Abstract

We present in this chapter, the performance study of a direct sequence-optical code-division multiple access (O-CDMA) link. In such systems, the main limitation is the multiple access interference (MAI). We investigate various schemes of receiver in the aim of improving the performances and mitigating MAI. Furthermore, we show the benefits of different techniques in regard to conventional ones. However, this system uses ultrashort light pulses that are sensible to the optical link parameters, especially the fiber chromatic dispersion. We have shown that when compensation dispersion devices are not deployed in the system, there is a trade-off between the limited dispersion effects and the MAI.

Keywords: PON, DS-OCDMA, multiple access interference, MAI, OOC, hard limiter, conventional correlation receiver, successive interference cancelation receiver (SIC), parallel interference cancelation receiver (PIC)

## 1. Introduction

The optical fiber offers a small footprint, a low attenuation, and especially a large bandwidth (estimated of the order of THz). However, the cost of a total redeployment of the optical fiber access network would be very important. In order to reduce these costs, it is possible to share the resource among several users, using a passive optical network (PON) type structure. In this case, it is necessary to set up multiple access techniques to differentiate the information associated with each user.

The two most widely used multiple access techniques for optical communications are time-division multiple access (TDMA) and wavelength-division multiple access (WDMA). These two techniques can constitute an economic brake, because the first requires the synchronization of all terminal equipment and the second one requires tunable wavelength filters to adapt to the desired wavelength. Another technique derived from radiofrequency systems has been envisaged for several decades for optical communications: code-division multiple access (CDMA) [1, 2].

In an OCDMA system, the manipulation of the signals can be considered either coherently or incoherently. In a coherent approach, the characteristics of the optical signal measured are amplitude and phase. This configuration requires having a local oscillator synchronized to the optical frequency on reception, which increases the cost of implementation. Since the light wave can be positive or negative, data spreading can be carried out using bipolar codes such as radiofrequency.

N Fð Þ ;W; λa; λ<sup>c</sup> ≤

r tðÞ¼ <sup>∑</sup><sup>N</sup>

• bkðÞ¼ <sup>t</sup> ∑∞

r tðÞ¼ ∑ N k¼1

provides three steps:

of the desired user.

decision variable

73

3. Single-user detection

• bð Þ<sup>k</sup>

delay of user k.

signals transmitted by all active users:

DOI: http://dx.doi.org/10.5772/intechopen.85860

<sup>i</sup>¼�<sup>∞</sup>bð Þ<sup>k</sup>

∑ ∞ i¼�∞

1 W

<sup>i</sup> is the ith data bit of kth user, and <sup>b</sup>ð Þ<sup>k</sup>

• PTb ð Þt is a rectangular pulse of duration Tb

∑ F�1 j¼0 mk i,j

many errors in a busy network.

user. As a result, they allow better performance.

3.1 Conventional correlation receiver (CCR)

F � 1 W � 1 F � 2

Direct Sequence-Optical Code-Division Multiple Access (DS-OCDMA): Receiver Structures…

The N output signals are directed via optical devices, such as star couplers, toward all receivers along the optical fiber. The received signal, r(t), is the sum of

<sup>W</sup> � <sup>2</sup> ��� <sup>F</sup> � <sup>λ</sup>

<sup>k</sup>¼<sup>1</sup>skð Þ <sup>t</sup> � <sup>τ</sup><sup>k</sup> where sk is the transmitted signal of kth user and <sup>τ</sup><sup>k</sup> is the

The transmitted signal sk is given by the following equation: skðÞ¼ t bkð Þt ckð Þt where

<sup>i</sup> PTb ð Þ <sup>t</sup> � iTb represents the data of the kth user

i n o∈{0,1}

� � where m<sup>k</sup>

The receiver is a critical part, because according to its structure and its adequacy to the considered codes, it will condition the performances of the system. Among the main receivers envisaged for optical CDMA, one can distinguish different types:

• Single-user receivers, for which only the knowledge of the code of the desired user is necessary. For these receivers, the interference generated by the other users is not taken into account and is considered as noise. As this interference increases significantly with the number of active users, these receivers make

required. These receivers are more complex than single-user receivers. They use knowledge of nondesired user codes to more reliably estimate the desired

In a DS-OCDMA system using a CCR receiver, the spread spectrum is achieved by directly multiplying a signature code sequence with the data to be transmitted. It

• At first, in reception, the receiver multiplexes the received signal with the code

• In the end, the decision variable will be compared to the value of the threshold

S of the decision-making block in order to make the estimated data.

• In this step, the multiplier output signal is reformatted via an integrator in order to evaluate the total power per bit. The output signal presents the

• Multiuser receivers, for which knowledge of the codes of other users is

PTC t � iTb � jTc � τ<sup>k</sup>

W � λ � � � �� � � � � � � � � : (1)

i,j <sup>¼</sup> <sup>C</sup><sup>k</sup>

<sup>j</sup> � bk

<sup>i</sup> : (2)

But most studies on optical CDMA are about inconsistent systems, much simpler and, therefore, less expensive. They are usually based on a modulation scheme called "intensity modulation-direct detection" (IM-DD), and it is the luminous intensity, positive quantity, which is the measured characteristic of the optical signal. Bipolar codes can no longer be used. Unipolar quasi-orthogonal codes are used.

This chapter is organized as follows: in Sections 1 and 2, a brief description on the state-of-the-art of the DS-OCDMA technique and its advantages over TDMA and WDMA are presented. In Section 3, a study of a DS-OCDMA system using a CCR and CCR with HL is reported. Section 4 is devoted to multiuser receivers such as SIC and PIC receivers. Section 5 is dedicated to the impact of chromatic dispersion on DS-OCDMA performances. Finally, conclusions are drawn in Section 6.

### 2. The optical CDMA network

Every user utilizes an amplitude shift keying (ASK) especially the on/off keying modulation to transmit all the binary data via a common optical fiber. The encoder impresses a sequence code upon the binary data (Figure 1). The sequence code is specific to each user, in order to be able to extract the data by correlation at the end receiver. For the data recovery, the received signal would be compared at first to the sequence code, and then to a threshold level at the comparator.

