quality-of-service differentiation by multilength variable-weight time-and-frequency-hopping optical...

Vol. 5, No. 8 / August 2006 / JOURNAL OF OPTICAL NETWORKING 611

Quality-of-service differentiation bymultilength variable-weight

time-and-frequency-hopping opticalorthogonal codes in optical

code-division multiple-accessnetworks

Chung-Keun Lee, Jinsoo Kim, and Seung-Woo Seo

School of Electrical Engineering and Computer Science, Seoul National University,Seoul 151-744, Korea,

[email protected], [email protected], and [email protected]

Received March 1, 2006; revised May 24, 2006; accepted June 7, 2006;published July 26, 2006 �Doc. ID 68418�

We consider multilength variable-weight time-and-frequency-hopping (TFH)codes for optical code-division multiple-access (CDMA) networks. To design aTFH code that can support differentiated requirements on transmission rates(TRs) and bit error rates (BERs), we propose a general construction methodfor two-dimensional optical CDMA code sequences with arbitrary code lengthsand variable code weights. The cross-correlation peak and the autocorrelationsidelobe of the proposed code are made to be at most “1” and “0”, respectively,for any pair of different-length and different-weight code sequences. We showthat the ratio between signal-to-interference ratios (SIRs) of the codes withdifferent weight and length can effectively be approximated by the ratio of thecode weights, regardless of the code lengths for the proposed codes. Therefore,independent adjustment of code length and weight can differentiate TR andBER for multiple service classes, and it simplifies code design for the servicedifferentiation. We also demonstrate BER performances under various condi-tions and corroborate that the proposed codes accomplish accurate and fine-tuned differentiation of TR and BER compared to the fixed-weight or fixed-length codes. © 2006 Optical Society of America

OCIS codes: 060.2330, 060.4510.

1. IntroductionThe optical code-division multiple access (OCDMA) technique has been considered asan attractive way to support many simultaneous users sharing a common channelresource. To alleviate multiple-access interference in optical code-division multiple-access (OCDMA) networks, Salehi introduced optical orthogonal codes (OOCs) thatare one-dimensional unipolar sequences [1]. By and large, the use of long codesequences can guarantee high autocorrelation and low cross correlation in OOC. Thusreducing the code lengths usually entails a reduction in the cardinality of the code. Away to resolve this problem is to use two-dimensional codes [2–4], in which codesequences are encoded in both time and frequency domains.

Future OCDMA networks are expected to support a variety of services such as con-comitant transmission of real-time video, voice, and multirate data. To support multi-media service within OCDMA networks, it is required to differentiate diverse qualityof service (QoS). QoS differentiation can usually be achieved by making a distinctionin transmission rates (TRs) and bit error rates (BERs). For instance, it is possible tovary the lengths or the weights of OOCs for the discrimination of TRs and BERs[5–7]. To overcome the excessive lengthening of OOCs, several two-dimensional codessuch as frequency-hopping and time-spreading codes have been constructed by Kwonget al. [8] using prime sequences and by Pu et al. [9] using algebraic congruent codes.In those studies, Kwong et al. divided service classes according to TRs by varying codelengths while BER analysis was not performed. The code proposed by Pu et al. cansupport only three service classes: one class has the highest TR, and the others havethe same TR and are differentiated by BERs. Inaty et al. proposed a differentiation

1536-5379/06/080611-14/$15.00 © 2006 Optical Society of America


scheme that gives trade-off between the TRs and BERs among service classes [10].They obtained high-transmission-rate codes by truncating one-coincidence sequencesgenerated by Bin’s algorithm [11], and they optimized the network throughout usingtransmission power control.

To accommodate accurate and fine-tuned differentiation of diverse service classes, itis required to design optical codes having variable code lengths and weights. In thispaper, we propose a generalized flexible code construction method to generate a time-and-frequency-hopping (TFH) code consisting of heterogeneous code sequences havingmultilengths and variable weights. The proposed method gives a TFH code satisfyingthe one-coincidence conditions; that is, the values of the periodic Hamming cross-correlation function do not exceed “1” for any pair of different-length and different-weight code sequences, and the autocorrelation sidelobe is “0”. Although these proper-ties limit the cardinality of the codes compared to that of the codes with looser auto-and cross-correlation constraints, they can provide the least multiple-access interfer-ence in OCDMA networks.

We show that the generated codes can provide delicate multimedia service differen-tiation in terms of both TR and BER. First, a sophisticated TR differentiation can berealized by differentiating the code length of the proposed codes, since the codesequences can have various lengths that are not restricted to prime multiples. For theBER differentiation, we make codes that have different weights so that the codesequences have different transmission power. By showing that the ratio betweensignal-to-interference ratios (SIRs) of any two service classes (codes) can effectively becharacterized by the ratio of the code weights regardless of the code lengths within acode and by demonstrating the numerical BER performance results under variousconditions, we corroborate that the proposed codes accomplish accurate and fine-tuneddifferentiation of TR and BER compared to the conventional fixed-weight or fixed-length codes. We also present an upper bound on the cardinality of the proposedcodes.

This paper is organized as follows. Section 2 is devoted to the generation of multi-length and variable-weight TFH codes. In Section 3 we show that the ratio of SIRs ofany two service classes can be approximated by the ratio of their weights regardless ofthe code lengths. TR and BER performances of the proposed codes under various con-ditions are shown in the same section. Finally, we conclude this paper in Section 4.

