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# Fourier Series

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1. Fourier Series Or How I Spent My Summer Vacation (the 2 weeks after the AP Exam) Kevin Bartkovich Phillips Exeter Academy

2. Background • Taylor Series • Polynomials • Derivatives • Equality of derivatives at a point • Fourier Series • Sines and cosines • Integrals • Equality of integrals over an interval of one period

3. How to determine coefficients Assume we are approximating a function f that is periodic with period for . We equate integrals over the period rather than derivatives at a point:

4. We can immediately solve for the constant term since all the sine and cosine terms integrate to 0, which yields so that

5. Strategy for other terms Multiply by cosx and integrate: Which yields

6. Why cos(mx)cos(nx) vanishes cos(mx + nx) = cos(mx)cos(nx) – sin(mx)sin(nx) cos(mx – nx) = cos(mx)cos(nx) + sin(mx)sin(nx) cos(mx+ nx) + cos(mx – nx) = 2cos(mx)cos(nx)

7. Likewise, we can multiply by sinx and integrate to find that • We can create similar integrals for all of the terms by multiplying by cos(kx) or sin(kx), in which all the terms integrate to 0 – except for cos2(kx) or sin2(kx) – which integrate to π.

8. General Form for Coefficients

9. Example: Square Wave Model a periodic square wave with amplitude 1 over the interval –π≤x ≤ π: This is an odd function, so its integral is 0; thus a0 = 0. Multiplying by coskx will also yield an odd function, so ak = 0 for all k.

10. On the other hand, multiplying by sinkx yields an even function that has an integral of 0 if k is even and 4/k if k is odd. Thus: bk= The Fourier Series is: Fourier series examples.xlsx

11. Example: Sawtooth Wave Suppose we create a Fourier Series of alternating sine curves: Fourier series examples.xlsx

12. Frequency Domain We can combine akcos(kx) +bksin(kx) into a single sinusoid, which can be written as Akcos(kx-φ), which has amplitude and phase shift

13. How to Find the kth Harmonic

14. Example: Noise Filter A Fourier Series allows us to transform a waveform from the time domain (amplitude vs. time) to the frequency domain (amplitude of the kth harmonic vs. k). Example: Filter out random errors in a signal composed of a sum of various sinusoids. Fourier series error filter.xlsx

15. Student Projects • Vibrato Fourier and vibrato.pptx • Cell phone transmissions Cellphones Effect on Sounds.pptx • Tides http://tidesandcurrents.noaa.gov/data_menu.shtml?stn=8423898 Fort Point, NH&type=Historic+Tide+Data

16. Thank You! http://faculty.kfupm.edu.sa/ES/akwahab/Frequency_Domain.htm