Abstract

We propose a new technique for mitigation of nonlinear distortion of broadband signals due to electrical/optical conversions and fibre transmission. This technique uses memory polynomials on signals on their original format. The performance improvement is assessed experimentally for an orthogonal frequency division multiplexing signal, with 750 MHz of bandwidth (centred at 1.5 GHz) and carrying 1.25 Gbps, transmitted along a wavelength division multiplexing long-reach passive optical network. It is shown that, in back-to-back, an error vector magnitude improvement of 4.1 dB and 11.8 dB can be achieved with digital pre-distortion and digital post-distortion, respectively. With 125 km of single-mode fibre transmission, these improvements are reduced to about 1 dB and 3 dB, respectively.

© 2015 Optical Society of America

1. Introduction

The quintuple-play (5th-play) service (consisting of telephony, Internet, video distribution, wireless service provision and home security/control applications) has been recently proposed to be provided along optical access networks [1, 2]. Recent high bitrate wireless standards, such as long-term evolution (LTE) [3], worldwide interoperability for microwave access (WiMAX) [4] and ultra-wideband (UWB) [5], use orthogonal frequency division multiplexing (OFDM) or a variant of OFDM. A bundle of these signals together with a OFDM-Gigabit Ethernet (OFDM-GbE) signal can be used to provide the 5th-play service using OFDM-GbE for fixed telephony and fixed high data-rate Internet, UWB for wireless high-definition audio-video distribution, LTE for mobile telephony and WiMAX for wireless Internet access and home security/control applications [6]. In [79], a solution to provide this OFDM-based 5th-play service to the user’s premises, in a cost-effective and future-proof way was proposed. In that solution, the wireless signals are centrally generated and transmitted in native format on radio-over-fibre (RoF) along a wavelength division multiplexing (WDM) long-reach passive optical network (LR-PON), and optical network units (ONUs) that are transparent to the wireless signals [79] are used at the end-users’ premises. However, OFDM signals are very sensitive to distortion in the transmission channel, such as linear distortion due to frequency-dependent power loss (FDPL) of filters/amplifiers or nonlinear distortion (NLD) caused by electro-optical (EO) and optical-electrical (OE) conversions combined with fiber dispersion. Linear distortion has been addressed using an equalizer and/or pre-emphasis (PE) [8, 9]. NLD of various natures (such as due to electrical amplifiers and EO-OE conversion) have been addressed in [1018] by linearising the transmission channel, S, with an approximate inverse system ∼ S−1. Several different approaches were attempted in order to implement system S−1 [1113]. These early implementations were all memoryless, meaning they only try to compensate for the instantaneous nonlinear behaviour of system S. However, as the bandwidth of the signal increases, the impact of memory effects on the nonlinearities of system S increases, therefore, reducing the accuracy that can be achieved with a memoryless implementation of system S−1 [14]. For this reason, a lot of research attention has been turned towards S−1 implementations using a digital distorter implemented with memory polynomials (MPs) [1419]. One advantage of these MP-related methods, compared to the method in [20], is the flexibility with which they can be adapted on real-time to variations in the parameters of system S. However, the S−1 implementations in [1418] are developed for narrow-band radio frequency (RF) signals, applying the digital distortion to the baseband equivalent of the bandpass RF signal. This approach, from here on called baseband-equivalent nonlinear impairment mitigation (BENIM), enables a reduction in the sampling frequency requirements of the digital distorter, which can be significant for narrow-band signals modulating high frequency RF-carriers. However, for a OFDM-GbE signal with a bandwidth of 750 MHz centred at 1.5 GHz [6], the signal bandwidth is comparable to the RF centre frequency. In this case, the reduction attained in the sampling frequency by using the BENIM is not that significant any more. Furthermore, the BENIM requires a pair of down- and up-converters at the input and output of the digital distorter, respectively. Alternatively, we propose applying the digital distortion directly to the upconverted signal, covering all the band from DC up to the highest frequency of the signal. This alternative approach is from here on referred to as broad baseband nonlinear impairment mitigation (BBNIM), and it is demonstrated experimentally in this paper considering the OFDM-GbE signal mentioned before.