2. Construction of Multilength Variable-Weight Time-and-Frequency-Hopping CodesIn this section, we explain how to generate TFH codes. To represent frequency-hopping codes with time-hopping patterns, we follow the conventional two-dimensional code representation [12] in the form of a q�L matrix, where q is thenumber of available frequency components and L is the code length. That is, wedenote a code sequence by a q�L matrix �xi,j�, where xi,j is an element at the �i+1�throw and the �j+1�th column with 0� i�q and 0� j�L. Here, xi,j is 1 if a positive pulseof wavelength (or frequency) index i occurs at the jth chip, and 0 otherwise. To sim-plify the matrix notation, we use the reduced two-dimensional coordinate representa-tion by the collection of the positions �i , j�, where xi,j is 1. For example, a code matrix

x =�1 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 1

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

�


can be reduced by representing the positions of “1”s in the matrix x, i.e., x= ��0,0� , �6,2� , �4,4�� when q=11, L=5, and weight w=3.

To deal with the case when two interfering codes have different lengths, we redefinethe periodic Hamming cross-correlation function for any two code matrices x and ywith their respective sizes q�Lx and q�Ly by

Hxy�� = �i=0

q−1

�j=0

Lx−1

xi,jyi,j+� �1�

for 0��Ly, where the operation on the second subscripts is performed by modulothe column size; that is, in this case the addition in the subscript of y is modulo Ly.Equation (1) means that the reference code is x and the Hamming cross correlation isevaluated for every time shift of y. If x and y are distinct, Eq. (1) represents the cross-correlation function. If they are identical it represents the autocorrelation function,and its peak should occur when �=0. To guarantee the performance of TFH codes,Hxy�� should be less than a specific value for every combination of x, y, and �. Thiscondition is necessary to suppress the interference between the multiple user codes inasynchronous communications, although it is very strict in that the Hamming correla-tion function is calculated for the all possible cyclic time shifts between the codesequences.

2.A. Generation of Constant Length and Weight Time-and-Frequency-Hopping CodesFirst, we present a generalized method to generate TFH codes with fixed length andweight. We consider a �w ,k ,d ,q� TFH code in which the number of available fre-quency components is q, the code length is L, and the code weight is w. We set k=L−w. Then k is the number of chips (time blanks) at which none of the frequencies aretransmitted. In addition, we require that distances between adjacent frequency indi-ces in every code sequence should be larger than a specified value d, which is calledthe code distance and is introduced to reduce interference between frequency compo-nents [11]. It is noted that the relationship between the parameters of �w ,k ,d ,q� TFHcodes and those of the �q�L ,w ,�a ,�c� multiwavelength OOCs is that L=w+k, �a=0,and �c=1 with an additional parameter for the minimum distance d, where �a is thepeak value of the autocorrelation sidelobe, and �c is the peak value of the cross corre-lation.

For the generation we first choose a distance sequence d= �d0 ,d1 , . . . , dw−1� and atime-hopping sequence b= �b0 ,b1 , . . . , bw−1�, satisfying

�j=0

w−1

dj � 0 �mod q�, �2�

which ascertains that code sequences are cyclic, and

�j=0

w−1

bj = k, �3�

respectively. We note that the elements in the distance sequence should satisfy d�dj�q−d+1 due to the distance constraint.

Distance and time-hopping sequences determine frequency-hopping and time-hopping patterns within a code, respectively. It is also noted that dj is the index differ-ence between jth and �j+1�th frequency indices, and bj is the number of chips betweenthe jth and �j+1�th frequency indices of the code sequence. Using a distance sequenced and a time-hopping sequence b, we define a generator sequence by

s�d,b� = ��fj,tj��j=0w−1 = ��f0,t0�,�f1,t1�, . . . , �fw−1,tw−1��, �4�

where ti+1= ti+bi+1 and fi+1= fi+di�mod q� for 0� i�w−1. For simplicity, we set�f0 , t0�= �0,0�.

Once a generator sequence s�d,b� is obtained, we can compose a TFH code of evencyclic shifts of the wavelength indices of the generator sequence; i.e., q code sequencesx for 0�h�q are generated by
h


xh = ��fj + h,tj��j=0w−1 = ��f0 + h,t0�,�f1 + h,t1�, . . . , �fw−1 + h,tw−1��, �5�

where the arithmetic in frequency indices is modulo q. To control the distance con-straint, we define an �L−1��w distance matrix G�d,b�= �Gi,j� by

Gi,j = dist�fn,fj� if tn = tj + i + 1 �mod L�,

� otherwise �6�

for 0� i�L−1 and 0� j�w, where the distance between frequency indices is definedby dist�a ,b�=b−a�mod q�. We consider the following two conditions:(I) All finite integers in the first column of G�d,b� are nonzero and are all distinct.(II) All finite integers in every row of G�d,b� are all distinct.

Condition I means that the frequency components in any code sequence are all dis-tinct and that autocorrelation sidelobes have vanished, i.e., Hxx��=0 for 0��Lx.Condition II is equivalent to the one-coincidence condition and means that cross-correlation peaks are no larger than 1, that is, Hxy��1 for 0��Ly and any pair ofdistinct code sequences x and y. When G�d,b� satisfies conditions I and II, we call�xh :0�h�q� a �w ,k ,d ,q� TFH code (for more details, refer to Kim et al. [13,14]). Forinstance, when w=3, k=4, d=0, and q=7, from a distance sequence d= �1,1,5� and atime-hopping sequence b= �0,1,3� we obtain a generator sequence s�d,b�

= ��0,0� , �1,1� , �2,3�� and a distance matrix

G�d,b� = �1 � �

� 1 �

2 � �

� � 5

� � 6

� 6 �

� .

