List of Figures
1.1 State transition diagram of a Markov chain with transition probabilities,
p
i, j
,wherei an d j ∈ [0,w]..............................................................................23
1.2 Typical relationship of two symmetric Gaussian conditional probability
density functions, p(x|0) an d p(x|1),whereZ
opt
th
is the optimal decision
threshold [21, 26]. ........................................................................................... 28
1.3 Gaussian error probability of (unipolar) 1-D prime codesand(bipolar)
Gold sequence s for various code length N..................................................... 30
1.4 Hard-limiting of 5 unipolar codewords of weight 5 with uneq ual pulse
height. ............................................................................................................. 32
1.5 State transition diagram of a Markov chain with transition probabilities,
p
i, j
,wherestatei represents that i pulse positions in the address code-
word (of weight w)ofahard-limitingreceiverarebeinghit[34]..................3 2
1.6 Gaussian, soft-limiting, and hard-limiting error probab ilities of the 1- D
prime codes for p = {13,17}......................................................................... 35
1.7 Cross-correlation functions between two 1-D codewords,10110010000
and 10011000100, u n der the chip-synchronous and chip-asynchronous
assumptions [21, 35, 37]. ................................................................................ 36
1.8 Soft-limiting error probabilities of the 2-D carrier-hopping prime codes
under the chip-synchronous and chip-asynchronous assumptions for
code length N = {49,121}and weight w = {7, 11}.......................................39
1.9 State transition diagram of the Markov chain with transition probabil-
ities p
i, j
,wherestatei represents i pulse sub-positions in the address
codeword (of weight w)ofahard-limitingreceiverarebeinghit[37]...........42
1.10 Hard-limiting error probabilities of the 2-D carrier-hopping prime codes
under the chip-synchronous and chip-asynchronous assumptions for
length N = {25,49} and weight w = {5, 7}...................................................45
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}.......................................................................................................46
2.1 A typical coding-based optical system model. ............................................... 55
2.2 Signal formats at various stages of an optical transmitter. ............................. 56
2.3 Tunable optical encoder in a parallel coding configuration [2, 33].................57
2.4 Tunable optical encoder in a serial coding configuration [33, 41, 42]. ...........58
2.5 Tunable incoherent optical encoder in an improved serialcodingconfig-
uration [33, 41, 42]. ......................................................................................... 59
2.6 Temporal phase coding: (a) tunable encoder; (b) tunable decoder [10–
13, 15, 16]. ....................................................................................................... 60
2.7 Spectral phase coding in free space [19–21, 35]............................................. 61
xi
xii List of Figures
2.8 Spectral phase coding in (a) waveguide and (b) fiber [21, 30, 34–36, 86].......62
2.9 Spectral amplitude coding in (a) waveguide and (b) fiber. ............................ 63
2.10 Spectral-amplitude differential receiver for the maximal-length sequence
(+1,+1,+1,−1 , −1,+1, −1),where
α
is the output ratio of the power
splitter, which is equal to 1 when orthogonal bipolar codes are in use
[16–18, 5 1, 52].................................................................................................64
2.11 Example of the spectral-amplitude decoding process with zero mutual
interference, where the maximal-length sequences of length7areused.......64
2.12 Spatial-temporal amplitude coding in multiple fibers and star couplers. .......66
2.13 Example of the encoding and decoding processes of 2-D binary pixels in
spatial-temporal amplitude coding, where black squares represent dark
pixels and the number in each square of the 2-D correlation functions
represents the darkness level after correlation. ...............................................67
2.14 Spectral-temporal-amplitude encoders in (a) waveguide and (b) fiber
[34, 36, 78, 81, 82]............................................................................................68
2.15 Timing diagrams of three types of multirate, multimediaservicessup-
ported by multilength coding..........................................................................70
2.16 Example of PPM coding with 4 symbols, represented by 4 PPMframes,
where T
b
is the bit period and R is the bit rate [70, 71, 73]. ............................72
2.17 Example of multicode keying with 4 symbols, representedby4distinct
codewords, where T
b
is the bit period and R is the bit rate [69, 75, 76]. .........72
2.18 Example of shifted-code keying with 4 symbols, represented by shifting
acodewordtooneofthefourtimepositions,whereT
b
is the bit period
and R is the bit rate [69, 74]. ...........................................................................73
2.19 Tunable transmitter and receiver for shifted-code keying with time-
shifted codewords. .......................................................................................... 73
2.20 Tunable transmitter and receiver for shifted-code keying with wavelength-
shifted codewords. .......................................................................................... 74
