This design shows the data flow of a mathematical operation called a 32-point complex Fast Fourier Transform (FFT). FFT operations are often used to analyze sounds and radio signals. The data flows from the 32 inputs at the top to the 32 outputs at the bottom. Sometimes the data divides, when it is needed twice. Where the data merges, it is added or (where a minus sign crosses one data path) subtracted. Each data path carries a complex number (or vector). The circles indicate where a vector is multiplied by a constant that rotates the vector, which is where most of the computational work occurs. The open circles indicate 90-degree rotations, where the multiplications can be done with a few additions (less work). At the top of the data flow diagram, the order of the inputs is re-arranged by what is called a bit-reversal permutation. The inputs are numbered 0 through 31 (in binary numbers, 00000 through 11111). We will explain the "bit-reversal permutation" with one example: Input 5 (binary 00101) moves over to position 20 (binary 10100) because 10100 is 00101 written in reverse (hence, "bit-reversal"). After the bit-reversal permutation, data pairs are processed by "butterfly" operations, which are those X-shaped patterns. There are five layers of butterflies, with 16 butterflies in each layer, for a 32-point FFT. One half of the first four layers is a 16-point FFT, and one quarter of the first three layers is an 8-point FFT, etc.