Since this distance matrix satisfies conditions I and II, we can generate seven codesequences xh= ��0+h ,0� , �1+h ,1� , �2+h ,3�� for 0�h�7.

Furthermore, according to Kim et al. [13,14], it is possible to obtain a TFH code oflarge cardinality by the usage of the combination of several distance and time-hoppingsequences. Indeed, we have the following distance and time-hopping sequences,

d�0� = �1,1,5�, d�1� = �2,2,3�, d�2� = �3,3,1�,

d�3� = �4,4,6�, d�4� = �5,5,4�, d�5� = �6,6,2�,

b�0� = �0,1,3�,

from which we get six generator sequences,

s�d�0�,b�0�� = ��0,0�,�1,1�,�2,3��,

s�d�1�,b�0�� = ��0,0�,�2,1�,�4,3��,

s�d�2�,b�0�� = ��0,0�,�3,1�,�6,3��,

s�d�3�,b�0�� = ��0,0�,�4,1�,�1,3��,

s�d�4�,b�0�� = ��0,0�,�5,1�,�3,3��,

s�d�5�,b�0�� = ��0,0�,�6,1�,�5,3��.

By the modulo 7 addition to these six generator sequences as in Eq. (5), we attain 7�6=42 code sequences. This cardinality saturates the bound that is derived in Sub-section 2.C.


The proposed code generation procedure is based on searching procedure, and itwould take much longer in expected time than the algebraic code generation algo-rithms. Even though it does not seem crucial because codes are usually generated inthe design of an OCDMA system prior to the usage, there is a way to abate theexhaustive search complexity in conjunction with the existing algebraic code genera-tion algorithms.

When q is prime, we can make use of prime sequences for choosing distancesequences so that the distance sequence d�i� becomes �i+1, i+1, . . . , i+1, �q− �w−1��i+1��mod q�� for 0� i�q−1. For example, when q=7, the distance sequence d�1�

becomes (2, 2, 2, 2, 1) for w=4 and (2, 2, 3) for w=3. Then we only need to find theproper time-hopping sequences. In addition, if the length and the weight of the codessatisfy

L � w�w − 1� + 1, �7�

which guarantees the existence of one-dimensional OOC patterns, we can use OOCpatterns as the time-hopping sequences. It is noted that when the length and theweight are fixed and the code length satisfies Eq. (7), the resulting code is the same asthe code proposed by Kwong et al. [16] except for the sequences with �a�0. WhenEq. (7) is not satisfied, full search for the time-hopping sequences is inevitable. How-ever, the total search space is not so large because the code length L is relativelysmall in this case.

When q is not prime, although the prime sequence is not applicable, we can stillmake use of OOC patterns for choosing time-hopping sequences when Eq. (7) is satis-fied. For example, we can use a predetermined OOC pattern b= �0,1,5,3� correspond-ing to w=4 and k=9 as a time-hopping sequence to construct a TFH code with param-eter �4,9,1,8�. Then we can find four distance sequences, d�0�= �2,3,6,5�, d�1�

= �3,6,5,2�, d�2�= �6,5,2,3�, and d�3�= �5,2,3,6�, from which a TFH code of cardinality32 is obtained.

When w=3, there is a way to reduce the generation complexity using OOC patternswhether q is prime or not. It requires the additional restriction that Eq. (7) be satis-fied, so that L and q should be the same or larger than 7. Following the OOC genera-tion algorithm proposed by Lee et al. [12], if the weights of OOC patterns are fixed tow, then the cross-correlation peak of the resulting two-dimensional codes becomes w−2. Hence when w=3, TFH codes satisfying conditions I and II can be generated fromOOC patterns without resorting to the exhaustive search.

The existence problem of the general OOCs is beyond the coverage of this paper,but there are some proven cases when the weight lies between 3 and 9 [17].

2.B. Generation of Multilength Variable-Weight Time-and-Frequency-Hopping CodesWe consider a multilength variable-weight TFH code having different lengths andweights. By assigning TFH codes with the same length and weight to a service class,it is possible to support multiple service classes with various TRs and BERs.

When code sequences of different lengths and weights are transmitted simulta-neously, multiple contiguous short code sequences and code sequences of large weightscan increase their cross correlation. Thus in the code design, we should be able to con-trol the cross correlations between two code sequences that belong to TFH codes of dif-ferent parameters. For this purpose, we have to extend the definition (6) of a distancematrix.

