1. Introduction

Optical Code Division Multiplexing (OCDM) provides a class of multiplexing technique other than Time Division Multiplexing (TDM) andWavelength Division Multiplexing (WDM) [1,2]. In Optical Code Division Multiple Access (OCDMA) [3–8], communication channels are created by allocating a unique codeword to each user that spreads each users signal in time, wavelength, phase, etc., so that multiple users can occupy the entire optical bandwidth simultaneously. Although there has been growing activity in OCDMA, little consideration has been paid to the application of OCDM to transport networks. An OCDM network has a teletraffic capacity larger than a WDM network, especially when bursty traffic is carried out [9]. OCDM is generally classified into incoherent and coherent regimes depending on the degree of coherence of the signal source. Incoherent OCDM uses unipolar code, in which an on-off keying modulation format is adopted [10]; on the other hand, coherent OCDM adopts carrier phase shifted optical chips pulse sequence [11–16]. The most significant advantage of the coherent scheme is the higher signal-to-interference-noise ratio, which is directly attributed to the superior orthogonality between the codes, which in turn yields higher processing gains. Most of the works in 1980s dealt with incoherent optical codes, but they have revealed the limited system performance. With the advent of optical device technology, the difficulties of optical phase encoding have been overcome in recent years [17].

Implementing the multiplexing in the optical layer, OCDM can further enhance bandwidth granularity of the WDM transport network and gives rise to the OCDM/WDM hybrid network. It is shown that, the spectrum efficiency of the OCDM/WDM could be twice as much of that with WDM only [18]. Bandwidth on one wavelength can be divided into small fractions labeled by optical codes, and assigned to different connections. In this way, provisioning fractional wavelength capacity is achieved by the marriage of OCDM and WDM. Different from the wavelength routing in a conventional WDM network, there are two dimensions of optical code and wavelength used for optical routing. It is called code/wavelength routing network [19–22].

In this paper we evaluate the performance of a bufferless OCDM/WDM Optical Packet Switch equipped with shared Wavelength Converters and in which the output packet contentions are solved in both code and wavelength domain. Two analytical models are proposed to dimension the switch resources. The first one allows us to dimension the number of optical codes supported on each wavelength so that the Packet Loss Probability due to both Multiple Access Interference (MAI) and beat noise is maintained under a threshold value. The second one allows us to dimension the number of Wavelength Converters so that the minimum packet loss probability due to output packet contentions is reached. By means of the proposed analytical models, the optimal parameters, in particular the number of wavelengths and the code length, are evaluated so that the best performance in terms of packet loss probability are reached for a fixed bandwidth. The organization of the paper is as follows. In Section 2 we describe the OCDM/WDM switch in question. In Section 3 we discuss the control algorithm used to schedule the packets and to decide which packets must be code and wavelength translated. The analytical models needed to evaluate the performance of the OCDM/WDM switch and to dimension the switch resources are introduced in Section 4. In Section 5 we discuss the effectiveness of the OCDM/WDM technique in solving output packet contentions. Our main conclusions and further research topics are discussed in Section 6.

2. WDM/OCDM Optical Packet Switch Architecture

We consider the WDM/OCDM optical packet switch with N Input/Output fibers (IF/OF) reported in Fig. 1. Each fiber supports M wavelengths denoted λ ₁, …,λ_M. On each wavelength up to F packets are carried out by using Optical Code Division Multiplexing. Let L be the code length and {OC_k,k = 1, …,F} the set of Optical Codes (OC) of a wavelength on which the packets can be carried out. An input (or output) channel is identified by the triple (i,λ_j,OC_k), where i (i = 1, …,N) identifies one of the input (or output) fibers, λ_j (j = 1, …,M) identifies one of the wavelengths on that IF (OF) and OC_k (k = 1, …,F) identifies one of the optical codes on that wavelength.

The operation mode of the architecture is synchronous, meaning that all arriving packets have a fixed size and their arrival on each input channel is synchronized on a time-slot basis [23]. The synchronization operation, not shown in Fig. 1, is realized by means of synchronizers located at the ingress of the switch.

