Suppose $f(t)$ is a continuous-valued, zero-mean stochastic signal with Gaussian autocorrelation (with some standard deviation $\sigma$). Suppose I then pass this signal through a step function, producing a new $\pm$1 signal $g(t)$ that has value +1 wherever $f(t) \ge 0$, and -1 wherever $f(t) < 0$. This signal is similar to a random telegraph signal, though perhaps with different statistics. My question is: does the autocorrelation of $g(t)$ have a simple form? If it were a random telegraph signal where the switching was described by a Poisson process, then I take it the autocorrelation would decay exponentially with the lag. But the switches between the two states (+1 and -1) don't seem to be independent here. Any guidance or references would be helpful.

Suppose $f(t)$ is a continuous-valued, zero-mean stochastic signal with Gaussian autocorrelation (with variance $\sigma^2$). Suppose I then pass this signal through a step function, producing a new $\pm$1 signal $g(t)$ that has value +1 wherever $f(t) \ge 0$, and -1 wherever $f(t) < 0$. This signal is similar to a random telegraph signal, though perhaps with different statistics. My question is: does the autocorrelation of $g(t)$ have a simple form? If it were a random telegraph signal where the switching was described by a Poisson process, then I take it the autocorrelation would decay exponentially with the lag. But the switches between the two states (+1 and -1) don't seem to be independent here. Any guidance or references would be helpful.

More generally, suppose f is composed with a "soft" step function, such as the error function $\mathrm{erf}(u)$. For instance, let $g(t)=\mathrm{erf}(\alpha f(t))$. For large $\alpha$, this approaches the case above. In general, what is the autocorrelation of $g$?