For low multiple access interference (MAI) and error probability, the chosen codes must have good correlation properties. In this chapter, we consider optical orthogonal codes (OOC) [1]. A class of codes is defined by (F, W, λa, λc) where F is the length of the sequences, W is the weight, and λ<sup>a</sup> and λ<sup>c</sup> the auto- and crosscorrelation constraints, respectively. The maximum number of users N in the OOC's class is defined as: N Fð Þ¼ ;W; <sup>1</sup>; <sup>1</sup> <sup>F</sup>�<sup>1</sup> W Wð Þ �1 j k.

The code signature, ck(t), of the kth user is ckðÞ¼ <sup>t</sup> ∑∞ <sup>j</sup>¼�∞<sup>c</sup> ð Þk <sup>j</sup> PTc t � jTc � � where PTc(t) is a unit rectangular pulse with duration of one chip Tc and c ð Þk j n o∈{0,1} is the jth value of the kth user spreading code. In general, in the case of λ<sup>a</sup> = λ<sup>c</sup> = λ, the number of users N is limited by the Johnson bound [1], given by the relation:

Figure 1. Synoptic scheme of a DS-OCDMA system.

Direct Sequence-Optical Code-Division Multiple Access (DS-OCDMA): Receiver Structures… DOI: http://dx.doi.org/10.5772/intechopen.85860

$$N(F, W, \lambda\_d, \lambda\_c) \le \left\lfloor \frac{1}{W} \left\lfloor \frac{F-1}{W-1} \left\lfloor \frac{F-2}{W-2} \left\lfloor -\dots - \left\lfloor \frac{F-\lambda}{W-\lambda} \right\rfloor - \dots - \right\rfloor \right\rfloor \right\rfloor. \tag{1}$$

The N output signals are directed via optical devices, such as star couplers, toward all receivers along the optical fiber. The received signal, r(t), is the sum of signals transmitted by all active users:

r tðÞ¼ <sup>∑</sup><sup>N</sup> <sup>k</sup>¼<sup>1</sup>skð Þ <sup>t</sup> � <sup>τ</sup><sup>k</sup> where sk is the transmitted signal of kth user and <sup>τ</sup><sup>k</sup> is the delay of user k.

The transmitted signal sk is given by the following equation: skðÞ¼ t bkð Þt ckð Þt where


$$r(t) = \sum\_{k=1}^{N} \sum\_{i=-\infty}^{\infty} \sum\_{j=0}^{F-1} m\_{i,j}^k P\_{T\_c} \left( t - iT\_b - jT\_c - \tau\_k \right) \text{ where } m\_{i,j}^k = \mathbf{C}\_j^k \cdot b\_i^k. \tag{2}$$

## 3. Single-user detection

signal measured are amplitude and phase. This configuration requires having a local oscillator synchronized to the optical frequency on reception, which increases the cost of implementation. Since the light wave can be positive or negative, data spreading can be carried out using bipolar codes such as radiofrequency.

But most studies on optical CDMA are about inconsistent systems, much simpler

This chapter is organized as follows: in Sections 1 and 2, a brief description on the state-of-the-art of the DS-OCDMA technique and its advantages over TDMA and WDMA are presented. In Section 3, a study of a DS-OCDMA system using a CCR and CCR with HL is reported. Section 4 is devoted to multiuser receivers such as SIC and PIC receivers. Section 5 is dedicated to the impact of chromatic dispersion on DS-OCDMA performances. Finally, conclusions are drawn in Section 6.

Every user utilizes an amplitude shift keying (ASK) especially the on/off keying modulation to transmit all the binary data via a common optical fiber. The encoder impresses a sequence code upon the binary data (Figure 1). The sequence code is specific to each user, in order to be able to extract the data by correlation at the end receiver. For the data recovery, the received signal would be compared at first to the

For low multiple access interference (MAI) and error probability, the chosen codes must have good correlation properties. In this chapter, we consider optical orthogonal codes (OOC) [1]. A class of codes is defined by (F, W, λa, λc) where F is the length of the sequences, W is the weight, and λ<sup>a</sup> and λ<sup>c</sup> the auto- and crosscorrelation constraints, respectively. The maximum number of users N in the OOC's

> W Wð Þ �1 j k

where PTc(t) is a unit rectangular pulse with duration of one chip Tc and

λ<sup>a</sup> = λ<sup>c</sup> = λ, the number of users N is limited by the Johnson bound [1], given by the

.

∈{0,1} is the jth value of the kth user spreading code. In general, in the case of

<sup>j</sup>¼�∞<sup>c</sup> ð Þk

<sup>j</sup> PTc t � jTc � �

sequence code, and then to a threshold level at the comparator.

The code signature, ck(t), of the kth user is ckðÞ¼ <sup>t</sup> ∑∞

and, therefore, less expensive. They are usually based on a modulation scheme called "intensity modulation-direct detection" (IM-DD), and it is the luminous intensity, positive quantity, which is the measured characteristic of the optical signal. Bipolar codes can no longer be used. Unipolar quasi-orthogonal codes are

used.

Multiplexing

c ð Þk j n o

relation:

Figure 1.

72

Synoptic scheme of a DS-OCDMA system.

2. The optical CDMA network

class is defined as: N Fð Þ¼ ;W; <sup>1</sup>; <sup>1</sup> <sup>F</sup>�<sup>1</sup>

The receiver is a critical part, because according to its structure and its adequacy to the considered codes, it will condition the performances of the system. Among the main receivers envisaged for optical CDMA, one can distinguish different types:


### 3.1 Conventional correlation receiver (CCR)

In a DS-OCDMA system using a CCR receiver, the spread spectrum is achieved by directly multiplying a signature code sequence with the data to be transmitted. It provides three steps:


The block diagram of the desired user's conventional receiver is shown in Figure 2.