 

Fig. 1. WDM/OCDM Optical Packet Switch.

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The WDM/OCDM optical packet switch illustrated in Fig. 1 performs the following operations: (i) the packets are wavelength demultiplexed and code decoded by means of one WDM demultiplexer and M · F OC decoders for each IF; (ii) the unit control, not shown in Fig. 1, processes the packet headers and according to the rules of the scheduling algorithm illustrated in Section 3, decides which packets have to be wavelength converted and which Output Wavelength Channels (OWC) and Output Optical Codes (OOC) are assigned to the packets to transmit; (iii) the Switching Fabric SF_c routes the packets towards either the bank of r Wavelength Converters (WC) or the Output Switching Fabrics SF_i (i = 1, …,N) according to decisions taken by the control unit; (iv) the converted packets are routed towards the SF_i (i = 1, …,N) where in output the packets are code and wavelength multiplexed by means of M · F OC coders and one WDM multiplexer.

One of the advantages of the proposed OPS is to reduce the number of WCs used because the control algorithm, as illustrated in Section 3, first tries to solve the contention in code domain by changing the packet code and only if this operation is unsuccessful, the contention is solved in wavelength domain by using one WC of the shared pool. This strategy, preferring the use of passive devices like OC decoders and coders rather than active elements like WCs, allows for the reduction of the switch cost.

3. Control Algorithm in WDM/OCDM Optical Packet Switches

A flow chart of the Scheduling Algorithm (SA) executed in each time-slot for the WDM/OCDM Optical Packet Switch (OPS) is illustrated in Figs. 2, 3. The SA starts with the Initialization phase in which some sets and variables are initialized:

After the Initialization phase, the control unit performs the packet scheduling operations. To allow for a fair assignment of the Wavelength Converters to the various OFs, these ones are randomly selected and for each of them the Code Conversion (CC) andWavelength Conversion (WC) phases are performed. In CC phase, illustrated in Fig. 2, the packets that can be directed without wavelength conversion are scheduled and the contentions are resolved by changing the code. In WC phase, described in Fig. 3, the packet contentions are resolved in the wavelength domain by using the WCs.

In CC phase, for each wavelength λ_j (j = 1, …,M), the control unit randomly schedules up to F packets, chosen among the packets arriving on wavelength λ_j, to be transmitted without wavelength conversion. The set Λ_{i, j} initially contains all the output channels on wavelength λ_j and optical codes OC_k (k = 1, …,F) which are used to forward without wavelength conversion up to F packets belonging to I_{i, j} set. The following actions are performed by the control unit for each wavelength λ_j: (i) one packet a and one output channel (i,λ_j,OC_k) are randomly selected from sets I_{i, j} and Λ_{i, j} respectively; (ii) the packet is scheduled to be forwarded on output channel (i,λ_j,OC_k); (iii) the sets I_{i, j} and Λ_{i, j} are updated by removing the elements a and (i,λ_j,OC_k) respectively. The CC phase complexity is O(F · M) because in the worst case at the least one operation for each output channel is performed.

In WC phase the control unit tries to forward with wavelength conversions the remaining packets on output channels not used in CC phase. The packets not scheduled and the free output channels for i − th OF, are stored in I_i and Λ_i respectively. The following actions are performed by the control unit until either I_i or Λ_i (or both) are empty and one WC is available: (i) one packet a and one output channel (i,λ_j,OC_k) are randomly selected from sets I_i and Λ_i respectively; (ii) the packet is scheduled to be forwarded on output channel (i,λ_j,OC_k); (iii) the sets I_i and Λ_i are updated by removing the elements a and (i,λ_j,OC_k) respectively; iv) the available number r_a of WCs is decremented by one. Before the end of WC phase, the control unit checks if there are packets in I_i which are yet to be scheduled and in this case they are discarded. The WC phase complexity is O(min(r,M · F)) because in the worst case at the least one operation is performed for each output channel until WCs are available.

Because the operations in CC and WC phases are performed for each OF, the computational complexity of the proposed scheduling algorithm equals O(N · F · M).

