Phase-Preserving Restrikes

An interesting third (nonlinear) input possibility is to add the input to the instantaneous magnitude of $z(n)$. This mode is useful for ``restriking'' a decaying filter oscillation in a phase-preserving manner. The result is that the amplitude of a decaying oscillation can be jumped without altering its phase at all.

Adding a real signal $u(n)$ to the magnitude of $z(n)$ is equivalent to scaling $z(n)$ by $g(n) = u(n)/r(n) + 1$, where $r(n)=\vert z(n)\vert$, to get

$$\left[\frac{u(n)}{r(n)}+1\right]z(n) = [u(n)+r(n)]e^{j\theta(n)}.$$

Since amplitude jumps can be perceived as ``clicks'' when they are too fast, it is useful to delay amplitude jumps until the next zero-crossing of the output $y(n)$. Waiting for a zero-crossing in $y(n)$ is equivalent to waiting for the phase of $z(n)$ to reach either 0 or $\pi$.

Waiting for phase 0 to scale the state $z(n)=x(n)$ by some gain factor $g$ is equivalent to adding an input impulse with amplitude $u(n)=(g-1)\vert z(n)\vert = (g-1)x(n)$ to $x(n)$ at that time. Therefore, another approach to achieving smooth filter restriking is to delay them until the next positive-going zero-crossing in the output signal $y(n)$.