Assuming that the user #1 is the desired user, the decoding part of the DS-OCDMA system is performed by correlation. The received signal is multiplied by the code of the desired user:

$$r\_{corr}(t) = r(t).c\_1(t) = \left(\sum\_{k=1}^{N} b\_k(t).c\_k(t)\right).c\_1(t) \tag{3}$$

$$r\_{corr}(t) = b\_1(t).c\_1(t) + \sum\_{k=2}^{N} b\_k(t).c\_k(t).c\_1(t). \tag{4}$$

At the output of the integration block, we will get the decision variable value Zð Þ<sup>1</sup> i

$$Z\_i^{(1)} = \int\_0^{T\_b} r\_{corr}(t)dt = \int\_0^{T\_b} \left(\sum\_{k=1}^N b\_k(t) \cdot c\_k(t)\right) \cdot c\_1(t)dt\tag{5}$$

$$\begin{split} Z\_i^{(1)}(t) &= \underbrace{\int\_0^{T\_b} b\_i^{(1)}(t) \cdot c\_1(t) dt}\_{i} + \underbrace{\int\_0^{T\_b} \left(\sum\_{k=2}^N b\_i^{(k)}(t) \cdot c\_k(t)\right) \cdot c\_1(t) dt}\_{ii} \\ &= \underbrace{\int\_0^{T\_b} b\_1(t) \cdot c\_1(t) dt}\_{0} + \int\_0^{T\_b} I(t) \cdot c\_1(t) dt}\_{0} \end{split} \tag{6}$$

3.2 Conventional correlation receiver with hard limiter (HL + CCR)

can be used. An ideal OHL function is defined as:

DOI: http://dx.doi.org/10.5772/intechopen.85860

BER as a function of the number of users for different lengths of the code.

[1, 3, 4].

Figure 4.

75

Receiver's performances as a function of number of users.

Figure 3.

In order to limit the power of interfering users, an optical hard limiter (OHL)

Direct Sequence-Optical Code-Division Multiple Access (DS-OCDMA): Receiver Structures…

g xð Þ¼ <sup>0</sup> if x<sup>&</sup>lt; <sup>1</sup>

So, if a received optical power x is bigger than or equal to 1, it will be clipped to 1. On the other hand, if a received optical power x is smaller than 1, it will be set to 0. Consequently, for S=W, an error can occur, if all the code chips are overlapped. Therefore, all other IAM configurations will be canceled. For the ideal chip synchronous case, the theoretical expression of the HL + CCR's error probability is given by

where 1 is the normalized optical power value of one chip.

1 if x≥ 1

(8)

where I tðÞ¼ <sup>∑</sup><sup>N</sup> <sup>k</sup>¼<sup>2</sup>bð Þ<sup>k</sup> <sup>i</sup> ðÞ� t ckð Þt :

The second term of this expression represents the multiple access interference (MAI) term due to all the nondesired users. It depends on the number of active users N and the OOC's correlation properties.

Then, a comparison of the decision variable with the threshold level S has been done. Recording the comparison result, an estimation of the transmitted bit, ^ b1ð Þt , is given. An error happened when the MAI term is greater than the threshold level S and the data ^ b1ð Þt are 0.

Analytical upper bound expression of the error probability has been demonstrated and presented by Salehi et al in [1]:

$$P\_{\rm es} \le \frac{1}{2} \sum\_{i=0}^{N-1} \binom{N-1}{i} \left(\frac{W^2}{2F}\right)^i \left(1 - \frac{W^2}{2F}\right)^{N-1-i} \tag{7}$$

Curves in Figure 3 show that the performances are degraded as the number of users increases. If the threshold level value, S, is chosen close to W, the errors number decreases.

Figure 2. The conventional correlation receiver (CCR).

Direct Sequence-Optical Code-Division Multiple Access (DS-OCDMA): Receiver Structures… DOI: http://dx.doi.org/10.5772/intechopen.85860

Figure 3. BER as a function of the number of users for different lengths of the code.

### 3.2 Conventional correlation receiver with hard limiter (HL + CCR)

In order to limit the power of interfering users, an optical hard limiter (OHL) can be used. An ideal OHL function is defined as:

$$\mathbf{g}(\mathbf{x}) = \begin{cases} \mathbf{0} & \text{if} \quad \mathbf{x} < \mathbf{1} \\ \mathbf{1} & \text{if} \quad \mathbf{x} \ge \mathbf{1} \end{cases} \tag{8}$$

where 1 is the normalized optical power value of one chip.

So, if a received optical power x is bigger than or equal to 1, it will be clipped to 1. On the other hand, if a received optical power x is smaller than 1, it will be set to 0. Consequently, for S=W, an error can occur, if all the code chips are overlapped. Therefore, all other IAM configurations will be canceled. For the ideal chip synchronous case, the theoretical expression of the HL + CCR's error probability is given by [1, 3, 4].

Figure 4. Receiver's performances as a function of number of users.

The block diagram of the desired user's conventional receiver is shown in

N k¼1

N k¼2

At the output of the integration block, we will get the decision variable value Zð Þ<sup>1</sup>

∑ N k¼1

> ∑ N k¼2 bð Þ<sup>k</sup>

I tð Þ� c1ð Þt dt

T ðb

0

T ðb

0

The second term of this expression represents the multiple access interference (MAI) term due to all the nondesired users. It depends on the number of active

Then, a comparison of the decision variable with the threshold level S has been

given. An error happened when the MAI term is greater than the threshold level S

Analytical upper bound expression of the error probability has been demon-

2F � �<sup>i</sup>

Curves in Figure 3 show that the performances are degraded as the number of users increases. If the threshold level value, S, is chosen close to W, the errors

<sup>1</sup> � <sup>W</sup><sup>2</sup> 2F � �<sup>N</sup>�1�<sup>i</sup>

done. Recording the comparison result, an estimation of the transmitted bit, ^

N � 1 i � � W<sup>2</sup>

T ðb

0

bkð Þt :ckð Þt � �

bkðÞ� t ckð Þt � �

> <sup>i</sup> ð Þ� t ckð Þt � �


:c1ð Þt (3)

� c1ð Þt dt (5)

� c1ð Þt dt

i

(6)

b1ð Þt , is

(7)

bkð Þt :ckð Þt :c1ð Þt : (4)

Assuming that the user #1 is the desired user, the decoding part of the DS-OCDMA system is performed by correlation. The received signal is multiplied

rcorrðÞ¼ t r tð Þ:c1ðÞ¼ t ∑

rcorrðÞ¼ t b1ð Þt :c1ðÞþt ∑

rcorrð Þt dt ¼

<sup>i</sup> ð Þ� t c1ð Þt dtþ


b1ðÞ� t c1ð Þt dt þ

Figure 2.

