Simulating noise with specific time character

Question

I am trying to generate synthetic data for a time-domain signal. Let's say my signal is a square wave and I have on top of it some random noise. I'll model the noise as Gaussian. If I generate the data as a vector of length N and then add to it random noise sampled from a normal distribution of mean 0 and width 1, I have a rough simulation of the situation I care about. However, this adds noise with characteristic timescale set by the sampling rate. I do not want this, as in reality the noise has a much longer timescale associated with it. What is an efficient way to generate noise with a specific bandwidth?

I've tried generating the noise at each sampled point and then using the FFT to cut out frequencies above a certain value. However, this severely attenuates the signal.

My idea was basically:

noise = normrnd(0,1);
f = fft(noise);
f(1000:end) = 0;
noise = ifft(f);

This kind of works but severly attenuates the signal.

Answer 1

It's pretty common just to generate white noise and filter it. Often an IIR is used since it's cheap and the noise phases are random anyway. It does attenuate the signal, but it costs nothing to amplify it.

You can also generate noise directly with an IFFT. In the example you give every coefficient in the output of fft(noise) is a Gaussian-distributed random variable, so instead of getting those coefficients with an FFT and zeroing out the ones you don't want, you can just set the ones you want and IFFT to get the resulting signal. Remember that the coefficents are complex, but the real and imaginary parts are independently Gaussion-distributed.