Appendix 1: Variance
The variance of the measured PS is defined as
$$ \text{var} \left\{ {S_{0} \left( f \right)} \right\} = \left\langle {\hat{S}_{0} \left( f \right)^{2} } \right\rangle - \left\langle {\hat{S}_{0} \left( f \right)} \right\rangle^{2} \equiv I - II, $$
(46)
where subscripts 0 stands for “measured with a record length T
g
” and the circumflex (hat) indicates estimate from a single record. In the following, we distinguish between a time average parameter T used to compute the true spectrum of the continuous velocity and the actual record length T
g
determined by the measured samples, see Fig. 13.
Let us look first at the second term above, II.
1.1 Second term (square of the mean)
We understand \( \hat{S}_{0} \left( f \right) \) as an estimate of the PS based on a finite record of length T
g
, see Fig. 13. We then get, using the direct method and the expression for the noise function PS derived above:
$$ \hat{S}_{0} \left( f \right) = \mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\tilde{u}_{0}^{{{\prime } }} \left( f \right)^{ * } \tilde{u}_{0}^{{{\prime } }} \left( f \right) $$
(47)
$$ \quad \quad \quad \;\; = \mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\int\limits_{ - T/2}^{T/2} {{\text{d}}t\int\limits_{ - T/2}^{T/2} {{\text{d}}t^{\prime}{\text{e}}^{{ - i2\pi f{\kern 1pt} \left( {t^{\prime } - t} \right)}} u^{\prime } \left( t \right)u^{\prime } \left( {t^{\prime } } \right)} } \otimes \frac{1}{{\nu \bar{N}}}\sum\limits_{{k,k^{\prime } }}^{N} {{\text{e}}^{{ - i2\pi f\tau_{{kk^{\prime } }} }} } $$
(48)
In the limit of T → ∞, the first term is the true spectrum. The convolution with the second term imposes the finite record on the result and generates the discrete version of the estimator:
$$ \hat{S}_{0} \left( f \right) = \frac{1}{{\nu \bar{N}}}\sum\limits_{{k,k^{\prime } }}^{N} {{\text{e}}^{{ - i2\pi f{\kern 1pt} \left( {t_{{k^{\prime } }} - t_{k} } \right)}} u^{\prime } \left( {t_{k} } \right)u^{\prime } \left( {t_{{k^{\prime } }} } \right)} $$
(49)
The ensemble mean of the spectral estimate is then
$$ S_{0} \left( f \right) = S_{{u^{\prime } }} \left( f \right) \otimes \frac{1}{{\nu \bar{N}}}\left\langle {\sum\limits_{{k,k^{\prime } }}^{N} {{\text{e}}^{{ - i2\pi f\tau_{{kk^{\prime } }} }} } } \right\rangle \equiv S_{{u^{\prime } }} \left( f \right) \otimes S_{g} \left( f \right) $$
(50)
where \( \tau_{{kk^{\prime } }} \equiv t_{k} - t_{{k^{\prime } }} \) are the measured delays between the record and a delayed copy.
