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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 Spring '09
 vogan
 Integrals, Fourier Series, Cos, Periodic function

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