Linear Envelopes

• A linear mapping of parameters can be determined from the equation for a line:
  $$y(t) = m t + b$$
  where $m$ is the slope of the line and $b$ is the point where the line crosses the vertical axis ($t=0$).

• The line slope is found by $m = (y_{\mbox{target}} - y_{\mbox{start}}) / (t_{\mbox{target}} - t_{\mbox{start}})$.

  Figure 7: A simple linear line segment envelope.

  

• In a system such as Matlab, we can plot this expression as $y(t) = m (t - t_0) + y_0$, where $t_0$ is the initial time value and $y_0$ is the value of $y$ at $t_0$.

• In an audio processing system, our sample rate is constant and this allows us to simply increment our parameter value by a fixed rate (i.e. slope) value at each time step:
  $$y[n] = m T_s + y[n-1],$$
  where $T_s$ is the sampling period.

• A C language implementation is shown below for each time step:

    if ( state != 0 ) {
      if (target > value) {
        value += rate;
        if (value >= target) {
          value = target;
          state = 0;
        }
      }
      else {
        value -= rate;
        if (value <= target) {
          value = target;
          state = 0;
        }
      }
    }
    return value;