parabolic function

List parabolic(Array y, int x)

Quadratic interpolation for estimating the true position of an inter-sample maximum when nearby samples are known.

• y : is a vector
• x : is an index point for that vector.

Return

vx, vy, the coordinates of the vertex of a parabola that goes through point x and its two neighbors.

References

1. "A few simple frequency estimation methods in Python". https://gist.github.com/endolith/255291#L38. Retrieved 2019-08-07.
2. "Quadratic Interpolation of Spectral Peaks". https://ccrma.stanford.edu/~jos/sasp/Quadratic_Interpolation_Spectral_Peaks.html. Retrieved 2019-08-07.

Examples

/* Defining a vector f with a local maximum at index 3 (= 6), find local
maximum if points 2, 3, and 4 actually defined a parabola.
*/
var f = Array([2, 3, 1, 6, 4, 2, 3, 1]);

print(parabolic(f, arrayMax(f)));
/* output:
[3.2142857142857144, 6.1607142857142856]
*/

Implementation

List parabolic(Array y, int x) {
  var xv =
      1 / 2.0 * (y[x - 1] - y[x + 1]) / (y[x - 1] - 2 * y[x] + y[x + 1]) + x;
  var yv = y[x] - 1 / 4.0 * (y[x - 1] - y[x + 1]) * (xv - x);
  return [xv, yv];
}