How to generate covariance matrix of coloured noise

Question

For standard gaussian white noise, the covariance matrix is a identity matrix. What about other coloured noises (generated from standard gaussian): Brown(red), Pink, Blue, Violet?

Additional details / thoughts: Preferably looking for a python code, since I don't have domain knowledge to understand the "process". Maybe generating 2D noise and then finding the covariance (numpy.cov) would work? I found https://stackoverflow.com/questions/67085963/generate-colors-of-noise-in-python to generate 1D noise, but I am not sure how to extend it to 2D.

Update for clarity: https://en.wikipedia.org/wiki/White_noise#White_noise_vector

the covariance matrix R of the components of a white noise vector w with n elements must be an n by n diagonal matrix, where each diagonal element Rii is the variance of component wi; and the correlation matrix must be the n by n identity matrix.

I want to find covariance matrix R of the components of coloured noise vector w with n elements.

Will generating enough noise vectors, stacking them as columns and finding covariance of the matrix gives the expected covariance matrix? This matches with what I expect from the covariance matrix: https://en.wikipedia.org/wiki/Estimation_of_covariance_matrices#Estimation_in_a_general_context

Answer 1

You can at least approximate colored noise by altering the spectrum of white noise. The following are some definitions of colored noise

\begin{equation} S_{pink}(f) = \frac{S_{0}}{f^{\alpha}} \end{equation}

where $\alpha$ is close to 1,

\begin{equation} S_{brown}(f) = \frac{S_{0}}{f^{2}}\end{equation}

\begin{equation} S_{blue}(f) = S_{0}f\end{equation}

You can see more here. I'm not a python person, so I'm not going to write python, but the basic procedure is something like: generate white noise $\rightarrow$ N-D FFT $\rightarrow$ N-D filter (e.g. for pink noise dot multiply the white noise spectrum by $\frac{1}{f}$) $\rightarrow$ N-D IFFT. This will give you an approximation of your desired colored noise.

For generating a covariance (correlation) matrix, the function you are likely looking for is spectrum's corrmtx, which is equivalent to Matlab's corrmtx. This will give you a correlation matrix estimate for 1-D data via time-averaging. However, you seemed confused on what the correlation matrix is and how to find it. For a data vector

\begin{equation}\underline{x}=\begin{bmatrix}x(0) & x(1) & \cdots & x(N-1)\end{bmatrix}^{T}\end{equation}

The correlation matrix is defined as

\begin{equation} R_{xx} = E\{\underline{x}\,\underline{x}^{H}\}\end{equation}

This means that the correlation matrix is the ensemble averaged outer product, and thus represents the complete set of statistical second moments of a data vector $\underline{x}$. Theoretically, for a vector of length $N$, the correlation matrix is an $N$-by-$N$ matrix. Estimating via time-averaging will typically decrease the size of the estimated correlation matrix.

Estimating the correlation matrix for 2-D data, for example, will result in a block diagonal matrix. Estimating correlation matrices of 2-D or higher dimensional data is pretty conceptually challenging, so I would start with 1-D and make sure you understand that.

EDIT 1: Question about Covariance Matrix vs. Correlation Matrix

Wikipedia's explanation doesn't seem right. The covariance matrix is defined as

\begin{equation}K_{xx} = E\{(\underline{x}-\mu_{x})(\underline{x}-\mu_{x})^{H}\}\end{equation}

where $\mu_{x}$ is assumed constant for a Wide-Sense Stationary random process. For noise to be white, it must be zero-mean. There are several stack exchange posts on this, I'll link this one. This means, for white noise,

\begin{equation} R_{xx} = K_{xx} = \sigma_{0}^{2}I\end{equation}

where $I$ is the identity matrix.

Practically speaking, any property that applies to the correlation matrix also applies to the covariance matrix, although the converse is not true. I find estimating covariance matrix to be more tedious and doesn't always provide a huge benefit, especially when dealing with higher dimensional data, but there very well could be practical use for it that I'm unaware of. To estimate it, you have to subtract off the mean.

Responding to your updated question, that is not how you would calculate the covariance matrix. You would still use the corrmtx function, but your data input would be the data with the mean subtracted off. If you wanted to mimic ensemble averaging, you could form an $N$-by-$M$ matrix that contains $M$ draws of an $N$ length random vector, where $M \gg N$. You would then take the outer product of each row and then average the $M$ outer products.