Mitigation of EO-OE conversion NLD in optical transmission systems has been already demonstrated in various papers, such as [16, 17]. However, even though they used the BENIM approach, these works have addressed only optical back-to-back (BtB) situations employing digital pre-distortion (DPD). In [20], a digital post-distortion (DPostD) scheme is demonstrated with fibre transmission, but, as explained before, the adopted solution does not provide the same flexibility of a MP-based approach.

The contribution of this paper is the experimental demonstration of BBNIM to mitigate EO-OE conversion NLD considering two MP-BBNIMs: (a) MP-DPD and (b) MP-DPostD. Although the proposed BBNIM can be applied to other types of optical networks, the demonstration in this paper is focused on BtB and with fibre transmission in a WDM LR-PON. As a result, the objectives of this paper are: (i) experimental demonstration of NLD mitigation in a WDM LR-PON by employing DPD and DPostD to an OFDM signal in native format, (ii) determine the signal performance improvement achieved with DPD and DPostD and (iii) assess how the improvements vary with the fibre length. The paper is organised as follows. In section 2, the principles of DPD and DPostD using MP are explained. In section 3, the characteristics of the signal and transmission system are described. In section 4, the experimental results in BtB and with fibre transmission are presented and discussed. In section 5, the conclusions are drawn.

2. Digital pre- and post-distortion using memory polynomials

One of the most general descriptions for a nonlinear system S is given by the Volterra series [10]. A discrete version of the Volterra series describing a nonlinear causal system S with finite memory, can be written as [14]

v[n]=p=1Pq1=0apqp=0aphp(q1,q2,,qp)m=1pu[nqm]
where u[nqm] is a delayed version by qm samples of the input signal u[n], v[n] are the samples of the output signal, hp(q1, q2,..., qp) is the p-th order discrete Volterra kernel (DVK) of system S, P is the highest nonlinear order of system S and ap is the maximum sample delay considered in the p-th order DVK. Equation (1) is a summation, weighted by the corresponding values of hp (q1, q2,..., qp), that can also be expressed in the form of a MP. The same is true for the inverse system S−1. Therefore, an approximate inverse of system S can be described using a MP. By assuming that the system S is time-invariant, several coefficients become redundant and the total number of coefficients can be significantly reduced [10]. In that case, a MP of order P is given by
u[n]=MP(w,v[n])=p=1Pq1=0apq2=0q1qp=0qp1wp(q1,q2,,qp)m=1pv[nqm].
where each factor wp (q1, q2,..., qp) corresponds to a coefficient of the MP. Hence, the coefficients of the MP are grouped in vector w and are from here on called the coefficients w. The number of w coefficients depends on the order and maximum delay per order in the MP. From Eq. (2), it is concluded that each wp kernel contributes with
Cp=mp=1apmp1=1mpm1=1m21,1pP
coefficients to the MP. Therefore, the total number of coefficients in vector w is C = C1+C2 + ... +CP. Equations (3) and (2) show that the type and number of coefficients are controlled by the maximum nonlinear order, P, and by the maximum delays in samples, considered in each order, a1 to aP. Since all MPs considered in this paper use the general structure shown in Eq. (2), the exact structure of a MP of order P in this work is defined using vector a= [a1 a2 ... aP]. For instance, a=[37 - 12 - -] corresponds to
MP(w,v[n])=q1=037w1(q1)v[nq1]+q1=012q2=0q1q3=0q2w3(q1q2,q3)v[nq1]v[nq2]v[nq3].
A comparison with [1418] shows that, while addressing the same nonlinearity order and sample-delay, the MP in this paper considers more combinations of products of v[nqm] than the MPs of the form x[n] · |x[nq]|k in [1418]. This is because, in [1418], several combinations were dropped as they were considered not to cause distortion within the narrow band of signal. In this paper, the signal is assumed to have a wideband and this simplification was not performed, resulting in the MP presented in Eq. (2).

