Wiener-Khinchin Theorem and Its Applications The Wiener-Khinchin theorem states that the Fourier transform of a signal's autocorrelation function equals its power spectrum. Autocorrelation measures how a signal correlates with itself over time, while the power spectrum represents the squared magnitude of its frequency components. This theorem is widely used in electronics and communication engineering, particularly in analyzing signals.
Proof of Wiener-Khinchin Theorem Using Fourier Transform To prove this theorem, start by expressing the autocorrelation function as an integral involving two instances of the signal multiplied together. By applying Fubini’s theorem to interchange integration limits and substituting variables for simplification, it becomes evident that one term corresponds to X(-ω), which is related to X(ω) through complex conjugation when dealing with real-valued signals. Ultimately, combining these results shows that |X(ω)|² (the modulus squared) equals the desired power spectrum.