To support m service classes, we shall build m TFH codes of different parameters.Without loss of generality we assume that every TFH code of each parameter has onegenerator sequence. We choose m distinct distance sequences d�0� ,d�1� , . . . , d�m−1� andm distinct time-hopping sequences b�0� ,b�1� , . . . , b�m−1�, where each distance sequenced�i�= �d0

�i� ,d1�i� , . . . , dwi−1

�i� � satisfies Eq. (2) and each time-hopping sequence b�i�

= �b0�i� ,b1

�i� , . . . , bwi−1�i� � satisfies Eq. (3) for i=0,1, . . . , m−1. Each pair of d�i� and b�i�

constitutes a generator sequence for �wi ,ki ,d ,q� TFH codes Ci for i=0,1, . . . , m−1.From these m pairs of d�i� and b�i�, we can generate m generator sequences s�i�

=s�d�i�,b�i�� following Eq. (4). We denote by G�i�=G�d�i�,b�i�� each distance matrix satisfy-ing conditions I and II. Then each Ci becomes a �wi ,ki ,d ,q� TFH code for i=0,1, . . . , m−1. To see the cross correlation between code sequences in the differentTFH codes, we expand the distance matrices G�i� to r�w extended distance matricesM�i�, defined by


M�i� = �Mj,k�i� � = �

G�i�

0

G�i�

0

G�i�

0

� ,

where r is the least common multiple of L0 ,L1 , . . . , Lm−1 and 0= �0,0, . . . , 0� is a vec-tor of size wi. We note that the repeat number of G�i�’s in M�i� becomes r /Li. For theone-coincidence condition between different classes the augmented matrix M= �M�0��M�1��¯ �M�m−1�� consisting of all the extended distance matrices should satisfythe following condition.

(III) All the nonzero finite integers in every row of the augmented distance matrixM should be distinct.

We note that condition III is an extension of condition II. Each service class codemay contain any number of generator sequences, and in this case condition III shouldalso be satisfied by the augmented matrix consisting of all the extended distancematrices corresponding to the generator sequences. The following theorem states thatcondition III is equivalent to the one-coincidence condition between code sequences ofunequal lengths.

Theorem 1 If condition III holds, then the cross-correlation peak Hxy�� betweenany pairs of code sequences x and y for 0��Ly is less than or equal to 1. Further-more, the converse holds.

Proof: First we prove that condition III is a sufficient condition for the cross corre-lation to have values less than or equal to 1. Let us suppose that Hx�i�x�j��1for some x0

�i�= ��f0�i� , t0

�i�� , �f1�i� , t1

�i�� , . . . , �fwi−1�i� , twi−1

�i� ��Ci, x0�j�= ��f0

�j� , t0�j�� , �f1

�j� , t1�j�� , . . . ,

�fwj−1�j� , twj−1

�j� ��Cj, and �. Then there are at least two collisions between x0�i� and the

time shift of x0�j�. So we have �fa

�i� , ta�i��= �fb

�j� , tb�j�+��mod r�� and �fc

�i� , tc�i��= �fd

�j� , td�j�

+��mod r�� for some a, b, c, and d, and by the definition of an extended distancematrix we also obtain Ml1,c

�i� =dist�fa�i� , fc

�i�� for ta�i�= tc

�i�+ l1+1�mod r� and Ml2,d�j�

=dist�fb�j� , fd

�j�� for tb�j�= td

�j�+ l2+1�mod r�. Since ta�i�− tc

�i�= tb�j�− td

�j��mod r�, we have l1= l2,which implies that Ml1,c

�i� =Ml2,d�j� . This contradicts condition III.

To prove the converse, let us suppose that Ml,u�i� =Ml,v

�j� , which is nonzero and finite,for some i, j, l, u, and v. We note that Ml,u

�i� =dist�fa�i� , fu

�i��= fu�i�− fa

�i��mod q� for ta�i�= tu

�i�+ l+1�mod r�, and Ml,v

�j� =dist�fb�j� , fv

�j��= fv�j�− fb

�j��mod q� for tb�j�= tv

�j�+ l+1�mod r�. We showthat two sequences x0

�i� and xh�j� collide at two time slots. Without loss of generality, we

may assume that tu� tv. Let x0= ��fc�j� , tc

�j�− �tv�j�− tu

�i��c=0wj−1 be the tu

�i�− tv�j� time shift of x0

�j�

and set h= fu�i�− fv

�j��mod q�. Then we have xh= ��fc�j�+ �fu

�i�− fv�j�� , tc

�j�+ �tu�i�− tv

�j��c=0wj−1. We will

consider the vth and the bth coordinates of xh. On the one hand, the vth coordinate ofxh becomes �fv

�j�+ �fu�i�− fv

�j�� , tv�j�+ �tu

�i�− tv�j��= �fu

�i� , tu�i��, and hence xh collides with x0

�i� attime slot tu

�i�. On the other hand, to see the time slot ta�i�, we consider the bth coordinate

of xh, �fb�j�+ �fu

�i�− fv�j�� , tb

�j�+ �tu�i�− tv

�j��. Its time component becomes l+1+ tu�i�= ta

�i��mod r�because tb

�j�− tv�j�= l+1�mod r�. The frequency component becomes fu

�i�−Ml,v�j� = fu

�i�−Ml,u�i�

= fa�i��mod q�, and hence xh and x0

�i� are in collision at time slot ta�i�, too. This completes

the proof. �

If C=�i=0m−1Ci satisfies conditions I and III, then we obtain Hxx��=0 for x in C and

0��Lx by condition I and Hxy��1 for x�y in C and 0��Ly by condition III[15]. We call such a set C a �w ,k ,d ,q� TFH code, where w= �w0 ,w1 , . . . , wm−1� andk= �k0 ,k1 , . . . , km−1�.

To find generator sequences, we proceed with the following procedures:Step 1 Choose distance sequences d�i� satisfying Eq. (2) and time-hopping

sequences b�i� satisfying Eq. (3) for 0� i�m.Step 2 Establish the extended distance matrices M�i� for 0� i�m.Step 3 Check whether each extended distance matrix satisfies condition I. If so,

then go to step 4. Otherwise, go to step 1.Step 4 Check whether the extended distance matrices satisfy condition III. If so,


then generate q code sequences for each generator sequence s�i� from Eq. (5) for 0� i�m. Otherwise, go to step 1.

