Cooley-Tukey algorithm (Butterfly algorithm)

One important fact that the Cooley-Tukey algorithm differs from other FFT algorithms is that the N factor can be randomly selected. In this way, n = r can be used. S The radix-r algorithm. The most popular algorithms are based on R = 2 or R = 4, and the simplest DFT can be implemented without any multiplication. For example, in S-level and r = 2, the following index ing result is:

Generally, the two-point DFT in the signal flowchart is drawn in the form of a butterfly chart. Figure 1 shows an 8-point conversion diagram. The signal flow chart has been simplified to the form of addition represented by arrows pointing to all nodes, while the constant coefficient multiplication is to add a factor representation to the arrow. The radix-r algorithm has logR(N) level, and each group has the same type of rotation factor.

Figure 1 Radix-2 Frequency extraction algorithm with a length of 8

From the signal flowchart, we can see that the computing can be completed "on-site", that is, the storage location used by the butterfly can be overwritten, because the data is no longer needed in the next calculation. The total multiplication of the rotation factor of the radix-2 transformation is:

Each two arrows has only one rotation factor.

Because the algorithm in Figure 1 first divides the original DFT into shorter DFT in the frequency domain, such an algorithm is called the decimation-in-frequency (DIF) algorithm. The typical input values appear in order, while the frequency value index is in reverse order. The table provides the feature values of the DIF Radix-2 algorithm.

Radix-2 FFT for table frequency Extraction

We can also construct an algorithm using time extraction (decimation h time, dit. In this case, the input sequence is separated first, and all the frequency values are displayed in order.

Figure 2 shows the necessary index transformation for the radix-2 and Radix-4 algorithms of index 41. The radix-2 algorithm requires the reverse of the bit order, that is, the reverse order of the bit. Radix-4 needs to first construct a two-digit "Number" and then reverse these numbers. Such an operation is called the reverse order of numbers.

Figure 2 reverse order of digits