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This paper fundamentally investigates the peak-to-average power ratio (PAPR) theory in all optical orthogonal frequency division multiplexing (OFDM) systems which employ intensity modulation-direct detection (IM-DD) scheme. We propose a low-complexity PAPR reduction scheme based on phase pre-emphasis. Simulations show that the proposed scheme brings about a 3.74 dB PAPR reduction and better nonlinear impairment tolerance in a 16×10Gb/s IM-DD all optical OFDM system.
©2009 Optical Society of America
1. Introduction
Optical OFDM has become a promising technique in long-haul and high-speed optical transmission systems, for its high spectral efficiency, relatively low bit rate and advanced robustness against chromatic dispersion and polarization mode dispersion [1–3]. Conventional optical OFDM systems utilize electronic fast Fourier transform (FFT) circuits and complicated digital to analog (D/A) converters to generate OFDM symbols, bringing about limited processing speed as well as high system cost. In order to eliminate these defects, many research studies [4–7] have been focused on optical implementation of Fourier transformation and constructing all optical OFDM systems. An optical discrete Fourier transformer (ODFT) was proposed for a 4×25Gb/s all optical OFDM system, eliminating requirements of 100Gb/s bandwidth for OFDM electronics of D/A converter and digital signal processors (DSP) [4]. 50×88.8Gb/s all optical OFDM transmission was demonstrated using a photonic integrated circuit [5]. 100Gb/s all optical OFDM transmission was demonstrated employing optical FFT elements based on coupler interferometry [6]. 4×10Gb/s all optical OFDM demultiplexer was successfully demonstrated using silica planar lightwave circuit (PLC) based optical FFT circuits [7]. Similar to conventional optical OFDM systems [8,9], high PAPR is a serious intrinsic cause of penalty in all optical OFDM systems as well. Moreover, it deteriorates nonlinear impairment in optical fibers. However, hardly any investigations are centered on the PAPR characteristics in all optical OFDM systems.
This paper studies the fundamental PAPR theory in all optical OFDM systems and illustrates the differences of PAPR characteristics between conventional optical OFDM systems and IM-DD all optical OFDM systems for the first time. It’s well-known that intensity modulation (IM) is a very simple and cost-effective modulation technology. Ref [4] demonstrated the transmission efficiency of a 4×25Gb/s IM-DD all optical OFDM system. For simplicity, this paper will be confined to IM-DD all optical OFDM systems. Our scheme is based on phase pre-emphasis and requires hardly any additional system or device complexity. Simulations show a 3.74 dB PAPR reduction and better nonlinear impairment tolerance in a 16×10Gb/s all optical OFDM system.
2. All optical OFDM system configuration
Configuration analysis of the all optical OFDM system is necessary for PAPR investigation. Fig. 1 depicts a 16×10Gb/s IM-DD all optical OFDM system, as an example. At the transmitter, a 160Gb/s serial data stream is converted into sixteen 10Gb/s parallel data streams by the serial to parallel convertor (S/P). Driven by a 10GHz clock signal, an electro-absorption modulator (EAM) generates a 10GHz phase-locked optical pulse train, with a duty cycle of 1/16, using the input continuous-wave (CW) laser. The optical pulse train is then split into 16 identical pulse trains, which are modulated by sixteen 10Gb/s parallel data streams by IM, respectively. The optical inverse discrete Fourier transformer (OIDFT) implements the optical inverse Fourier transformation to generate all optical OFDM symbols. After transmission, the ODFT rebuilds optical signals in 16 sub-carriers at the receiver. 16 photodetectors (PDs) demodulate data in every sub-carrier after EAM sampling. The parallel to serial processor (P/S) combines sixteen 10Gb/s parallel electronic data streams into a 160Gb/s serial data stream. Because all optical OFDM systems need no electronic FFT or D/A circuits, the limitation of processing speed and high cost of electronic circuits are eliminated.
There are mainly two kinds of feasible ODFTs and OIDFTs. One is based on a phase shifter array and an optical delay line array, proposed by Kyusang Lee [4]. The main idea of this scheme is to divide the overall discrete Fourier transformation (DFT) into multiple phase-shifting and optical delay operations. In this kind of ODFT, the phase-shiftings are implemented by phase shifters and the optical delays are implemented by delay lines. The other is based on silica PLC technology [7,10]. This scheme is proposed on the basis of the similarity of transfer functions between 2×2 DFT and 2×2 directional couplers. For example, Fig.2 depicts a PLC-based OIDFT with 4 sub-carriers, owing to the page layout limitation. The OIDFT with 16 sub-carriers can be constructed in the same way [10]. Figure 2 depicts the core layer of the OIDFT circuit. It consists of four directional couplers and nine phase shifters (in yellow). The phase-shifting value of every phase shifter is Π/2. In Fig. 2, α0, α1, α2, α3 are the amplitudes of four parallel input lights, and β0, β1, β2, β3 are the DFT of α0, α1, α2, α3. The light beams of β0, β1, β2, β3 are coupled together after different time delays. The lengths of delay lines (dL, 2dL, 3dL) are determined by bit rate of every sub-carrier [7]. It is worth mentioning that the two phase-shifters (in red) is only related with PAPR reduction, not necessary in the latter scheme. The PLC-based OIDFT is easy to achieve using current silica PLC technology.
