# Error Propagation through an IFFT

I'm not sure how to approach this problem. I will describe what I am trying to do, and what some of my matlab outputs look like. I'm not trained in signal processing or anything, so please be patient reading.

I have a fourier transform of some signal, and its frequency domain covariance matrix. For example, say my signal $x[f]$ is $100$ points long. Its covariance matrix $\Sigma$ is $100 \times 100$. I noticed that its diagonal is real, and when I plot it, it looks like a PSD. All the off diagonal elements are complex.

What I wish to do, is take the IFFT of the signal $x[f] \rightarrow x[t]$, and then plot $x[t]$ with error bars from the covariance $\Sigma$. Is this possible? After I take the IFFT of $x[f]$, what do I need to do to $\Sigma?$

Maybe it helps you to note that the FFT or IFFT with length $N$ can be realized by multiplication with the Fourier matrix $F$, or its conjugate transpose divided by $N$ respectively.
If you have the covariance matrix $\Sigma$ of your signal in frequency domain, the covariance matrix of the signal after IFFT (or multiplication with $F^H/N$) is then given by the usual way to compute the covariance matrix of a linear transform, i.e.
$$\Sigma' = 1/N^2 \; F^H \Sigma F$$
Note the prefactor $1/N^2$ is just because the IFFT is given by $F^H / N$.