Page 543 - Fiber Optic Communications Fund

524 Fiber Optic Communications

As discussed in Section 11.6.1, the tap weights for the next iteration should be chosen in a direction opposite
to the gradient vector,
∗
W [k] (n+1) = W [k] (n) + x,out [n − k]e [n]Δ, (11.165)
xx
xx
x
∗
W [k] (n+1) = W [k] (n) + y,out [n − k]e [n]Δ. (11.166)
xy
x
xy
Similarly, the tap weights W [k] and W [k] are altered as
yy
yx
W [k] (n+1) = W [k] (n) + ∗ [n − k]e [n]Δ, (11.167)
yy yy y,out y
∗
W [k] (n+1) = W [k] (n) + x,out [n − k]e [n]Δ. (11.168)
yx
yx
y
Example 11.5
Show that
exp (̂x) ⋅ exp (−̂x)= I, (11.169)
where ̂x is any operator and I is an identity operator.
Solution:
Expanding exp (±̂x) in a Taylor series, we find
̂ x ⋅ ̂x
exp (̂x)= I + ̂x + +··· (11.170)
2!
̂ x ⋅ ̂x
exp (−̂x)= I − ̂x + +··· (11.171)
2!
Now consider the product
( ̂ x ⋅ ̂x ) ( ̂ x ⋅ ̂x )
exp (̂x) ⋅ exp (−̂x)= I + ̂x + +··· ⋅ I − ̂x + +···
2 2
( I ⋅ ̂x ⋅ ̂x ̂ x ⋅ ̂x ⋅ I )
= I +(̂x ⋅ I − I ⋅ ̂x)+ − ̂x ⋅ ̂x + +···
2 2
= I. (11.172)

Exercises
11.1 Explain the phase increment algorithm for IF estimation.

11.2 Discuss the phase-unwrapping techniques used in phase compensation.

11.3 Write a computer program to compensate for IF and laser phase noise in a back-to-back configuration
with the following parameters: transmitter laser linewidth = 5 MHz, LO linewidth = 10 MHz, f =
IF
200 MHz, symbol rate = 25 GSym/s, modulation = NRZ-QPSK. Determine the optimum block size.
11.4 Discuss the advantages and disadvantages of CD compensation in the time domain and the frequency
domain.