3. PAPR theory in IM-DD all optical OFDM systems
In this section, a review of PAPR theory in conventional optical OFDM systems will be given. After that, the differences in IM-DD all optical OFDM systems will be illuminated.
In conventional optical OFDM systems, the base band equivalent time-domain signal xn can be expressed by Eq. (1).
where N is the number of sub-carriers, and Xk denotes the kth modulated phase shift keying (PSK) or quadrature amplitude modulation (QAM) symbol. Xk can be written as Xk=ak+jbk, where ak and bk indicate the real component and imaginary component of Xk, respectively. Assuming that ak and bk are independent with each other, the statistical characteristics of ak and bk are given in Eq. (2) and Eq. (3).
where E and D are expectation operator and variation operator, respectively. σ2 is the variation of ak and bk. Setting xn=x n,I+jx n,Q, x n,I and x n,Q respectively indicate the real component and imaginary component of xn. Calculating Eq. (1)~(3) and applying the central limit theorem for large N [11], the probability distributions of x n,I and x n,Q follow the Gaussian distribution. Therefore, the complementary cumulative distribution function (CCDF) of PAPR can be written as Eq.(4) [12].
Nevertheless, Eq. (4) is only applicable to conventional optical OFDM systems with PSK or QAM mapping, because Eq.(3) will be unestablished in IM-DD all optical OFDM systems.
In the IM-DD all optical OFDM system shown in Fig.1, no PSK or QAM are employed. It is clear that 16 incident lights of the OIDFT have identical initial phase, because they originates from a same phase-locked pulse train. It should be noticed that there will be initial phase differences between the 16 incident lights, owing to the different transmission lengths from the EAM to the OIDFT. However, these initial phase differences are time-invariant and are available for PAPR investigation once the system is set up. For simplicity, the initial phases of the 16 incident lights are set zero. Under this condition of IM scheme, Xk follows 0–1 distribution. Consequently, Eq.(2) and (3) should be modified to Eq. (5) and (6).
According to probability theorem, summations of independent non-identical distributed random variables have analytical probability distribution functions only if they satisfy the Lindeberg condition [11]. Unfortunately, x n,I and x n,Q don’t satisfy the Linderberg condition any more. So they have no analytical probability distribution functions. Consequently, Eq. (4) is not suitable for IM-DD all optical OFDM systems. Numerical simulation is the only way to investigate PAPR characteristics under this condition.
4. A novel simple PAPR reduction scheme
High PAPR is a major drawback in optical OFDM systems. It deteriorates nonlinear impairment in optical fibers. Many research studies have been focused on PAPR reduction schemes [8,9]. Most of them are related with those optical OFDM systems which utilize electronic FFT circuits. However, they may be ineffective or unfeasible in all optical OFDM systems. Hardly any reports are centered on PAPR reduction scheme in all optical OFDM systems. Between conventional optical OFDM systems and IM-DD all optical OFDM systems, the differences of PAPR characteristics are notable. We propose a novel PAPR reduction scheme for the latter based on phase pre-emphasis, which requires hardly any additional system or device complexity.
Equation (1) implies that every OFDM symbol contains N components (x 0,x 1,…,x N-1), lining up in sequence in time domain. Because Xk follows 0–1 distribution, it’s clear that and |xn| reaches a maximum when n=0. This means that Xk (k=0,1,…,N-1) are multiplexed by the same phase factor. Because the value of |xn| always reaches its maximum when n=0, PAPR will be reduced if the in-phase condition (when n=0) is eliminated. In our PAPR reduction scheme, different phase pre-emphasises are introduced and Eq. (1) is converted to Eq. (7) and (8).
where φk is the kth phase pre-emphasis. Chosen suitable values for φk, the in-phase condition (when n=0) can be eliminated and PAPR will be reduced.
Simulation proves the effectiveness of the proposed PAPR reduction scheme. In order to get suitable values for φk, a simulation tool is developed using the Matlab package. The simulation is related with an ODFT that is based on a phase shifter array and an optical delay line array. Fig.3 depicts the diagram of PAPR simulation. Random inputs (x 0, x 1,…, x 15) are multiplexed by phase pre-emphasis values (φ 0,φ 1,…,φ 15). After ODFT, PAPR values are calculated for different chooses of φk. The optimized values for φk are obtained when the PAPR is the minimal. The optimized φk is . 65000 points are used in the simulation. Fig.4 depicts the comparison of the probability distributions of PAPR between IM-DD all optical OFDM systems with and without phase pre-emphasis, where X-axis represents the PAPR value and Y-axis represents the corresponding possibility. Without phase pre-condition, the probability of PAPR as high as 10dB is larger than 0.02, and the most possible PAPR is 9dB. The remarkable characteristics are that the PAPR in IM-DD all optical OFDM systems can only be discrete values. The underlying cause is implied in Eq. (1). As aforementioned, |xn| reaches the maximum when n=0, and Xk follows 0–1 distribution. Consequently, the maximum of |xn| can only be an integral number, varying from 1 to 16. Moreover, the expectation of |xn| is determined by Xk, so that the average power of all optical OFDM symbols changes slightly. Therefore, PAPR can only be discrete values. Since the PAPR distribution here differs greatly from those in conventional optical OFDM systems, they cannot have similar probability distribution functions, which is theoretically proved in section 3. What is more, Fig.4 shows that PAPR characteristics improve significantly when appropriate phase pre-emphasis is applied. Fig.4 illustrates evidently that the possibility of PAPR larger than 7dB is almost zero, and the most possible PAPR is less than 5dB. Under this condition, PAPR changes continuously. This is because |xn| doesn’t always reach the maximum when n=0, owing to the influence of phase pre-emphasis.
