Orthogonality of subcarriers
 Note that each subcarrier has exactly an integer number of
cycles in the interval T, and the number of cycles between
adjacent subcarriers differs by exactly one.
 This property accounts for the orthogonality between the
subcarriers, shown as follows:

t s T
ts

k
exp(  j 2 (t  t s ))  s (t ) dt
T
t s T
ts
Ns
1
2
k
i
exp(  j 2 (t  t s ))  d N s exp( j 2 (t  t s )) dt
i
T
T
N
2
i  s
2

Ns
1
2

N
i  s
2
d
i
Ns
2

t s T
ts
exp( j 2
ik
(t  t s )) dt  d N s T
k
T
2
Wireless Communication Technologies 2.5.2
1
Orthogonality of subcarriers
 The orthogonality of the different OFDM subcarriers can also
be demonstrated in the spectrum.
 The spectrum of a single symbol is a convolution of a group
of Dirac pulses located at the subcarrier frequencies with the
spectrum of a square pulse that is one for a T-second period
and zero otherwise.
f1  1Hz
1
 The amplitude spectrum of
the square pulse is equal to
Sinc (fT ), which has zeros
for all frequency f that are
an integer multiple of 1/T,
as shown in Figure.
0.8
f 2  2 Hz
0.6
f 3  3Hz
0.4
f 4  4 Hz
0.2
0
-0.2
0
2
4
Wireless Communication Technologies 2.5.2
2