Download Optical Orthogonal Code Division Multiple Access System, Part II

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO. 8, AUGUST 1994
2592
Optical Orthogonal Code-Division Multiple-Access
System-Part 11: MultibWSequence-Period
OOCDMA
Hyuck M. Kwon, Member, IEEE
1. The possible number of cyclic shifts is the length of a
sequence. Hence, log, F bits where F is sequence-length, can
be transmitted per sequence-period. Furthermore, the bit error
probability can be improved significantly compared to onebidsequence-period transmission because the final decision is
not based on a fixed threshold but on the peak correlation
position.
The goal of this paper is to derive the bit error probability
of the multibits transmission per sequence-period OOCDMA
system, using Abshire's Gaussian approximation [8] of the
APD output (as it was used for the analysis of the onebidsequence-period OOCDMA system in the presence of APD
and thermal noise [7]).
It has been shown [ 2 ] ,[3] that an optical hard-limiter before
the receiver correlator can reduce significantly the interference
effect for the ideal one-bidsequence-period OOCDMA system
in the absence of thermal and APD noise. However, in the
presence of thermal noise and APD noise, it was shown
that the improvement from using a hard-limiter for the onebidsequence-period transmission OOCDMA system is not sigI. INTRODUCTION
nificant [7].The second goal of this paper is to study the optical
N THE OOCDMA code of References [1]-[7], only one hard-limiter role in the presence of thermal and APD noise
bit of user's data is transmitted per sequence-period, and a for the multibitshequence-period transmission OOCDMA systhreshold is employed for the final bit decision. The system tem. In this paper the chip-synchronous OOCDMA system is
introduced in this paper achieves a higher system throughput considered. Section I1 described the multibits per sequenceas well as a better bit error probability than a one-bidsequence- period transmission OOCDMA system model. Section I11
period OOCDMA system, by mapping multibits of user data presents the analysis of the no hard-limiter case. Section IV
into a shifted version of the OOCDMA sequence. The shifted presents the analysis of the multibits/sequence-period transversion can be transmitted and detected by correlating the mission OOCDMA system with hard-limiter, while Section V
received signal in parallel with locally generated shifted ver- provides numerical results and Section VI gives conclusions.
sions of a user's sequence and using a maximum-correlation- The appendix derives the probability density functions of the
location (instead of using a threshold) in the final decision, total interference contribution to the accumulated final decision
because the autocorrelation between a shifted and an unshifted random variable at the correlation time and noncorrelation
version of an OOCDMA sequence is bounded by I and also times. These probability densities are used for the analyses
the cross-correlation between a shifted version of a sequence in Sections I11 and IV.
and any shift of another OOCDMA sequence is bounded by
Abstract- In a recently proposed optical orthogonal code
division multiple-access (OOCDMA) system, one bit of user's
data is transmitted per sequence-period, and a threshold is
employed for the final bit decision. In this paper, a system that
can transmit multibits per sequence-period is introduced, and
avalanche photodiode (APD) noise, thermal noise, and interference, are included. This system, derived by exploiting orthogonal
properties of the OOCDMA code sequence and using a maximum search (instead of a threshold) in the final decision, is
log, F times higher in throughput, where F is sequence-period.
For example, four orders of magnitude are better in bit error
probability at -56 dBW received laser power, with F = 1000
chips, 10 "marks" in a sequence, and 10 users of 30 Mb/s data
rate for one-bit/sequence-period and 270 Mb/s data rate for
multibitdsequence-periodsystem. Furthermore,an exact analysis
is performed for the log,F bitdsequence-period system with a
hard-limiter placed before the receiver, and its performance is
compared to the performance without hard-limiter, for the chipsynchronous case. The improvement from using a hard-limiteris
significant in the log, F bitdsequence-period OCCDMA system
(while it is not in a one-bit/sequence-period OOCDMA system
in [7]).
I
Paper approved by M. S. Goodman, the Editor for Optical Switching of
the IEEE Communications Societv. ManuscriDt received Februarv 8. 1991:
revised July 10, 1992 and Mar& 12, 1993,'This work was supported by
NASNJohnson Space Center under Contract NAS9-17900. This paper was
Dresented in art at the Intemational Conference on Communications. Denver.
kolorado, Juke 23-26, 1991, and Lasers and Electro-Optics Society (LEOS)
Summer Topical on Optical Multiple Access Networks, Monterey, CA, July
25-27, 1990.
The author is with the Department of Electrical Engineering, Wichita State
University, Wichita, KS 67206 USA.
IEEE Log Number 9401954.