The w coefficients are obtained with a proper estimator. The first estimation of the optimum w coefficients is done from the input and output signals of system S (u and v signals). This estimation uses a linear least-square error minimization [14,16] and is performed in four steps: (1) capturing a sequence of Ns samples of the input and output signals of the system S, (2) affecting the captured output signal, v[n], by gLF1 (inverse of the low frequency gain of system S) to obtain v′[n], (3) constructing the matrix V from signal v′[n] and (4) calculating the vector w containing the value of the w coefficients from matrix V and signal u[n].

The matrix V is comprised by Ns rows (one per captured sample) and C columns (one per w coefficient) and it is given by

V=[V1C1V2C2VPCP]}Ns
where the generic Vp is calculated using v′ according to
Vp=[(v[0])p(v[1])(v[0])p1(v[ap])p1v[1ap](v[ap])p(v[1])p(v[0])(v[1])p1(v[1ap])p1v[2ap](v[1ap])p(v[Ns1])p(v[Nsap])p].
The coefficients w are obtained from vector w, which is given by [14,16]
w=(VHV)1VHu,
where u= [u[0], u[1],...u[Ns − 1]]T, (.)T denotes the transpose operation and (.)H denotes the complex conjugate transpose operation.

Under the assumption that the system S is practically time-invariant, that means the values w are valid for some time, but not forever. At some time instant, it may be necessary to update the values of w. This update can be done as indicated in [14]. As a result, the values of (i + 1)-th estimate of w are given by

wi+1=wi+μ(VHV)1VH(uVwi),
where μ is the relaxation constant.

When applying the MP defined in Eq. (2) to mitigate the distortion of system S, there are two possibilities regarding the placing of digital distorter (implemented by the MP): (i) ahead of the system S or (ii) following the system S. In the first case, the compensation is achieved by pre-distortion, and in the second case, by post-distortion [14]. The working principles of DPD and DPostD are illustrated in Fig. 1.

 

Fig. 1. Implementation schemes of digital (a) pre-distortion and (b) post-distortion, using a digital signal processor (DSP) programmed with a memory polynomial (MP), adjusted by the coefficients w, to compensate for the distortion introduced by the transmission system S. Remark: the signals u and v in implementations (a) and (b) are different.

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Figure 1 shows that, depending on the MP placement, different signals are acted upon by the MP and different signals may be obtained at the output of the system S+MP (observe signals v[n] and y[n] in Figs. 1(a) and 1(b), respectively) if the system S is not invertible or the noise effects cannot be neglected. When the system is invertible and the noise can be neglected, the use of DPostD leads to the same output signal as DPD.

Figure 1 shows that the MP is implemented in a digital signal processor (DSP). For a practical transmission scenario, this MP has to run in real-time. Therefore, the memory requirements of the MP, the calculation load of running the MP and the used sampling rate have to be carefully considered. The MP requires memory for the coefficients and for the delayed samples. Therefore, the total memory bits requirement of the MP is C + max(a1, a2,..., aP) times the number of memory bits used to store each number. Many publications that use MPs [14,1618,21,22] consider between some tens and a few hundred coefficients and a maximum delay of a few tens of samples. We consider up to 350 coefficients and maximum delay of 37 samples because early studies have shown that considering more coefficients or longer delays does not improve significantly the signal performance. The total calculation load to obtain the MP output is the total load of: (1) calculating the products between the input samples, (2) multiplying the results by the coefficients and (3) sum all these. This requires: (1) m=2P(m1)Cm multiplications, (2) C multiplications and (3) C − 1 additions. The update of coefficients w is mathematically much more complex than obtaining the MP output, since matrix inversions are required (see Eq. (8)). However, the time available to perform the coefficient update is quite long (see section 4). For this motive, the calculation load due to the coefficients update is not as significant as the load of calculating the MP output. The sampling frequency depends on the bandwidth of the signals that are processed by the MP. The typical signal bandwidths considered in [14, 1618, 21, 22] are in the order of tens of MHz and, therefore, the sampling frequency is in the order of few hundred Msamples/s. In order to avoid aliasing effects and focus the work on NLD mitigation, the sampling frequency in our case is set to 20 Gsamples/s.