For example, we are to construct a multilength variable-weight TFH code withparameters q=13, w= �w0 ,w1 ,w2�= �4,3,2�, k= �k0 ,k1 ,k2�= �6,2,3�, and d=1, so thatit can support three different service classes. Applying the above procedure, we canfind three distance sequences satisfying Eq. (2), d0= �2,5,9,10�, d1= �3,4,6�, and d2= �6,7�, and three time-hopping sequences satisfying Eq. (3), b0= �6,0,0,0�, b1= �2,0,0�, and b2= �3,0�. Using Eq. (4) we obtain three generator sequences:

s�0� = ��0,0�,�2,7�,�7,8�,�3,9��,

s�1� = ��0,0�,�3,3�,�7,4��,

s�2� = ��0,0�,�6,4��.

From these we can calculate their respective extended distance matrices M�0�, M�1�,and M�2�, and the resulting augmented matrix becomes

M = �M�0��M�1��M�2�� = �� 5 9 10

� 1 6 �

� 11 � �

� � � �

� � � �

� � � �

2 � � �

7 � � 12

3 � 8 4

0 0 0 0

�� 4 6

� 10 �

3 � �

7 � 9

0 0 0

� 4 6

� 10 �

3 � �

7 � 9

0 0 0

�� 7

� �

� �

6 �

0 0

� 7

� �

� �

6 �

0 0

� .

As the distance matrix M satisfy conditions I and III, 13 �=q� code sequences are gen-erated from each generator sequence, and 39 code sequences can be obtained byevenly shifting all the frequency compoments in the generator sequences, as can beseen in Table 1. They altogether constitute a ((4, 3, 2), (6, 2, 3), 1, 13) TFH code.

Admitting that the code generation procedure of the multilength variable-weightTFH code requires a more complex searching procedure in finding generatorsequences than the fixed-length fixed-weight case, we can apply the similarapproaches presented in the previous subsection. Also, there are useful existing codegeneration algorithms for some special cases. When q is prime, we can adopt the gen-eration algorithm of multiple-length fixed-weight prime sequences [8] so that the dis-tance sequence dj

�i� becomes �i+1, i+1, . . . , i+1, �q− �wj−1��i+1��mod q�� for 0� i�q−1 and 0� j�m. Then we have to find the time-hopping patterns based on the exist-ing OOC generation algorithms. When the code length is fixed and weight is variable,the time-hopping sequence b can be generated from the pairwise balanced designs[18,19] or from the packing design with a partition of the elements in the projectivegeometry [19], which produce variable-weight OOCs with constant length. However,these approaches are only applied to the cases when the code length is fixed or thecode weight is fixed. When both values are variable, we have to perform a full searchprocedure. In this case, it is important to calculate the expected number of code

Table 1. Code Sequences of a ((4,3,2), (6,2,3), 1,13) Time-and-Frequency-Hopping Code

Sequences i=0 i=1 i=2

s0�i� ((0,0), (2,7), (7,8), (3,9)) ((0,0), (3,3), (7,4)) ((0,0), (6,4))

s1�i� ((1,0), (3,7), (8,8), (4,9)) ((1,0), (4,3), (8,4)) ((1,0), (7,4))

s12�i� ((12,0), (1,7), (6,8), (2,9)) ((12,0), (2,3), (6,4)) ((12,0), (5,4))


sequences to confine the searching space based on the cardinality bound on the codes.Hence we present a cardinality issue of the proposed codes in Subsection 2.C.

Disregarding the complexity of the code generation, the proposed constructionmethod for the TFH code generation has no restriction on the code lengths and thecode weights, and thus on the number of service classes. These flexible conditions onthe code parameters allow refined differentiation in TR and BER, as is investigated indetail in Section 3.

2.C. Cardinality of Multilength Variable-Weight CodesWe discuss the cardinality bound of �w ,k ,d ,q� TFH codes. First, we consider the casewhen there is one service class with the �w ,k ,d ,q� TFH code. Let g be the number ofgenerator sequences in the service class, so that the number of code sequences N=gq.Then the number of nonzero finite integers in the augmented distance matrix isgw�w−1� and is bounded by �q−2d−1�+ �q−1��L−2�, because the first finite nonzerointeger of every column in the augmented distance matrix should be larger than dand smaller than q−d, while the finite nonzero integers in the other L−2 rows canhave q−1 different values based on condition (II). Thus the cardinality bound of asingle service class will be

Nw�w − 1� � q��q − 2d − 1� + �q − 1��L − 2��. �8�

We note that the inequality (8) can be applied to TFH codes without time-hoppingpatterns, that is, one-coincidence frequency-hopping (FH) codes [11]. In this case wehave L=w, and Eq. (8) can be rewritten as

Nw �q�q − 2d − 1�

w − 1+

q�w − 2��q − 1�

w − 1. �9�

To compare this bound with the bound obtained by Bin [11] for one-coincidence FHcodes,

Nw � q�q − 2d − 1� + d�d + 1�, �10�

we subtract the right-hand side of Eq. (10) from that of Eq. (9). We assume that w�2 to avoid the trivial cases. Then since q−2d−1�0, we get

d

w − 1�2q�w − 2� − �d + 1��w − 1��,

which is bounded below by 3d�d+1��w−2� / �w−1��0 when w�2 and is negative whenw=2. Thus it can be seen that the Bin’s bound (10) is tighter than Eq. (9) when L=w�2. Even though our bound is generally looser than the Bin’s bound, it can dealwith the codes with time-hopping patterns, which is impossible by the Bin’s bound.Thus it is more useful to calculate the cardinality bound of the general two–dimensional OCDMA codes.