Multiplexing

by the code of the desired user:

Zð Þ<sup>1</sup> <sup>i</sup> ¼

Zð Þ<sup>1</sup> <sup>i</sup> ðÞ¼ t

where I tðÞ¼ <sup>∑</sup><sup>N</sup>

and the data ^

number decreases.

Figure 2.

74

T ðb

0

bð Þ<sup>1</sup>

T ðb

0

T ðb

0

<sup>i</sup> ðÞ� t ckð Þt :

¼

<sup>k</sup>¼<sup>2</sup>bð Þ<sup>k</sup>

users N and the OOC's correlation properties.

b1ð Þt are 0.

The conventional correlation receiver (CCR).

strated and presented by Salehi et al in [1]:

Pes ≤ 1 <sup>2</sup> <sup>∑</sup> N�1 i¼0

$$P\_{ehl} = \frac{1}{2} \binom{W}{S} \prod\_{i=0}^{S-1} \left( 1 - \left( 1 - \frac{W^2}{2F} \right)^{N-1-i} \right) \tag{9}$$

Figure 4 shows the comparison between theoretical performances of the CCR and HL + CCR as a function of the active users number N, for F = 1000, W = 7, and S = 7. It is well verified that the HL + CCR provides better performance than the CCR.

### 4. Multiuser detection

In order to obtain better performances than those obtained by single-user detection, multiuser detection has been studied for OCDMA links [5–7]. Indeed, this type of detection, already used for wireless CDMA, has proved its effectiveness in reducing the impact of interference on performance.

The advantage of multiuser detection over single-user detection is the knowledge of nondesired user codes that allows finer evaluation of the interference present in the received signal. As a result, the data are better detected.

These detectors operate in two main stages:


In the CDMA systems, we can distinguish between two types of multiuser detectors:


### 4.1 Optical successive interference cancelation

The aim of the proposed method is the estimation of interference from interfering users. Once the IAM is determined, it is deducted from the received signal before detecting the desired user. The optical successive interference cancelation structure is shown in Figure 5. We assume that the desired user is the user n°# 1.

The next step is the detection and estimation of the desired user data, for example, user 1, by conventional correlation method. The suppression of the undesired users could take until N�1 stages. At the output of the successive stages block, we obtain such signal before the conventional receiver for the user 1:

Direct Sequence-Optical Code-Division Multiple Access (DS-OCDMA): Receiver Structures…

DOI: http://dx.doi.org/10.5772/intechopen.85860

b ð Þ N <sup>i</sup> cNð Þt

b ð Þ N

N j¼2 ^ bð Þ<sup>j</sup> <sup>i</sup> cjð Þt

<sup>i</sup> cjðÞ�t ∑

N j¼2 b ð Þj <sup>i</sup> � ^ b ð Þj i 

<sup>i</sup> c1ð Þþt ∑

N j¼2 ^ bð Þ<sup>j</sup> <sup>i</sup> cjð Þt

<sup>i</sup> cNðÞ�<sup>t</sup> ^

b ð Þ N�1 <sup>i</sup> cN�<sup>1</sup>ð Þt

� cjð Þt

(11)

sN�<sup>1</sup>ðÞ¼ <sup>t</sup> sN�<sup>2</sup>ðÞ�<sup>t</sup> ^

Figure 5.

77

Principle of the optical-SIC receiver.

…

¼ ∑ N j¼1 b ð Þj

¼ b ð Þ1

<sup>¼</sup> sN�<sup>3</sup>ð Þ�<sup>t</sup> ^

¼ r tð Þ� ∑

The optical-SIC receiver has the knowledge of all the active users' codes patterns, and we assume that all the users have the same transmitting energy. So, there are no strongest interfering signals [8].

The first step provides the estimation ^ bð Þ <sup>N</sup> <sup>i</sup> , for example, of the data of the Nth nondesired user by application of the traditional correlation method.

Then, the transmitting signal of the Nth user is reproduced and removed from the received signal r(t). We call the signal received after the subtraction, s1(t), which referred to output signal of the first cancelation stage (Cs = 1):

$$\begin{split} \boldsymbol{\sigma}\_{1}(t) &= \boldsymbol{r}(t) - \hat{\boldsymbol{b}}\_{i}^{(N)} \boldsymbol{c}\_{N}(t) \\ &= \hat{\boldsymbol{b}}\_{i}^{(1)} \boldsymbol{c}\_{1}(t) + \sum\_{j=2}^{N-1} \hat{\boldsymbol{b}}\_{i}^{(j)} \boldsymbol{c}\_{j}(t) + \left( \boldsymbol{b}\_{i}^{(N)} - \hat{\boldsymbol{b}}\_{i}^{(N)} \right) \boldsymbol{c}\_{N}(t) \end{split} \tag{10}$$

Figure 5. Principle of the optical-SIC receiver.

Pehl <sup>¼</sup> <sup>1</sup> 2

in reducing the impact of interference on performance.

• Successive interference cancelation receiver (SIC).

• Parallel interference cancelation receiver (PIC).

4.1 Optical successive interference cancelation

are no strongest interfering signals [8].