11. SYSTEM MODEL
F~~ N transmitter and receiver pairs, the optical signal
can be written as
(baseband) Of the nth
s n ( t ) = Pc,(t - /;'Ic),
0
5t 5T
= FT,
(1)
for multibits transmission per sequence-period where P is a
user's received laser power and cn(t ),'!!:I
is the 1: shift
0090-6778/94$04.00 0 1994 IEEE
KWON OPTICAL ORTHOGONAL CDMA SYSTEM-PART
I1
2593
version of the nth user signature sequence cn(t)corresponding
to a block of log, F bits of the nth user’s (0, 1) binary data.
The form of signature signal c n ( t ) is a train of (0, 1) binary
rectangular pulses with chip interval T, and length F .
The received signal is modeled as
N
WTICAL FIBER ONE
CHIP DELAY LOOP
N
l 3 28
APD
...
(TAPCONNECTIONS)
APD
...
n=l
where T, is the associated delay for a given receiver. We
assume that the detection system is synchronous with the first
user and all delays are relative to the first user delay (where
TI = 0) as in the Part I of this paper [7]. Furthermore, there is
no loss of generality in assuming T~ E (0, T ) for 2 5 n 5 N
since we are concerned only with the time delays modulo
symbol time T .
Fig. l(a) shows a block diagram of the multibitdsequenceperiod transmission system OOCDMA receiver of the no
hard-limiter case for user 1, using the passive optical tapped
delay lines [9]-[lo]. Fig. l(b) is another implementation
equivalent to Fig. l(a), using the active optical device. Fig. l(a)
is used for the illustration of the receiver below, and Figure
l(b) will be used in Sections I11 and IV for the analysis.
There are F - 1 optical fiber chip delays to implement F
correlators in parallel. For illustration, the signature sequence
length F is chosen to be 32. Let “mark’ and “space” mean
“1” and “0,” respectively, in a signature sequence. The marks
are located on 0, 9, 12, and 27 chip positions in the first
user’s unshifted signature sequence. Optical fiber delay lines
incoherently combine tapped signals, resulting simply in the
summation of optical power. The tap connections are indicated
in Fig. l(a) to generate F shift versions of the first user’s
signature sequence, cl(t - W,),
1 = 0 , l , . . ‘ , F - 1. For
example, the tap connections are (0, 9, 12, 27) for correlation
with c l ( t ) , (1, 10, 13, 28) for correlation with q ( t - T,),
and (31, 8, 11, 26) for correlation with cl(t - ( F - 1)T,) =
cl(t - 31T,). The background light n b ( t ) is added before
the APD. Each of the F number of APDs converts incident
photons to photoelectrons with quantum efficiency 7,and each
of the F correlators integrates the APD output over the code
symbol interval T = FT, and takes a sample every code
symbol time interval T to produce the correlation output,
e.g., Zl(1) at the Zth correlator. The receiver is assumed to
be synchronized with the transmitter at each symbol decision
time. The integrated output is
1I‘
Zl(1) = -
APD ( ~ ( t ) ~-l lT,)
(t
(a)
N
11’ 2;
(b)
Fig. 1. (a) Multibitskequence-period transmission system OOCDMA pa
ssive receiver for user 1 with an unshifted signature sequence of marks on (0,
9, 12, 27) chip positions and F = 32 sequence length. The F correlations
are performed in parallel. (b) Multibitdsequence-period transmission system
OOCDMA active receiver for user 1 with an unshifted signature sequence of
marks on (0, 9, 12, 27) chip positions and F = 32 sequence length. The F
correlations are performed in parallel.
and chooses “1” if Z1 is greater than a threshold, Th, and
chooses “0” otherwise, see [7, Fig. 21.