Now we extend this bound to the case when there are multiple service classes with�w ,k ,d ,q� multilength variable-weight TFH codes. Let gi be the number of generatorsequences in the ith service class, so that the number of code sequences in the ith ser-vice class is Ni=giq for i=0,1, . . . , m−1. Then the number of nonzero finite integersin the augmented extended distance matrix is �i=0

m−1giwi�wi−1�r /Li and is bounded by�q−2d−1�+ �q−1��r−2� because of the same reason in the single service class case andcondition III. Thus we obtain

�i=0

m−1

Niwi�wi − 1�r

Li� q��q − 2d − 1� + �q − 1��r − 2��. �11�

Especially when d=0, this bound reduces to

�i=0

m−1

Niwi�wi − 1�r

Li� q�q − 1��r − 1�.

This seems very similar to the bound derived by Kwong et al. [8] However, this boundis tighter. It is because the proposed TFH codes exclude the cases that each codesequence uses only one frequency component (conventional one-dimensional OOCs) oronly one time slot.


Some examples of cardinality bounds together with code generation parameters arelisted in Table 2, where the asterisk mark �� denotes the case when the optimal car-dinality bound is achieved. Here the code distances are fixed to zero to see whetherthe maximum cardinality is obtained in the inequality (11).

3. Performance Analysis and DiscussionWe show the BER differentiation method for the proposed codes and compare the BERperformances under the various conditions to demonstrate the service differentiationamong multiple service classes.

3.A. Method for Differentiating the Bit-Error-Rate PerformanceTo illustrate the differentiation in terms of BER, we analyze the BER performance ofthe proposed codes.

For 0� i�m, if there are ni users in the ith service class equipped with a�wi ,ki ,d ,q� TFH code, then the BER performance can be approximated to

BERi = Q��SIRi

2� �12�

by using the Gaussian approximation [5,10]. Here, Q�x�=1/�2�x� exp�−s2 /2�ds and

SIRi is the signal-to-interference ratio of the ith service class. Following the hypoth-esis that the most important interference of the OCDMA system is the multiple userinterference, we concentrate on the multiple access interference, and the physicalparameters such as thermal noise, dark current, and beat noise are not considered.Then the SIRi becomes

wi2

�l=0

l=m−1

�nl − il��il2

and can be approximated by

SIRi =wi

2

�l=0

l=m−1

nl�il2

�13�

for sufficiently large nl’s, where il is 1 when l= i and 0 otherwise, �il2 is the mean vari-

ance of the cross-correlation values between the pair of ith class codes and lth class

Table 2. Examples of the Generation Conditions and the CardinalitiesWhen d=0a

m q w0 w1 w2 k0 k1 k2 N0 N1 N2 Cardinality

1 7 3 4 42 42*

1 7 3 10 84 84*

1 8 4 3 16 282 7 3 3 2 12 21 35 3N0+N1=98*

2 7 4 2 2 2 7 21 4N0+N1=772 8 4 2 0 2 8 32 6N0+N1=842 11 3 2 10 11 110 330 3N0+N1=660*

3 13 4 3 2 6 2 3 13 13 13 3N0+3N1+N2=351

3 23 9 6 3 36 24 12 23 23 46 72N0+45N1+18N2=22517

3 31 8 6 4 12 14 16 31 31 62 28N0+15N1+6N2=8835

aIn this table, m is the number of service classes and q is the number of available frequency components. wi and Ni standfor the code weight of the ith service class and the number of code sequences in the ith service class, respectively. The aster-isk mark �*� represents the case when the total number of codes meets the cardinality bound.


codes. Now we set Hx�i�x�j�� as the average cross-correlation value for any pair ofcodes in service classes i and j. Then

�ij2 = E��Hx�i�x�j��2� − E2��Hx�i�x�j��,

where E��Hx�i�x�j�� is the time-shift averaged value of Hx�i�x�j�� and

E��Hx�i�x�j�� = pij =1

2

wi�gjwj − ij�

�gjq − ij�Lj�

wiwj

2qLj.

Here, pij is the probability of chip collision between the address code that belongs tothe ith service class and the incoming code sequence that belongs to the jth serviceclass (2 in the denominator means the equiprobability of sending 0 and 1). Since theproposed TFH codes maintain the one-coincidence property between any time-shiftedversion of the different-length code sequence pairs, the variance reduces to

�ij2 = pij�1 − pij�,

and Eq. (13) becomes

SIRi =wi

2

�j=0

j=m−1

njpij�1 − pij�

.

Because the function Q�x� is monotonic, it is sufficient to compare the values SIRiinstead of the values of BERi for the comparison of the BER performances of eachclass code. To approximate the relative BER performances between the ith and the jthservice classes, we let Rij=SIRi /SIRj be the ratio of SIRi to SIRj. To consider its depen-dence on the code parameters, we describe �ij

2 in Eq. (12) as a function of wi, wj, Lj,and the number of frequency components q.

