Consider a vector $\boldsymbol{X}$ of $N$ equispaced points whose autocorrelation $C(k)$ is to be estimated:
\begin{equation} C(k)=\frac{1}{N} \sum_{t=0}^{N-k-1} X_t\cdot X_{t+k} \end{equation}
It is well-known that this can be computed efficiently using Fourier transforms:
\begin{equation} C(k) = \mathcal{F}^{-1}\big\{\mathcal{F}\{\boldsymbol{X}\}\cdot\overline{\mathcal{F}\{\boldsymbol{X}\}}\big\} \end{equation}
where $\mathcal{F}\{.\}$ is the Fourier transform and $\overline{\mathcal{F}\{.\}}$ is its complex conjugate. However in the case that the denominator of $C(k)$ is not a constant $N$ but rather dependent on $k$:
\begin{equation} C(k)=\frac{1}{N-k} \sum_{t=0}^{N-k-1} X_t \cdot X_{t+k} \end{equation}
Can this also be computed using Fourier transforms? If so can you provide guidance on how to do this?