111. MULTI~ITS/SEQUENCE-PERIOD
OOCDMA SYSTEMWITHOUT
HARD-LIMITER
+ n b ( t ) ) dt,
1=0, l , . - . , F - l
t
l q F BITS OUTPUT
(3)
at the correlation time 1 = O , . - . , F - 1, for a multibits
transmission per sequence-period where APD(z) is the APD
output due to the number of incident photons z. The receiver
performs F correlations in parallel and selects the correlation
location I* at which Z l ( l * )is maximum, and decodes 1* into
the corresponding log, F bits (Fig. 1) (while the receiver for
the one-bit transmission per sequence-period performs one
correlation at the proper correlation time I* and yields 21,
Let the first user receiver be the desired receiver. The desired
receiver is looking for the correlation location at which the
accumulated output Zl is maximum among F accumulated
outputs 21(0), . . . , & ( F - 1).Let E* be the proper correlation
location. Then the accumulated output at the proper correlation
location Zl(Z*)will be maximum when there is no noise and
no interference in the channel. If the channel is operating in
noise or interference, any of the accumulated outputs at the
noncorrelation times, Z l ( l ) ,E # E * , 1 = O , . . . , F - 1, can be
larger than the accumulated output at the proper correlation
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO. 8. AUGUST 1994
2594
location Zl(Z*). Then, a symbol (which carries log, F data
bits) is in error. Hence, the probability of symbol error is
Ps(e)= Pr [Zl(l*) is not maximum among
Z 1 ( 0 ) , . . . , Z 1 ( F - 111. (4)
FCORREUTOR SEOUENCESFOR USER 1 RECEIVER
cl“’
This symbol error probability can be found from 1 minus the
symbol correct probability P,(C); P s ( E ) = 1 - P,(G). The
symbol is correct if the accumulated output at the proper correlation time is maximum. Thus, the symbol correct probability
can be written as
P J C ) = pr
n
[l=:;:l*
{z1(1*)
2 zl(l))
]
(5)
where the notation fl means the intersection of events. The
accumulated outputs Zl(l), I = 0, . . . , F - 1, are independent
because photons are incident upon the avalanche photodiode
independently. The Z1
(1) are, however, not identical because
they depend on the interference pattern. The accumulated outputs after the APD, Zl(Z), 1 = 0, . . . ! F - 1, are assumed to be
Gaussian random variables by using Abshire’s approximation
[8]. This assumption is an asymptotic approximation and its
region of the validity is given in [8]. The mean and variance
of Zl(l) can be computed with the APD parameters if the
interference contribution 11( I ) is known. With these mean
and variance, the symbol correct probability in (5) can be
computed.
Fig. 2 illustrates an example of correlation values Zl(1) at
the first receiver for an ideal (noiseless) channel in Fig. l(b)
when there are only two users with unit laser power, and user 1
and user 2 are sending sequences cl(t) and c2(t-3Tc). For this
ideal case, the accumulated output 21( I ) at the noncorrelation
time 1, 1 # I * ! I = 0, ... , F - 1, is the cyclic correlation
value between the received signal and the desired receiver
sequence, and it is the total undesired interference plus the
desired signal contribution, I l ( l ) .For example, &(I) = 11(1)
is 2 at I = 12 because the first mark of the local sequence
cl(t - 12Tc) is hit by the third mark of the second user’s
transmitted signature sequence c2(t - 3Tc), and the second
mark of the local sequence c1( t- 12Tc) is hit by the third mark
of the first user’s transmitted signature sequence c1 (t), and the
third and the fourth mark of the local sequence cl(t - 12Tc)
are not hit.
The desired receiver’s accumulated output Zl(I*) at the
correlation time I* for the ideal channel is the number of
marks in the user signature sequence K plus the total undesired interference contribution 11(1*). For example, in Fig.
2, Zl(l* = 0) = K 11(1*) is 4 because the total number of
marks in the desired sequence K is 4 and the total undesired
interference contribution 11( I * ) is 0.
The unconditional symbol-correct probability in (5) can
be expressed by averaging the conditional symbol-correct
probabilities over the interference pattems. However, there are
too many interference pattems to count, and in (5) there are
too many random variables 21( I ) which need to compare with
Zl(l*). For large F , e.g., F = 1000, the direct calculation of
the unconditional symbol-correct probability is not feasible.
Fortunately, however, a lot of 21(1) within the F accumulated
+
0
Cl
5
10
15
20
25
30 32
(I - 31 T,)
IIIIIIII
0
5
I
10
I I I I I I I I I I I I I
15
20
25
I l l
3U 32
Fig. 2. An example of correlation values Zl(Z) at the first receiver for an
ideal channel. User 1 and user 2 are sending sequences c1 (t) and cz (t - 3Tc),
respectively. Active optical device implementation in Fig. l(b) is assumed.
The associated relative delay 72 for user 2 is 0. The shifts of the sequence
corresponding to the log, F bits are 1; = 0 and 1; = 3 for user 1 and user
2, respectively.
outputs are identical because the interference contribution
11(1) repeats many times. This reduces the F number of
comparisons in (5) significantly.
For example, in Fig. 2, 11(1) = 0 repeats nine times,
Il(1)= 1 repeats 16 times, and I1(1) = 2 repeats six times
within F = 32 number of correlations. Let n, denote the
number of repetitions of an off-peak correlation 11 (1) within
a sequence period F for which Il(Z)is equal to m. The Il(1)
can be between 0 and 2N - 1 for N users. In the appendix,
the probability density function of 11( I ) denoted by p ~ ~ ( q ( m ) ,
1 # 1*, I = 0, ‘ . . , F - 1, is derived. Using this probability
density function of 11(1),the average number of repetitions n,
for 11( I ) = m within a sequence period F is approximated by
nm = ( F - 1)Prob(11(1) = m ) = ( F - l)pIl(l)(m). (6)
In (6), ( F - 1) is used instead of F because the proper
correlation time 1’ is excluded in the counting of repetitions.