Then the second term above is
$$ II = \left[ {S_{{u^{\prime}}} \left( f \right) \otimes S_{g} \left( f \right)} \right]^{2} $$
(51)
1.2 First term (mean square)
We understand \( \left\langle {\hat{S}_{0} \left( f \right)^{2} } \right\rangle \) as the ensemble average of the product of two copies of the same measured PS. By taking the Fourier transform of four copies of the same velocity record sampled at t, t′, t″, t′′′ spanning the same time interval {−T/2, T/2} where we let T go to infinity, we obtain:
$$ \begin{aligned} \left\langle {\hat{S}_{0} \left( f \right)^{2} } \right\rangle & = \mathop {\lim }\limits_{T \to \infty } \frac{1}{{T^{2} }}\left\langle {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\left[ {{\text{d}}t{\text{d}}t^{\prime } {\text{d}}t^{\prime \prime } {\text{d}}t^{\prime \prime \prime } \;{\text{e}}^{{ - i2\pi f\left[ {\left( {t^{\prime } - t} \right) + \left( {t^{\prime \prime \prime } - t^{\prime \prime } } \right)} \right]}} } \right.} } } } } \right. \\ & \left. {\left. { \quad \cdot \left[ {u^{\prime } \left( t \right)\;\frac{1}{\nu }\sum\limits_{k}^{N} {\delta \left( {t - t_{k} } \right)} \;u^{\prime } \left( {t^{\prime } } \right)\;\frac{1}{\nu }\sum\limits_{k^{\prime }}^{N} {\delta \left({t^{\prime} - t_{{k^{\prime } }} } \right)} \;u^{\prime } \left( {t^{\prime \prime } } \right)\;\frac{1}{\nu }\sum\limits_{{k^{\prime \prime } }}^{N} {\delta \left( {t^{\prime \prime } - t_{{k^{\prime \prime } }} } \right)} \;u^{\prime } \left( {t^{\prime \prime \prime } } \right)\;\frac{1}{\nu }\sum\limits_{{k^{\prime \prime \prime } }}^{N} {\delta \left( {t^{\prime \prime \prime } - t_{{k^{\prime \prime \prime } }} } \right)} \;} \right]} \right]} \right\rangle, \\ \end{aligned} $$
(52)
or rearranging and using independence between velocity and sampling,
$$ \begin{aligned} \left\langle {\hat{S}_{0} \left( f \right)^{2} } \right\rangle & = \mathop {\lim }\limits_{T \to \infty } \frac{1}{{T^{2} }}\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\left[ {{\text{d}}t{\text{d}}t^{\prime } {\text{d}}t^{\prime \prime } {\text{d}}t^{\prime \prime \prime } \;{\text{e}}^{{ - i2\pi f\left[ {\left( {t^{\prime } - t} \right) + \left( {t^{\prime \prime \prime } - t^{\prime \prime } } \right)} \right]}} \left\langle {u^{\prime } \left( t \right)u^{\prime } \left( {t^{\prime } } \right)u^{\prime } \left( {t^{\prime \prime } } \right)u^{\prime}\left( {t^{\prime \prime \prime } } \right)} \right\rangle } \right.} } } } \\ & \left. { \quad \cdot \frac{1}{{\nu^{4} }}\left\langle {\sum\limits_{{k,k^{\prime } ,k^{\prime \prime } ,k^{\prime \prime \prime } }}^{N} {\delta \left( {t - t_{k} } \right)\delta \left( {t^{\prime } - t_{{k^{\prime } }} } \right)\delta \left( {t^{\prime \prime } - t_{{k^{\prime \prime } }} } \right)\delta \left( {t^{\prime \prime \prime } - t_{{k^{\prime \prime \prime } }} } \right)} } \right\rangle } \right] \\ \end{aligned} $$
(53)
This can be expressed as a convolution of the velocity part and the sampling function part:
