Periodogram

The periodogram is an estimate of the spectral density of a time series, obtained by computing the squared magnitude of its discrete Fourier transform. For a real-valued sequence x0, x1, ..., xN−1, the two-sided periodogram at frequency fk = k/N is P(fk) = (1/N) |∑n=0 to N−1 xn e−i 2π fk n|^2. Since real signals have symmetric spectra, a one-sided periodogram is often used for frequencies in [0, fs/2], where fs is the sampling rate. The periodogram is simple and nonparametric, but it is biased and its variance does not decrease with more data unless averaging is applied.

To improve performance, several variants are employed. The Welch method divides the data into overlapping blocks, applies a window to each block, computes a block periodogram, and averages the results to reduce variance and leakage. The multitaper method uses multiple orthogonal tapers to obtain several independent estimates and combines them to further reduce variance and spectral leakage. In practice, window choice and normalization affect leakage and resolution.

For unevenly sampled data, the Lomb-Scargle periodogram generalizes the concept and provides a statistically robust estimate of the power spectrum, widely used in astronomy and other fields where regular sampling is difficult.

Interpretation centers on identifying periodic components: peaks in the periodogram suggest dominant frequencies in the signal. Caution is required due to spectral leakage, the impact of sampling rate and windowing, and the presence of noise. The periodogram remains a foundational tool in spectral analysis with applications across physics, engineering, geoscience, climate science, and neuroscience.