Unipolar codes for optical spectral-amplitude CDMA systems based on combinatorial designs

Unipolar codes for optical spectral-amplitude CDMA systems based on combinatorial designs  K. Cui, M. S. Leeson and E. L. Hines School of Engineering, University of Warwick, Coventry, CV4 7AL, United Kingdom E-mail: {K.Cui, M.S.Leeson, E.L.Hines}@warwick.ac.uk. Abstract: A novel class of optical signature codes based on combinatorial designs is proposed for optical spectral-amplitude code-division multiple-access (CDMA) systems. It is applicable to both synchronous and asynchronous incoherent optical CDMA and is compatible with both frequency-hopping and time-spreading schemes. Simplicity of construction, larger code cardinality, and larger flexibility in cross-correlation (CC) control make the proposed code family an interesting candidate for future optical CDMA applications that require a large number of simultaneous users. It has been shown that the system performance can be significantly improved by using the proposed codes with ideal in-phase CC in preference to Hadamard codes.

1

Introduction Driven by the rapid increasing demands of communication bandwidth, multiplexing becomes essential

in both wireless radio and fibre-optic networks. Recently, optical code-division multiple-access (CDMA), adapted from spread spectrum techniques, has emerged as a promising candidate in replacing of time-division multiplexing (TDM) and wavelength-division multiplexing (WDM) technologies [1], [2]. Code-based network access could potentially simplify the network control and management with enhanced information security. The efficient multiple-access protocol also allows many users to access the fibre channel asynchronously and simultaneously without delay and scheduling [2], [3]. However, this scheme is eventually noise-limited due to multiuser interference (MUI) from correlation between different user codes. MUI results in the encoded signals overlapping and corrupting each other, and thus introduces bit errors that degrade the system performance. Optical CDMA systems employing the spectral-amplitude coding of broadband sources are now receiving more attention because this method is ideally MUI-free and can be simply constructed without using any sampling techniques or aliasing cancellation [4], [5]. Since the coding is performed in the

spectral domain and does not rely on direct-sequence encoding, the spreading gain is independent of the bit rate. Thus, spectral-amplitude coding has extreme promise for service differentiation in future access networks, where users with different bit rates and quality of service (QoS) are accommodated simultaneously [3], [6]. For ideal MUI cancellation, optical spectral-amplitude CDMA systems commonly

employ

code

sequences

with

fixed

in-phase

cross-correlation

(CC).

Let

X  x0 , x1 ,..., x N 1  and Y   y0 , y1 ,..., y N 1  be two different code sequences with in-phase CC defined N 1

as  XY   xi yi   [7], [8]. The code sets have ideal in-phase CC when  is equal to one, as this is the i 0

minimum value that can be achieved. This proposal can avoid the limitations of unipolar codes and does not require the complexity of coherent systems [5]. In this paper, we present a novel class of optical signature codes based on mutually orthogonal Latin squares (MOLS) [9], [10], a special case of balanced incomplete block designs (BIBD). The padding method [8], [11] (i.e. adding some ‘0’s and ‘1’s to each codeword) is then applied to relax the correlation constraint. This new code family not only possesses ideal in-phase CC but also exists for a much wider range than other typical spreading sequences. Two major advantages of our proposed scheme are its large flexibility in control of code cardinality and its considerable availability of weight-length combinations. The remainder of this paper is organized as follows. Section 2 introduces the principle of spectral-amplitude coding and basic system structure. The combinatorial code construction and evaluation are given in Section 3. In Section 4, the system analysis and numerical results are presented. Finally, conclusions are drawn in Section 5.

2

Spectral-amplitude coding scheme The ideal optical spectral-amplitude CDMA system [5], [7] is shown in Fig. 1. For each bit of data,

encoding is performed by directly modulating the optical signal with wide spectral content. The broadband optical pulse with spectral distribution A(v) is launched into the spectral encoder when the data bit is “1” and no pulse is launched when it is “0” [7]. The shape of A(v) is a sequence of spectral pulses arranged according to the unipolar code used. The encoded optical pulses from all users are combined at a star coupler and transmitted over the optical fibre link. At the desired receiver, the superimposed signal is divided by a 1 :  splitter and then decoded by two decoders with complementary decoding functions. Note that the notation Av  indicates the pulse pattern with complementary spectral distribution of A(v) . The original data can be recovered by using subsequent balanced detection.

