Continuous noise representation

Question

I am new to stochastic processes (and actually Mathematica too) and there are many things that I still didn't fully understand yet so please forgive me if I say something wrong.

What I am trying to do is fairly simple. I have a sinusoidal signal and I want to add some noise to it, so that I get a "mixed signal" of the form $$ S(t) = \sin(\omega t) + aN(\sigma,t)$$

where N is a Gaussian white noise with variance $\sigma^2$ and $a$ is an amplitude coefficient (noise amplitude).

I had been searching for the proper way to do this with Mathematica, and from what I had found, most people do this by first constructs a list of values of the signal function (in my case, $\sin(\omega t))$ and then separately constructs another list filled with the noise and finally add the two together to get the mixture. But what I really want to build is something behave like a function, instead of just a plain, discrete list of values, so that I can extract the value of my noisy signal at any arbitrary time and do all sort of things like plotting the signal (continuously) and even put it in a differential equation. But I am not so sure about how to do that.

Speaking of stochastic differential equation, I had also tried ItoProcess to solve a differential equation involving Gaussian white noise with arbitrary amplitude and variance (i.e $\frac{dx}{dt} = f(x,t) + aN(\sigma,t)$ ), but I got very confused about how it works (perhaps this is because of my lack of understanding in the actual Ito Process in mathematics). I am expecting to have two parameters to specify the noise I put in my DE, one for the amplitude and the other for the variance, but the actual ItoProcess has only one parameter associated with the noise (the "diffusion"). I had tried to google for an explanation (both the mathematical and mathematica's Ito Process), but I am still not very sure what is it and how do what I intended to do.

Like I said, I have very little knowledge in the mathematics of stochastic processes and there may be something fundamentally wrong with my understanding (I am saying this because I am feeling like I am...and I have no time to learn it all from the beginning), so I decide to ask my question directly instead of just searching around on the web and heading nowhere.

Any input or suggestion will be highly appreciated. Thank you.

Answer 1

For white noise, you can use WhiteNoiseProcess. This is just a process, so you need to pass it to RandomFunction to get actual white noise data from t=0 to t=tmax. So...

RandomFunction[WhiteNoiseProcess[σ], {0, tmax}]

However, this only returns a discrete set of values. To make it continuous, you need to pass it to Interpolation. A snag is that RandomFunction returns TemporalData, whereas Interpolation requires a conventional list of values. To convert the TemporalData to a normal table of values, you use Normal, so...

Normal[RandomFunction[WhiteNoiseProcess[σ], {0, tmax}]]

A further snag is that the table you want is nested one layer down. To extract it, write...

Normal[RandomFunction[WhiteNoiseProcess[σ], {0, tmax}]][[1]]

This gives you a table of time-value pairs, which you can now pass to Interpolation, like so...

Interpolation[Normal[RandomFunction[WhiteNoiseProcess[σ], {0, tmax}]][[1]]]

You now have your continuous noise function N[σ,t], which is defined between 0 and tmax.

You can plot this like so...

tmax = 10;
ω = 6;
σ = 1;
noise = Interpolation[Normal[RandomFunction[WhiteNoiseProcess[σ],{0, tmax}]][[1]]];
Plot[Sin[ω t] + noise[t], {t, 0, tmax}]

In the above, the RandomFunction only samples the white noise every second. You should be able to sample it more frequently by including a dt, i.e. according to the documentation, RandomFunction[proc,{tmin,tmax,dt}], but this doesn't seem to work for WhiteNoiseProcess.

A workaround, is to scale the time in the RandomFunction and in the Interpolation by the sample rate, as follows...

tmax = 10;
samplesPerSec = 10;
ω = 6;
σ = 1;
noise = Interpolation[Normal[RandomFunction[WhiteNoiseProcess[σ], {0, tmax*samplesPerSec}]][[1]]];
Plot[Sin[ω t] + noise[t*samplesPerSec], {t, 0, tmax}]