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Fourier series and frequency analysis. Integrate numerically: Trapezoid rule. Orthogonality of sines , cosines. S ines and cosines are orthogonal functions.
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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)
Fourier series • Any periodic signal can be decomposed to a sum of sines and cosines.
Orthogonal functions Function space F(t) is the known function “Normal” vector space F is known vector
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
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.
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).
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 • …