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

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172.3.3 Example Let’s return to the three data points we used in thePiecewise Linear <strong>and</strong> Lagrange examples. Suppose we wanted to do the secondmethod to create a Cubic Spline on these data points.Cubic Spline <strong>Interpolation</strong> has notable advantages <strong>and</strong> disadvantagesover the previous two methods we looked at. The advantage to Cubic Splines isthe high level of smoothness within the curve, <strong>and</strong> that the function being apiecewise function of degree 3 ensures controlled behavior within theinterpolation. This allows us to get a better estimate of where these missingpoints lie when executed on a digital sound file. The disadvantage with thistechnique is that it’s very computationally intensive, <strong>and</strong> therefore in terms ofexecution time it’s rather slow. Despite its advantages, Cubic Spline interpolationperformed disappointingly in the interpolation of sound. The results from testingthis method will be in section 3.3.
18Chapter 3: ResultsFor the results of testing contained herein, to demonstrate each of thehypotheses given in the previous examples, let our first test sample be definedby the equation f(x)=3sin(x). This is an example of a perfect sound wave(meaning a constant tone, volume, <strong>and</strong> pitch is assumed). We will do threedifferent samplings on this equation, each time increasing the sample rate. We’llstart by just sampling the zeros <strong>and</strong> the extrema of the function (so that oursample domain is 0, π/2, π, 3π/2, 2π), then the second time through the halfwaypoints will be added in (so our sample domain will be 0, π/4, π/2, 3π/4, π, 5π/43π/2, etc.). In our last sample, we’ll double the sampling rate again, <strong>and</strong> add inthe halfway points between those points.Our second test sample used a r<strong>and</strong>om number generator to generate 11y values between -10 <strong>and</strong> 10 for x values of 0 through 10. I then connected thesepoints by h<strong>and</strong> to make a sound wave (note with 11 data points this sound wouldbe of extremely short duration: about 4 milliseconds of sound), but this is morethan enough to demonstrate the behavior of an actual sound wave.3.1 Piecewise Linear ResultsOur first low sample rate returned very inaccurate interpolation resultsusing this method. As the only end points for our lines were the extrema <strong>and</strong>zeros of our function, this gave a very inaccurate interpolation of the data, asseen in the diagram on the next page. The distances between the interpolants<strong>and</strong> the actual sound wave might not seem like much at first glance, but when
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 12 and 13: 11It’s obvious from Figure 2.1.2
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 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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