The first step provides the estimation ^

<sup>s</sup>1ðÞ¼ <sup>t</sup> r tðÞ� ^

¼ ^ bð Þ<sup>1</sup>

These detectors operate in two main stages:

4. Multiuser detection

Multiplexing

estimated interference.

detectors:

76

W S � �Y S�1

i¼0

<sup>1</sup> � <sup>1</sup> � <sup>W</sup><sup>2</sup>

Figure 4 shows the comparison between theoretical performances of the CCR and HL + CCR as a function of the active users number N, for F = 1000, W = 7, and S = 7. It is well verified that the HL + CCR provides better performance than the CCR.

In order to obtain better performances than those obtained by single-user detection, multiuser detection has been studied for OCDMA links [5–7]. Indeed, this type of detection, already used for wireless CDMA, has proved its effectiveness

The advantage of multiuser detection over single-user detection is the knowledge of nondesired user codes that allows finer evaluation of the interference present in the received signal. As a result, the data are better detected.

1. estimating all or part of the interference present in the received signal.

2. detecting the desired user data after subtracting from received signal the

In the CDMA systems, we can distinguish between two types of multiuser

The aim of the proposed method is the estimation of interference from interfer-

bð Þ <sup>N</sup>

Then, the transmitting signal of the Nth user is reproduced and removed from the received signal r(t). We call the signal received after the subtraction, s1(t),

<sup>i</sup> cjðÞþ<sup>t</sup> <sup>b</sup>ð Þ <sup>N</sup>

<sup>i</sup> � ^ b ð Þ N i � �

<sup>i</sup> , for example, of the data of the Nth

cNð Þt

(10)

ing users. Once the IAM is determined, it is deducted from the received signal before detecting the desired user. The optical successive interference cancelation structure is shown in Figure 5. We assume that the desired user is the user n°# 1. The optical-SIC receiver has the knowledge of all the active users' codes patterns, and we assume that all the users have the same transmitting energy. So, there

nondesired user by application of the traditional correlation method.

which referred to output signal of the first cancelation stage (Cs = 1):

N�1 j¼2 ^ bð Þj

bð Þ <sup>N</sup> <sup>i</sup> cNð Þt

<sup>i</sup> c1ð Þþt ∑

2F � �N�1�i !

(9)

The next step is the detection and estimation of the desired user data, for example, user 1, by conventional correlation method. The suppression of the undesired users could take until N�1 stages. At the output of the successive stages block, we obtain such signal before the conventional receiver for the user 1:

$$\begin{aligned} s\_{N-1}(t) &= s\_{N-2}(t) - \hat{b}\_i^{(N)} c\_N(t) \\ &= s\_{N-3}(t) - \hat{b}\_i^{(N)} c\_N(t) - \hat{b}\_i^{(N-1)} c\_{N-1}(t) \\ &\dots \\ &= r(t) - \sum\_{j=2}^N \hat{b}\_i^{(j)} c\_j(t) \\ &= \sum\_{j=1}^N b\_i^{(j)} c\_j(t) - \sum\_{j=2}^N \hat{b}\_i^{(j)} c\_j(t) \\ &= b\_i^{(1)} c\_1(t) + \sum\_{j=2}^N \left( b\_i^{(j)} - \hat{b}\_i^{(j)} \right) \cdot c\_j(t) \end{aligned} \tag{11}$$

### Multiplexing

### 4.1.1 First cancelation stage

We consider that noise perturbation is less significant than interference contribution, so the performances are linked only to MAI.

The decision variable for the desired user #1 is Zð Þ<sup>1</sup> <sup>i</sup> <sup>¼</sup> Wbð Þ<sup>1</sup> <sup>i</sup> þ I<sup>1</sup> þ A<sup>1</sup> where:

• <sup>I</sup><sup>1</sup> <sup>¼</sup> <sup>∑</sup><sup>N</sup> k¼3 Ð Tb <sup>0</sup> b ð Þk <sup>i</sup> ckð Þt c1ð Þt dt is The MAI at the output of the other users

$$\bullet \ A\_1 = \left( b\_i^{(N)} - \hat{b}\_i^{(N)} \right) \int\_0^{T\_b} c\_N(t) c\_1(t) dt \text{ term linked to the cancellation of the user } \star \text{N.} $$

The cancelation term A1 can only take two values 0 and �1. Indeed, as we have conventional receivers on this stage, there are only errors for the data bð Þ <sup>N</sup> <sup>i</sup> equal to 0, so:

$$A\_1 = \begin{cases} \mathbf{0} & \text{si } (b\_i^{(N)} = \hat{b}\_i^{(N)} \text{ (if no error in detection )}) \\ -\mathbf{1} & \text{if error in detection} \end{cases}$$

The error probability is

$$P\_{\epsilon\_1} = \frac{1}{2} P\_{\epsilon\_{10}} + \frac{1}{2} P\_{\epsilon\_{11}}.\tag{12}$$

:

prob A<sup>1</sup> ¼ �1=b

prob A<sup>1</sup> <sup>¼</sup> <sup>0</sup>=bð Þ<sup>1</sup>

hence

Pe<sup>11</sup> <sup>¼</sup> <sup>1</sup> 2 W<sup>2</sup>

> ¼ 1 2 W<sup>2</sup>

79

<sup>þ</sup> <sup>1</sup> � <sup>1</sup> 2 W<sup>2</sup>

desired user (n ° 1) has sent the data "0":

<sup>i</sup> ≥S1=b

<sup>¼</sup> prob I<sup>1</sup> <sup>þ</sup> <sup>A</sup><sup>1</sup> <sup>≥</sup> <sup>S</sup>1=bð Þ<sup>1</sup>