$$ \begin{aligned} \left\langle {\hat{S}_{0} \left( f \right)^{2} } \right\rangle & = \mathop {\lim }\limits_{T \to \infty } \frac{1}{{T^{2} }}\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {{\text{d}}t{\text{d}}t^{\prime } {\text{d}}t^{\prime \prime } {\text{d}}t^{\prime \prime \prime } \;{\text{e}}^{{ - i2\pi f\left[ {\left( {t^{\prime } - t} \right) + \left( {t^{\prime \prime \prime } - t^{\prime \prime } } \right)} \right]}} \left\langle {u^{\prime } \left( t \right)u^{\prime } \left( {t^{\prime } } \right)u^{\prime } \left( {t^{\prime \prime } } \right)u^{\prime } \left( {t^{\prime \prime \prime } } \right)} \right\rangle } } } } \\ & \quad \otimes \mathop {\lim }\limits_{T \to \infty } \frac{1}{{\nu^{2} \bar{N}^{2} }}\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {{\text{d}}t{\text{d}}t^{\prime } {\text{d}}t^{\prime \prime } {\text{d}}t^{\prime \prime \prime } \;{\text{e}}^{{ - i2\pi f\left[ {\left( {t^{\prime } - t} \right) + \left( {t^{\prime \prime \prime } - t^{\prime \prime } } \right)} \right]}} } } \;} } \left\langle {\sum\limits_{{k,k^{\prime } ,k^{\prime \prime } ,k^{\prime \prime \prime } }}^{N} {\delta \left( {t - t_{k} } \right)\delta \left( {t^{\prime } - t_{{k^{\prime } }} } \right)\delta \left( {t^{\prime \prime } - t_{{k^{\prime \prime } }} } \right)\delta \left( {t^{\prime \prime \prime } - t_{{k^{\prime \prime \prime } }} } \right)} } \right\rangle \\ \end{aligned} $$
(54)
1.3 Velocity part
To proceed with the velocity part, we assume fourth order Gaussian statistics:
$$ \begin{gathered} \mathop {\lim }\limits_{T \to \infty } \frac{1}{{T^{2} }}\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {{\text{d}}t\,{\text{d}}t^{\prime } {\text{d}}t^{\prime \prime } {\text{d}}t^{\prime \prime \prime } \,{\text{e}}^{{ - i2\pi f\left[ {\left( {t^{\prime } - t} \right) + \left( {t^{\prime \prime \prime } - t^{\prime \prime } } \right)} \right]}} } } } } \hfill \\ \quad \quad \quad \quad \quad \cdot \left[ {\left\langle {u^{\prime } \left( t \right)u^{\prime } \left( {t^{\prime } } \right)} \right\rangle \cdot \left\langle {u^{\prime } \left( {t^{\prime \prime } } \right)u^{\prime } \left( {t^{\prime \prime \prime } } \right)} \right\rangle + \left\langle {u^{\prime } \left( t \right)u^{\prime } \left( {t^{\prime \prime } } \right)} \right\rangle \cdot \left\langle {u^{\prime } \left( {t^{\prime } } \right)u^{\prime } \left( {t^{\prime \prime \prime } } \right)} \right\rangle + \left\langle {u^{\prime } \left( t \right)u^{\prime } \left( {t^{\prime \prime \prime } } \right)} \right\rangle \cdot \left\langle {u^{\prime } \left( {t^{\prime } } \right)u^{\prime } \left( {t^{\prime \prime } } \right)} \right\rangle } \right] \hfill \\ \end{gathered} $$
(55)
1.4 First velocity term in mean square
$$ \boxed1 = \mathop {\lim }\limits_{T \to \infty } \frac{1}{{T^{2} }}\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {{\text{d}}t\;} } {\text{d}}t^{\prime } \;{\text{d}}t^{\prime \prime } \;{\text{d}}t^{\prime \prime \prime } \;} } {\text{e}}^{{ - i2\pi f\left[ {\left( {t^{\prime } - t} \right) + \left( {t^{\prime \prime \prime } - t^{\prime \prime } } \right)} \right]}} \left\langle {u^{\prime } \left( t \right)u^{\prime } \left( {t^{\prime } } \right)} \right\rangle \left\langle {u^{\prime } \left( {t^{\prime \prime } } \right)u^{\prime } \left( {t^{\prime \prime \prime } } \right)} \right\rangle $$
(56)
Split into product of two integrals
$$ \boxed1 = \mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {{\text{e}}^{{ - i2\pi f\left( {t^{\prime } - t} \right)}} \left\langle {u^{\prime } \left( t \right)u^{\prime } \left( {t^{\prime } } \right)} \right\rangle \,{\text{d}}t\,{\text{d}}t^{\prime } \cdot } } \mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {{\text{e}}^{{ - i2\pi f\left( {t^{\prime \prime \prime } - t^{\prime \prime } } \right)}} \left\langle {u^{\prime } \left( {t^{\prime \prime } } \right)u^{\prime } \left( {t^{\prime \prime \prime } } \right)} \right\rangle \,{\text{d}}t^{\prime \prime } \,{\text{d}}t^{\prime \prime \prime } } } $$