4. Resource dimensioning in WDM/OCDM Optical Packet Switches

We illustrate the analytical models to dimension the resources in WDM/OCDM Optical Packet Switch. Two analytical models are proposed: the first one allows the dimensioning of the number F of codes so that the Packet Loss Probability P^noise_loss due to both Multiple Access Interference (MAI) and beat noise on each wavelength is limited to a given threshold value; the second one allows the dimensioning r of the WCs so that the Packet Loss Probability P^opc_loss due to packet contentions is minimized. We will evaluate P^noise_loss and P^opc_loss in Sections 4.1 and 4.2 respectively as a function of the following traffic and switch parameters: (i) N is the number of IF/OF; (ii) M is the number of wavelengths; (iii) F is the number of codes carried out on each wavelength; (iv) p is the offered traffic offered to each input channel; (v) L is the code length; (vi) H is the packet length.

 

Fig. 2. Scheduling Algorithm in WDM/OCDM Optical Packet Switch; Initialization and Conversion Code phases are described.

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4.1. Evaluation of the Packet Loss Probability due to MAI and beat noise

The evaluation of the Packet Loss Probability P^noise_loss for a target user will be carried out under the assumptions that coherent OCDM techniques are considered and the coding is based upon optical amplitude, where each chip in a code sequence can have a phase 0 or π with a Binary-Phase-Shift-Keying (BPSK) scheme [3, 9, 10, 21]. Gold Codes are assumed that leads to a maximum number of codes equal to L+2 [24, 25]. In evaluating P^noise_loss we take into account both MAI and beat noise modeled as gaussian processes [26].

Due to the synchronous operation mode of the switch, we can express P^noise_loss by conditioning to the number of packets arriving on any input wavelength. We can write:

 

Fig. 3. Scheduling Algorithm in WDM/OCDM Optical Packet Switch; Wavelength Conversion phase is described.

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wherein P^noise,i_loss is the Packet Loss Probability due to MAI and beat noise when there are i packets arriving on any input wavelength. If we assume that an error occurring in a single bit of the packet due to MAI and beat noise results in whole packet error, the following expression can be given for P^noise,i_loss:

where P^noise,i_bit is the probability that a bit is corrupted from MAI and beat noise when i packets are arriving on any input wavelength. If we assume that interfering users transmit bit ‘0’ and ‘1’ with equal probability $\frac{1}{2}$ , the following expression holds for P^noise,i_bit:

where P^{noise,i,′target′}_bit,h is the probability that a bit transmitted by a target user is corrupted for MAI and beat noise when exactly h interfering users transmit bit ‘1’.

P^{noise,i,′target′}_bit,h is evaluated in Appendix A and its expression is given by:

wherein:

The following expressions for $m_{Z_{0,h}}$ , $m_{Z_{1,h}}$ , ${\sigma }_{Z_{0,h}}$ , ${\sigma }_{Z_{1,h}}$ , th_i have been evaluated in Appendix A:

ℜ and T_c being the detector responsitivity and the chip duration respectively. The dependence on signal, MAI noise, Primary Beat noise and Secondary Beat noise [29] have been indicated in Eqs. (5)–(8). Finally by inserting Eqs. (5)–(9) in Eq. (4) and by using Eqs. (1)–(3) we can evaluate P^noise_loss.

4.2. Evaluation of the Packet Loss Probability due to packet contentions

The evaluation of the Packet Loss Probability P^opc_loss due to packet contentions will be evaluated under the assumption that packet arrivals on the N · M · F input wavelength channels are independent. We also assume uniform and symmetric traffic that is packet arrivals on each input channel occur with probability p and a packet has the same probability $\frac{1}{N}$ to be directed to any given OF. The uniform and symmetric traffic scenario is assumed because it is the one requiring the highest number of WCs and thus the highest conversion cost [27]. The explanation of this is illustrated in the following: (i) the packet loss is either due to the lack of Output Channels or to the lack of WCs and WCs dimensioning must be performed so that the second loss event is negligible with respect to the first one; (ii) the uniform and symmetric traffic assumption involves the lowest packet loss due to lack of Output Channels and then according to the point (i), the highest number of WCs.

