b shows the block diagram of the receiver for each user n . At the receiver’s front end, we use an optical amplifier with a power gain of G in order to compensate the losses of the optical system. Note that we can use the 

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... recent years, tremendous progress has been made in the development of optical amplifiers for optical communication networks. There are various types of optical amplifiers, namely the semiconductor optical amplifiers, the erbium-doped fibre amplifiers (EDFA), the Raman fibre amplifiers and the phase sensitive amplifiers. The optical amplifiers have revolutionised optical communications [1]. In optical fibre communication systems, optical amplifiers do not require high-speed electronic circuitry, and are transparent to data rate and format, which dramatically reduces cost. Currently, optical communication technology is moving from point-to-point systems to optical networks. Optical amplifiers can play an important role at many places in all-optical networks. In a transport layer, there are in-line amplifiers, power amplifiers and receivers’ preamplifiers. In optical cross connect and wavelength add-drop functions, amplifiers are used for loss compensation. The EDFA provides high-gain, high-power and low- noise figures. Several optical channels can be amplified simultaneously within the EDFA in a single optical fibre [2, 3]. On the other hand, coherent ultra-short light pulse CDMA (CULP CDMA) or femtosecond CDMA is one of the earliest coherent techniques that were proposed during the late 1980s. The system performance of this scheme is evaluated in [4] with the assumptions of a random binary spreading sequence and an ultra-fast photodetector, which is able to output the instantaneous power of the pulse. Similar to the other multiple access systems, the success of system operation depends on the effective reduction of the BER. Channel error correction is an appropriate technique for this purpose. In most of the previously employed channel coding methods, error- correcting codes are externally applied to the CDMA link [5 – 10]. A disadvantage of an externally coded fibre-optic CDMA scheme is the extra bandwidth required compared with the uncoded system. In our recent paper [11], we have applied an internal coding method to the CULP CDMA system in order to improve the performance of the system. In fact, the scheme considered in [11], called internally coded time-hopping (TH) CULP CDMA, is a multichannel CDMA in which the optical CDMA and TH multiple access techniques have been combined using an error control code, like super-orthogonal encoder, in order to have the advantages of both multiple access schemes along with the performance enhancement because of the underlying error control code. The complexity of the TH-CULP CDMA is not significantly higher than that of the conventional OCDMA employing coherent spectral encoding and decoding schemes; that is, the decoding of error correcting codes is done entirely in the electronic domain and, thus, using an internal channel encoder / decoder would not necessarily result in an undue increase of optical system complexity. We have evaluated the performance of the internally coded TH-CULP CDMA system considering the effects of the multiple access interference (MAI) and thermal noise for both soft and hard decoders. The bit error rate (BER) has been evaluated using a Chernoff bound and saddle point approximation. It has been shown in [11] that using internal coding results in significant reduction of the BER of the system without decreasing the bandwidth efficiency, compared with the conventional uncoded CULP CDMA system. In this paper, according to the important role of optical amplifiers in compensating the losses of the optical system, we evaluate the performance of the internally coded TH-CULP CDMA system using the optical amplifier and compare the results with those of the conventional CULP CDMA scheme. We assume a preamplifier at the input of the receiver, which compensates the losses because of the spectral encoder, decoder and optical fibre path. In the internally coded TH-CULP CDMA system introduced in [11], each bit’s duration is divided into N s frames. Two PN sequences are assigned to each user, named PN 1 and PN 2. In each bit interval, based on the PN 1 sequence component and the channel encoder output, one of the N s frames is selected, in which the spectral phase-coded pulse is transmitted. For the channel encoder, a super- orthogonal convolutional (SOC) code is used as its path-generating function (which is required for the performance evaluation) is available. The PN 2 sequence determines the phase mask pattern that is used to encode the phase of the ultra-short pulse in the spectral domain, that is, the ultra-short pulse