Fourier series and frequency analysis. Integrate numerically: Trapezoid rule.

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Presentation transcript:

Fourier series and frequency analysis

Integrate numerically: Trapezoid rule

Orthogonality of sines, cosines

Sines and cosines are orthogonal functions In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle. The word comes from the Greek ὀρθός (orthos), meaning "straight", and γωνία (gonia), meaning "angle". (wikipedia)mathematicsvectorsperpendicularGreek

MATLAB Demo

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

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

Use orthogonality

Coefficients

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

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

FFT Demo

Questions?

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 Environment Weather balloon – atmospheric temperature profile Lakes - (you can rent boats/canoes for Lake Waban). Weather station (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 … Possible examples: