x(t) = a_{0}  + a_{1} cos (w_{o}t + q_{1}) + a_{2} cos (2w_{o}t + q_{2}) 
+ ... + a_{N} cos (Nw_{o}t + q_{N}) 
where the fundamental frequency
w_{o}
is 2p /T rad/sec,
the amplitude coefficients
a_{1}, ..., a_{N}
are nonnegative, and the radian
phase angles satisfy 0 £ q_{1
}, ..., q_{N}
< 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 a_{n} vs. n)
and phase spectrum (a plot of
q_{n} vs.
n) are shown. (If the dccomponent is negative,
a_{0} < 0,
then a_{0} 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 dccomponent 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.