3. Experimental setup description

One of the objectives of this paper is to assess the improvement of signal performance achieved by using DPD or DPostD in a WDM LR-PON. In this paper, we consider the WDM LR-PON described in [6]. In order to focus the study on the DPD/DPostD demonstration in this network, we assume insignificant cross-talk between ONUs and communication directions, and consider only one ONU and downstream direction. Accordingly, a scheme of the experimental setup is shown in Fig. 2.

 

Fig. 2. Experimental setup employed to evaluate the performance improvement achieved by mitigating the NLD in the transmission system S using DPD and DPostD. The necessary MP, w coefficient estimator and signal (de)/modulator are implemented in the digital domain by MATLAB programs.

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The transmission system in Fig. 2 works as follows. The computer uses a MAT-LAB program to generate u[n] samples of the OFDM-GbE signal. The OFDM-GbE signal is a custom defined OFDM signal [9,23]. It is comprised by 128 sub-carriers, from which, 81 are used to carry data, 8 are used as pilot sub-carriers for the equaliser at the receiver and the remaining 39 sub-carriers are nulled. The OFDM-GbE signal has an OFDM symbol duration of 128 ns and covers a band going from of about 1.125 GHz to about 1.875 GHz. The cyclic prefix duration is 4 ns and the data sub-carriers are modulated using 4-quadrature amplitude modulation (QAM), what results in a total raw bit-rate of about 1.25 Gbps carried by the OFDM-GbE signal. The OFDM-GbE electrical signal is generated from the samples of u[n] by the Tektronix arbitrary waveform generator (AWG) 7122B operating at 20 Gsamples/s. The amplitude of the electrical signal at the AWG output is increased using an electrical amplifier, Ap. An electrical low-pass filter (LPF) is used to limit the noise power at the optical modulator input. A continuous-wave laser provides an optical carrier with a wavelength of 1552.52 nm. The optical carrier is modulated using a Mach-Zehnder modulator (MZM) (characterized by Vπ =3.6 V, voltage required to produce a phase shift of π between the MZM arms), biased at its quadrature point. The modulated optical carrier is passed to an optical multiplexer (Mux), and its output is launched into the feeder fibre with an optical power of −5.3 dBm. At the other end of the feeder fibre, the optical signal reaches the remote node (RN), where it is amplified by an erbium-doped fiber amplifier (EDFA). The optical signal at EDFA output is passed to an optical demultiplexer (Demux) and the optical signal at the Demux output is launched into the distribution fibre. The gain of the EDFA is adjusted to 21.5 dB, hence compensating for the losses of the feeder fibre and optical Demux, resulting in an optical power launched into the distribution fibre of −5.3 dBm. At the end of the distribution fibre, the optical signal reaches the ONU where it is photodetected by a PIN-transimpedance amplifier (TIA). The photodetected OFDM-GbE signal is captured by the Agilent Infiniium digital storage oscilloscope (DSO) 81204A at 20 Gsamples/s and the resulting samples v[n] are passed to the computer running MATLAB. This computer is loaded with the necessary MATLAB programs to perform the signal processing. The high sample-rate operations, such as OFDM-GbE signal demodulation, signal performance calculation and DPD/DPostD application are done offline. The low-rate operation of updating the coefficients is done in real-time.

