Create sine function with random increase in phase

Question

I am trying to simulate a signal that randomly increases its phase, so far I have tried two thing but neither worked. I usually use matlab but I want to learn some Mathematica so I thought I would try this in Mathematica.

My first try was

times = Table[i, {i, 0, 2, 0.05}];
function[source_, t_][fiIN_] := 
    With[{fi = fiIN + 0.01*RandomReal[]},source*(1 + Sin[2 Pi*t + fi])];

Where I wanted to feed fi back into fiIN for each subsequent t value (in the list times). I did not know how to makes this work though so I went on with my second try:

fiupdate[fi_] := fiupdate[fi - 1] + 0.01*RandomReal[]; 
fiupdate[1] = 0;
fitimes = Range[Length[times]];

However this function does not remember the value of RandomReal[] for the earlier steps, so fiupdate[10] could be smaller than [9] or [8]. Also when using this function I get an error:

fis = fiupdate[fitimes];
$RecursionLimit::reclim: Recursion depth of 1024 exceeded

Im not sure how to make this work. Any help is appreciated. Thank you!

Answer 1

Your problem can be reduced to creating an increasing function, phase, and then use Sin[t + phase[t]].

Here is one way to do this by interpolating a sorted list of random numbers:

tmax = 40;
phase = Interpolation[Sort[RandomReal[10, tmax]]];
Plot[phase[t], {t, 1, tmax}]
Plot[{Sin[t], Sin[t + phase[t]]}, {t, 1, tmax}]

Answer 2

You can use Sin[t + phase[t]] where phase[t] is Interpolation of a random process.

This random process can be implemented by Accumulate

tmin = 0;
tmax = 10;
dt = 0.1;

phase = Interpolation[
   Transpose@{ Range[tmin, tmax, dt], 
     Accumulate@RandomReal[1.0, Floor[(tmax - tmin)/dt + 1]]}, 
   InterpolationOrder -> 1];

Plot[phase[t], {t, tmin, tmax}]

or by PoissonProcess

phase = Interpolation[
   RandomFunction[
     PoissonProcess[1/dt], {0, Floor[(tmax - tmin)/dt + 1]}]["Path"], 
   InterpolationOrder -> 1];

Plot[phase[t], {t, tmin, tmax}]

Answer 3

One way will be to use Nest or something like that...

tmax = 20;
times = Table[i, {i, 0, tmax, 0.05}];
fi = With[{dist = .8}, (*here you can control how big jump is possible between phases*)
     NestList[RandomReal[{#, # + dist}] &,RandomReal[],-1 + Length@times]];
phaseIncrese =Transpose@{times, (1 + Sin[2 Pi*#1 + #2]) & @@@ (Transpose[{times,fi}])};
With[{(f = Interpolation@phaseIncrese)},
     Plot[{1 + Sin[2 Pi*t], f[t]}, {t, 1, 4}, Frame -> True]
]