Using the observation that Il(1) = m repeats nm times
on average within F number of correlations, see (6), the F
number of comparisons in (5) can be significantly reduced to
at most 2N - 1 comparisons, because the sequence length F is
much larger than the number of users N and the 11( I ) can be
between 0 and 2N - 1 for N users. The conditional symbolcorrect probability given 1,( I * ) can be approximated as
m=O
where the fact that all Zl(Z) are statistically independent is
used and
is the product notation.
If 1l(l) is equal to m, then m marks arrive at A, incident
photon arrival rate, and K N - m spaces arrive at A,/M,
incident photon arrival rate for the desired receiver’s accumulated output Zl(1). Thus, the mean and variance of Zl(I) in a
practical (nonideal) channel are, respectively
n
KWON OPTICAL ORTHOGONAL CDMA SYSTEM-PART I1
2595
IV. MULTIBITS/SEQUENCE-PERIOD
OOCDMA SYSTEMWITH HARD-LIMITER
and
+
+ F(A6 + I6/e)] + F(TcIs/e + '&)
0 2 ( m )= G2FeT,[mA, ( K N - m ) A s / M ,
(9)
where the parameters A, Ab, e, &/e, M e , G, I,, F,, and &f
are described in Part I of this paper. See the details in [7, Table
I] for parameters chosen in this paper.
At the proper correlation time I*, the contribution of the
desired signal and the contribution of the total undesired users
to Zl(l*) are the total weight of a signature sequence K
and Il(Z*), respectively. Hence, total K
I1(Z*) marks at
A, incident photon arrival rate are accumulated, and K N ( K 11(Z*))spaces at A,/M, incident photon arrival rate
are accumulated for the desired receiver's accumulated output
Zl(l*).Thus, the mean and variance of Zl(l*) are from (8)
and (9) with m = K Il(Z*), respectively. The conditional
symbol-correct probability given Il(Z*) and Il(E)= m with
n, repetitions is
+
+
+
{Pr[Z1(1*)2 Zl(l), I l ( q = m I I1(l*)]}nm
nm
= [ S-m
m P , , ( l . ) ( a ) ~ ~ P l l ( r ) ( B434
)
(10)
m = 0, 1,...,2N- l,Il(Z*)= 0, 1 , . . . , 2 N - 2 where
pzl ( I : ) ( a ) and pzl ( 1 ) (B) are Gaussian probability density
functions. After the change of the integral variables, the
conditional symbol-correct probability given I1(Z*) in (7) can
be written as
P,(C IIl(l*)) = E,
= Q(
(1 - 4(z, Il(l*)>nm
4 K + Il(l*)>.
+ P ( K + Il(l*)) - 4 4 )
4m)
[
rol I]
JJ (1 - q(z,
~ i ( ~ * ) } ~ m
(15)
(12)
where & ( a ) is the integral of the normal Gaussian density
from a to infinity. The unconditional probability of a correct
symbol decision can be expressed as
P . ( c ) = Erl(i*) E,
The receiver with the hard-limiter has the same configuration as Fig. 1 except a hard-limiter is placed before the optical
tapped-delay line, Fig. 1(a), or before the sequence-despreader,
Fig. l(b). The response of the optical hard-limiter is described
in Part I of this issue. Let i k ( E ) denote the number of hits on
the kth mark of the locally generated sequence cl(t - ZT,)by
N signals' marks, k = 1, 2 , . . . , K , I = 0, 1, 2 , . . . ,F - 1.
Let i(E) = (il(Z), i 2 ( l ) , . . . ,Z K ( I ) ) be the interference pattern
for the correlation with cl(t - lT,), 11( I ) the total summation
of i k ( l ) over k from 0 to K , and li(Z)l the number of nonzero
elements in i(Z).(For example, in Fig. 4 of Part I in this paper
for I = 0 case, K = 3, Il(Z = 0) = 4, i(Z = 0) = (0, 3, l ) ,
and li(l = 0)l = 2.) Then the performance of the system
with hard-limiter for multibitdsequence-period depends on
li(Z)l as well as Il(l), while the system performance without
hard-limiter depends only on 11( I ) .
For one-bithequence-period transmission, the conditional
probability given 11 that the interference pattern i(l = 0) at
the proper correlation time has m nonzero elements, m =
1, 2,. . . , minimum ( K , II), was derived in Part I of this
paper. (See (25) of Part I.) For multibitdsequence-period
transmission, the conditional probability given I1(1) that the
interference pattern i(Z) at any noncorrelation time I # I*,
E = 0, 1, 2 , . . ' , F - 1, has m nonzero elements, becomes (25)
of Part I with 11 = 11(Z).