Due to the uniform and symmetric traffic assumption, the Packet Loss Probability P^opc_loss may be evaluated by considering the packet loss occurring at any OF in the following denoted target. In particular by conditioning to the number i (i = 0,1, …, r) of WCs still available when the OF target is selected according to the scheduling algorithm reported in Figs. 2, 3, we can write:

wherein:

The probabilities P^opc,OF,i_loss (i=0,1, …, r) may be computed according to a procedure illustrated in [27] where these probabilities are evaluated for a Multi-Fiber Optical Packet Switch using both wavelength and space domain to solve output packet contentions.

The probabilities p_Q(i) (i = 0,1, …, r) can be evaluated by taking into account that: (i) the control algorithm illustrated in Section 3, selects the OFs in a random way and hence the probability that the target OF is the j − th OL (j = 1, …,N) selected is equal to $\frac{1}{N}$ ; (ii) when the OF target is the j − th OF selected, i (i = 1, …, r) WCs are available for the target OF if the first j − 1 OFs selected need (r − i) WCs; conversely WCs are not available for the OF target if the first j − 1 OFs selected need a number of WCs greater than r. According to the following remarks and denoting with W_j and $p_{W_j}\left(i\right)$ the number of conversions required by the first j selected OFs and the W_j’s p.m.f. respectively, we can write:

The probabilities $p_{W_j}\left(i\right)$ are evaluated by assuming the number of conversions required in the various OFs statistically independent. Notice that in reality this independence assumption does not hold in fact: (i) the number of conversions required in an OF depends on the number of packets directed to the OF; (ii) the number of packets arriving on a specific wavelength and directed to the various OFs are statistically dependent and negatively correlated [28] in fact their sum is not larger than N · F and every additional packet destined to a specific OF reduces the likelihood of packets directed to an other OF. Although this dependence, it is shown in [28] that it is very slight and can be neglected especially when the traffic offered to the switch is low. According to the observations above and denoting with ⊗ the convolution operator, we can express $p_{W_j}\left(i\right)$ for j = 1, …,N − 1 as follows:

wherein:

W and p_W(i) (i = 0, …,F · M) are the number of conversions required by any OF and the W’s p.m.f. respectively. The probabilities p_W(i) are evaluated in [27].

In conclusions, by means of Eqs. (10)–(13), we are able to evaluate the Packet Loss Probability P^opc_loss due to packet contentions.

5. Numerical Results

This section will be devoted to cope with the following issues: (i) sensitivity analysis of the resource dimensioning of an OCDM/WDM switch; (ii) evaluation of optimal parameters for the design of OCDM/WDM switches so that the best performance in terms of Packet Loss Probability P^opc_loss is obtained.

The Packet Loss Probability P^noise_loss due to MAI, Primary Beat Noise (PBN) and Secondary Beat Noise (SBN) [29] is reported in Figs. 4–6 versus the number F of OCs supported on each wavelength. The offered traffic p is varied from 0.2 to 1 and the packet length H equals 500 bytes. The code length L is chosen to be 511, 1023, 2047 in Figs. 4–6 respectively. To evaluate P^noise_loss, we have used the model introduced in Section 4.1.

 

Fig. 4. Packet Loss Probability due to MAI and beat noise as a function of the number F of Optical Codes. The code length is L=511. The traffic parameters are H=500 bytes and p varying from 0.2 to 1.

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The dimensioning F of used Optical Codes is reported in Fig. 7 for L=1023 and 2047. In particular the greatest value F is determined so that P^noise_loss is lower than or equal to the threshold Packet Loss Probability P^noise,th_loss = 10⁻⁹. These values have been referred to as F ₁₀₂₃ and F ₂₀₄₇ for the cases L=1023 and L=2047 respectively.

