MIT18_03S10_c20

MIT18_03S10_c20 - 18.03 Class 20, March 19, 2010 Periodic...

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18.03 Class 20 , March 19, 2010 Periodic signals, Fourier series [1] Periodic functions [2] Sines and cosines [3] Parity [4] Integrals [1] Periodic functions: for example the heartbeat, or the sound of a violin, or innumerable electronic signals. I showed an example of somewhat simplified waveforms of a violin and a flute. A function f(t) is "periodic" if there is L > 0 such that f(t+2L) = f(t) for every t . 2L is a "period." So strictly speaking the examples given are not periodic, but rather they coincide with periodic functions for some period of time. Our methods will accept this approximation, and yield results which merely approxmimate real life behavior, as usual. The constant function is periodic of every period. Otherwise, all the periodic functions we'll encounter have a minimal period, which is often called THE period. Any "window" (interval) of length 2L determines the function. You can choose the window as convenient. We'll often use the window [-L,L] . (This is one reason why chose to write the period as 2L.) [2] Sine and cosines are basic periodic functions. For this reason a natural period to start with is 2L = 2\pi : L = pi . We'll use the basic window [-pi,pi] . Question: what other cosines have period 2pi ? 1. cos (t), cos(t/2), cos(t/3), . .. 2. cos(pi t), cos(2 pi t), .
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MIT18_03S10_c20 - 18.03 Class 20, March 19, 2010 Periodic...

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