(57)
$$ \boxed1 = S_{{u^{\prime } }} \left( f \right)^{2} $$
(58)
1.5 Second velocity term in mean square
$$ \boxed2 = \mathop {\lim }\limits_{T \to \infty } \frac{1}{{T^{2} }}\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {{\text{d}}t\;} } {\text{d}}t^{\prime } \;{\text{d}}t^{\prime \prime } \;{\text{d}}t^{\prime \prime \prime } \;} } {\text{e}}^{{ - i2\pi f\left[ {\left( {t^{\prime } - t} \right) + \left( {t^{\prime \prime \prime } - t^{\prime \prime } } \right)} \right]}} \left\langle {u\prime \left( t \right)u^{\prime } \left( {t^{\prime \prime } } \right)} \right\rangle \left\langle {u^{\prime } \left( {t^{\prime } } \right)u^{\prime } \left( {t^{\prime \prime \prime } } \right)} \right\rangle $$
(59)
Shuffle variable in exponent to match covariance terms:
$$ \boxed2 = \mathop {\lim }\limits_{T \to \infty } \frac{1}{{T^{2} }}\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {{\text{d}}t\;} } {\text{d}}t^{\prime } \;{\text{d}}t^{\prime \prime } \;{\text{d}}t^{\prime \prime \prime } \;} } {\text{e}}^{{ - i2\pi f\left[ {\left( {t^{\prime \prime } - t} \right) + \left( {t^{\prime \prime \prime } - t^{\prime } } \right) + 2t^{\prime } - 2t^{\prime \prime } } \right]}} \left\langle {u^{\prime } \left( t \right)u^{\prime } \left( {t^{\prime \prime } } \right)} \right\rangle \left\langle {u^{\prime } \left( {t^{\prime } } \right)u^{\prime } \left( {t^{\prime \prime \prime } } \right)} \right\rangle $$
(60)
We then again have a product of two independent integrals, but with additional phase shifts:
$$ \boxed2 = \mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {{\text{e}}^{{ + i4\pi f\,t^{\prime \prime } }} {\text{e}}^{{ - i2\pi f\left( {t^{\prime \prime } - t} \right)}} \left\langle {u^{\prime } \left( t \right)u^{\prime } \left( {t^{\prime \prime } } \right)} \right\rangle {\text{d}}t\,{\text{d}}t^{\prime \prime } \cdot } } \mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {{\text{e}}^{{ + i4\pi ft^{\prime } }} {\text{e}}^{{ - i2\pi f\left( {t^{\prime \prime \prime } - t^{\prime } } \right)}} \left\langle {u^{\prime } \left( {t^{\prime } } \right)u^{\prime } \left( {t^{\prime \prime \prime } } \right)} \right\rangle {\text{d}}t^{\prime } \,{\text{d}}t^{\prime \prime \prime } } } $$
(61)
Introducing τ″ ≡ t″ − t and τ′′′ ≡ t′′′ − t′, we get
$$ \boxed2 = \mathop {\lim }\limits_{T \to \infty } \int\limits_{0}^{T} {{\text{e}}^{{ + i4\pi f\,t^{\prime \prime } }} {\text{d}}t^{\prime\prime}\int\limits_{0}^{T} {{\text{e}}^{{ - i2\pi f\tau^{\prime \prime } }} C_{{u^{\prime}}} \left( {\tau^{\prime \prime } } \right){\text{d}}\tau^{\prime \prime } \cdot \mathop {\lim }\limits_{T \to \infty } } } \int\limits_{0}^{T} {{\text{e}}^{{^{{ + i4\pi f\,t^{\prime } }} }} {\text{d}}t^{\prime } \int\limits_{0}^{T} {{\text{e}}^{{ - i2\pi f\tau^{\prime \prime \prime } }} C_{{u^{\prime } }} \left( {\tau^{\prime \prime \prime } } \right){\text{d}}\tau^{\prime \prime \prime } } } $$
(62)
$$ \boxed2 = \left[ {\delta \left( f \right) \cdot S_{{u^{\prime } }} \left( f \right)} \right]^{2} = 0 $$
(63)
1.6 Third velocity term in mean square