<sup>¼</sup> prob A<sup>1</sup> ¼ �1=bð Þ<sup>1</sup>

þ prob A<sup>1</sup> ¼ 0=b

prob I<sup>1</sup> ≥S1=bð Þ<sup>1</sup>

prob A<sup>1</sup> ¼ �1=bð Þ<sup>1</sup>

prob A<sup>1</sup> ¼ 0=b

prob I<sup>1</sup> ≥S<sup>1</sup> þ 1=b

� �

Pe<sup>10</sup> <sup>¼</sup> prob Zð Þ<sup>1</sup>

ð Þ1 <sup>i</sup> ¼ 1 � � <sup>¼</sup> prob bð Þ <sup>N</sup>

DOI: http://dx.doi.org/10.5772/intechopen.85860

<sup>i</sup> � ^ b ð Þ N <sup>i</sup> ¼ �1=b

<sup>L</sup> prob Zð Þ <sup>N</sup>

<sup>L</sup> f S<sup>ð</sup> <sup>N</sup> � <sup>1</sup>; <sup>N</sup> � <sup>2</sup>; <sup>N</sup> � <sup>2</sup>Þ � <sup>f</sup>ð Þ <sup>0</sup>; <sup>S</sup><sup>1</sup> � <sup>W</sup>; <sup>N</sup> � <sup>2</sup>

<sup>L</sup> f S<sup>ð</sup> <sup>N</sup> � <sup>1</sup>; <sup>N</sup> � <sup>2</sup>; <sup>N</sup> � <sup>2</sup>Þ � <sup>f</sup>ð Þ <sup>0</sup>; <sup>S</sup><sup>1</sup> � <sup>W</sup>; <sup>N</sup> � <sup>2</sup>

In the same way, we calculate the probability of error in the case where the

<sup>i</sup> ¼ 0

� � � prob I<sup>1</sup> <sup>≥</sup>S<sup>1</sup> <sup>þ</sup> <sup>1</sup>=bð Þ<sup>1</sup>

� � <sup>¼</sup> f Sð Þ <sup>1</sup> <sup>þ</sup> <sup>1</sup>; <sup>N</sup> � <sup>2</sup>; <sup>N</sup> � <sup>2</sup>

� � <sup>¼</sup> f Sð Þ <sup>1</sup>; <sup>N</sup> � <sup>2</sup>; <sup>N</sup> � <sup>2</sup>

2 W<sup>2</sup>

> 2 W<sup>2</sup>

<sup>i</sup> ¼ 0

ð Þ1 <sup>i</sup> ¼ 0 � � � prob I<sup>1</sup> <sup>≥</sup>S1=<sup>b</sup>

<sup>i</sup> ¼ 0

<sup>i</sup> ¼ 0 � � <sup>¼</sup> <sup>1</sup>

ð Þ1 <sup>i</sup> ¼ 0 � � <sup>¼</sup> <sup>1</sup> � <sup>1</sup>

ð Þ1 <sup>i</sup> ¼ 0

<sup>L</sup> f Sð Þ <sup>N</sup> � <sup>1</sup>; <sup>N</sup> � <sup>2</sup>; <sup>N</sup> � <sup>2</sup> � �

> ð Þ1 <sup>i</sup> ¼ 0

� �

<sup>¼</sup> <sup>W</sup><sup>2</sup> L � 1 2 prob ^ b ð Þ N <sup>i</sup> ¼ 1=b

¼ 1 2 W<sup>2</sup>

¼ 1 2 W<sup>2</sup>

� � <sup>¼</sup> <sup>1</sup> � prob A<sup>1</sup> ¼ �1=<sup>b</sup>

<sup>¼</sup> <sup>1</sup> � <sup>1</sup> 2 W<sup>2</sup>

<sup>i</sup> ¼ 1

ð Þ1 <sup>i</sup> ¼ 1

ð Þ N <sup>i</sup> ¼ 0=b

� �

� �

� prob <sup>ð</sup> Tb

<sup>i</sup> ¼ 0

� fð Þ 0; S<sup>1</sup> � W � 1; N � 2

<sup>i</sup> ¼ 0

;

� �

ð Þ1 <sup>i</sup> ¼ 0 � �

<sup>L</sup> f Sð Þ <sup>N</sup>; <sup>N</sup> � <sup>2</sup>; <sup>N</sup> � <sup>2</sup>

<sup>L</sup> f Sð Þ <sup>N</sup>; <sup>N</sup> � <sup>2</sup>; <sup>N</sup> � <sup>2</sup>

ð Þ1 <sup>i</sup> ¼ 1

<sup>i</sup> <sup>¼</sup> <sup>1</sup> et bð Þ <sup>N</sup>

0

cNð Þt c1ð Þt dt ¼ 1 � �

;

(17)

: (18)

(19)

� �

Direct Sequence-Optical Code-Division Multiple Access (DS-OCDMA): Receiver Structures…

<sup>i</sup> <sup>≥</sup>SN=bð Þ<sup>1</sup>

� �

<sup>L</sup> f Sð Þ <sup>N</sup> � <sup>1</sup>; <sup>N</sup> � <sup>2</sup>; <sup>N</sup> � <sup>2</sup>

ð Þ1 <sup>i</sup> ¼ 1

<sup>L</sup> f Sð Þ <sup>N</sup> � <sup>1</sup>; <sup>N</sup> � <sup>2</sup>; <sup>N</sup> � <sup>2</sup>

$$\begin{aligned} \text{where } P\_{\epsilon\_{10}} &= \text{prob}\left(\hat{b}\_{i}^{(1)} = \mathbf{1}/b\_{i}^{(1)} = \mathbf{0}\right) \text{ and } P\_{\epsilon\_{11}} = \text{prob}\left(\hat{b}\_{i}^{(1)} = \mathbf{0}/b\_{i}^{(1)} = \mathbf{1}\right) \\\\ P\_{\epsilon\_{11}} &= \text{prob}\left(Z\_{i}^{(1)} < S\_{1}/b\_{i}^{(1)} = \mathbf{1}\right) \\\\ &= \text{prob}\left(W + I\_{1} + A\_{1} < S\_{1}/b\_{i}^{(1)} = \mathbf{1}\right) \end{aligned} \tag{13}$$

$$P\_{\epsilon\_{11}} = \text{prob}\left(A\_{1} = -1/b\_{i}^{(1)} = \mathbf{1}\right) \cdot \text{prob}\left(I\_{1} < S\_{1} - W + 1/b\_{i}^{(1)} = \mathbf{1}\right) \tag{14}$$

$$\begin{aligned} \text{(14)}\\ \text{(15)}\\ \text{(17)} \end{aligned} \tag{15}$$

$$prob\left(I\_1 < \mathbf{S}\_1 - \mathbf{W} + \mathbf{1}/b\_i^{(1)} = \mathbf{1}\right) = \sum\_{i=0}^{S\_1 - W} \mathbf{C}\_{N-2}^i \left(\frac{\mathbf{W}^2}{2L}\right)^i \left(1 - \frac{\mathbf{W}^2}{2L}\right)^{N-2-i}.\tag{15}$$

