Fast Fourier Transform Implementation in Chaotic Time Series Algorithm

This program implements Fast Fourier Transform (FFT) algorithms specifically designed for chaotic time series analysis, focusing on computing average period and power spectrum. The algorithm utilizes efficient FFT computation techniques (typically using radix-2 or split-radix methods) to transform chaotic time domain data into frequency domain representations. Through average period calculation implemented via autocorrelation analysis and peak detection algorithms, the system identifies repeating patterns and periodic characteristics within chaotic sequences. The power spectrum computation employs periodogram or Welch's method implementations, revealing energy distribution across different frequency components using magnitude-squared FFT outputs. These analytical results, achieved through optimized FFT libraries (such as FFTW or custom Cooley-Tukey implementations), provide crucial insights into chaotic system behaviors and enable frequency-domain characterization for trend prediction applications. The implementation includes proper windowing functions (Hamming/Hanning) and zero-padding techniques to enhance spectral resolution while maintaining computational efficiency through optimized butterfly operations and memory allocation strategies.