Data Interpolation and Its Effects on Digital Sound Quality - McMurry ...

Chapter 2: <strong>Data</strong> <strong>Interpolation</strong> Techniques<strong>Data</strong> interpolation is a means of taking two points <strong>and</strong> fitting a function inbetween them to estimate the location of a third point in between the two givenpoints. There are several different means of data interpolation, however in thisthesis we will take a look at three selected methods to be applied to our soundsamples. These are Piecewise Linear <strong>Interpolation</strong>, Lagrange Polynomial<strong>Interpolation</strong>, <strong>and</strong> Cubic Spline <strong>Interpolation</strong>.92.1 Piecewise Linear <strong>Interpolation</strong>Perhaps the simplest <strong>and</strong> quickest of all known interpolation methods isthe Piecewise Linear <strong>Interpolation</strong> method. If you recall the midpoint formula fromhigh school algebra for finding the point exactly halfway between two givenpoints, this is very similar. Piecewise Linear <strong>Interpolation</strong> calculates the slope<strong>and</strong> y-intercept of the particular line that passes through two points to allow oneto estimate the location of any point chosen between the two points.2.1.1 Definition Given two known points A := (x 0 , y 0 ) <strong>and</strong> B := (x 1 , y 1 ) thelinear interpolant is defined as the straight line between the two points. For avalue x in the interval (x 0 , x 1 ), the value y along the straight line is given from theequationy y x x0 0y1y x x0 1 0This can then be solved for y to yield