Comparison of Power Spectrum Estimation using FFT Method and Wiener-Khinchin Theorem via Autocorrelation Function

We can perform a comparative analysis of power spectrum estimation using both FFT-based methods and the Wiener-Khinchin theorem approach through autocorrelation functions. The FFT method implementation typically involves applying a windowing function to the time-domain signal, followed by zero-padding if necessary, and then computing the squared magnitude of the FFT coefficients to obtain the power spectral density. This approach directly transforms the signal from time domain to frequency domain, revealing power distribution across different frequency components through algorithms like Welch's method or periodogram estimation. The Wiener-Khinchin theorem approach utilizes autocorrelation functions where we first compute the autocorrelation sequence of the signal using either time-domain convolution or frequency-domain multiplication (via FFT for efficiency). The theorem establishes that the power spectrum equals the Fourier transform of the autocorrelation function. Implementation typically involves calculating the autocorrelation using functions like xcorr() in MATLAB or numpy.correlate() in Python, followed by FFT application to obtain the power spectral density. By comparing these two methodologies, we can conduct in-depth research on power spectrum characteristics. The FFT method offers computational efficiency and direct frequency analysis, while the autocorrelation approach provides better statistical properties and reduced variance in spectrum estimation. This comparative study enables more accurate conclusions about signal characteristics, stationarity assumptions, and optimal method selection based on specific application requirements.