The average probability over Il(Z) that the interference
pattern i(Z) at noncorrelation time has m nonzero elements,
0 5 m 5 K , can be written as
(13)
where EIl~I.~[f(ll(Z*))]
is the average of f ( I l ( I * ) )over the
random variable Il(l*). The probability density of 11(Z*) is
derived in the Appendix. The unconditional probability of
a wrong symbol decision P,(E) is 1 - Ps(C). Each usersignature sequence has the F number of cyclic shifts, and
they are orthogonal from the autocorrelation property of an
OOCDMA sequence. Each shift of a user sequence corresponds to a symbol which carries the log, F bits information.
Hence, the bit error probability is that of the typical F-ary
orthogonal modulation which can be written as
This probability can be computed using (25) of Part I in this
paper with I1 = 11(1) and the pdf of the 11( I ) in the Appendix.
The noiseless off-peak correlation value 21( I ) at noncorrelation time I after the hard-limiter is equal to the number of
nonzero elements in the interference pattern i(Z),1 # I*, 1 =
0, 1,. . . ,F - 1. In similarity to the derivation of the symbolcorrect probability for the no hard-limiter case (5)-(13), a
lot of correlations Zl(l) within F accumulated outputs are
identical random variables because the noiseless off-peak li(Z)I
repeats many times. Let nm denote the number of repetitions
of the noiseless off-peak li(Z)I within a sequence period F
for which the off-peak correlation value after the hard-limiter
li(Z)l is equal to m. Then the average number of n, within
a sequence-period F is
nm = (F - l)Pr(li(I)l = m).
(16)
In (16), ( F - 1) is used instead of F because the proper
correlation time Z* is excluded in the counting of repetitions.
Even if the kth mark of the locally generated sequence
cl(t - ZT,)were not hit by any mark of N users, the output
2596
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO. 8. AUGUST 1994
of the hard-limiter at the kth mark position of c1( t - ET,) can
be unit power in a practical optical receiver for many users as
discussed in the Part I of this issue. The total N users' spaces
contribution N P / M e to the kth mark of the desired user signal
c l ( i - lT,), can be larger than the unit laser chip-pulse power
P if the number of users N is larger than the extinction ratio
M e . In this case (Le., N 2 M e ) the hard-limiter clips the
intensity back to the unit laser chip-pulse power. In the other
case (Le., N < Me), the output of the hard-limiter is zero.
,
such an indicator function which is shown in
Let i ~ denote
(26) of the Part I of this issue. For the li(E)I = 0 case (Le.,
none of marks of c l ( t - ET,) hit by N users' mark), K . z~~
pulses after the hard-limiter are incident upon the APD with
A, photon arrival rate. For 0 < li(E)I = m 5 K case, m marks
of cl(t - ET,) hit by N users' marks, and ( K - m ) marks of
c1(t - ET,) hit by N users' spaces). Hence, m ( K - m) .i ~
pulses after the hard-limiter are incident upon the APD with
A, photon arrival rate.
At correlation time 1 = 1*, for any interference pattern i ( l * ) ,
K pulses after the hard-limiter are incident upon the APD with
As photon arrival rate because the optical light power intensity
is larger than or equal to the unit power intensity due to the
signal presence at any "mark" interval of c l ( t - E*T,).
The final symbol decision random variables Zl(1) after
integration, E = 0, 1,. . . , F - 1 (including correlation time E*),
are approximated to be Gaussian random variables. Their mean
and variances are as follows: a) At correlation time 1 = 1*,
the mean and variance are independent of the interference
statistics Il(E*)because the hard-limiter outputs K pulses for
any Il(E*),and are, respectively, (8) and (9) with m = K
and with deletion of the term containing extinction ratio Me;
b) At noncorrelation time 1 # E*, E = 0, 1 %... ,F - 1, for
0 5 li(l)l = m 5 K case, the mean and variance are from (8)
and (9), respectively, with m replaced by m ( K - m ) . ZM,
and deletion of the term containing extinction ratio Me.
A symbol which carries log, F data bits is in error if any
of Zl(E), 1 # 1*, E = 0, l , . . . , F - 1, is larger than Zl(E*).
In similarity to the derivation of (13) for the no hard-limiter
case, the symbol-correct probability with hard-limiter can be
written as
+
+
Lo
K
PS(C) = E,
]
(1 - q(z))""
(17)
where EZ[f(z)]
is the average of f(z)over the normal
Gaussian random variable z with zero mean and unit variance,
and
where Q(a) is the integral of the normal Gaussian density
from a to infinity, p and 0 are from (8) and (9) with deletion
of the term containing extinction ratio M e , n, is from (16),
and Z M , is from (26) of the Part I. The bit error probability
is from (14) with P,(C) in (17). Note that the number of
products in (17) is K because the maximum noiseless offpeak correlation can be K due to the hard-limiter's function,
while the maximum noiseless off-peak correlation for the no
hard-limiter case can be 2 N - 1 as shown in (13).
