Correlation of neighboring periodogram powers ($\mathrm{Cov}(|Z_k{|}^2,|Z_l{|}^2)=|C_{kl}{|}^2+|P_{kl}{|}^2$)

Correlation of neighboring periodogram powers describes dependence between squared spectral magnitudes estimated from the same random record. Their dependence is not determined by the sign of amplitude covariance. A negative correlation between complex Fourier amplitudes can produce a positive correlation between their powers, changing the precision obtained by averaging nearby spectral values.

Let $Z_k$ be zero-mean jointly Gaussian complex Fourier estimates. Define their complex covariance and nonconjugated second moment byand put $I_k=|Z_k{|}^2$. Applying Isserlis's theorem to the four factors gives three pairings:Subtracting the product of the means therefore yieldsIn particular, Gaussian power covariance is nonnegative even when a complex amplitude covariance is negative. Also $\mathrm{Var}(I_k)=C_{kk}^2+|P_{kk}{|}^2$; ignoring the pseudo-covariance of a complex random variable changes the answer when that term is present.

Consider independent sample noise with independent Gaussian real and imaginary components of equal variance. Its nonconjugated moments vanish, so $P=0$ after any deterministic complex linear transform. If the normalized bin variances are all $V$, thenFor the periodic Hann window with $N\ge 5$ on the original $N$-point Fourier grid, noise covariance between windowed Fourier bins gives amplitude covariance ratios $-2/3$ at separation one and $1/6$ at separation two. The corresponding power correlations are $4/9$ and $1/36$. More distant nonconjugate bins have zero covariance in that particular model.

An original calculation averages $M$ consecutive powers without wrapping around the Fourier grid. Assume $N\ge M+2$ so that the block does not acquire additional cyclic neighbor pairs. Counting the $M-1$ adjacent pairs and $\max (M-2,0)$ second-neighbor pairs givesFor $M=2$, the variance is $13V^2/18$, rather than the independent-value result $V^2/2$. For $M=10$, it is $0.18444V^2$, rather than $0.10V^2$. A variance-equivalent independent-power count is the reciprocal of this normalized variance: about $5.42$ for those ten bins. For $M\ge 2$, the bracket simplifies to $35M/18-1$, approaching roughly $18M/35$ independent powers for large blocks in sufficiently long records.

Real-valued Gaussian input has conjugate Fourier symmetry and can have nonzero $P_{kl}$, especially for mirrored frequencies and endpoint bins. A deterministic spectral component also invalidates the zero-mean calculation unless its additional mean-dependent terms are included. Colored noise, taper convention and detrending change $C$ and $P$; the formulas above do not make all measured spectra follow one correlation pattern.

A periodogram may be expressed as a squared spectrum or a power spectral density. SciPy documents their window-dependent conversion, one-sided folding and optional detrending explicitly.^[ref] A common positive scaling does not change the correlation coefficient, but combining unequal scalings or folding conjugate values requires carrying those transformations through the covariance.

A proposed check generates independent records under the stated noise model, calculates bin powers with the same window, and compares empirical pair covariances and block-average variance with the predictions. Independent records provide a different averaging resource from additional bins or zero-padded frequency points in one record. Report which values were averaged and their covariance before assigning an independent-sample precision gain.