$$ \boxed3 = \mathop {\lim }\limits_{T \to \infty } \frac{1}{{T^{2} }}\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {{\text{d}}t\;} } {\text{d}}t^{\prime } \;{\text{d}}t^{\prime \prime } \;{\text{d}}t^{\prime \prime \prime } \;} } {\text{e}}^{{ - i2\pi f\left[ {\left( {t^{\prime } - t} \right) + \left( {t^{\prime \prime \prime } - t^{\prime \prime } } \right)} \right]}} \left\langle {u^{\prime } \left( t \right)u^{\prime } \left( {t^{\prime \prime \prime } } \right)} \right\rangle \left\langle {u^{\prime } \left( {t^{\prime } } \right)u^{\prime } \left( {t^{\prime \prime } } \right)} \right\rangle $$
(64)
Shuffle variable in exponent:
$$ \boxed3 = \mathop {\lim }\limits_{T \to \infty } \frac{1}{{T^{2} }}\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {{\text{d}}t\;} } {\text{d}}t^{\prime } \;{\text{d}}t^{\prime \prime } \;{\text{d}}t^{\prime \prime \prime } \;} } {\text{e}}^{{ - i2\pi f\left[ {\left( {t^{\prime \prime \prime } - t} \right) + \left( {t^{\prime } - t^{\prime \prime } } \right)} \right]}} \left\langle {u^{\prime } \left( t \right)u^{\prime } \left( {t^{\prime \prime \prime } } \right)} \right\rangle \left\langle {u^{\prime } \left( {t^{\prime } } \right)u^{\prime } \left( {t^{\prime \prime } } \right)} \right\rangle $$
(65)
We then again have a product of two independent integrals:
$$ \boxed3 = S_{{u^{\prime } }} \left( f \right)^{2} $$
(66)
Thus the sum of the three terms becomes
$$ \boxed1 + \boxed2 + \boxed3 = 2S_{{u^{\prime } }} \left( f \right)^{2} $$
(67)
Subtracting the square of the mean, we get for the velocity part of the variance:
$$ \text{var}_{\text{velocity}} \left\{ {S_{0} \left( f \right)} \right\} = S_{{u^{{^{\prime } }} }} \left( f \right)^{2} , $$
(68)
which is the standard expression for the variance of the PS of a stationary random variable.
1.7 Sampling function part
We can now consider the expression for the sampling function:
$$ \frac{1}{{\nu^{2} \bar{N}^{2} }}\mathop {\lim }\limits_{T \to \infty } \int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {{\text{d}}t{\text{d}}t^{\prime } {\text{d}}t^{\prime \prime } {\text{d}}t^{\prime \prime \prime } \;{\text{e}}^{{ - i2\pi f\left[ {\left( {t^{\prime } - t} \right) + \left( {t^{\prime \prime \prime } - t^{\prime \prime } } \right)} \right]}} } } \;\;} } \sum\limits_{{k,k^{\prime } ,k^{\prime \prime } ,k^{\prime \prime \prime } }}^{N} {\delta \left( {t - t_{k} } \right)\delta \left( {t^{\prime } - t_{{k^{\prime } }} } \right)\delta \left( {t^{\prime \prime } - t_{{k^{\prime \prime } }} } \right)\delta \left( {t^{\prime \prime \prime } - t_{{k^{\prime \prime \prime } }} } \right)} $$
(69)
Using the delta functions one by one, we simply get:
$$ \frac{1}{{\nu^{2} \bar{N}^{2} }}\sum\limits_{{k,k^{\prime } ,k^{\prime \prime } ,k^{\prime \prime \prime } }}^{N} {{\text{e}}^{{ - i2\pi f\left[ {\left( {t_{{k^{\prime } }} - t_{k} } \right) + \left( {t_{{k^{\prime \prime \prime } }} - t_{{k^{\prime \prime } }} } \right)} \right]}} } $$
(70)
or, defining \( \tau_{{kk^{\prime } }} \equiv t_{{k^{\prime } }} - t_{k} \) and \( \tau_{{k^{\prime\prime}k^{\prime\prime\prime}}} \equiv t_{{k^{\prime\prime\prime}}} - t_{{k^{\prime\prime}}} \),
$$ \frac{1}{{\nu^{2} \bar{N}^{2} }}\sum\limits_{{k,k^{\prime},k^{\prime\prime},k^{\prime\prime\prime}}}^{N} {{\text{e}}^{{ - i2\pi f\left[ {\tau_{{kk^{\prime}}} + \tau_{{k^{\prime\prime}k^{\prime\prime\prime}}} } \right]}} } $$
(71)
We can now see how the sampling function picks out individual velocity samples through the convolution with the exponentials.
