Power Spectral Densities

Consider a signal, $n(t)$, containing Gaussian noise and dimensions $U$, which may be voltage, strain, etc. We define the (one sided) power spectral density, $S(\vert f\vert)$, of this signal by the equation

$$\left\langle\tilde{n}(f) \tilde{n}^\ast(f')\right\rangle = \frac{1}{2}S\left(\left\vert f\right\vert\right)\delta(f-f).$$ (16.16)

The total power in a signal is independent of whether it is computed in the time or the frequency domain (Parseval's theorem). The power in a signal in the interval $(0,T)$ is given by

$$P = \frac{1}{T} \int_{0}^{T} dt \left\vert h(t)\right\vert^2 = \int_{0}^{f_c} df S\left(\left\vert f\right\vert\right).$$ (16.17)

For discretely sampled quantities we have

$$\left\langle\tilde{n}(f_k) \tilde{n}^\ast(f_{k'})\right\rang... ...h{S\left(\left\vert f_{k}\right\vert\right)}\delta(f_k-f_{k'})$$ (16.18)

which gives

$$\left\langle\tilde{n}_k \tilde{n}_{k'}^\ast\right\rangle = ... ...uremath{S\left(\left\vert f_{k}\right\vert\right)}\delta_{kk'}$$ (16.19)

which defines in terms of the discrete frequency domain quantities. Parsevals theorem becomes

$$\Delta t \sum_{j=0}^{N-1} \vert h_j\vert^2 = \sum_{k=0}^{[N/2]} S\left(\left\vert f_k\right\vert\right),$$ (16.20)

the power spectral density having units of $\mathrm{time}\times U^2$. The definition in equation [] is equivalent to that in the standards document T010095^16.1:

$$\ensuremath{S\left(\left\vert f_{k}\right\vert\right)}= \lef... ... \tilde{h}_{N-k} \vert^2 \right] & k\neq 0. \end{array}\right.$$ (16.21)