Notice as the decrease in offered traffic p allows a reduction of the MAI and beat noise and consequently an increase in the supported number F of OCs. The increase in code length L allows the ratio of the signal power to the MAI and beat noise variance to be increased and the support of a higher number F of OCs. As shown in Fig. 7, for p=0.8, F equals 6 and 14 for L=1023,2047 respectively. Obviously that is paid with an increase in used bandwidth which is around twice when the code length changes from L=1023 to L=2047.

 

Fig. 5. Packet Loss Probability due to MAI and beat noise as a function of the number F of Optical Codes. The code length is L=1023. The traffic parameters are H=500 bytes and p varying from 0.2 to 1.

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In Fig. 7 we also report the ratio F ₂₀₄₇/F ₁₀₂₃ of the number of OCs for L=2047 to the number of OCs for L=1023. Notice as the ratio F ₂₀₄₇/F ₁₀₂₃ is always greater than two, that is F ₂₀₄₇ > 2 · F ₁₀₂₃. The relation follows from the advantages of the statistical multiplexing according to which if a x Hz bandwidth is available, a greater total number of users can be supported when optical codes with length L=2047 are used on the entire band with respect to the case in which the available bandwidth is divided up in two x/2 Hz width parts and shorter optical codes with length L=1023 are used on each sub-band. Starting from the value 1 and by decreasing the offered traffic p, the statistical multiplexing gain is greater and greater and for this reason the ratio F ₂₀₄₇/F ₁₀₂₃ increases.

Next we compare the switch performance in terms of Packet Loss Probability P^opc_loss for different values of the pair (M,L) and when the same bandwidth is allocated for the entire WDM/OCDM signal. This involves that when L is increased, in order to avoid Inter-Symbol Interference (ISI) problems [6–24, 26–30], the bandwidth of each OCDM signal has to be proportionally increased and the number M of wavelengths supported has to be reduced. Then to guarantee no ISI problems and the use of the same bandwidth for the entire WDM/OCDM signal, the performance of the switch OCDM/WDM is evaluated by maintaining fixed the product L · M.

The Packet Loss Probability P^opc_loss due to output packet contentions is reported in Fig. 8 as a function of the number r of used WCs for H=500 bytes, N=8 and p=0.6,0.7,0.8. Two case studies are considered with the code length L equal to 1023 and 2047. As earlier mentioned, the increase in code length L from 1023 to 2047 leads to double the bandwidth allocated to each OCDM signal in order to overcome ISI problems. Because we perform the comparison of the OCDM/WDM switch when the same bandwidth is allocated to the entire WDM/OCDM signal, the increase of L from 1023 to 2047 leads to halve the number M of wavelengths from 16 to 8.

 

Fig. 6. Packet Loss Probability due to MAI and beat noise as a function of the number F of Optical Codes. The code length is L=2047. The traffic parameters are H=500 bytes and p varying from 0.2 to 1.

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The number F of used OCs for each wavelength is chosen so that the threshold Packet Loss Probability P^noise,th_loss due to MAI and beat noise is fixed equal to 10⁻⁹. For instance that leads for p=0.7 to the use of a number F of OCs equal to 6 and 15 for L=1023, 2047 respectively. From Fig. 8, we can notice that all of the sketched curves have the same trend, decreasing as r increases up to a threshold value r_th for which the packet loss probability saturates: this saturation value for the probability denotes the packet loss probability of an OCDM/WDM switch using a full set of WCs, and therefore represents the Packet Loss probability due to lack of output channels. The r_th value denotes the lowest number of WCs needed in an OCDM/WDM switch to reach the same Packet Loss Probability of an OCDM/WDM switch using a full set of WCs. The WC dimensioning is less severe as the number M of wavelengths decreases. For instance when p=0.6, r_th equals 30 and 46 when M equals 8 and 16 respectively. This is to be expected, as the lower M is, the higher F and hence the probability that a contending packet can be forwarded on the same arriving wavelength by changing the optical code and without the use of a WC.

