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Fourier series and frequency analysis

Fourier series and frequency analysis. Integrate numerically: Trapezoid rule. Orthogonality of sines , cosines. S ines and cosines are orthogonal functions.

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Fourier series and frequency analysis

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  1. Fourier series and frequency analysis

  2. Integrate numerically: Trapezoid rule

  3. Orthogonality of sines, cosines

  4. Sinesand cosines areorthogonal functions • In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle. The word comes from the Greekὀρθός (orthos), meaning "straight", and γωνία (gonia), meaning "angle". (wikipedia)

  5. MATLAB Demo

  6. Fourier series • Any periodic signal can be decomposed to a sum of sines and cosines.

  7. If f(t) is given, what are a’s and b’s?

  8. Use orthogonality

  9. Coefficients

  10. Orthogonal functions Function space F(t) is the known function “Normal” vector space F is known vector

  11. FFT • Fast Fourier Transform • N log(N) operations rather than N2-matters when N is big • Classic algorithms requires 2n points • Uses complex notation • Often use windowing on real data

  12. FFT Demo

  13. Questions?

  14. Project • Fits within the theme “The natural world” • Teams of 4 (nominally) • All students must be in the same section (class time will be used to work) • You will submit project preferences, we will form the teams. • Class after spring break will be focused on project, still may be some lectures.

  15. Other things • May use “raw” sensors and build your own circuits, or you may buy commercial sensors. • May leverage labs we have already done. • May focus more on the experiment, the data, the analysis –or- more on building something interesting. • May focus more on making a finished “product” (though must stay within spirit of the class). • Each team will get a budget. • Option of doing a PCB if needed (can share with another team).

  16. The natural world Possible examples: • Environment • Weather balloon – atmospheric temperature profile • Lakes- (you can rent boats/canoes for Lake Waban). • Weatherstation (wind, sun, rain). • Underwater ROV • ….. • Bio-instrumentation • EMG – like EKG but senses muscle activity. • EEG (Brain waves) • Pulse oximeter – all analog, calibrated. • Biomechanics (accelerometers on yourself) • Wii balance board/force plate • Heart rate by measuring your weight change • …

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