Formulas: Special considerations for materiel categories/modes

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

Last Updated 29 Oct 2026

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Formula 1

\begin{aligned} X(f, T) & =\int_{0}^{1} x(t) e^{-j 2 \pi f t} d t \\ -\infty<f & <\infty\end{aligned}

Formula 2

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

Formula 3

The finite Fourier transform always exists for well defined $x(t), 0 \leq t \leq T<\infty$ .

x(t), 0 \leq t \leq T<\infty

Formula 4

\mathrm{p}(\mathrm{x})=\left(\sigma_{\mathrm{x}} \sqrt{2 \pi}\right)^{-1} \exp \left[-\frac{\left(\mathrm{x}-\mu_{\mathrm{x}}\right)^{2}}{2 \sigma_{\mathrm{x}}^{2}}\right] \quad-\infty<\mathrm{x}<\infty

Formula 5

defined for - $\infty < f < \infty$ , the inverse Fourier transform of X(f) is defined as a real valued function of time,

\infty < f < \infty

Formula 6

System superposition: $y_{1}+y_{2}=h\left(x_{1}\right)+h\left(x_{2}\right)=h\left(x_{1}+x_{2}\right)$

y_{1}+y_{2}=h\left(x_{1}\right)+h\left(x_{2}\right)=h\left(x_{1}+x_{2}\right)

Formula 7

\widehat{\mathbf{m}}(t)=\frac{1}{N} \sum_{i=1}^{N} \mathbf{x}_{i}(t) \quad 0 \leq t \leq T

Formula 8

For an ensemble {xi(t): $0 \leq t \leq T < \infty, i=1,2, N$ } of N time history

0 \leq t \leq T < \infty, i=1,2, N

Formula 9

records with a mean, $\mu(t)$ , and a standard deviation, $\sigma(t)$ , $0 \leq t \leq T$ , where the mean $\mu(t)$ is estimated by

0 \leq t \leq T

Formula 10

varying envelope that begins at zero and ends at zero over a certain period of time, $T < \infty$ .

T < \infty

Formula 11

f_0 < f

Formula 12

for some $f_0$ and $f_0 < f$ where $f_0 \approx \frac{1}{T}$ .

f_0 \approx \frac{1}{T}

Formula 13

For an ensemble ${x_i(t): 0 \leq t \leq T < \infty, i = 1,2, N}$ of N time history records with

{x_i(t): 0 \leq t \leq T < \infty, i = 1,2, N}

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DEFINITIONS