,
-70
-65
-60
-55
-50
-45
40
RECEIVED LASER POWER (dEW, P
Fig. 3. Bit error probability versus received laser power P with the number
of users N as a parameter, for F = 1000, log, F bitskquence-period,
and chip-synchronous OOCDMA system without hard-limiter. (The maximum
number of "marks" I< was used for given N and F.)
V. NUMERICAL
RESULTS
Each user's data bit rate in this numerical results is 30
Mb/s and log, F x 30 Mb/s (270 M for F = 1000) for onebidsequence-period and log, F bitdsequence-period transmission, respectively.
A. MultibitdSequence-Period, No Hard-Limiter
Fig. 3 shows the bit error probability versus received laser
power, with the number of users N as a parameter, for
F = 1000, using the maximum number of marks K for given
N and F (see [7, eq. (5)]). A chip-synchronous multibits per
sequence-period transmission OOCDMA system without hardlimiter can support up to N = 13 users with 0.799 x lo-''
bit error probability (Fig. 3) (while a chip-synchronous onebidsequence-period transmission OOCDMA system without
hard-limiter can support up to N = 9 users with 0.278 x lo-'
bit error probability ([7, Fig. 81)) using F = 1000 chips at
the received laser power P = -55 dBW. To achieve lo-' bit
error probability, a chip-synchronous log, F bits per sequenceperiod OOCDMA system is more than 3 dB better than a
chip-synchronous one-bidsequence-period OOCDMA system
(compare Fig. 3 with [7, Fig. 81). This is because at the final
decision the maximum peak correlation within a sequence
period is taken and no threshold is used, while the correlation
at the correlation time (1 = 0) is compared to a threshold in a
one-bidsequence-period transmission. The APD noise, thermal
noise, leakage currents, and background light may increase the
probability of false decision by comparing the decision value
with the threshold in a one-bidsequence-period transmission
even though the optimum threshold is employed.
B. MultibitdSequence-Period, With Hard-Limiter
It was reported in [7] that, for a one-bidsequence-period
OOCDMA system in the presence of APD and thermal noise,
the improvement from using a hard-limiter before the receiver
is not significant. However, the hard-limiter in a multibits
KWON OPTICAL ORTHOGONAL CDMA SYSTEM-PART
2597
I1
lN,Kl=(2.22), (3.18). 14.16). 15.14). l8.13l. (7.12). (B,?ll.
(S,lIl,(~O,lO),
(17.81, 118.7). (20.71
-70
-70
-65
-60
-55
-50
-45
-40
RECEIVED LASER WWER ( a ~ w )P.
Fig. 4. Bit error probability versus received laser power P with the number
of users N as a parameter, for F = 1000 chips, log, F bitslsequence-period,
and chip-synchronous OOCDMA system with hard-limiter. (The maximum
number of “marks” K was used for given N and F.)
per sequence-period OOCDMA system improves the performance significantly. See Fig. 4 ( F = 1000 case), and
especially note the results for the number of users N between 10 and 20. For example, the bit error probability
of the multibits/sequence-period OOCDMA system without
(Fig. 3) but the bit
a hard-limiter is about 0.124 x
error probability of the multibitdsequence-period OOCDMA
system with a hard-limiter is 0.2481 x
(Fig. 4) for
F = 1000 chips, N = 20 users, and P = -55 dBW.
The reason for such a big improvement in the multibits
transmission per sequence-period system compared to the
one-bidsequence-period system is because the conditional bit
error probability given “0” bit transmission (i.e., signal pulse
absence, see [7, eq. (27)]) dominates over the conditional
bit error probability given “1” bit transmission (see [7, eq.
( 2 8 ) ] ) in the one-bidsequence-period system. There is no
signal-absent case in the multibitdsequence-period system.
An OOCDMA signature sequence or its shifted version is
always transmitted in the multibits/sequence-period system
while the maximum correlation Zl(Z*) is searched and its
position is used to decode multibits in the multibitdsequenceperiod system. Thus, significant improvement from using a
hard-limiter can be achieved for the multibitdsequence-period
system.
A chip-synchronous multibits per sequence-period transmission system with a hard-limiter can support up to N = 17 users
with 0.448 x
bit error probability using F = 1000 chips
(Fig. 4) at the received laser power P = -55 dBW.