We report in Table 1 the values of P^opc_loss for r=8, 78 that is in the case studies in which few and many WCs are used. For each case study we report P^opc_loss for p=0.6, 0.7, 0.8, and (M,L)={(16,1023),(8,2047)}. From Table 1 we can notice that when few WCs are employed, the lowest P^opc_loss is reached when fewer wavelengths are used that is in the case (M,F)=(8,2047). For instance in the case p=0.7, P^opc_loss = 2.75 · 10⁻³ when (M,F)=(8,2047). The result is not amazing, because when few WCs are used, the packet loss is smaller if fewer wavelength conversions are needed. That occurs when fewer wavelengths and a greater number F of OCs is used. Even when many WCs are used, the lowest P^opc_loss is reached when fewer wavelengths are used. In this case P^opc_loss has already reached the saturation value that depends on the total number M · F of available output channels. P^opc_loss is as much smaller as M · F increases. As previously mentioned, for L increasing, the statistical multiplexing gain leads to higher M · F and consequently smaller values of P^opc_loss are obtained as shown in Table 1. For example for p=0.7, P^opc_loss equals 1.28 · 10⁻⁶ for (M,F)=(8,2047).

 

Fig. 7. Dimensioning of the number F of optical codes versus p for H=500 bytes, P^noise,th_loss = 10 ⁻⁹ and L=1023, 2047. The ratio of the number of optical codes for L=1023 to the number of optical codes for L=2047 is also reported.

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The numerical results show that due to a statistical multiplexing effect, better performance is obtained when fewer wavelengths and more length code are used. Due to the statistical multiplexing, the effect is especially evident when low traffic is offered to the OCDM/WDM Optical Packet Switch. In this traffic scenario, impairments due to MAI and beat noise is reduced and a greater number of optical codes can be supported. The advantages is twofold: the reduction of the Packet Loss Probability due to output packet contentions and the saving of wavelength converters used. Obviously the increase in code length L is paid with an increase in complexity of both laser transmitters and OCDM coders/decoders. Nevertheless notice that the use of holographic techniques allows us to fabricate compact and low cost Super-Structured Fiber Bragg Gratings (SSFBG) able to generate much long codes [31].

Table 1. Packet Loss Probability P^opc_loss due to output packets contentions versus the number M of wavelengths and the code length L for r=8, 78, N=8, H=500 bytes and P^noise,th_loss=10⁻⁹, p=0.6, 0.7, 0.8

View Table

 

Fig. 8. Packet Loss Probability P^opc_loss due to output packet contentions as a function of the number r of used WCs for p=0.6, 0.7, 0.8, N=8, H=500 bytes and (M,L)={(8,2047),(16,1023)}. The threshold Packet Loss Probability P^noise,th_loss due to MAI and beat noise is 10⁻⁹.

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6. Conclusion

In this paper, the performance in Packet Loss Probability due to output packet contentions of an OCDM/WDM switch is investigated. The blocking performance arising from the limited number of output channels and the number of converters is analyzed through uniform traffic models. The proposed analytical models allowed us to dimension the switch resources and to suitably choose the number of wavelengths and the code length so that the best performances in terms of packet loss probability are reached for a fixed bandwidth of the WDM/OCDM signal. The numerical results show that due to a statistical multiplexing effect, better performance is obtained when fewer wavelengths and more length code are used. Due to the statistical multiplexing, the effect is especially evident when low traffic is offered to the OCDM/WDM Optical Packet Switch.

Appendix A: Evaluation of P^{noise,i,′target′}_bit,h (i = 2 …,F;h = 1, …, i − 1)

The receiver model of a target user is reported in Fig. 9 [26, 29]. Three different kinds of noise sources should be taken into account: MAI noise arising from the network, beat noise at the detector and electrical receiver noise (thermal and shot noise). The bandwidth of the receiver is limited to the chip rate and thus is equivalent to an integration over one-chip interval and a thresholder. In this paper, we will focus on beat noise (both Primary and Secondary Beat) and MAI, which are the two most important performance limitations. Other receiver noises such as shot and thermal noise will be discussed in succeeding paper. In this case, if chip-rate squarelow photodetector is used, the output signal Z_a,h from the integrator when the ‘target’ user transmits bit a (a = 0,1) and exactly h interfering users transmit bit 1 is given by the following expression [26, 29]:

 

Fig. 9. OCDM Receiver model and noise sources.

