Periodograms and spectral estimation

Take n real observations indexed from zero through n minus one, and subtract a specified mean to obtain z. Our transform divides the Fourier sum by the square root of two pi n; its squared magnitude is the angular periodogram. At Fourier frequencies omega-j equals two pi j divided by n, Parseval's identity says that the sum of periodogram ordinates times the frequency spacing equals the average squared centered observation. Frequencies beyond pi can be represented as negative frequencies modulo two pi.

A real observation vector has conjugate Fourier coefficients at opposite frequencies, so their periodogram ordinates are identical rather than independent pieces of evidence. Subtracting the sample mean changes only the zero-frequency Fourier coefficient on this exact grid. It forces that coefficient to zero, but does not establish that the population spectrum vanishes near zero. At frequencies between Fourier bins, mean removal and the finite observation window can influence the transform more broadly.

Consider iid centered Gaussian white noise with variance sigma-squared, with its known zero mean used in the transform. At any interior positive Fourier bin, excluding zero and the Nyquist frequency when present, the periodogram divided by the angular density has exactly an exponential distribution of mean one. Distinct such nonconjugate bins are independent. Zero and Nyquist coefficients are real and instead yield a one-degree-of-freedom squared-normal distribution, so the interior formula must not be applied to those endpoints.

For an interior bin the variance of the raw ordinate equals the square of its mean, regardless of sample size. Its relative standard deviation is one. Averaging m distinct interior bins of this white-noise model leaves the mean unchanged and divides the variance by m. This is an exact finite-sample calculation for this Gaussian model, not a universal assertion that neighboring periodogram ordinates of an arbitrary dependent series are independent or have identical expectations.

For a colored spectrum, a local average combines frequencies whose population densities may differ. Enlarging the averaging window reduces variance but can blur narrow peaks and introduce bias. A common asymptotic design increases the number of averaged bins while shrinking their bandwidth: m-n tends to infinity and m-n divided by n tends to zero. Consistency then additionally requires suitable smoothness, dependence and moment assumptions; the bandwidth rule alone does not prove it for every stationary process.

A finite observation window also spreads frequency energy through its spectral window. Tapering changes that window, often reducing sidelobe leakage while widening the central lobe. This leakage concerns finite-window frequency resolution, a different issue from using future data in forecast validation. Parametric spectral estimation offers another tradeoff: a fitted ARMA density can be smooth and interpretable, but a badly chosen model can hide genuine structure. Inspect normalization, smoothing scale and model assumptions before assigning meaning to a striking peak.