-65
-60
-55
-50
-45
-40
RECEIVED LASERPOWER (aBw). P
(A) Chipsync, on%bdseq.-pend. no hard.lmlt8r
(8) Chlpasync , one-blUseq .pBdd, no hsrMmller
(C). Chipsync, one.blll.W,
mth harbllmhsl
(D). Chipsync.. lop2 F bildsw -psrbd. no harcNmller
(E) Chip-sync.,k q p F bdpireq.-ped, wdh haia.limitw
Fig. 5. Bit error probability versus received laser power P for five
OOCDMA systems considered with N = 20 users K = 7 “marks”
and F = 1000 chips. (Each user’s data bit rate is 30 Mbls for
one-bitkequence-period transmission system, and 270 Mbits per second
for log, F bitslsequence-period system.)
with ( K , F ) = (7, 1000). First, for received laser power P
less than -54 dBW and F = 1000, the one-bidsequenceperiod transmission system can be more than 3 dB better
than a log, F bitshequence-period transmission system with
no hard-limiter (compare curve A with D in Fig. 5 ) at lo-’
bit error probability. However, in general, for received power
larger than -54 dBW (which is more interesting because
bit error probability is small), a log, F bitdsequence-period
system can be more than 6.4 dB better at lop7 bit error
probability, as well as log, F bits higher throughput, then the
one-bidsequence-period system for the no hard-limiter case.
Second, in a one-bidsequence-period OOCDMA system, the
improvement from using a hard-limiter before the receiver is
not significant (compare curve A with C in Fig. 5, or columns
A with C in Table I). However, in a multibitdsequence-period
OOCDMA system, the hard-limiter before the receiver can
improve the performance significantly (compare curves D with
E in Fig. 5 , or columns D and E in Table I). More than 3
dB improvement in power for ( F , N , K ) = (1000, 20, 7) in
Fig. 5 can be achieved, or four more users in Table I can share
the optical channel at bit error probability less than
for
( F , K ) = (1000, 10) if the hard-limiter is placed before the
receiver.
VI. CONCLUSION
A log, F bits/sequence-period OOCDMA system was introduced where F is the length of a sequence. The system was
C. Comparisons of One-Bit and Multibits
compared with a one-bidsequence-period system for the chipTransmissiodSeq-Period
synchronous case in the presence of APD and thermal noise.
Table I summarizes the maximum number of users that In addition, the multibitdsequence-period OOCDMA system
can be supported with less than lop9 bit error probability with a hard-limiter placed before the receiver was analyzed
at the received laser power P = -55 dBW. Fig. 5 compares, and compared to the system without hard-limiter. A log, F
the performance of five systems (considered in both Part 1 bits/sequence-period system can be more power-efficient for
and Part I1 of this paper) when supporting N = 20 users lo-’ bit error probability, as well as log, F bits higher
IEEE TRANSACTIONS ON COMMUNICATIONS. VOL. 42, NO. 8, AUGUST 1994
2598
F-7
”-
Number of
Chips, F
F = 1000
Number of users,
B
C
11
9
A
9
D
13
E
17
throughput, than the one-bithequence-period system for the
no hard-limiter case. In a one-bivsequence-period OOCDMA
system, the improvement from using a hard-limiter before the
receiver is not significant. However, in a multibitdsequenceperiod OOCDMA system, the hard-limiter before the receiver
can improve the performance significantly.
In this Appendix, first the pdf of the total undesired interference contribution, 1 1 (Z*), to the desired receiver’s accumulated
output Zl(Z*) at correlation time I* is derived. Second, the
total undesired interference plus the desired signal contribution
11(1) to the desired receiver’s accumulated output Zl(1) at a
noncorrelation time 1, Z # 1*, 1 = 0, 1,.. . F - 1, is derived.
The statistics 11(1*)and Il(Z), 1 # I*, are different. The
Il(1) can be between 0 and 2N - 1 because N users are
considered for 11(1) calculation, while Il(Z*) can be 0 and
2 N - 2 because the desired signal is not counted as an interferer
and ( N - 1) users are influential for 11(1*)
calculation at the
proper correlation time.
~
A. P d f O f 1 1 ( 1 * )
Let 71 be 0 and let T, be the relative delay of user n
to user 1. In addition, let,:Z -1 and
be corresponding
shifts to the log, F bits transmission of user n for segment
(7, - F 5 j I r,) and segment (7, < j 5 r,
F - l),
respectively, 0 5 T, 5 F - 1. See Fig. 6 for an illustration.
Let Ip)(Z*)be user n’s interference contribution to the desired
receiver’s accumulated output at a correlation time l*. Then it
is equal to the cyclic cross-correlation between user 1’s and
user n’s signal, and it is bounded by 2
+
7,
- r,
Fig. 6. User n’s interference contribution to the user 1-receiver’s accumulated output at the correlation time I*. The T~ is the relative delay of user n to
user 1, and I* is the corresponding shift to the log, F bits transmission of user
are corresponding shifts
1 for segment (0 5 j 5 F - 1).The 1:. - and I:,
to the log, F bits transmission of user n for segment (7, - F 5 j 5 T ~
and segment (7, < j 5 T,
F - 1). respectively.
+
the autocorrelation orthogonal property but not the crosscorrelation orthogonal property between user signals.’