2.21 Wavelength-aware hard-limiting detector for 2-D wavelength-time codes
[91].................................................................................................................. 75
2.22 2-D tunable wavelength-time encoder with in-fiber FBGs [81]. ....................77
2.23 Tunable AWG-based encoder for 1-D spectral (phase and amplitude)
coding and 2-D spectral-temp oral coding.......................................................78
2.24 Tunable AWG-based optical balanced receivers for 1-D spectral (phase
and amplitude) coding and 2-D spectral-temporal coding. ............................80
3.1 Examples of (a) autocorrelation functions of the original p rime code-
word C
1
and ( b) cr oss-correlation fu n ction s of C
1
and C
2
,bothover
GF(7) and of length 49 and weight 7, f or the tr ansmission of data-bit
stream 1011001110 in OOK........................................................................... 92
3.2 Error probabilities of the original prime codes over GF(p) for N = p
2
and w = p = {11, 17}.....................................................................................95
List of Figures xiii
3.3 Examples of (a) autocorrelation functions of the extended prime code-
word E
2
and ( b) cross-correlation function s o f E
2
and E
3
,bothover
GF(5) and of len gth 45 and weight 5 , for the transmission of data-bit
stream 1011001110 in OOK...........................................................................98
3.4 Error probabilities of the extended prime codes over GF(p) for N =
p(2p −1 ) and w = p = {7,11}....................................................................100
3.5 Chip-synchronous, hard-limiting error probabilities of the original and
extended prime codes over GF(p) for N = p
2
and N = p(2p −1),re-
spectively, and w = p = { 7,11}...................................................................101
3.6 Error probabilities of the generalized prime codes over GF(p) for N =
p
k+1
and w = p = 11 and k = {1,2}............................................................106
3.7 Error probabilities of the 2
n
prime codes over GF(p) for N = p
2
, w = 2
n
,
p = {11, 19},andn = 3................................................................................116
3.8 Error probabilities of the (N, w, 1,1) and (N,w, 1,2) OOCs for N =
{73,241} and w = {3, 4,5}..........................................................................123
4.1 Examples of (a) autocorrelation functions of the synchronous prime
codeword C
1,2
;(b)cross-correlationfunctionsofC
1,2
and C
1,4
(from the
same subset); and (c) cross-correlation functions of C
1,2
and C
2,1
(from
two different subsets), all over GF(7) and of length 49 and weight 7, for
the transmission of data-bit stream 1011001110 in OOK. The vertical
grid lines indicate the in-phase positions...................................................... 131
4.2 Chip-synchronous, soft-limiting error probabilities of the asynchronous
and synchronous prime codes over GF(p) for w = p = {11,17,23}...........133
4.3 Gaussian error probabilities of the asynchronous and synchronous prime
codes over GF(p) for w = p = {11,17,23}.................................................135
4.4 Tree structure of the synchronous multilevel prime codesoverGF(p) of
aprimep and a positive integer n,whereΦ is the code cardinality in
each subset. ................................................................................................... 137
4.5 Chip-synchronous, soft-limiting error probabilities of the synchronous
trilevel prime codes and the asynchronous prime codes, both over GF(p),
for w = p = {7, 11}......................................................................................14 2
4.6 Example of three types of services supported by O-TDMA andsyn-
chronous O-CDMA in OOK......................................................................... 145
5.1 A carrier-hopping prime codeword of weight 5 with 5 wavelengths and
35 time slots. Each dark square indicates the chip (time) location and
transmitting wavelength of an optical pulse in the codeword.......................151
5.2 Chip-synchronous, soft-limiting error probabilities of the carrier-
hopping prime codes with k = 2, L = w,andN = p
1
p
2
for p = p
1
=
p
2
= {11,13,17}and w = {p −2 , p}...........................................................154
5.3 Chip-synchronous, soft-limiting error probabilities of the carrier-
hopping prime codes and the 1-D prime codes (in hybrid WDM-coding),
both with L = w = p and N = p
2
for p = {7,11}........................................155
xiv List of Figures
5.4 Tree structure of the multilevel carrier-hopping prime codes over GF(p)
of a prime p and positive in tegers n and k,where
λ
c
is the maximum
periodic cross-correlation function and Φ is the code cardinality in each
subset. ...........................................................................................................158
5.5 Chip-synchronous, hard-limiting error probabilities of the multilevel
carrier-hopping prime co d es with k = 1andN = 89 for n = {1, 2,3}
and L = w = {11,14,17}..............................................................................163
5.6 Chip-synchronous, hard-limiting error probabilities of the trilevel
carrier-hopping prime codes with L = w = 7andN = p
k
for p =
{7,11,13, 17} and k = {1,2,3 }....................................................................164
5.7 Chip-synchronous, hard-limiting error probabilities of the bilevel
carrier-hopping prime codes with k = 1, L = w = 8, and N = 23, op-
timized as a function of K............................................................................. 165
