior overall numerical performance may be obtained by right-shifting after every other butterfly stage [8], which corresponds to dividing both the forward and inverse FFT by (i.e., is implemented by half as many right shifts as dividing by ). Thus, in fixed-point, numerical performance can be improved, no extra work is required, and the normalization work (right-shifting) is spread equally between the forward and reverse transform, instead of concentrating all right-shifts in the inverse transform. The NDFT is therefore quite attractive for fixed-point implementations. Because signal amplitude can grow by a factor of 2 from one butterfly stage to the next, an extra guard bit is needed for each pair of subsequent NDFT butterfly stages. Also note that if the DFT length is not a power of , the number of right-shifts in the forward and reverse transform must differ by one (because is odd instead of even).
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