Fig.1 Ideal optical spectral-amplitude CDMA system,

   w    [5], [7] Define the complement of sequence  X  by

X  whose elements are obtained from

X by xi  1  xi

 

[4]. The in-phase CC between sequence X and Y is therefore: N -1

N 1

i 0

i 0

 XY   x i yi   1  xi  yi  w  

(1)

The discrimination of interference from any user having sequence Y can then be achieved by intended receivers that compute  XY   w     XY , where w denotes the code weight [7]. However, the phase-induced intensity noise (PIIN) that is due to the intensity fluctuation of incoherent sources severely degrades the system performance when the value of  is large. Therefore, signature codes with ideal

in-phase CC (i.e.   1 ) become attractive since the noise effects can be effectively suppressed and hence, a higher signal to noise ratio (SNR) results.

3 Code construction based on mutually orthogonal Latin square (MOLS) A Latin square (LS) of side n is defined as an arrangement of n symbols that each symbol occurs exactly once in each row and exactly once in each column [9], [10]. Orthogonality is one of the important concepts in Latin squares. Two Latin squares L1n  ai , j  and L2n  bi , j  are defined as orthogonal if the





ordered pairs ai , j , bi , j are distinct for all i and j . A set of squares L1n , L2n ,..., Lnn is mutually orthogonal, or a set of MOLS, if they are orthogonal in pairs [9]. Let N n  be the maximum number of Latin squares in a set of MOLS of side n. If Q= pm is a prime power (p is a prime number and m  1 ), then N Q  is equal to Q  1 [9], [12]. The Q  1 MOLS of side Q can be constructed through the use of finite field arithmetic [13], [14]. Let  0  0, 1 ,  2 ,...,  n1 be the elements of GF Q  . The set of MOLS Ln is given by Ln  i   j

(2)

where 0  i, j  Q  1 ; 1    Q  1 ; and the algebra is performed in Galois field. In the case of Ln of side 4, there are at most 3 mutually orthogonal Latin squares, i.e. N 4  3 . The generalized construction over GF 4 is then derived as follows. Recall the addition table for a degree-one polynomial when  1  1 , and fill in the i, j  th position according to the modular arithmetic. The following code matrix L14 is then obtained from GF 4 and shown in Table 1.

Table 1: Addition Table for MOLS over GF 4

The other two MOLS can be also constructed using the algorithm as (2) L24  i   2 j

and

L34  i   3 j

(3)

that is

0 2 2 L4   3  1

1 2 3 3 0 1 2 1 0  0 3 2

(4)

0 3 3 L4   1  2

1 2 3 2 1 0 0 3 2  3 0 1

(5)

and





The array L14，L24，L34 is mutually orthogonal and known as a set of MOLS of side 4 [12]. Each element in a MOLS is then denoted row by row and another n  n integer lattice C formed using the linear mapping C i, j   Q * i  j . The numbers C i, j  are referred to as cell labels, i.e.

0 1 2 3 4 5 6 7  C 4,4    8 9 10 11   12 13 14 15

(6)

The initial sets of codewords are generated by reading off the positions of integers in C corresponding to L4 as illustrated in Figure 2.

Fig. 2 Code construction of MOLS

The corresponding design for GF 4 with the resolvability classes is given in Table 2. Each cell represents a codeword, with integers denoting the positions of optical ‘1’s in the code set. For example, the parallel class  1   1 contains the points with labels {0,5,10,15} , {1，4，11，14} , {2，7，8，13} and {3,6,9,12} , respectively. The corresponding codewords are (1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1), (0 1 0 0 1 0 0

0 0 0 0 1 0 0 1 0), (0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0) and (0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0) respectively. Notice that the codewords in the same class are orthogonal and do not interfere with each other, while suffering the same amount of interference from any other groups.

Table 2: Original code family based on MOLS

However, these original MOLS classes as in [10], [14] do not have fixed CC and cannot be used for the conventional optical spectral-amplitude system with balanced detection [5], [7]. Therefore, a flexible padding method is proposed to make the entire code family have unity CC and enlarge the code cardinality. Any t 1  t  Q  1 group of code sets can be grouped together and padded with sub-blocks of length t . Note that the padding order has to be unique for the same class but different from other groups. These t sequences s'   are constructed by using the sample mapping as shown in Fig. 3.

1, s '     0 ,

if   p z  else

(7)

where z is the index of code groups z  1,..., t  and pz  is the random selected chip position in the padding sequences.

Fig. 3 Graphical representation of the mapping method

The value of t is determined by network capacity, i.e. t approaches Q  1 in the full load case and vice versa in that it is set to be low when only small number of active users is required to access. The shorter length of padding blocks reduces relative delays between pulse positions in traditional delay-line systems [15] and implies an increasing chip-rate. The proposed coding can then offer improved security and permit the optical CDMA system to operate at higher bit-rates. As a result, a set of Q 2  t t  3 number sequences, denoted by st  , is obtained as shown in Table 3.

