Page 422 - Fiber Optic Communications Fund

Channel Multiplexing Techniques 403

where T is the symbol period and d n0 is a constant. Let f , f , … , f be the carrier frequencies. The carrier
2
s
N
1
exp (i2f t) is modulated by the data stream d , n = 1, 2, … , N and then the modulated signals are combined
n
n
to obtain the transmitted signal.
N
∑
s (t)= d (t) exp (i2f t). (9.72)
tr n n
n=1
At the receiver, the received signal is multiplexed by the bank of local oscillators and integrators. In the
absence of channel distortion and noise effects, the signal after the integrator can be written as
1 T s
̂ m T ∫ 0 s (t) exp (−i2f t)dt, m = 1, 2, … , N (9.73)
d =
m
tr
s
N
1 T s ∑
= d (t) exp [i2(f − f )]dt. (9.74)
n
n
m
T ∫ 0 n=1
s
The integrator is nothing but a low-pass filter and the terms oscillating at frequencies f − f , n ≠ m do not
n m
contribute significantly if f − f is larger than 1∕T . Therefore, a significant contribution comes only from
n m s
the d.c. term corresponding to n = m in Eq. (9.74), leading to
̂ (9.75)
d ≅ d .
m m 0
One of the disadvantages of this approach is that frequency separation between carriers should be sufficiently
large so that the contributions from the cross-terms at frequencies f − f , n ≠ m in Eq. (9.74) are small. This
n m
leads to excessive bandwidth requirements. Besides, the transmitter and receiver require a bank of analog
oscillators, product modulators, and filters, increasing the complexity of the architecture.
The bandwidth can be utilized efficiently if the carriers are orthogonal. Suppose we choose the carrier
frequencies such that
{
1 T s 1if m = n
m
n
T ∫ 0 exp [i2(f − f )t]= 0 otherwise. (9.76)
s
Now, the carriers are said to be orthogonal over the interval [0, T ].If
s
f = m∕T , m = 1, 2, … , N, (9.77)
s
m
it can easily be verified that Eq. (9.76) is satisfied. Therefore, the carrier frequencies should be integral multi-
ples of the symbol rate (= 1∕T ). For example, the first carrier is a sinusoid with period T , the second carrier
s s
is a sinusoid with period T ∕2, and so on. Using Eqs. (9.71) and (9.77), Eq. (9.72) may be rewritten as
s
N ( )
∑ i2nt
s (t)= d exp , 0 < t < T . (9.78)
tr n 0 T s
n=1 s
If we discretize the time interval
t = kΔt, k = 1, 2, … , N, (9.79)
where Δt is the sampling interval with
NΔt = T , (9.80)
s
Eq. (9.78) is modified as
N
∑ [ i2kn ]
s (kΔt) ≡ s = d exp , k = 1, 2, … , N. (9.81)
k
tr
n 0
n=1 N