As shown in Fig. 2, the digital input of the transmission system is the AWG input, which is limited to a maximum value, sM (a sample reaching sM generates the maximum AWG output voltage). For this reason, the input signal peak amplitude, sp, is measured as a percentage of sM. A graphical representation of the transmission system nonlinearity can be obtained by transmitting the OFDM-GbE signal along the experimental setup and plotting the output signal vector v′ (normalised v), as a function of the input signal vector u, as shown in Fig. 3.

 

Fig. 3. Input-output characteristics of system S, of the experimental setup of Fig. 2 in optical back-to-back, for sp of (a) 10%, (b) 30% and (c) 100%.

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Figure 3 shows the input-output characteristic of system S, obtained in the transmission system of Fig. 2 in optical BtB. The characteristics shown in Fig. 3 result from the combination of several effects. The individual contribution of each effect was assessed from preliminary tests, in which the input-output characteristic was obtained with simpler versions of the transmission system (such as only AWG+DSO, and without optical and electro-optical components). From these tests, it was concluded that:

  1. the sine-like shape in the input-output characteristic is caused by the characteristic of the quadrature-biased MZM and, as a consequence, a non-injective behaviour occurs for MZM inputs exceeding Vπ/2, which correspond to u[n] amplitudes exceeding 0.3. By other words, the system is non-invertible for sp>30% because of the MZM.
  2. the saturation of the electrical amplifier Ap limits the signal excursion at MZM input. Hence, for sp>30%, the v′ excursion does not follow an amplitude reduction according to a sine-curve, but rather produces a minor amplitude reduction.
  3. the thickness of the characteristic of system S grows with sp, because of FDPL and noise added by devices such as the AWG, electrical amplifier Ap and TIA.

4. Experimental results and discussion

In this section, the a-vectors (selected from a list of more than 2900 a-vectors) that result in better signal performance are presented, the NLD mitigation is demonstrated, and the signal performance improvement is assessed. The signal performance is measured as the error vector magnitude (EVM) of the data transmission, which consists of 10 different data sequences (each comprised by 1424800 samples) transmitted one at a time along the experimental setup. The EVM is then calculated as the average of the EVMs (in linear units) over the 10 data sequences and three noise runs per each data sequence.

Early experiments demonstrated that the parameters of the experimental setup fluctuate along time. In order to focus the work on the performance improvements of DPD/DPostD, it was decided to reduce the degradation caused by use of outdated coefficients. For this motive, the coefficients w were updated in real-time (by a MATLAB program) every 30 seconds using Eq. (8) and considering μ =0.1, so that the update rate is significantly faster than the fluctuations observed in the setup. Regarding the choice of the MP, early studies showed that considering even-order Volterra kernels and/or Volterra kernels of order higher than 5 in the MP of the inverse transmission system of Fig. 2 did not contribute to improve significantly the signal performance. For this reason, it is considered that P ≤ 5 and a=[a1a3a5] in Eq. (2). It is also considered that Ns=70 ksamples as in [14].

4.1. Experimental results in optical back-to-back

The best performing a-vectors, for several sp values and for the experimental setup of Fig. 2 in optical BtB, are presented in Table 1.

Tables Icon

Table 1. a-vectors, number of coefficients, C, and corresponding number of multiplications per sampling interval that are required to calculate the output samples of the best performing MP for DPD and DPostD in the experimental setup of Fig. 2 in optical BtB, as a function of peak and RMS amplitudes of the input signal (expressed as percentage of sM).

Table 1 shows that, for sp values between 10% and 25% (system S is invertible, see Fig. 3), the a-values selected for DPD and DPostD, are approximately the same, but, when the sp increases above 30% (system S is non-invertible, see Fig. 3), the complexity of the MP increases/(decreases) when DPostD/(DPD) is used. The reason for this is because, in the presence of saturation/clipping, the MP in the DPostD needs higher order components and a higher variety of combinations to properly expand the clipped/saturated amplitude peaks. However, with DPD, expanding the amplitude peaks at the input of a non-invertible system that clips those peaks does not overcome the clipping. For this motive, with DPD, the MP complexity does not increase with sp.