5.8 Chip-synchronous, hard-limiting bit error probabilities of the carrier-
hopping prime codes of L = w = 10 and N = 97 in OOK and 2
m
-ary
shifted-code keying for m = {1, 2,3,4}........................................................170
5.9 Chip-synchronous, hard-limiting bit error probabilities of the time-
shifted carrier-hopping prime codes of L = w = 10 and N = 97 in 2
3
-ary
shifted-code keying for x = {0, 1,2,3}.........................................................172
5.10 SE versus bandwidth-expansion factor LN of the 2
m
-ary wavelength-
shifted and OOK carrier-hopping prime codes, where L = w = 10, m =
{1,2, 3},andP
e
≈ 10
−9
................................................................................ 176
5.11 Number of simultaneous users K versus bandwidth-expansion factor LN
of the 2
m
-ary wavelength-shifted and OOK carrier-hopping prime codes,
where L = w = 10, m = {1,2, 3,4},andP
e
≈ 10
−9
....................................177
5.12 Chip-synchronous, soft-limiting error probabilitiesoftheextended
carrier-hopping prime codes for various pr ime num b er s and k = {2,3}......183
5.13 Chip-synchronous, soft-limiting error probabilitiesoftheexpanded
carrier-hopping prime codes and the car r ier-hopping prime codes in h y -
brid WDM-coding with p
$
groups of wavelengths.......................................188
5.14 Soft-limiting error probabilities of the expanded carrier-hopping prime
codes with and without the chip-synchron o us assumption for w = {3,5 },
p
$
= {3,5,7},andN = {23,41, 83} ............................................................. 191
5.15 Chip-synchronous, hard-limiting error probabilitiesofthequadratic-
congruence car r ier-hopping prime codes and the carrier-hopping prime
codes of both L = w = 9andN = {19,47,103}..........................................195
5.16 Chip-synchronous, hard-limiting error probabilitiesofthequadratic-
congruence carrier-hopping prime codes of L = w = 9andN = 103 in
OOK and multicode keying for m = {1, 2,3,4}...........................................199
5.17 Chip-synchronous, hard-limiting error probabilitiesofthequadratic-
congruence carrier-hopping prime codes of L = w = 9andN = 103 in
shifted-code keying and multicode keying for m = {1, 2,3}........................ 200
List of Figures xv
5.18 Chip-synchronous, hard-limiting error probabilitiesofthequadratic-
congruence carrier-hopping prime codes in 2
m
-ary multicode keying
and the carrier-hopping p r ime codes in OOK, of both L = w = 10 and
N = {289,529},form = {3,4 }....................................................................201
5.19 Chip-synchronous, hard- and soft-limiting error probabilities of the (p ×
N, w, 1,1) prime-permuted codes with the time-spreading (N,w,1, 1)
OOCs for p = {11,13,23}, N = {41,61},andw = {4, 5,6}.......................211
5.20 Chip-synchronous, hard-limiting error probabilitiesofthe(p ×
N, w, 2,2) prime-permuted codes with the time-spreading (N,w,2, 1)
OOCs for p = {5,7,17}, N = {13,19,27,199},andw = {5,6,7 }.............214
5.21 Chip-synchronous, hard-limiting error probabilitiesofthe(p ×
N, w, 2,2), (p ×N,w,1, 2),and(p ×N, w,1,1) prime-permuted codes for
p = {5,11,13,17}, N = {13,41,73,111},andw = {3,4,5, 7,9}...............217
5.22 A 2-D prime-permuted codeword obtained by putting (a) the multi-
wavelength codewords C
0
, C
2
,andC
4
,fromthemaximal-lengthse-
quences in Table 5.8, onto nonzero time slots of the time-spread ing
(7,3,1, 1) OOC cod eword 1011000 and (b) the multiwavelength code-
words C
0
, C
1
, C
2
,
C
3
,andC
4
,fromthemodifiedmaximal-lengthse-
quences in Table 5.10, onto time slots of the time-spreading Barker se-
quence (+1 , +1,+1, −1,+1),whereL is the number of wavelengths,
which depends on the length of the multiwavelength codewords. ............... 219
5.23 Chip-synchronous, hard-limiting error probabilitiesoftheOOK-based
(p ×N, w, 1,1) prime-permuted codes with the (N, w, 1,1) OOCs for var-
ious N, w,andp............................................................................................226
5.24 Chip-synchronous, soft-limiting error probabilitiesoftheCIK-based
(L × N,w,1,1 ) prime-permuted codes with the time-spreading Barker
sequence of length N = w and the OOK-based (L ×N,w,1,1 ) prime-
permuted codes with the time-spreading (N,w,1, 1) OOCs for L = p + 1
and various N, w,andp................................................................................229
5.25 Chip-synchronous and chip-asynchronous, Gaussian error probabilities
of the CIK-based (L ×N,w,1, 1) prime-permuted codes with the Gold
sequences for L = p + 1andN = w = p = {7,31,63}................................233
5.26 Chip-synchronous, hard-limiting error probabilitiesofthe(p ×
N, w, 2,2) QC-permu ted codes with the time-spreading (N,w,2,2) OOCs
for various p, N,andw.................................................................................2 3 9
5.27 Chip-synchronous, hard-limiting error probabilitiesofthe(p ×
N, w, 2,2) QC-permuted codes and the (p ×N,w,1, 1), (p ×N, w, 1,2),
and (p ×N,w,2, 2) prime-permuted codes for various p , N,andw.............240
6.1 Chip-synchronous, soft-limiting error probabilities of the asynchronous
and synch r o nous original carrier-hopping prime codes f o r k = 1, L =
w = 10, and N = p
1
= {37,101}..................................................................256
6.2 Tree structure of the synchronous multilevel carrier-hopping prime
codes over GF(p) of a prime p an d positive integers n an d k,where
Φ is the code cardinality in each subset........................................................258
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