Table 3: Padded-orthogonal codes for Q=4

The ‘padded-orthogonal codes’ have the interesting characteristics that: (1) each code sequence has Q 2  t elements that can be divided into Q  1 groups, with each group containing only one ‘1’ chip; (2)

the in-phase CC between any two distinct sequences is exactly one. Furthermore, the weight-length combinations are significantly increased by the randomness of the chip position in the padding order. This code family thus exists for a much wider range than other traditional signature sequences. The number of possible combinations is given by t ! and so for the full set of the GF 4 example above, up to 120 codewords are possible. The proposed codes can be used in a time-spreading asynchronous optical CDMA system with the addition of more complex dynamic threshold estimation. Nevertheless, they do not perform well because the undesirable out-of-phase CC may be as high as the code weight. Therefore, it is recommended that they are employed in the synchronous or frequency-encoded scheme, in which the code synchronization is always reserved. In the spectral-amplitude systems, temporal synchronisation of the code sequences is then not required, i.e. it is inherently code synchronous. [7]. The ideal periodic correlation can be achieved at each synchronized time and cyclic-shifts of codewords are permitted. This results in an increased number of accommodated users in comparison with asynchronous schemes, conditional on using spreading codes with the same length.

4

Performance analysis and code comparison The interference effects of incoherent light sources are considered according to the theoretical analysis

in [5], [7], [16]. The variance of photocurrent due to the detection of ideal unpolarized thermal light, which is generated by spontaneous emission, can be expressed as 2 2 2 I 2   PIIN   shot   thermal

 I 2 B c  2eIB  4k BTr B RL

(8)

where e is the electronic charge, I is average photocurrent, B is the electrical equivalent noise bandwidth of the receiver,  c is coherence time of source, k B is Boltzmann’s constant, Tr is the absolute temperature of receiver noise and R L is the load resistance. The source coherence time  c can be expressed in terms of its source power spectral density (PSD) Gv  as [17]

c 





0

G 2 v dv

  G v dv   0 

2

(9)

A term c k i  denotes the i th element of the k th padded-orthogonal codes and d k is the data bit of

k th user. The PSD at photodetector 1 and 2 of the n th receiver during one bit period can then be written as [5]

G1 v  

K N -1 Psr d k  ck i cn i   1   v k 1 i 0

  v  N  2i  2  u v  v0  2N   

v   N  2i    u v  v0  2N   and N -1   Psr K G2 v   d k  ck i cn i  1   v  k 1 i 0

  v  N  2i  2  u v  v0  2N   

(10)

v   N  2i    u v  v0  2N  

(11)

respectively, where Psr is the effective power of a broadband source at the receiver, v is the optical source bandwidth, K is the number of active users (  Q 2  Q ), the code length is given by

N  Q 2  Q  1 , and u v  is the unit step function. The photocurrent I is therefore given by 



0

0

I  I1  I 2     G1 v dv    G2 v dv



K    Psr  K    Psr  wd   d   1   d k  kN 1     k 1, k  n  N 1     k 1, k  n 



Q  1d n    Psr Q 2  Q  11   

(12)

where  denotes the photodiode responsivity given by   e hvc .

Here,  is the quantum

efficiency, h is Planck’s constant, and v c is the central frequency of the original broadband optical pulse. The signal from the desired user is then given by the difference of the photodiode outputs as I1  I 2 . As a result, it is clear that the multiuser noise can be completely eliminated. When all the active users are transmitting ‘1’ bits in an equiprobable system, using the average value of



K

c i   K Q and the correlation properties, the total noise variance can be expressed as:

k 1 k

IT

2

 I1

2

 I2

2

 I th

2

 BI 1  c1  BI 2  c 2  2eB I 1  I 2   2



2

4 K BTr B RL

2  2 K  1  Q  4 K BTr B B 2 Psr Q  1K  K  1    Q  K   eBPsr  2  2 RL  Q  Q  1  2v Q 2  Q  1  Q





(13)

In the limit of high power, the shot and thermal noise can be neglected, giving the following SNR:

 I  I 2  2v    1 22    I T  BK Q  1  K  1  Q  K   Q  Q2  

(14)

It is clear that when K is large enough, the SNR due to the intensity noise limit, , will increase as Q increases. System improvement of bit-error rate (BER) through an increase in Q is then observed, where the PIIN is efficiently suppressed. Thus, (12) and (13) give an average SNR limit as (15) [6], where BER can be estimated by PE 

SNRmin 

I 2  I1 2 IT

2



1 erfc 2





SNRmin 8 .

