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What is the simplest formula of some noise signal?


where t is time.

What is the name of a noise, which power spectral density is gaussian?


Actually I need a function which can be easily calculated over time in Wolfram Mathematica. As for now I generated 1000 pseudo random numbers uniformly distributed from -1 to 1 and the use Interpolation[]. But firstly I think this signal is not "good" and also it is calculated for a long time.

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Matlab would be better. It has specific function for whatever distribution to generate random sequences. This can simulate any noisy process. – Jon Feb 7 '12 at 7:59
Thanks! Which function(s) to use in Matlab (for example)? – Suzan Cioc Feb 7 '12 at 8:59
For a Gaussian, you could use normrnd or also random should do the work. But this can be done with whatever distribution. E.g. normrnd(0,1,1,100) will generate 100 random number, with a Gaussian distribution, in a vector of 100 elements having mean 0 and variance 1. You can take 100 time steps and you have simulated your noise. – Jon Feb 7 '12 at 9:04
The same can be done in Mathematica also. I was thinking you are saying Matlab has dedicated functions to generate various noise signals. – Suzan Cioc Feb 7 '12 at 9:45
But this is a noise signal: It is sampled at discrete times. – Jon Feb 7 '12 at 10:15

This question isn't really sufficiently well defined to allow an answer. Do you really want a formula, or an algorithm? What does "simplest" mean?

What statistical properties do you want your noise to have? E.g., is it OK if it repeats after a long time? Do you need its frequency spectrum to have some desired degree of smoothness? Do you care if its spectrum goes to exactly zero above some cut-off-frequency? Do you only want some way to generate a hissing sound for a video game? -- in that case all you care about is that it sounds like a hiss, and you actually want it to repeat after a set time so that you can loop it without creating an audible click.

One possible "simple" answer would be $\hat{f}$, where $f$ is the desired spectrum in the frequency domain and the hat denotes a (discrete or continuous) Fourier transform. Whether that is the kind of "simple" you want is up to you to define. Another kind of "simple" would be something efficiently implementable in software, in which case you might want something like a pseudorandom number generator with its output fed into a digital filter.

One way to see that your question is really much more complex than you might imagine is that it is not very different from the question of how to generate pseudorandom numbers. There is a truly gigantic literature on that topic, with no single solution that is best for all applications.

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See update. I have no very defined constraints, so I wish just any solution of your taste. For example, how to generate white noise? Is this the same as a hiss? – Suzan Cioc Feb 7 '12 at 7:11

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