The IA1)(Z*)is a random variable which can be 0, 1, or 2.
The probability maSS function of 1:’) ( I ) is derived below.
Assume that the total number of marks of a user sequence
is K and the marks are uniformly distributed in a sequence
period. Then, the probability that a mark of cn(F - r,
j -,:Z -1) hits a mark of c l ( j - Z*), for 0 5 j 5 T,, is
( K T , / F ) ~ / T , .This is true because for each mark of KT,/F
marks in 0 5 j 5 r, interval of the user n’s signature
sequence cn(F - T,
j - l:, -1), there are Kr,/F marks
in 0 5 j 5 T , interval of the first user’s signature sequence
c l ( j - 1’) where the overlaps could take place. (See Fig.
6.) Furthermore, with KT,/F marks of user n’s signature
sequence over 0 I j 5 r, interval, there would be a
total ( K T , / F ) ~possible overlaps. The probability of each
individual overlap is equal to 1 / ~ ,, since marks of a user
sequence are uniformly distributed. Similarly, the probability
that a mark of c n ( j - T, -,:Z o) hits a mark of c l ( j - Z*), for
rn 1 5 j 5 F - 1, is ( K ( F - T , ) / F ) ~ / ( F- r,).
The probability of any delay 7, is 1/F. Let T denote the
average probability over T , that lA1)(l*)= 2 (i.e., both a
mark of cn(F - T, + j - 1:, -1) hits a mark of c l ( j - 1*), for
0 5 j 5 r,, and a mark of c n ( j - r, - Z:o,)
hits a mark of
c l ( j - 1*), for 7 ,
1 5 j 5 F - 1). Then, it is given by
+j
,:Z
-
+
+
+
F-1
=
-1)
-
-
l * ) c n ( j - 7 , - 1:$)
I 2,
j=T,+l
1 =O,...,F-l,
(All
because each term of the right-hand side of (Al) is bounded by
1 from the cross-correlation property [7, eq. (4)]).Therefore,
in general, a multibit transmission system does maintain
Pr (7,) Pr ( 2 hits
r,
F-1
Cl(j
P r [ p ( l * ) = 21
T
j=O
+
u
+
APPENDIX
IF)(Z*)= c e l ( j - Z*)c,(F
F
1 7),
=o
K 4F2- 1
F4 6
Let q be the average probability over r, that lA1)(Z*)= 1,
(Le., either (not both) a mark of cn(F - r,
j - l:? -1)
hits a mark of c l ( j - I * ) , for 0 5 j 5 T,, or a mark of
c n ( j - T , - Z:o,)
hits a mark of c l ( j - Z*), for r, 1 5 j 5
+
’
+
In the log, F bits/sequence-period transmission system, even though the
cyclic cross-correlation is not orthogonal, the optical orthogonal CDMA
(OOCDMA) terminology is used in this paper.
)
K W O N OPTICAL ORTHOGONAL CDMA SYSTEM-PART
F
q
11
2599
- 1). Then, it is given by
interference plus the desired signal contribution at any noncorrelation time 1 is the N-fold convolutions of pdf of 1:’) in
(A7) and pdf of i.i.d. IL1)in (A5), n = 2, 3 , . . . ,N , which
can be written as
Pr[IL1)(E*)= 11
G
F-1
=
Pr(T,)Pr(lhit
r,
17,)
=o
S(2(N - 1) - i - j - a )
Let p be the average probability over
Then
T,
that IL1)(E*)= 0.
S(2N - 1 - 2 - j
-
a).
K2
p G 1-q-r=
1-
7.
(A8)
(A4)
REFERENCES
Hence, the probability mass function of lA1)(Z*)can be expressed as
~~:i)(~.)(a)
= p6(a)
+ @ ( a - 1)+ rS(a - 2)
(A5)
where 6 ( x ) is the Dirac delta function.
The pdf of the total undesired interference contribution
Il(E*) to the desired receiver’s accumulated output Zl(E*)
at the correlation time Z*, is the ( N - 1)-fold convolutions
of pdf of identical independent distributed (i.i.d.) Iil)(E*),
n = 2, 3 , . . . , N , which can be written as
. S(2(N - 1) - i - j
- a ) . (A6)’
B. P d f O f I l ( 1 )
At any noncorrelation time, E # I * , E = 0, 1,. . . , F - 1, the:
statistics of the desired signal contribution I:’)(E) to Z1( E ) are:
=p6(a)
+ (1- p)S(a: - 1),
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(A7)
and the statistics of the undesired user n’s interference con-. Hyuck M. K~~~ (s’82-~’84) for a photograph and biography, please
tribution is given in (A5), 2 5 n 5 N . The total undesired the May 1994 issue of this TRANSACTIONS, p. 2126.
see
- I