Let's consider we have a continuous random signal ${ t \in ] - \infty \,;\, + \infty [ \mapsto b (t)}$. We assume this signal to be stationary, so that when ensemble-averaged, one may introduce the function ${ t \mapsto C (t) }$, such that \begin{equation} \langle b (t_{1}) \, b^{*} (t_{2}) \rangle = C (t_{1} \!-\! t_{2}) \, , \; (1) \end{equation} which only depends on the time-lag ${ t_{1} \!-\! t_{2} }$. Introducing the continuous Fourier transform as \begin{equation} \begin{cases} \displaystyle \widehat{f} (\omega) = \!\! \int_{- \infty}^{+ \infty} \!\!\!\! \mathrm{d} t \, f (t) \, e^{i \omega t} \, , \\ \displaystyle f (t) = \frac{1}{2 \pi} \!\! \int_{- \infty}^{+ \infty} \!\!\!\! \mathrm{d} \omega \, \widehat{f} (\omega) \, e^{- i \omega t} \, , \end{cases} \end{equation} we may rewrite the previous relation as \begin{equation} \langle \widehat{b} (\omega_{1}) \, \widehat{b}^{*} (\omega_{2}) \rangle = 2 \pi \, \delta_{\rm D} (\omega_{1} \!-\! \omega_{2}) \, \widehat{C} (\omega_{1}) \, , \; (2) \end{equation} or equivalently \begin{equation} \widehat{C} (\omega_{1}) = \frac{1}{2 \pi} \!\! \int \!\! \mathrm{d} \omega \, \langle \widehat{b} (\omega_{1}) \, \widehat{b}^{*} (\omega_{2})\rangle \, . \; (2b) \end{equation} I am interested in determining the smooth autocorrelation function ${ \omega \mapsto \widehat{C} (\omega) }$ from various empiric realisations. However, the main difficulty is that I only know the functions ${ t \mapsto b (t) }$ on time intervals of finite size, i.e. ${ t \in [0 , \Delta T] }$, where ${\Delta T}$ is imposed by the finite durations of my experiments.
In order to estimate the smooth function ${ \omega \mapsto \widehat{C} (\omega) }$, I therefore tried to proceed as follows. Because the function ${ t \mapsto b (t) }$ is only defined on an interval of finite size, I may define its Fourier series as \begin{equation} b (t) = \sum_{n \in \mathbb{Z}} \widehat{b_{n}} \, e^{- i \omega_{n} t} \, , \; (3) \end{equation} where ${ \omega_{n} = (2 \pi n)/ \Delta T }$ and where the coefficients $\widehat{b_{n}}$ are defined as \begin{equation} \widehat{b_{n}} = \frac{1}{\Delta T} \!\! \int_{0}^{\Delta T} \!\!\!\!\!\! \mathrm{d} t \, b (t) \, e^{ i \omega_{n} t} \, . \; (4) \end{equation} At this stage, it is crucial to note that the coefficients $\widehat{b_{n}}$ do no really depend on ${ \Delta T }$, i.e. if one considers an interval of length ${ 2 \!\times\! \Delta T }$, one will obtain a similar value for the coefficients $\widehat{b_{n}}$ (the longer the time interval, the better the estimation). Assuming that the Fourier series from equation (3) may be repeated for ${ t \in ] - \infty \, ; \, + \infty [ }$, one may perform a continous Fourier transform of equation (3) so as to obtain \begin{equation} \widehat{b} (\omega) = 2 \pi \sum_{n} \widehat{b_{n}} \, \delta_{\rm D} (\omega \!-\! \omega_{n}) \, . \; (5) \end{equation} Injecting this decomposition in (2b), one may finally obtain \begin{equation} \widehat{C} (\omega) = 2 \pi \sum_{n} \widehat{C_{n}} \delta_{\rm D} (\omega \!-\! \omega_{n}) \, , \; (6) \end{equation} where the coefficients $\widehat{C_{n}}$ are given by \begin{equation} \widehat{C_{n}} = \langle \widehat{b}_{n} \sum_{n'} \widehat{b}_{n'}^{*} \rangle \, . \; (7) \end{equation} As detailed after equation (4), one can note that the coefficients $\widehat{C}_{n}$ do not really depend on ${ \Delta T }$, since the larger ${ \Delta T }$, the better their estimation. However, a major flaw of the expression (7) of the autocorrelation is that it is not a smooth function, i.e. the function ${ \omega \mapsto \widehat{C} (\omega) }$ takes the form of a discrete Dirac comb. Of course, knowing the function ${ n \mapsto \widehat{C_{n}} }$, i.e. knowing the function ${ \omega \mapsto \widehat{C_{n}} }$ for a discrete set of values ${ \omega \!=\! \omega_{n} }$, I could construct a smooth interpolated function ${ \omega \mapsto \widehat{C_{n}} (\omega) }$. BUT, this would not lead me to have a smooth autocorrelation function ${ \omega \mapsto \widehat{C} (\omega) }$, which is what I need for my physical application. It seems I am missing something fundamental w.r.t continuous Fourier transform and discrete Fourier series, but I am unable to determine how I should proceed to estimate a smooth autocorrelation function ${ \omega \mapsto \widehat{C} (\omega) }$ (which should of course in the end be independent of $\Delta T$) This should have a link with window function, but I am not sure of the best way to proceed...
Stated very shortly, my question is therefore the following one :
How can I estimate the smooth autocorrelation $\omega \mapsto \widehat{C} (\omega)$ from equation (2), while knowing the function ${ t \mapsto b(t) }$ only for ${ t \in [ 0 ; \Delta T ] }$, i.e. on intervals on finite size, for which I can't compute continuous Fourier transforms?
(I already asked the same question on Physics.SE, but figured this question may rather benefit from being asked on M.SE. If duplicates should be deleted, I will of course do it!)