We define the function

$$f(a,b,k) = \sum\_{i=a}^{b} \mathcal{C}\_k^i \left(\frac{\mathcal{W}^2}{2L}\right)^i \left(1 - \frac{\mathcal{W}^2}{2L}\right)^{k-i}.\tag{16}$$

f(a,b,k) = 0 si a > b then prob I<sup>1</sup> <S<sup>1</sup> � W þ 1=b ð Þ1 <sup>i</sup> ¼ 1 � � <sup>¼</sup> <sup>f</sup>ð Þ <sup>0</sup>; <sup>S</sup><sup>1</sup> � <sup>W</sup>; <sup>N</sup> � <sup>2</sup> : Similarly, on the one hand, we have prob I<sup>1</sup> <S<sup>1</sup> � W=b ð Þ1 <sup>i</sup> ¼ 1 � � <sup>¼</sup> <sup>f</sup>ð Þ <sup>0</sup>; <sup>S</sup><sup>1</sup> � <sup>W</sup> � <sup>1</sup>; <sup>N</sup> � <sup>2</sup> : On the other hand, we have

Direct Sequence-Optical Code-Division Multiple Access (DS-OCDMA): Receiver Structures… DOI: http://dx.doi.org/10.5772/intechopen.85860

$$\begin{split}prob\left(A\_{1} = -1/b\_{i}^{(1)} = \mathbf{1}\right) &= prob\left(b\_{i}^{(N)} - \dot{b}\_{i}^{(N)} = -\mathbf{1}/b\_{i}^{(1)} = \mathbf{1}\right) \cdot prob\left(\int\_{0}^{T\_{N}} c\_{N}(t)c\_{1}(t)dt = \mathbf{1}\right) \\ &= \frac{W^{2}}{L} \cdot \frac{1}{2} prob\left(\dot{b}\_{i}^{(N)} = \mathbf{1}/b\_{i}^{(N)} = \mathbf{0}/b\_{i}^{(1)} = \mathbf{1}\right) \\ &= \frac{1}{2}\frac{W^{2}}{L} \cdot prob\left(Z\_{i}^{(N)} \ge S\_{N}/b\_{i}^{(1)} = \mathbf{1} \text{ at } b\_{i}^{(N)} = \mathbf{0}\right) \\ &= \frac{1}{2}\frac{W^{2}}{L} f(S\_{N} - \mathbf{1}, N - 2, N - 2) \\ \phantom{\left(A\_{1} = 0, \left|b\_{i}^{(1)} = 1\right.\right)} = \mathbf{1} - prob\left(A\_{1} = -1/b\_{i}^{(1)} = \mathbf{1}\right) \\ &= \mathbf{1} - \frac{W^{2}}{L} f(S\_{N} - \mathbf{1}, N - 2, N - 2) \end{split} \tag{17}$$

;

hence

4.1.1 First cancelation stage

k¼3 Ð Tb <sup>0</sup> b ð Þk

ð Þ N <sup>i</sup> � ^ b ð Þ N i � � <sup>Ð</sup> Tb

The error probability is

where Pe<sup>10</sup> <sup>¼</sup> prob ^

<sup>A</sup><sup>1</sup> <sup>¼</sup> <sup>0</sup> si <sup>ð</sup><sup>b</sup>

b ð Þ1 <sup>i</sup> <sup>¼</sup> <sup>1</sup>=bð Þ<sup>1</sup>

Pe<sup>11</sup> <sup>¼</sup> prob A<sup>1</sup> ¼ �1=bð Þ<sup>1</sup>

prob I<sup>1</sup> <sup>&</sup>lt; <sup>S</sup><sup>1</sup> � <sup>W</sup> <sup>þ</sup> <sup>1</sup>=bð Þ<sup>1</sup>

We define the function

f(a,b,k) = 0 si a > b

prob I<sup>1</sup> <S<sup>1</sup> � W=b

78

then prob I<sup>1</sup> <S<sup>1</sup> � W þ 1=b

� �

On the other hand, we have

Similarly, on the one hand, we have

ð Þ1 <sup>i</sup> ¼ 1

<sup>þ</sup> prob A<sup>1</sup> <sup>¼</sup> <sup>0</sup>=bð Þ<sup>1</sup>

� �

(

• <sup>I</sup><sup>1</sup> <sup>¼</sup> <sup>∑</sup><sup>N</sup>

Multiplexing

• A<sup>1</sup> ¼ b

0, so:

We consider that noise perturbation is less significant than interference

<sup>i</sup> <sup>¼</sup> Wbð Þ<sup>1</sup>

<sup>0</sup> cNð Þt c1ð Þt dt term linked to the cancelation of the user #N.