In order to demonstrate the linear distortion mitigation achieved by DPD and DPostD with an invertible system, the sp amplitude is set to 10% (corresponding to RMS voltage at MZM input of about 120 mV) and the power spectral density (PSD) at DSO output is obtained and presented in Fig. 4.

 

Fig. 4. PSDs obtained at DSO output with the experimental setup in optical BtB with sp=10%, without BBNIM (red line) and with BBNIM (black line). In (a), DPD; in (b), DPostD.

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Figure 4 shows up to 2 dB of in-band frequency distortion without BBNIM, that are compensated for when DPD or DPostD are used. With DPostD, the noise floor is severely affected. This happens because the DPostD is applied to the signal+noise during the process of distortion compensation. The presence of adjacent spectral regrowth (ASR) is not visible, suggesting the system is linear.

In order to demonstrate the NLD mitigation with an invertible system, the sp amplitude is set to 25% and the PSD at DSO output is obtained and presented in Fig. 5. Figure 5(a) shows that, without BBNIM, the presence of about 2.4 dB of ASR, caused by NLD, around the spectrum of the OFDM-GbE signal is visible as well as 1.8 dB of in-band frequency distortion. When DPD is used, Fig. 5(a) shows the mitigation of the ASR around the spectrum of the OFDM-GbE signal and compensation of the frequency distortion inside the spectrum of the OFDM-GbE signal while achieving the same signal level. When DPostD is used, Fig. 5(b) shows that the signal spectrum is equally flat and has the same level as in Fig. 5(a), but the noise floor is severely affected (as for sp=10%, see Fig. 4). These results demonstrate that, for an invertible system, the DPostD is mitigating the NLD and frequency distortion as effectively as the DPD. However, for a non-invertible system, this is no longer true, as it is demonstrated in the PSDs, obtained when sp=50%, presented in Fig. 6.

 

Fig. 5. PSDs obtained at DSO output with the experimental setup in optical BtB with sp=25%, without BBNIM (red line) and with BBNIM (black line). In (a), DPD; in (b), DPostD.

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Fig. 6. PSDs obtained at DSO output with the experimental setup in optical BtB with sp=50%, without BBNIM (red line) and with BBNIM (black line). In (a), DPD; in (b), DPostD.

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Compared to when no BBNIM is used, Fig. 6(a) shows that, with DPD, the signal spectrum is flatter (reduction of frequency distortion by 1.2 dB), but the ASR around the OFDM-GbE signal spectrum has an up to about 3 dB higher PSD. When DPostD is used, Fig. 6(b) shows that the signal spectrum is flatter than with DPD (reduction of frequency distortion by 1.8 dB), but the ASR around the OFDM-GbE signal spectrum is reduced by 10–11 dB. These results demonstrate that, for a non-invertible system, the DPD is not so capable of mitigating the NLD as the DPostD is.

The spectral analysis showed that the NLD mitigation achieved by DPD and DPostD depends significantly on the signal amplitude. In order to assess how this translates into signal performance improvement, the EVM is obtained for the sp amplitudes presented in Table 1. In Fig. 7, the EVM is shown as a function of the signal level at the output of the system, VDSO.

 

Fig. 7. EVM of OFDM-GbE signal as a function of RMS voltage at DSO output, transmitted along the experimental setup of Fig. 2 in optical back-to-back when using DPD, DPostD or no BBNIM.

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Figure 7 shows that, for the original signal without BBNIM, the lowest EVM is −28.5 dB for a VDSO of 11 mV. Compared to this situation, the EVM improvements achieved by using DPD and DPostD are summarized in Table 2.

