I'd like to generate a random discrete-time signal that is band-limited to some bandwidth B (by means of a digital filter, ie in MATLAB). The catch is that I'd like this signal to have an arbitrary PDF, for example, a uniform distribution.
If I start with a uniform distribution, and then filter to band-limit the signal to B, I end up with something that's basically gaussian. Since my band-limiting FIR filter has a large number of taps (say 128), this makes sense given the Central Limit Theorem, as I am basically summing a number of IID random variables.
What I don't understand, is if it is possible to now transform this signal to a new distribution, such as uniform, while maintaining the band-limitedness. I understand that I can transform a RV from one PDF to another by basically integrating/mapping over the CDF (hand-wavy, it's been a while!), but I believe this significantly changes the frequency content of the band-limited signal. For example, if I have a gaussian distribution, and I want to transform to a uniform, I basically map values "near the mean" of the gaussian to something "closer to the max" in the uniform RV. But now the frequency content has changed, and my spectrum will be smeared.
Does what I'm describing make sense, and has anyone looked at something similar? Perhaps what I need to do is look at transforming the signal as a vector, not a scalar (ie treat it as a random process, not individual random variables). But this is where I get a little out of my depth.