46 Optical Coding Theory with Prime
general, the error probabilities of both assumptions improve as w or N increases
due to higher autocorrelation peaks or lower hit probabilities. However, their error
probabilities get worse as K increases because of stronger mutual interference. As
expected, the performance of the chip-asynchronous case is always better than the
chip-synchron o us case because the latter gives the perfo r mance upper boun d. The
difference in their error probabilities increases with w or N and is fo und to be about
two to three orders of magnitude in this example. To validate the accuracy of the
hard-limiting, chip-asynchronous analysis, the results ofcomputersimulationare
also plotted in the figure and found to closely match the chip- asyn chronous curves.
The computer simulation is performed similar to that of Figure 1.8, but nonzero am-
plitude levels of the multiplexed signal at any time instant is clipped to the same level
in order to emulate the hard-limiting operation.
5 10 15 20 25 30 35 40 45 50
10
−12
10
−11
10
−10
10
−9
10
−8
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
Error probability P
e,asyn
Number of simultaneous users K
Simulation (hard limiting)
Soft limiting
Hard limiting
(w,N ) = (5,25)
(5,25)
(7,49)
(7,49)
FIGURE 1.11 Chip-asynchronous error probabilities of the 2-D carrier-hopping prime codes
with and without hard-limiting for length N = {25, 49} and weight w = {5,7}.
Figure 1.11 plots the soft- and hard-limiting error probabilities [from P
e,asyn
in
Equation (1.5) and P
e,asyn,hard
in Equation (1.6), respectively] of the 2-D carrier-
hopping prime codes in Section 5.1 against the number of simultaneous users K un-
der the chip-asynchronous assumption. The same code parameters as in Figure 1.10
are used. As expected, the har d-limiting case always resultsinbetterperformance
than the soft-limiting case, and the difference in their error probabilities is about
three to four orders of magnitude in this example.
Fundamental Materials and Tools 47
1.8.7 Spectral Efficiency
In addition to error pro bability, spectral efficiency (SE) isanotherfigureofmeritfor
comparing optical codes. SE considers data transfer rate R
bit
(in bits/s), number of
simultaneous users K (for a given error probability), number of wavelength channels
L (in Hz), and number of time slots (or code length) N as a whole, and is gener ally
defined as [38–43]
SE (bits/s/Hz)=
Aggregated data transfer rate
Total transmission bandwidth
=
KR
bit
LΛ
=
KR
chip
LNΛ
where Λ is the wavelength spacing per wavelength channel, includingguardbands,
and R
chip
= NR
bit
is the chip rate. The target is to obtain the SE as large as possible
for high efficiency o r good bandwidth utilization. The definition assumes the use of
2-D codes, but also applies to 1-D codes by setting L = 1.
1.9 SUMMARY
In this chapter, various linear algebraic tools that are essential to th e construction
and analyses of bipolar and un ipolar (also kn own as coheren t an d incoher en t) o pti-
cal cod es were reviewed and developed. The definitions of Hamming distance and
weight, correlation functions, and cardina lity upper bound, which are useful for code
classifications, were established. The concept and application of the Markov chain
in optical coding theory were introduced. The use of Gaussianapproximationto
simplify performance analysis of unipolar and bipolar codeswasstudied.Theper-
formances of optical codes in a soft- and hard-limiting receiver were analyzed using
combinatorial methods. The hard - limiting receiver was shown to per form better than
the soft-limiting receiver. Finally, the performances of optical codes in both types
of receivers without the classical pessimistic chip-synchronization assumption were
investigated. Reflecting the actual effect of mutual interference, the more realistic
chip-asynchron o us assumption was shown to give m ore accurate code performance.
The theoretical analytical models were validated by computer simulations.
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