= Correlation of neighboring periodogram powers
{title2=$\operatorname{Cov}(|Z_k|^2,|Z_l|^2)=|C_{kl}|^2+|P_{kl}|^2$}
<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 by
$$
C_{kl}=\mathbb E[Z_k\overline{Z_l}],\qquad P_{kl}=\mathbb E[Z_kZ_l],
$$
and put $I_k=|Z_k|^2$. Applying <Isserlis's theorem> to the four factors gives three pairings:
$$
\mathbb E[Z_k\overline{Z_k}Z_l\overline{Z_l}]
=C_{kk}C_{ll}+|C_{kl}|^2+|P_{kl}|^2.
$$
Subtracting the product of the means therefore yields
$$
\boxed{\operatorname{Cov}(I_k,I_l)=|C_{kl}|^2+|P_{kl}|^2.}
$$
In particular, Gaussian power covariance is nonnegative even when a complex amplitude covariance is negative. Also $\operatorname{Var}(I_k)=C_{kk}^2+|P_{kk}|^2$; 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 $V$, then
$$
\operatorname{Corr}(I_k,I_l)=\frac{|C_{kl}|^2}{V^2}.
$$
For the periodic <Hann window> with $N\ge5$ 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 \b[$4/9$ 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 gives
$$
\operatorname{Var}\left(\frac1M\sum_{k=1}^M I_k\right)
=\frac{V^2}{M^2}\left[M+\frac89(M-1)+\frac1{18}\max(M-2,0)\right].
$$
For $M=2$, 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 $M\ge2$, the bracket simplifies to $35M/18-1$, approaching roughly $18M/35$ independent powers for large blocks in sufficiently long records.
\Image[/hann-periodogram-power-averaging.png]
{title=Correlated periodic <Hann window> powers give less <variance> reduction than independent spectral powers}
{description=Exact finite-block <variance> and <variance>-equivalent independent-power count for Gaussian complex white sample noise on the original N=64 grid. Blocks contain one to 32 consecutive bins, so the plotted calculation has no extra wraparound pairs. Each <variance> is normalized by its model’s single-power <variance>. The lines are analytical predictions, not experimental observations.}
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.<SciPy 1.16.2 documentation>[scipy.signal.periodogram, scaling, return_onesided and detrend parameters; Notes]{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.
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