**Description:** Consider the 0 and 1 step function f(x) as shown in my seminar description above. We compute the Fourier series as follows.
f(x)= 0 0<x<pi/2
= 1 pi/2<x<pi
f(x)= ao/2+summation an cosnwx+summation bn sinnwx
w=2pi/T where T is time
T=pi
Therefore w=2pi/pi
=2
ie time taken by 0 and 1
w=frequency
ao =2/T( (integration 0 dx limit 0 to pi/2)+ (integration 1 dx limit
pi/2 to pi))
=2/pi(pi-pi/2)limit pi/2 to pi
=1
an=2/T ((integration 0 cos2nx dx limit 0 to pi/2)+( integration 1 cos2nx
dx limit pi/2 to pi))
=2/pi (sin2nx limit pi/2 to pi)
=2/pi2n(sin2npi-sinnpi)
sin2npi=0 sin npi=0
=0
bn=2/T((integration 0 sin2nx dx limit 0 to pi/2)+(integration
1 sin2nx dx limit pi/2 to pi))
=2/pi n(-cos2nx limit pi/2 to pi))
=1/npi(-cos2npi+cosnpi)
cos npi=(-1)power n
cos 2npi=1
=1/n
pi (-1+(-1)power n)
bn=if n=1,3,5 ie odd
= -2/pi(sin 2 pi+ sin 6 pi/3 + sin 10 pi/5 . . .
. .
f(x)= 1/2 – 2/pi ( sin 2 pi+ sin 6 pi/3 + sin 10 pi/5 . . . .
. . . . infinity
now the function f(x) which is determined by the Fourier series
will overlap the digital function f(x) as shown in above graph and
it will superpose it. Thus the digital signal is converted into sinusoidal
electrical signal and data is send from one computer to the other by electrical
cables. |