c20 - 18.03 Class 20, March 24, 2006 Periodic signals,...

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18.03 Class 20, March 24, 2006 Periodic signals, Fourier series [1] Periodic functions: for example the heartbeat, or the sound of a violin, or innumerable electronic signals. I showed an example of violin and flute. A function f(t) is "periodic" if there is P > 0 such that f(t+P) = f(t) for every t . P 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 approximate 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 P determines the function. You can choose the window as convenient. We'll often use the window [-P/2,P/2] . t cos(t) is NOT periodic. [2] Sine and cosines are basic periodic functions. For this reason a natural period to start with is P = 2\pi . We'll use the basic window [-pi,pi] . Question: what other sines and cosines have period 2pi ? Answer: cos(nt) and sin(nt) for n = 2, 3, . .... Also, cos(0t) = 1 (and sin(0t) = 0 ). These are "harmonics" of the "fundamental" sinusoids with n = 1 . If f(t) and g(t) are periodic of period P then so is af(t) + bg (t) . So we can form linear combinations. There is a standard notation for the

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c20 - 18.03 Class 20, March 24, 2006 Periodic signals,...

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