x(t) = a0 | + a1 cos (wot + q1) + a2 cos (2wot + q2) |
+ ... + aN cos (Nwot + qN) |
where the fundamental frequency
wo
is 2p /T rad/sec,
the amplitude coefficients
a1, ..., aN
are non-negative, and the radian
phase angles satisfy 0 £ q1
, ..., qN
< 2p.
To explore the Fourier series approximation, select a labeled signal,
use the mouse to sketch one period of a signal, or use the mouse to
modify a selected signal. Specify the number of harmonics, N,
and click "Calculate." The approximation will be shown in red.
In addition, the magnitude spectrum
(a plot of an vs. n)
and phase spectrum (a plot of
qn vs.
n) are shown. (If the dc-component is negative,
a0 < 0,
then |a0| is shown in
the magnitude spectrum and an angle of
p radians
is shown in the phase spectrum.) To see a table of the coefficients, click
"Table."
Suggested Exercises:
1. Sketch a signal that has a large fundamental frequency component, but small
small dc-component and small higher harmonics.
2. Sketch a signal that has large dc and fundamental frequency components, but small
higher harmonics.
3. Sketch a signal that has small dc and fundamental frequency components, but
large second harmonic.
4. Sketch a signal that has a small fundamental frequency component, but
large dc component and large second harmonic.
5. Describe how you would construct
a signal that has small dc and fundamental frequency components, but
large second, third, and fourth harmonics.
Original applet by Steve Crutchfield, update by Hsi Chen Lee.
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