Formulas: SISO Auto and Cross-Spectral Definitions Review

MIL-STD-810H, Environmental Engineering Considerations and Laboratory Tests

Last Updated 29 Oct 2026

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Formula 1

\begin{aligned} X(f) & =X(f, T)=\int_{0}^{T} x(t) e^{-j 2 \pi f t} d t \\ Y(f) & =Y(f, T)=\int_{0}^{T} y(t) e^{-j 2 \pi f t} d t\end{aligned}

Formula 2

G_{x x}(f)=2 \lim _{T arrow \infty} \frac{1}{T} E\left[|X(f, T)|^{2}\right]

Formula 3

Estimates of $G_{x x}(f), G_{y y}(f)$ and $G_{x y}(f)$ as computed over a "finite time" interval are defined as:

\begin{aligned} \tilde{G}_{x x}(f)=S_{x x}(f) & =\frac{2}{T}\left[\left|X(f, T)\right|^{2}\right] \\ \tilde{G}_{y y}(f)=S_{y y}(f) & =\frac{2}{T}\left[\left|Y(f, T)\right|^{2}\right] \\ \tilde{G}_{x y}(f)=S_{x y}(f) & =\frac{2}{T}\left[X^{*}(f) Y(f)\right] \end{aligned}

Formula 4

and will have a discrete spectral resolution of $B_{e} \approx \Delta f=\frac{1}{T}$ .

B_{e} \approx \Delta f=\frac{1}{T}

Formula 5

and will have a discrete spectral resolution of $B_{e} \approx \Delta f=\frac{1}{T}$ . Employment of $S_{x x}(f), S_{y y}(f)$ and $S_{x y}(f)$ will generally be unacceptable due to the large random error associated with the "raw" estimate. In practice, the random error is reduced, (refer to paragraph 6.1, reference d, for a detailed error discussion), by computing an ensemble of $n_{d}$ different averages of length T to obtain a "smooth" estimate defined as:

\begin{aligned} \hat{G}_{x x}(f) & =\frac{2}{n_{d} T} \sum_{i=1}^{n_{d}}\left[\left|X(f, T)\right|^{2}\right] \\ \hat{G}_{y y}(f) & =\frac{2}{n_{d} T} \sum_{i=1}^{n_{d}}\left[\left|Y(f, T)\right|^{2}\right] \\ \hat{G}_{x y}(f) & =\frac{2}{n_{d} T} \sum_{i=1}^{n_{d}}\left[X^{*}(f) Y(f)\right] \end{aligned}