Since the samples are uncorrelated (Poisson sampling), we can write
$$ \frac{1}{{\nu^{2} \bar{N}^{2} }}\sum\limits_{{k,k^{\prime } ,k^{\prime \prime } ,k^{\prime \prime \prime } }}^{N} {{\text{e}}^{{ - i2\pi f\left[ {\tau_{{kk^{\prime } }} + \tau_{{k^{\prime \prime } k^{\prime \prime \prime } }} } \right]}} } = \left( {\frac{1}{{\nu \bar{N}}}\sum\limits_{{k,k^{\prime } }}^{N} {{\text{e}}^{{ - i2\pi f\tau_{{kk^{\prime } }} }} } } \right)^{2} $$
(72)
1.8 Final result
Thus, for the total variance
$$ \begin{aligned} \text{var} \left\{ {\hat{S}_{0} \left( f \right)} \right\} & = \mathop {\lim }\limits_{T \to \infty } \int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {\int\limits_{ - T/2}^{T/2} {{\text{d}}t{\text{d}}t^{\prime } {\text{d}}t^{\prime \prime } {\text{d}}t^{\prime \prime \prime } \;{\text{e}}^{{ - i2\pi f\left[ {\left( {t^{\prime } - t} \right) + \left( {t^{\prime \prime \prime } - t^{\prime \prime } } \right)} \right]}} \left\langle {u^{\prime } \left( t \right)u^{\prime } \left( {t^{\prime } } \right)u^{\prime } \left( {t^{\prime \prime } } \right)u^{\prime}\left( {t\prime \prime \prime } \right)} \right\rangle } } } } \\ & \quad \otimes \left\langle {\frac{1}{{\nu \bar{N}}}\sum\limits_{{k,k^{\prime } }}^{N} {{\text{e}}^{{ - i2\pi f\tau_{{kk^{\prime } }} }} } } \right\rangle^{2} \\ \end{aligned} $$
(73)
$$ \quad \quad \quad \quad = \left[ {S_{{u^{\prime}}} \left( f \right) \otimes \left\langle {\frac{1}{{\nu \bar{N}}}\sum\limits_{{k,k^{\prime } }}^{N} {{\text{e}}^{{ - i2\pi f\tau_{{kk^{\prime } }} }} } } \right\rangle } \right]^{2} $$
(74)
$$ \quad \quad \quad \quad = \left[ {\frac{{\overline{{u^{\prime 2} }} }}{\nu } + S_{{u^{\prime } }} \left( f \right) \otimes \left\langle {\frac{1}{{\nu \bar{N}}}\sum\limits_{{k \ne k^{\prime } }}^{N} {{\text{e}}^{{ - i2\pi f\tau_{{kk^{\prime } }} }} } } \right\rangle } \right]^{2} $$
(75)
$$ \quad \quad \quad \quad = \left[ {\frac{{\overline{{u^{\prime 2} }} }}{\nu } + S_{{u^{\prime } }} \left( f \right) \otimes \sin \! \text{c}^{2} \left( {2\pi f{\kern 1pt} T_{g} } \right)} \right]^{2} $$
(76)
$$ \quad \quad \quad \quad = \left( {\frac{{\overline{{u^{\prime 2} }} }}{\nu }} \right)^{2} + 2\frac{{\overline{{u^{\prime 2} }} }}{\nu }S_{{u^{\prime}}} \left( f \right) \otimes \sin \! \text{c}^{2} \left( {2\pi f\,T_{g} } \right) + S_{{u^{\prime } }} \left( f \right)^{2} \otimes \sin \! \text{c}^{4} \left( {2\pi f\,T_{g} } \right) $$
(77)
The convolution with the sinc-squared is just the usual convolution with the frequency window corresponding to a rectangular time window {0, T
g
}.
For an infinite record, T
g
→ ∞, we find
$$ \text{var} \left\{ {S_{0} \left( f \right)} \right\} = \left( {\frac{{\overline{{u^{\prime 2} }} }}{\nu }} \right)^{2} + 2\frac{{\overline{{u^{\prime 2} }} }}{\nu }S_{{u^{\prime } }} \left( f \right) + S_{{u^{\prime } }} \left( f \right)^{2} = \left[ {\frac{{\overline{{u^{\prime 2} }} }}{\nu } + S_{{u^{\prime } }} \left( f \right)} \right]^{2} . $$
(78)
