The radix-2 decimation-in-frequency algorithm rearranges the discrete Fourier transform (DFT) equation into two parts: computation of the even-numbered discrete-frequency indices Xk X k for k=024…N-2 k 0 2 4 … N 2 (or X2r X 2 r as in Equation 2) and computation of the odd-numbered indices k=135…N-1 k 1 3 5 … N 1 (or X2r+1 X 2 r 1 as in Equation 3)

The mathematical simplifications in Equation 2 and Equation 3 reveal that both the even-indexed and odd-indexed frequency outputs Xk X k can each be computed by a length- N2 N 2 DFT. The inputs to these DFTs are sums or differences of the first and second halves of the input signal, respectively, where the input to the short DFT producing the odd-indexed frequencies is multiplied by a so-called twiddle factor term W N k =ⅇ-ⅈ2πkN W N k 2 k N . This is called a decimation in frequency because the frequency samples are computed separately in alternating groups, and a radix-2 algorithm because there are two groups. Figure 1 graphically illustrates this form of the DFT computation. This conversion of the full DFT into a series of shorter DFTs with a simple preprocessing step gives the decimation-in-frequency FFT its computational savings.

Whereas direct computation of all NN DFT frequencies according to the DFT equation would require N2 N 2 complex multiplies and N2-N N 2 N complex additions (for complex-valued data), by breaking the computation into two short-length DFTs with some preliminary combining of the data, as illustrated in Figure 1, the computational cost is now

New Operation Counts
• 2N22+N=N22+N2 2 N 2 2 N N 2 2 N 2 complex multiplies
• 2N2N2-1+N=N22 2 N 2 N 2 1 N N 2 2 complex additions

This simple manipulation has reduced the total computational cost of the DFT by almost a factor of two!

The initial combining operations for both short-length DFTs involve parallel groups of two time samples, xn x n and xn+N2 x n N 2 . One of these so-called butterfly operations is illustrated in Figure 2. There are N2 N 2 butterflies per stage, each requiring a complex addition and subtraction followed by one twiddle-factor multiplication by W N n =ⅇ-ⅈ2πnN W N n 2 n N on the lower output branch. It is worthwhile to note that the initial add/subtract part of the DIF butterfly is actually a length-2 DFT! The theory of multi-dimensional index maps shows that this must be the case, and that FFTs of any factorable length may consist of successive stages of shorter-length FFTs with twiddle-factor multiplications in between. It is also worth noting that this butterfly differs from the decimation-in-time radix-2 butterfly in that the twiddle factor multiplication occurs after the combining.

The same radix-2 decimation in frequency can be applied recursively to the two length- N2 N 2 DFTs to save additional computation. When successively applied until the shorter and shorter DFTs reach length-2, the result is the radix-2 decimation-in-frequency FFT algorithm.

The full radix-2 decimation-in-frequency decomposition illustrated in Figure 3 requires M=log2N M 2 N stages, each with N2 N 2 butterflies per stage. Each butterfly requires 11 complex multiply and 22 adds per butterfly. The total cost of the algorithm is thus

Computational cost of radix-2 DIF FFT
• N2log2N N 2 2 N complex multiplies
• Nlog2N N 2 N complex adds

This is a remarkable savings over direct computation of the DFT. For example, a length-1024 DFT would require 10485761048576 complex multiplications and 10475521047552 complex additions with direct computation, but only 51205120 complex multiplications and 1024010240 complex additions using the radix-2 FFT, a savings by a factor of 100 or more. The relative savings increase with longer FFT lengths, and are less for shorter lengths. Modest additional reductions in computation can be achieved by noting that certain twiddle factors, namely W N 0 W N 0 , W N N 2 W N N 2 , W N N 4 W N N 4 , W N N 8 W N N 8 , W N 3 N 8 W N 3 N 8 , require no multiplications, or fewer real multiplies than other ones. By implementing special butterflies for these twiddle factors as discussed in FFT algorithm and programming tricks, the computational cost of the radix-2 decimation-in-frequency FFT can be reduced to

• 2Nlog2N-7N+12 2 N 2 N 7 N 12 real multiplies
• 3Nlog2N-3N+4 3 N 2 N 3 N 4 real additions

The decimation-in-frequency FFT is a flow-graph reversal of the decimation-in-time FFT: it has the same twiddle factors (in reverse pattern) and the same operation counts.

It is important to note that, if the input data are in order before beginning the FFT computations, the outputs of each butterfly throughout the computation can be placed in the same memory locations from which the inputs were fetched, resulting in an in-place algorithm that requires no extra memory to perform the FFT. Most FFT implementations are in-place, and overwrite the input data with the intermediate values and finally the output.