Then Rij becomes

Rij =wi

2

�l=0

l=m−1

nlpil�1 − pil� �wj

2

�l=0

l=m−1

nlpjl�1 − pjl�

and is rewritten as

�l=0

l=m−1

wj2Rijnlpil�1 − pil� = �

l=0

l=m−1

wi2nlpjl�1 − pjl�;

hence

�l=0

l=m−1

wj2Rijnlpil�1 − pil� − wi

2nlpjl�1 − pjl� = 0.

Applying the derived pij, we get

�l=0

l=m−1

nl�wj2Rij

wiwl

2qLl�1 −

wiwl

2qLl� − wi

2wjwl

2qLl�1 −

wjwl

2qLl�� = 0,

and it can be rewritten as

wiwj �l=0

l=m−1 nlwl

2qLl�wjRij�1 −

wiwl

2qLl� − wi�1 −

wjwl

2qLl�� = 0.

Setting wl /2qLl=�l, we get

wiwj �l=0

l=m−1

nl��wjRij − wi��l − wiwj�Rij − 1��l2� = 0,


and since �l2 1, it can be reduced to

wjRij − wi = 0;

therefore

Rij =wi

wj. �14�

From this result, we can deduce that if we set wi�wi+1 for the �w ,k ,d ,q� TFHcode, where 0� i�m−1, it is possible to differentiate the BER performance of the ser-vice classes using the code weights without controlling the code length.

3.B. Performances ComparisonsTo confirm the analysis in the previous subsection, we compare the BER performancesunder the various conditions. BER values are calculated under the assumption of theGaussian distribution of multiple user interference (MUI).

First, we concentrate on the BER performances of the constant weight codes. We fixthe weight of each service class to 8 and change the code lengths as shown in Fig. 1.The total number of simultaneous users is fixed to 30. Among the various user distri-butions, we plot the cases when N0=6,12,18, respectively. According to the result ofthe previous subsection, the BER performances of different service classes are almostthe same, although the TRs of service classes are set to different rates such as1/20:1/30:1/40=6:4:3. This implies that the differentiation between the BER perfor-mances is not mainly dependent on the code lengths for the proposed codes. Instead,the weight is considered as a crucial factor to differentiate BER performances. We canalso note that as N0 decreases and N2 increases, the BER performances of all the ser-vice classes are improved. This is because when the code weight is fixed and the codelengths are variable, shorter codes can produce more “1” pulses during a fixed timeinterval than longer codes can, which induces more interference to the OCDMA net-work.

We also investigate the effect of different weights. To eliminate the effect of differ-ent code length, we fix the code length to 20, while weights in the codes are changedto 8, 6, and 4, respectively. Figure 2 shows that the codes with larger weights give bet-ter BER performance than the codes with smaller weights, as expected from Eq. (14).It is noted that the distinct differentiation of the BER performance is always guaran-teed under any user distribution, unlike the fixed-weight case. We can also observe

Fig. 1. BER performances of three service classes when the code weights are fixed andthe code lengths are varied.


the same trend of BER change as in Fig. 1, because when the code length is fixed,codes with larger weights can generate more interfering pulses than the codes withsmaller weight can.

From these two performance results, we can infer that the fixed-weight multilengthFH codes [8], which have the same autocorrelation and cross-correlation properties asthe proposed TFH codes, can only provide TR differentiation and are not able to sup-port distinguished BER differentiation.

We emphasize that the proposed TFH code can make a service class with a betterBER to have a higher TR than the other service classes have, by designing the TFHcodes assigned to that service class to have the larger weights and the shorter codelengths. This differentiation policy is not possible in the truncating method [10] with-out additional devices for the power control, because the code sequences do not con-tain time blank patterns that can be used to adjust the code length without increasingthe weights.

Now we consider the change of the BER performances under the various user dis-tributions. Although the differentiation of the BER performances between the serviceclasses is guaranteed when weight is differentiated, as shown in Fig. 2, the overallBER performance varies as the user distribution changes. For example, when N0=6,the BER of service class 0 varies from 10−8 to 10−11 as N2 increases. So if we design anOCDMA network to maintain constant BER performance under any user distribution,we should control the overall interference independent of that.

To do this, let us call wi /Li=D the weight density and apply pij�wiwj /2qLi to thedenominator term of SIRi in Eq. (13), which represents the MUI. Then it becomes

�j=0

m−1

nj

wjD

2q �1 −wjD

2q � ,

which is constant for any user distribution �n0 ,n1 , . . . ,nm−1�. Hence SIRi is onlyrelated to wi, and thus the BER performance can be equalized.

In Fig. 3, the weight density of the code is fixed at 0.2. As the weights in the codeare different, distinct BER differentiation is obtained; the BER of service class 0 isapproximately 10−11, it is 10−7 for service class 1, and 10−4 for service class 2. Also, theoverall BER performance under various user distributions remains nearly constant,unlike the previous results in Figs. 1 and 2 where the weight densities of the codesare all different. Hence if the purpose of the code design is mainly on maintainingBER performance under any user distribution, equal weight densities of the codes

Fig. 2. BER performances of three service classes when the code lengths are fixed andthe code weights are varied.


should be considered. In this situation, however, we cannot design the service classeshaving higher BER and higher TR, because the codes with larger weights requirelonger code lengths.

Summarizing the above results, we can conclude that the proposed code can sup-port more sophisticated service differentiation, especially in terms of TR and BERcompared to the fixed-weight code generation algorithm [8] and the code truncatingmethod [10].

