MixedRadixFFTAlgorithmen

MixedRadixFFTA refers to a Fast Fourier Transform (FFT) algorithm that can handle input sequences whose lengths are composite numbers, meaning they can be factored into smaller integers. Traditional FFT algorithms, such as the Cooley-Tukey algorithm, are often most efficient when the input length is a power of two. Mixed-radix FFT algorithms generalize this by breaking down a composite-length DFT (Discrete Fourier Transform) into smaller DFTs of varying sizes, corresponding to the factors of the composite length.

The core idea behind mixed-radix FFT is to apply the principle of divide and conquer to the DFT definition. If the length N of the DFT can be expressed as a product of factors, say N = N1 * N2, then the N-point DFT can be reformulated as a sequence of N2 N1-point DFTs and N1 N2-point DFTs. The "mixed-radix" aspect arises when these factors N1 and N2 are not necessarily powers of the same base, leading to DFTs of different sizes.

The computational efficiency of a mixed-radix FFT algorithm depends on the choice of factorizations and the efficiency of the smaller DFTs. While it can be more complex to implement than a simple power-of-two FFT, it offers flexibility for signal processing applications where data lengths are not powers of two. This can be advantageous in scenarios where data acquisition or processing naturally results in composite lengths, avoiding the need for padding or truncation that might otherwise be required to use a power-of-two FFT. The algorithm aims to minimize the total number of complex multiplications and additions required.