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

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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
10y = y 0 + (x – x 0 )yx y x1 01 0Which is the formula for the linear interpolation in the interval (x 0 , x 1 ) (Bradie387).Figure 2.1.2 an example of Piecewise Linear <strong>Interpolation</strong> Using Example 2.1.32.1.3 Example Suppose we wanted to do a Piecewise Linear <strong>Interpolation</strong>on the points A = (-1, 3), B = (3, 4), <strong>and</strong> C = (6, -2). First we calculate theinterpolant between A <strong>and</strong> B, the results of the calculation give us the following:yy43 3 ( x1) 3 1x 134 , which is the line that connects the points A <strong>and</strong> B.We now do a similar step to calculate the interpolant between the points B <strong>and</strong> C:24y 2 ( x3) 6 3y 2x 4, which is the line connecting B <strong>and</strong> C.
Page 1: Data Inter
Page 6 and 7: 5aren’t close enough to capture t
Page 8 and 9: 7audio conversion. Second, it has t
Page 12 and 13: 11It’s obvious from Figure 2.1.2
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Page 16 and 17: 152.3.1 Example Suppose we want to
Page 18 and 19: 172.3.3 Example Let’s return to t
Page 20 and 21: 19put in perspective that it’s so
Page 22 and 23: 213.2 Lagrange ResultsFor our examp
Page 24 and 25: 23section, no sound wave is exactly
Page 26 and 27: 25However, in the very high harmoni
Page 28 and 29: 27AppendicesAppendix A: Program Hea
Page 30 and 31: 29namespace Maths{namespace <strong
Page 32 and 33: 31namespace Interpolation</
Page 34 and 35: 33double dx = x - m_x[1];return m_y

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