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

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 2

3. Definition or 3

4. 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

5. We can immediately solve for the constant term since all the sine and cosine terms integrate to 0, which yields so that 5

6. Strategy for other terms Multiply by cosx and integrate: Which yields 6

7. 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) = 2 cos(mx)cos(nx) 7

8. 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 cos 2 (kx) or sin 2 (kx) – which integrate to π. 8

9. General Form for Coefficients 9

10. 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 a 0 = 0. Multiplying by coskx will also yield an odd function, so a k = 0 for all k. 10

11. 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 =bk = The Fourier Series is: Fourier series examples.xlsx 11

12. Example: Sawtooth Wave Suppose we create a Fourier Series of alternating sine curves: Fourier series examples.xlsx 12

13. Frequency Domain We can combine a k cos(kx) + b k sin(kx) into a single sinusoid, which can be written as A k cos(kx - φ), which has amplitude and phase shift 13

14. How to Find the kth Harmonic 14

15. 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

16. Student Projects Vibrato Fourier and vibrato.pptx Cell phone transmissions Cellphones Effect on Sounds.pptx Tides http://tidesandcurrents.noaa.gov/data_menu.shtml?st n=8423898 Fort Point, NH&type=Historic+Tide+Data 16

17. Thank You! http://faculty.kfupm.edu.sa/ES/akwahab/Frequency_Domain.htm 17