Figure 5 depicts the differences between CCDFs, where X-axis represents the PAPR0 value and Y-axis represents the possibility when PAPR is larger than PAPR0. In the blue curve, there are some components parallel to the X-axis. This indicates that the possibility of some PAPR0 is zero, consistent with the aforementioned discrete PAPR values. 3.74dB PAPR reduction is achieved when the possibility of PAPR larger than PAPR0 is 0.001. To sum up, the proposed PAPR reduction scheme is effective in IM-DD all optical OFDM systems.
The implementation of the proposed PAPR reduction scheme in IM-DD all optical OFDM systems needs hardly any additional system or device complexity. In an IM-DD system, initial signal phases will not affect the detection, so no modification is necessary for the receiver. In all optical OFDM systems employing OIDFTs based on a phase shifter array and a delay line array, the only modification for PAPR reduction is to change the values of the phase shifter array according to the phase pre-emphasis, without any device complexity. In all optical OFDM systems using silica PLC-based FFTs, only a few phase shifters are needed for PAPR reduction. As shown in Fig.2, only two additional phase shifters (in red) are needed. Considering that the circuit already contains nine phase shifters, the additional two can be produced synchronously during fabrication, without obvious additional complexity. In all, the proposed PAPR reduction will introduce negligible device or system complexity.
5. 16×10Gb/s IM-DD all optical OFDM application
We utilize VPI TransmissionMaker™ V7.6 to simulate a 16×10Gb/s IM-DD all optical OFDM system, the configuration of which is shown in Fig.1. In this simulation, the ODFT and OIDFT are based on a phase shifter array and a delay line array. The optical signal eye diagrams transmitted into the fiber are given in Fig.6 and Fig.7. Under the condition of no phase pre-emphasis, the eye diagram shows a power peak at the central of an OFDM symbol, as shown in Fig.6. This is consistent with the aforementioned PAPR characteristics in IM-DD all optical OFDM systems. When phase pre-emphasis is applied, no obvious power peak appears in the eye diagram, as shown in Fig.7. This means that with phase pre-emphasis, the peak power is dispersed to the whole OFDM symbol cycle. As a result, the corresponding PAPR will be significantly lower.
Figure 8 depicts the bit error rate (BER) performance versus average transmitted power, under three different conditions: transmission without phase pre-emphasis, transmission with phase pre-emphasis and linear transmission. The related simulation parameters are given in table 1. In Fig.8, X-axis represents the average optical signal power transmitted into fibers. Y-axis represents lg(BER). It is obvious that the proposed PAPR reduction scheme brings high tolerance of nonlinear impairment in fibers. If the required BER is 1×10-9, the maximum transmitted power is enhanced by 0.9mW, which is an increase of 39%. If the required BER is 1×10-12, the maximum transmitted power is enhanced by 0.75mW, which is an increase of 42%. The frequency space between sub-channels is typically as low as tens of GHz in all optical OFDM systems. This will result in serious cross phase modulation (XPM) and four wave mixing (FWM) effects in optical fibers. Further studies are needed to reduce XPM and FWM effects, in order to enhance system performance.
Table 1. Simulation parameters
6. Conclusions
All optical OFDM is a promising technique in long-haul and high-speed optical transmission systems. High PAPR is a serious intrinsic defect in all optical OFDM systems. This paper fundamentally investigates PAPR theory in IM-DD all optical OFDM systems. In IM-DD all optical OFDM systems, the number of subcarriers is relatively small, so that the analytical PARP model based on the central limit theorem is not applicable unlike conventional optical OFDM systems. As a result, numerical simulation is the only approach. An effective and low-complexity PAPR reduction scheme is proposed. Simulation shows a 3.74dB PAPR reduction. The proposed PAPR reduction scheme is applied in a 16×10Gb/s IM-DD all optical OFDM system. Simulation indicates that the maximal transmitted powers increase by 39% and 42%, when the BER requirements are 1×10-9 and1×10-12, respectively. Therefore, system tolerance of nonlinear impairment increases significantly, owing to the contribution of the proposed PAPR reduction scheme.
Acknowledgement
This work was supported by the National Basic Research Program of China (973 Program:2010CB328300), China National Science Foundation Project (under granted:60772013) and the 863 high technology plan(2009AA03Z408).
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