Colored noise in SDE

Question

I want to numerically study the behavior of a system described by a set of differential equations in the presence of colored noise. It seems that the standard procedure is to use the Langevin equation: $ y=fdt+gdX; dX=-aXdt+bdW$, where $W$ is the Wiener process.

The spectral density function of $X$ is $1/f^2$ type. What about the spectral density function of $dX$? If my MATLAB calculations are correct, in this case the power density increases with the frequency. If this is true, how can I introduce $1/f$ noise in my system?

Answer 1

You cannot create $1/f$ noise as solutions to an SDE. The best you can do is to take a vector $\{\xi_k\}_{k=-N}^N$ of i.i.d. random variables, divide $\xi_k$ by $\sqrt k$, and take the Fourier transform of that. This will be your $1/f$ noise $X$...