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

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11It’s obvious from Figure 2.1.2 that Piecewise Linear <strong>Interpolation</strong> <strong>and</strong> ourexample are not very precise at all. Using straight lines to connect the pointsresults in a very jagged connection, <strong>and</strong> the chances of missing points lying onstraight lines between points are very unlikely. I highly suspect that this will proveto be detrimental to interpolating data points in a sound file as the wave willflatten out to close to a straight line that does not accurately reproduce sound.We will show this in chapter 3 section 1 when the results are analyzed.2.2 Lagrange Polynomial <strong>Interpolation</strong>Polynomial <strong>Interpolation</strong> is a method of fitting a single non-linearpolynomial to a set of data points. Upon determining the number <strong>and</strong> location ofthe points involved, the Polynomial <strong>Interpolation</strong> method creates a polynomialwith a degree one less than the number of points in the data set that passesthrough each of the given points (e.g. with 3 data points a quadratic is created,with 4 data points a cubic is created, etc.). There are several different forms ofthe Interpolating Polynomial, but this section will focus on one: The LagrangeForm of the Interpolating Polynomial.2.2.1 Definition Given a set of k+1 points in the Cartesian coordinatesystem, the interpolation polynomial is given by the linear combination:kj0yjjl ( x)Where l j (x) is the combination of Lagrange basis polynomials as follows:
12lj( x)ki0,ijxxixjxiThe result is a polynomial with degree k. (Bradie 341)Figure 2.2.2 the data set in example 2.1.3 with the 2 ndpolynomial.degree interpolation2.2.3 Example Suppose we want to perform a polynomial interpolation onthe points used in the Piecewise Linear example. We would first calculate all ofthe Lagrange basis polynomials as follows:x3 x6 1l x x x4 7 2820( ) * ( 9 18)x1 x6 1l x x x4 3 12x1 ( x3) 1 2 2l 2( x) * ( x 2x 3)7 3 2121( ) * ( 5 6)The interpolation polynomial is then the sum of the corresponding y value timeseach of the basis polynomials:
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 10 and 11: Chapter 2: Data <s
Page 14 and 15: 131 1 1L x x x x x x x28 3 212-.321
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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