is multiplied by the PN 2 sequence in the frequency domain. The encoded signal is broadened and becomes noise-like in the time domain. At the receiver’s front end, we have an optical amplifier with a power gain of G . Then, the spectral decoder compensates the phase offset simply by multiplying the amplified signal by the desired user’s PN 2 sequence in the spectral domain. The spectral decoder’s output enters the Viterbi decoder of the underlying convolutional code to recover the user’s data bit. We consider a soft input decoder and evaluate the multiple access performance of the system using saddle point approximation. We consider an additive noise model for the optical amplifier [12– 15]. We compare the results with those of the conventional CULP CDMA system with and without an amplifier. The rest of the paper is organised as follows. Section 2 describes the system. In Section 3, we provide the performance analysis of the internally coded TH-CULP CDMA systems with an optical amplifier. In Section 4, some numerical results are presented and finally in Section 5, the paper is concluded. Fig. 1 shows the block diagram of the TH-CULP CDMA system with an optical amplifier. Similar to the TH-CULP CDMA system, introduced in [11], the bit duration, that is, T b , is divided into N s frames. Let the duration of each frame be T f , then we have T b 1⁄4 N s T f . Two PN sequences, namely PN 1 and PN 2, are assigned to each user. The components of the PN 1 and PN 2 sequences are i.i.d integer-valued È random É variables with uniform distributions on 0, 1, . . . , N À 1 and f 2 1, 1 g , respectively. The uncoded ultra-short pulse has a flat spectrum with a bandwidth of W , which is divided into N 0 chips and, thus, each chip has a bandwidth of V 1⁄4 W = N 0 . The phase of each chip is adjusted independently by multiplying the uncoded pulse spectrum by a phase mask whose pattern is determined by the PN 2 sequence components. A pseudo- random phase mask transforms the incident ultra-short pulse into a low-intensity pseudo-noise burst. This pulse is transmitted in a frame that is determined by the output of the channel (SOC) encoder and PN 1 sequence components at the corresponding bit interval; that is, the bit stream of the user is applied to the SOC encoder with the constraint length K , which generates 2 K À 2 different symbols. The encoder output in each bit interval is then added to the PN sequence’s component at that interval in mod N s , where N s 1⁄4 2 K À 2 . The result specifies one of the N s frames in which the spectrally encoded pulse is transmitted. optical amplifier in different stages of the system. For instance, it can be placed in the transmitter, at the output of the laser source, or after the spectral encoder. Also, it can be used at the spectral decoder’s input or output. However, it is shown in [13] that the amplifier has the best effect when it is placed at the spectral decoder’s input (in the receiver) or at the spectral encoder’s output (in the transmitter). Therefore we use it as a preamplifier at the spectral decoder’s input. Also, we assume that L 1 and L 3 are losses because of the spectral encoder and decoder, respectively, and L 2 is the power loss of the optical fibre path. Note that these losses were neglected in [11]. Fig. 2 shows the block diagram of CULP CDMA with optical amplifier. After the optical amplifier, a typical spectral decoder is used to compute the correlation of the received signal with the desired signal by multiplying the received signal in the PN 2 sequence of the desired user in the spectral domain. Then, there is an ultra-fast detector that measures the intensity of the output in each frame time of each bit interval. Then, the underlying convolutional decoder uses these measurements for the branch metric calculations of the related trellis diagram, the details of which will be given later. Fig. 3 demonstrates the schematic timing diagrams of a typical transmitted signal and the received signals before and after the spectral decoder of the desired user. As can be realised from this before the spectral decoder, the signals in all frames are noise-like, whereas, after the spectral decoder, the phase shifts of the desired signal are removed and the coherent ultra-short pulse of the user is reconstructed. However, for the undesired signals, the phase shifts are rearranged but not removed, and the signals at the output of the decoder remain low-intensity pseudo-noise bursts. For the performance evaluation, we assume that the first user is the desired user, which sends all-zero sequences. For the convolutional code, only the lower and upper bounds on the BER are analytically available, which are computed using the path-generating function of the code. The path-generating function in our application is computed as ...