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(14)$$Z_{a,h}=\stackrel{\mathrm{signal}}{\overbrace{\frac{1}{2}a\Re T_c{S}_{\mathrm{ac}}^2}}+\stackrel{\mathrm{MAI\; noise}}{\overbrace{\frac{1}{2}\Re T_c\sum _{j=1}^h{S}_{\mathrm{cc},j}^2}}+\stackrel{\mathrm{Primary\; Beat\; noise}}{\overbrace{a\Re T_c\sum _{j=1}^h{S}_{\mathrm{ac}}{S}_{\mathrm{cc},j}\cos \Delta {\phi }_{d,j}}}+\stackrel{\mathrm{Secondary\; Beat\; noise}}{\overbrace{\Re T_c\sum _{j=1}^{h-1}\sum _{u=j+1}^h{S}_{\mathrm{cc},j}{S}_{\mathrm{cc},u}\cos \Delta {\phi }_{j,u}.}}$$

wherein:

• ℜ is the responsitivity of the detector;
• T_c is the chip duration;
• a is the variable that assumes value 1 if target user transmits bit ‘1’, otherwise assumes value 0;
• S_ac is the autocorrelation of the sequence associated to the target user; if Gold codes are used we have S_ac = L [24]:
• S_{cc, j} is the cross-correlation between the sequences associated to the target user and the j − th interfering user; as it will be later more clear in order to evaluate P^{noise,i,′target′}_bit,h, we need to calculate the S_cc,j ’s average quadratic value $m_{S_{\mathrm{cc},j}^2}$ and the S ² _{cc, j} ’s variance ${\sigma }_{S_{\mathrm{cc},j}^2}^2$ ; if Gold codes are used, these terms can be evaluated by taking into account that the random variables S_{cc, j} (j = 1, …,h) are identically distributed with probabilities [24]:

  (15)$$S_{\mathrm{cc}}=\{\begin{array}{cc}-1+t& \text{with probability}\frac{\alpha }{L}\\ -1& \text{with probability}\\ -1-t& \text{with probability}\frac{\gamma }{L}\end{array}\frac{\beta }{L}$$

  with n = log ₂(L+1), $t=2^{\frac{n+1}{2}}$ , $\alpha =2^{n-2}+2^{\frac{n-3}{2}}$ , β = 2ⁿ − 2^n-1 -1, $\gamma =2^{n-2}+2^{\frac{n-3}{2}}$ . According to Eq. (15) after some algebra we obtain the following expressions for $m_{S_{\mathrm{cc}}^2}$ and ${\sigma }_{S_{\mathrm{cc}}^2}^2$ :

  (16)$$m_{S_{\mathrm{cc}}^2}=L+1-\frac{1}{L}{\sigma }_{S_{\mathrm{cc}}^2}^2=L^2+2L-\frac{2}{L}-\frac{1}{L^2}.$$

• φ_{d, j} = φ_d − φ_j, φ_j,u = φ_j − φ_u with φ_d and φ_s (s = 1, …,h) being the carrier phase of the target and s − th interfering user respectively. In coherent systems we can assume the random variables φ_d,j and φ_j,u as uniformly distributed during [−π,π] [26].

The probabilities P^{noise,i,′target′}_bit,h (i = 2 …, F;h = 1, …, i − 1) are evaluated under the following assumptions:

• the ‘target’ user transmits bit ‘0’ and ‘1’ with equal probability $\frac{1}{2}$ ;
• the interfering MAI and beat noises are modeled as Gaussian noise [26, 29].

