I've been trying to determine the effect of performing the cross-correlation operator on two windows signals.

For example: suppose we have two Sinc functions. The Sinc function is defined as $\frac{sin(x)}{x}$. The cross-correlation of these two signals is

\begin{align} R(\tau) &= \int_{-\infty}^{\infty} \overline{\textrm{sinc}(\pi t)} \cdot \textrm{sinc}(\pi\left(t + \tau\right))\space dt \\ &= \textrm{sinc}(\pi t) \end{align}

However, what would happen if this Sinc function is windowed by a windowing function $w(t)$ of length $L$? For the simplest case, what would happen if $w(t)$ is a boxcar function?

I've thought about this myself, and here is my reasoning: since the cross-correlation between two boxcar functions is equal to a triangular function, I was thinking this might be equal to: $\textrm{sinc}(\pi t) \cdot \textrm{tri}(t)$. Where the triangular function is non-zero between $[-L, L]$.

However, I am not able to proof that this is indeed the correct solution, and when I try this in the discrete domain, in code, I get a Sinc function that deviates from this solution.


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