<sup>i</sup> ð Þ if no error in detection

and Pe<sup>11</sup> <sup>¼</sup> prob ^

<sup>i</sup> ¼ 1

� �

<sup>i</sup> ckð Þt c1ð Þt dt is The MAI at the output of the other users

The cancelation term A1 can only take two values 0 and �1. Indeed, as we have

conventional receivers on this stage, there are only errors for the data bð Þ <sup>N</sup>

ð Þ N <sup>i</sup> <sup>¼</sup> ^ bð Þ <sup>N</sup>

�1 if error in detection

Pe<sup>1</sup> <sup>¼</sup> <sup>1</sup> 2 Pe<sup>10</sup> þ 1 2

<sup>i</sup> ¼ 0

<sup>i</sup> <sup>&</sup>lt; <sup>S</sup>1=bð Þ<sup>1</sup>

¼ prob W þ I<sup>1</sup> þ A<sup>1</sup> <S1=b

¼ ∑ S1�W i¼0 Ci N�2

> W<sup>2</sup> 2L � �<sup>i</sup>

¼ fð Þ 0; S<sup>1</sup> � W � 1; N � 2 :

� �

� �

Pe<sup>11</sup> <sup>¼</sup> prob Zð Þ<sup>1</sup>

� �

� �

<sup>i</sup> ¼ 1

<sup>i</sup> ¼ 1

b i¼a Ci k

ð Þ1 <sup>i</sup> ¼ 1

<sup>i</sup> ¼ 1

f að Þ¼ ; b; k ∑

� �

<sup>i</sup> þ I<sup>1</sup> þ A<sup>1</sup> where:

:

ð Þ1 <sup>i</sup> ¼ 1

Pe<sup>11</sup> : (12)

� �

<sup>i</sup> ¼ 1

b ð Þ1 <sup>i</sup> ¼ 0=b

ð Þ1 <sup>i</sup> ¼ 1

� �

� �

ð Þ1 <sup>i</sup> ¼ 1

<sup>1</sup> � <sup>W</sup><sup>2</sup> 2L � �<sup>N</sup>�2�<sup>i</sup>

� prob I<sup>1</sup> <sup>&</sup>lt;S<sup>1</sup> � <sup>W</sup> <sup>þ</sup> <sup>1</sup>=bð Þ<sup>1</sup>

W<sup>2</sup> 2L � �<sup>i</sup>

<sup>1</sup> � <sup>W</sup><sup>2</sup> 2L � �<sup>k</sup>�<sup>i</sup>

¼ fð Þ 0; S<sup>1</sup> � W; N � 2 :

� prob I<sup>1</sup> <S<sup>1</sup> � W=b

<sup>i</sup> equal to

(13)

(14)

: (15)

: (16)

contribution, so the performances are linked only to MAI. The decision variable for the desired user #1 is Zð Þ<sup>1</sup>

$$\begin{aligned} P\_{\epsilon\_{11}} &= \frac{1}{2} \frac{W^2}{L} f(\mathbf{S}\_N - 1, N - 2, N - 2) \cdot f(\mathbf{0}, \mathbf{S}\_1 - W, N - 2) \\\\ &+ \left( 1 - \frac{1}{2} \frac{W^2}{L} f(\mathbf{S}\_N - 1, N - 2, N - 2) \right) \cdot f(\mathbf{0}, \mathbf{S}\_1 - W - 1, N - 2) \ . \end{aligned} \tag{18}$$
 
$$= \frac{1}{2} \frac{W^2}{L} f(\mathbf{S}\_N - 1, N - 2, N - 2) \cdot f(\mathbf{0}, \mathbf{S}\_1 - W, N - 2)$$

In the same way, we calculate the probability of error in the case where the desired user (n ° 1) has sent the data "0":

Pe<sup>10</sup> <sup>¼</sup> prob Zð Þ<sup>1</sup> <sup>i</sup> ≥S1=b ð Þ1 <sup>i</sup> ¼ 0 � � <sup>¼</sup> prob I<sup>1</sup> <sup>þ</sup> <sup>A</sup><sup>1</sup> <sup>≥</sup> <sup>S</sup>1=bð Þ<sup>1</sup> <sup>i</sup> ¼ 0 � � <sup>¼</sup> prob A<sup>1</sup> ¼ �1=bð Þ<sup>1</sup> <sup>i</sup> ¼ 0 � � � prob I<sup>1</sup> <sup>≥</sup>S<sup>1</sup> <sup>þ</sup> <sup>1</sup>=bð Þ<sup>1</sup> <sup>i</sup> ¼ 0 � � þ prob A<sup>1</sup> ¼ 0=b ð Þ1 <sup>i</sup> ¼ 0 � � � prob I<sup>1</sup> <sup>≥</sup>S1=<sup>b</sup> ð Þ1 <sup>i</sup> ¼ 0 � � (19) prob I<sup>1</sup> ≥S<sup>1</sup> þ 1=b ð Þ1 <sup>i</sup> ¼ 0 � � <sup>¼</sup> f Sð Þ <sup>1</sup> <sup>þ</sup> <sup>1</sup>; <sup>N</sup> � <sup>2</sup>; <sup>N</sup> � <sup>2</sup> prob I<sup>1</sup> ≥S1=bð Þ<sup>1</sup> <sup>i</sup> ¼ 0 � � <sup>¼</sup> f Sð Þ <sup>1</sup>; <sup>N</sup> � <sup>2</sup>; <sup>N</sup> � <sup>2</sup> prob A<sup>1</sup> ¼ �1=bð Þ<sup>1</sup> <sup>i</sup> ¼ 0 � � <sup>¼</sup> <sup>1</sup> 2 W<sup>2</sup> <sup>L</sup> f Sð Þ <sup>N</sup>; <sup>N</sup> � <sup>2</sup>; <sup>N</sup> � <sup>2</sup> prob A<sup>1</sup> ¼ 0=b ð Þ1 <sup>i</sup> ¼ 0 � � <sup>¼</sup> <sup>1</sup> � <sup>1</sup> 2 W<sup>2</sup> <sup>L</sup> f Sð Þ <sup>N</sup>; <sup>N</sup> � <sup>2</sup>; <sup>N</sup> � <sup>2</sup> ;

### Multiplexing

hence

$$\begin{split} P\_{\epsilon\_{10}} &= \frac{1}{2} \frac{\mathcal{W}^2}{L} f(\mathcal{S}\_N, N-2, N-2) \cdot f(\mathcal{S}\_1 + 1, N-2, N-2) \\ &+ \left( 1 - \frac{1}{2} \frac{\mathcal{W}^2}{L} f(\mathcal{S}\_N, N-2, N-2) \right) \cdot f(\mathcal{S}\_1, N-2, N-2) \end{split} \tag{20}$$