According to the above assumptions we can write:

(17)$$P_{\mathrm{bit},h}^{\mathrm{noise},i,\prime \mathrm{target}\prime }=\frac{1}{4}\mathrm{erfc}\left(\frac{{\mathrm{th}}_i-m_{Z_{0,h}}}{\sqrt{2}{\sigma }_{Z_{0,h}}}\right)+\frac{1}{4}\mathrm{erfc}\left(\frac{m_{Z_{1,h}}-{\mathrm{th}}_i}{\sqrt{2}{\sigma }_{Z_{1,h}}}\right).$$

The average values Z_a,h (a=0,1) are obtained from Eq. (14)

(18)$$m_{Z_{a,h}}=\frac{1}{2}a\Re T_c{L}^2+\frac{1}{2}\Re T_c\sum _{j=1}^h{m}_{S_{\mathrm{cc},j}^2}+a\Re T_{c}L\sum _{j=1}^h{m}_{S_{\mathrm{cc},j}}{m}_{\cos {\phi }_{d,j}}+\Re T_c\sum _{j=1}^{h-1}\sum _{u=\mathrm{j}+1}^h{m}_{S_{\mathrm{cc},j}}{m}_{S_{\mathrm{cc},u}}{m}_{\cos {\phi }_{j,u}}.$$

It is possible to prove that the components in Eq. (14) are uncorrelated. For this reason we can write the following expression for ${\sigma }_{Z_{a,h}^2}$ (a=0,1):

(19)$${\sigma }_{Z_{a,h}}^2=\frac{1}{4}h{\Re }^2{T}_c^2{\sigma }_{S_{\mathrm{cc}}^2}^2+a{\Re }^2{T}_c^2{L}^2\sum _{j=1}^h{m}_{S_{\mathrm{cc},j}^2}{m}_{\cos {\phi }_{d,j}^2}+{\Re }^2{T}_c^2\sum _{j=1}^{h-1}\sum _{u=j+1}^h{m}_{S_{\mathrm{cc},j}^2}{m}_{S_{\mathrm{cc},u}^2}{m}_{\cos {\phi }_{j,u}^2}.$$

By substituting Eq. (16) and S_ac = L in Eqs. (18)–(19) and by taking into account that the random variables φ_d,j and φ_j,u are uniformly distributed during [−π,π], we obtain for $m_{Z_{a,h}}$ , ${\sigma }_{Z_{a,h}}^2$ (a=0,1):

(20)$$m_{Z_{a,h}}=\frac{1}{2}a\Re T_c{L}^2+\frac{1}{2}h\Re T_c\left(L+1-\frac{1}{L}\right).$$

(21)$${\sigma }_{Z_{a,h}}^2={\Re }^2{T}_c^2\left(\frac{1}{4}h\left(L^2+2L-\frac{2}{L}-\frac{1}{L^2}\right)+\frac{1}{2}{\mathrm{ahL}}^2\left(L+1-\frac{1}{L}\right)+\frac{1}{4}h\left(h-1\right){\left(L+1-\frac{1}{L}\right)}^2\right).$$

The variable th_i appearing in Eq. (17) is the receiver threshold. In the case of gaussian noise the expression of the optimum threshold has been evaluated [26] and it needs the knowledge of the number h of active interfering users to the receiver. That is not the case. For this reason by assuming that the number i of scheduled packets is known we set as threshold th_i the one obtained when the average number $\frac{1}{2}\left(i-1\right)$ of active interfering users are considered. The following expression holds:

(22)$${\mathrm{th}}_i=\frac{{\sigma }_{Z_{0,\frac{1}{2}\left(i-1\right)}}{m}_{Z_{1,\frac{1}{2}\left(i-1\right)}}+{\sigma }_{Z_{1,\frac{1}{2}\left(i-1\right)}}{m}_{Z_{0,\frac{1}{2}\left(i-1\right)}}}{{\sigma }_{Z_{0,\frac{1}{2}\left(i-1\right)}}+{\sigma }_{Z_{1,\frac{1}{2}\left(i-1\right)}}}.$$

Finally by inserting Eqs. (20)–(22) in Eq. (17) we can evaluate P^{noise,i,′target′}_bit,h.

Acknowledgments

The work described in this paper was carried out with the support of the BONE-project (“Building the Future Optical Network in Europe”), a Network of Excellence funded by the European Commission through the 7th ICT-Framework Programme.

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