periodogrammes

A periodogram is a nonparametric estimator used in spectral analysis to estimate the distribution of power of a signal across frequencies. It is derived from time-domain data by computing a Fourier transform of the signal and squaring the magnitude. In many texts, the periodogram is described as an estimate of the power spectral density (PSD) of the underlying process.

For a finite sequence x[n], n = 0,...,N-1, the classical periodogram at frequency f is Pxx(f) = (1/N) |X(f)|^2, where X(f) is the discrete-time Fourier transform of x[n]. Often, frequencies are taken at the DFT bins, so Pxx[k] = (1/N) |X[k]|^2 with X[k] = sum_{n=0}^{N-1} x[n] e^{-i 2 pi k n / N}. The periodogram provides a straightforward, data-driven view of how signal power is distributed over frequency but is known to be biased and inconsistent as an estimator of the true PSD for finite data. It can exhibit spectral leakage due to the finite observation window and can have high variance.

Numerous refinements address these issues. Windowing (for example, Hanning, Hamming, Blackman) reduces leakage, and smoothing or averaging across segments (Welch’s method) lowers variance. The multitaper method uses multiple orthogonal tapers to improve variance and bias properties. For unevenly sampled data, the Lomb–Scargle periodogram adapts the approach to fit sinusoids to the observations, a common tool in astronomy.

Periodograms are widely used in engineering, geophysics, astronomy, and neuroscience to detect periodic components, study stationarity, and characterize noise, with careful interpretation of bias, variance, and statistical significance.