# Simulating two signals with a desired coherence

I'm trying to simulate two frequency-domain signals with a desired coherence, and I'd like to check if my idea is right.

To simulate two time-domain signals with a desired correlation, we can use the following method:

Suppose $\Sigma$ is the desired correlation. Let $X \sim N(0,1)$.

Then if we take

$X_1 = \Sigma^{1/2} X$,

$X_1$ will have the desired correlation:

$\mathbb{E} [X_1 X_1^{\dagger}] = \mathbb{E}[\Sigma^{1/2} X (\Sigma^{1/2} X)^{\dagger}] \\ = \mathbb{E}[\Sigma^{1/2} X X^{\dagger} \Sigma^{1/2 \dagger}] \\ = \Sigma^{1/2} \mathbb{E} [X X^{\dagger}] \Sigma^{1/2} \\ = \Sigma$

I'm trying to extend this to the frequency domain, after applying a Fourier transform to $X$. Let $X^F$ be the Fourier transform of $X$, ie. $X^F = WX$ where $W$ is the DFT matrix. Let the desired coherence be $S$. To use an idea similar to that used to simulate a desired correlation, I choose

$X_1^F = S^{1/2} S_{XX}^{-1/2} X^F$

where $S_{XX}$ is the cross-spectrum of $X^F$. Then

$\mathbb{E}[X_1^F X_1^{F\dagger}] = S^{1/2} S_{XX}^{-1/2} \mathbb{E}[X^F X^{F\dagger}] S_{XX}^{-1/2 \dagger} S^{1/2 \dagger} \\ = S^{1/2} S_{XX}^{-1/2} S_{XX} S_{XX}^{-1/2} S^{1/2} \\ = S$

I wrote a Python program to try this out but I didn't get a coherence close to my desired $S$, so I'm hoping to check whether there are any conceptual errors (otherwise it might be a numerical issue?). In particular, I'm not sure if $S_{XX}^{-1/2}=S_{XX}^{-1/2 \dagger}$ in general.

• how many terms are you averaging? – Stanley Pawlukiewicz Aug 13 '18 at 22:10
• X is a 5 by 1000 matrix of Gaussian white noise. It's supposed to simulate 1000 samples of activity from 5 sensors. I also tried 10000 samples. – Anon Aug 13 '18 at 23:06