Autospectrum

Autospectrum, or auto-spectral density, is a concept in signal processing that describes how the power of a signal or random process is distributed across frequency. For a real-valued, second-order stationary process X(t), the autocorrelation function is R_xx(τ) = E[X(t) X(t+τ)]. The autospectrum S_xx(f) is the Fourier transform of R_xx(τ). By the Wiener-Khinchin theorem, S_xx(f) and R_xx(τ) contain the same information about the signal's structure, with S_xx(f) representing the expected power at frequency f.

In discrete-time form, with r_xx[m] = E[x[n] x[n+m]], the autospectrum is given by S_xx(ω) = sum over m of r_xx[m] e^{-j ω m}, or equivalently by applying a frequency-domain estimator to the data. In practice, the autospectrum is often interpreted as the power spectral density (PSD) of the signal.

Estimating the autospectrum from finite data typically involves computing a periodogram, and often using averaging techniques such as Bartlett’s method, Welch’s method, or multitaper approaches to reduce variance and improve reliability. Key considerations include stationarity assumptions, windowing effects, and the presence of noise or trends, which can bias the estimate and obscure true frequency content.

Applications of autospectrum analysis span engineering and science, including communications, vibration analysis, acoustics, geophysics, and biomedical signal processing. It helps identify dominant frequencies, resonances, and the spectral structure of signals, and can be extended to cross-spectral analysis when comparing two signals.