I have a trigonometric function $g(x)$ and I'm interested in how its frequency changes as $|x|$ increases. When I plot the function I can see that the frequency increases with $|x|$, but is there a way to show this formally?
This is the function:
$$g(x) = \sin(2\pi(x-a)b \ \cos(2\pi(x-a)c)) + d.$$
[Edit] In a previous question I asked about he periodicity of this function and accepted an answer. However, knowing that the function is not periodic does not really tell me anything about the frequency. Just that the function values don't repeat. I am interested in the behaviour of the frequency, because I use this function in an application in swarm intelligence (known as Angle Modulated Particle Swarm Optimization (PSO)) to generate potential solutions to binary problems. Basically, the function is sampled at regular intervals $x = (0, 1, ..., n)$. At each sample position, the function value is mapped to a binary bit as follows: if $g(x) > 0 \rightarrow 1$, else if $g(x) \leq 0 \rightarrow 0$. Now, if the frequency increases indefinitely as $x$ increases, this has implications for the algorithm when $n$ is large, because of the way a search is carried out in PSO. So, although this question is closely related to my previous question, there are other implications.