A Musical Time-Varying Filter Example

Note, however, that a gain $g$ may vary with time independently of $x$ to yield a linear time-varying filter. In this case, linearity may be demonstrated by verifying

$$g(n) \left[ \alpha \cdot x_1(n) + \beta \cdot x_2(n)\right] = \alpha \cdot [g(n)\cdot x_1(n)] + \beta\cdot[g(n)\cdot x_2(n)]$$

to show that both scaling and superposition hold. A simple example of a linear time-varying filter is a tremolo function, which can be written as a time-varying gain, $y(n)=g(n)x(n)$. For example, $g(n) = 1 + \cos[2\pi (4)nT]$ would give a maximally deep tremolo with 4 swells per second.