4. ConclusionsIn this paper, we proposed a multiclass TFH code generation scheme that can con-struct the codes with multiple code lengths and variable weights in a flexible mannerto support multimedia service differentiation. The proposed code construction methodallows codes to have diverse code lengths and weights. We have shown that it is pos-sible to diminish the search space by using algebraically generated sequences as thedistance and time-hopping patterns, i.e., OOCs and prime sequence patterns asexplained in Section 2. However, the existence of a multilength variable-weight TFHcode with arbitrary parameters is still recondite.

We also showed that the ratio of the SIR between any two service classes can beapproximated by the ratio of the weights in the code independent of the code lengths.So it is possible to design the codes with arbitrary TR and BER by adjusting theweight and length independently.

Finally, we presented the BER performance under various situations and concludedthat the proposed code can provide the exact and flexible differentiation of TR andBER compared to the fixed-weight or fixed-length codes.

AcknowledgmentThis work was supported by the Brain Korea 21 project. Lee and Seo were also sup-ported partly by the Basic Research Program of the Korea Science and EngineeringFoundation and by the University IT Research Center project.

Fig. 3. BER performances of three service classes when weight densities are fixed.


References1. S. V. Salehi, “Code division multiple-access techniques in optical fiber networks—Part I:

Fundamental principles,” IEEE Trans. Commun. 37, 824–833 (1989).2. S. V. Salehi, “Performance comparison of multiwavelength CDMA and WDMA+CDMA for

fiber optic networks,” IEEE Trans. Commun. 45, 1426–1434 (1997).3. S. Peng and Y. Hu, “Two-dimensional optical CDMA differential system with prime/OOC

codes,” IEEE Photon. Technol. Lett. 13, 1373–1375 (2001).4. E. S. Shivaleela, K. N. Sivarajan, and A. Selvarajan, “Design of a new family of two-

dimensional codes for fiber-optic CDMA networks,” J. Lightwave Technol. 16, 501–508(1998).

5. S. V. Maric, O. Moreno, and C. J. Corrada, “Multimedia transmission in fiber-optic LANsusing optical CDMA,” J. Lightwave Technol. 14, 2149–2153 (1996).

6. W. C. Kwong and G.-C. Yang, “Design of multilength optical orthogonal codes for opticalCDMA multimedia networks,” IEEE Trans. Commun. 50, 1258–1265 (2002).

7. I. B. Djordjevic, B. Vasic, and J. Rorison, “Design of multiweight unipolar codes formultimedia optical CDMA applications based on pairwise balanced designs,” J. LightwaveTechnol. 21, 1850–1856 (2003).

8. W. C. Kwong and G.-C. Yang, “Frequency-hopping codes for multimedia services in mobiletelecommunications,” IEEE Trans. Veh. Technol. 48, 1906–1915 (1999).

9. T. Pu, Y. Q. Li, and S. W. Yang, “Research of algebra congruent codes used in two-dimensional OCDMA system,” J. Lightwave Technol. 21, 2557–2564 (2003).

10. E. Inaty, H. M. H. Shalaby, P. Fortier, and L. A. Rusch, “Multirate optical fast frequencyhopping CDMA system using power control,” J. Lightwave Technol. 20, 166–177 (2002).

11. L. Bin, “One-coincidence sequences with specified distance between adjacent symbols forfrequency-hopping multiple access,” IEEE Trans. Commun. 45, 408–410 (1997).

12. S.-S. Lee and S.-W. Seo, “New construction of multiwavelength optical orthogonal codes,”IEEE Trans. Commun. 50, 2003–2008 (2002).

13. J. Kim, C.-K. Lee, S.-W. Seo, and B. Lee, “Frequency-hopping optical orthogonal codes witharbitrary time blank patterns,” Appl. Opt. 41, 4070–4077 (2002).

14. C.-K. Lee, J. Kim, and S.-W. Seo, “Generation and performance analysis of the frequency-hopping optical orthogonal codes with arbitrary time blank patterns,” in Proceedings ofIEEE International Conference on Communications, Vol. 4 (Institute of Electrical andElectronics Engineers, 2001), pp. 1275–1279.

15. C.-K. Lee, J. Kim, and S.-W. Seo, “Multi-length time-and-frequency-hopping codes formultimedia service differentiation,” in Proceedings of IEEE International Conference onCommunications, Vol. 3 (Institute of Electrical and Electronics Engineers, 2005), pp.1603–1607.

16. W. C. Kwong, G.-C. Yang, V. Baby, C.-S. Bres, and P. R. Prucnal, “Multiple-wavelengthoptical orthogonal codes under prime-sequence permutations for optical CDMA,” IEEETrans. Commun. 53, 117–123 (2005).

17. F.-R. Gu and J. Wu, “Construction of two-dimensional wavelength/time optical orthogonalcodes using difference family,” J. Lightwave Technol. 23, 3642–3652 (2005).

18. I. B. Djordjevic, B. Vasic, and J. Rorison, “Design of multiweight unipolar codes formultimedia optical CDMA applications based on pairwise balanced designs,” J. LightwaveTechnol. 21, 1850–1856 (2003).

19. F.-R. Gu and J. Wu, “Construction and performance analysis of variable-weight opticalorthogonal codes for asynchronous optical CDMA systems,” J. Lightwave Technol. 23,740–748 (2005).

quality-of-service differentiation by multilength variable-weight time-and-frequency-hopping optical...

Documents