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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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 Winter '08
 Staff
 Fourier Series

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