$f_n(x) = \left\lfloor \frac{\sin(2\pi (x / n + 1/ 4) + 1 }{2}\right\rfloor$ and related

Question

$f_n(x) = \left\lfloor \frac{ \sin(2\pi (\frac{x}{n} + \frac{1}{4})) + 1}{2}\right \rfloor = 1 \iff x = kn$ and $ f_n(x) = 0 \iff x \neq kn$. Let $g_n(x)$ be what's within the floor brackets. Then $f_{n_1} \odot \dots \odot f_{n_r}$ where $\odot = $ boolean and, is equal to $\left\lfloor \frac{g_{n_1} + \dots + g_{n_r}}{r} \right\rfloor$ so we have the formula $f_{n_1 \cdots n_r}(x) =\left \lfloor \frac{g_{n_1}(x) + \dots + g_{n_r}(x)}{r}\right \rfloor$. Is this true and can we say anything else?

Answer 1

\We can also say $f_{n_1 \cdots n_r}(x) = \prod_{i=1}^r f_{n_i}(x) = \odot_{i=1}^r f_{n_i}(x)$.

Also for boolean or we have $f_{n_1}(x) \oplus \dots \oplus f_{n_r}(x) = \lfloor \max\{g_{n_1}(x), \dots, g_{n_r}(x)\}\rfloor$