Q 2  2 Psr

2

Q

2



 Q 1

2

 2 K  1  Q  4 K B Tr B B 2 Psr Q  1K  K  1    Q  K   eBPsr  2  2 RL 2v Q 2  Q  1  Q   Q  Q  1 2



(15)



As w  for Hadamard codes is always equal to 2 (regardless of code length), the SNR for Hadamard codes is independent of code length [8]. Therefore, Hadamard codes can be compared with other sequences in Table 4 as a reference, although different code lengths are used.

Table 4: Code comparison for optical spectral- amplitude CDMA system

The performance analysis of our proposed coding and the previously investigated finite geometries [14], modified quadratic congruence (MQC) [16] and the Hadamard codes [5] is illustrated in Fig. 4. The system parameters are: v = 2.5 THz (equivalent to a 20 nm line-width), B = 311 MHz (bit rate 622 Mbits-1) at an operating wavelength of 1550 nm. It is apparent that padded-orthogonal codes result in lower bit error probabilities and offer much larger flexibility in code cardinality compared with the other sequences. This is attributed to their lower mean and standard deviation values of their cross-correlation spectrum [18]. Furthermore, it can be observed that for a given bit error probability, the proposed code family is able to support more users. Taking the typical BER of 10-9 for optical systems, it can support approximately 70 users in contrast to only about 55 for finite geometries, 50 for modified quadratic congruence and 16 for Hadamard codes.

0

10

-2

10

Bit Error Rate (BER)

-4

10

-6

10

-8

10

Hadamard Padded-Orthogonal Codes (Q = 16) Finite Geometries (p = 15, s = 1) M. Quadratic Congruence (p = 13)

-10

10

-12

10

0

50

100 150 200 Number of active users (K)

250

300

Fig.4 Bit-error rate versus the number of active users

In conventional coded optical CDMA systems, the Gaussian approximation as in [18], [19] is a standard method to analyze the system performance without knowledge of the code structure. However, only the effect crosstalk interference is considered and account is not taken of the detector noise. The performance comparison with the spectral-amplitude scheme is shown in Figure 5, when Q = 13. The

padded-orthogonal codes perform better in the latter approach, because the effect of MUI becomes minimal and the interference originated from incoherent sources is significantly alleviated. The proposed analysis then gives more accurate results than the Gaussian approximation.

0

10

-2

Probability of Error (Pe)

10

-4

10

-6

10

-8

10

-10

10

Gaussian approximation Optical spectral-amplitude CDMA

-12

10

0

20

40

60 80 100 120 140 Number of simultaneous users (K)

160

180

200

Fig.5 Performance comparison of Gaussian approximation and proposed analysis

The system capacity versus number of simultaneous users is also shown in Fig. 6 for different cases using Hadamard codes with N = 256, Finite Geometries with p = 5 and padded-orthogonal codes employing Q = 5 or Q = 11. At a given system performance requirement, the network capacity is determined by the possible number of simultaneous users multiplied by the bit rate per user [8]. As the required bit error rate is 10-9, the figure shows that the capacity of the system using our proposed signature codes is larger than that when using other options. Furthermore, the capacity will increase with larger values of Q.

2

System Capacity (Gbits/s)

10

1

10

Hadamard (N = 256) Finite Geometries (p = 5, s = 1) Padded-Orthogonal Codes (Q = 5) Unipolar Coding Proposed (Q = 11)

0

10

0

10

20

30 40 50 60 70 Number of simultaneous users (K)

80

90

100

Fig. 6 System capacity versus number of simultaneous users at 10-9 BER

5

Conclusion A novel class of optical spreading sequences, termed ‘padded-orthogonal codes’ has been introduced.

The proposed codes can be easily constructed in an algebraic way over GF(Q) for each prime power Q. Therefore, they exist for a much wider number of integers than prime codes [11], [19], MQC and Hadamard codes. Moreover, they offer considerable greater flexibility in the control of code cardinality and correlation constraint. In the presence of MUI and PIIN, numerical results show that BER improvements of several orders or magnitude compared with existing codes. Furthermore, for a 10-9 BER, the new codes can accommodate a minimum of 25% more users than previous codes and some 250% more than Hadamard codes. This translates to between three and ten times the network capacity for more than just a few users, demonstrating the considerable promise of the new codes.

6

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