Tables Icon

Table 2. Optimum EVM and maximum EVM improvements, and corresponding RMS voltage at DSO input, with DPD and DPostD, for optical BtB.

Table 2 shows that, with DPD and DPostD, the optimum EVM improves by 1.6 dB and 3.1 dB, respectively. However, it is also shown that higher EVM improvements can be achieved, namely: (i) for VDSO=17.5 mV, the DPD improves the EVM by about 4 dB (from about −25 dB to −29 dB) and, (ii) for VDSO=24.4 mV, the DPostD improves the EVM by about 12 dB (from about −19 dB to −31 dB).

4.2. Experimental results with single-mode fibre transmission

The results in BtB are interesting to provide a comparison basis and to demonstrate the principle, but a practical system has several tens of km of optical fibre. The presence of the optical fibre changes the characteristics of the transmission system by introducing propagation losses and chromatic dispersion (CD). In order to assess the improvement achieved by DPD or DPostD for various fibre lengths, the a-values are re-optimized for several sp values and for the experimental setup of Fig. 2 with total standard single-mode fibre (SSMF) lengths of 75, 100 and 125 km. The results are presented in Table 3.

Tables Icon

Table 3. a-vectors of the best performing MPs for DPD and DPostD in the experimental setup of Fig. 2 for total fibre lengths of 75, 100 and 125 km, as a function of sp.

Table 3 shows that, for sp=20%, the increase of the fibre length leads to the reduction of the complexity of the MPs. This is because of the signal-to-noise ratio (SNR) reduction with the fibre length. As a result, the compensation effort of the MP is focused on compensating for the FDPL to improve the SNR of the more attenuated sub-carriers. To achieve this, a MP of first order is sufficient. Table 3 shows also that, for sp>20%, the MPs do not vary substantially with the fibre length. This indicates that, for the highest sp values, the NLD detected by the w coefficients estimation and during the MP selection is not overpowered by noise. It can also be seen that, when only the OFDM-GbE signal is used, the NLD detected by these systems does not vary significantly with the fibre length.

When using DPD or DPostD (with the corresponding MPs from Table 3), the EVMs of the OFDM-GbE signal transmitted along the experimental setup of Fig. 2 with total fibre lengths of 75, 100 and 125 km are presented in Fig. 8, as a function of VDSO. For comparison, the signal performance obtained with no BBNIM and with PE are also presented. This PE implements the estimated inverse transfer-function of the transmission channel [8, 9]. The signal amplitudes considered in Fig. 8 are the same ones as used in Table 3.

 

Fig. 8. EVM of OFDM-GbE signal (○) without BBNIM, (□) with DPD, (⋄) with DPostD and (◂) with PE, as a function of RMS voltage at DSO output, transmitted along the experimental setup of Fig. 2 with (a) 75 km, (b) 100 km and (c) 125 km of total fibre length.

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Consistent with the BtB results shown in Fig. 7, Fig. 8 shows that the EVM improvement achieved with DPostD is higher than with DPD. However, the magnitude of the improvement decreases with the fibre length. Namely, for 125 km of SSMF length, the maximum EVM improvement achieved with DPD and DPostD is about 1 dB and 3 dB, respectively. This is caused by a SNR reduction, due to the increase of the losses in the transmission fibre combined with the signal-power saturation of the MZM. These two effects lead to the increase of the impact of noise relative to the effect of NLD on signal performance and, therefore, reduces the EVM improvements achieved by NLD mitigation for longer fibre lengths. Furthermore, a sufficiently low SNR mitigates the FDPL compensation abilities of DPD or DPostD. This is shown for a fibre length of 125 km in Fig. 8, where the PE achieves an improvement of 2.0 dB or more (with VDSO values below 1.3 mV), compared to the EVM attained without BBNIM, with DPD or DPostD. This may be attributed to the different FDPL estimation methods that are used. While the FDPL estimate for the BBNIMs (DPD or DPostD) is obtained from the transmitted/received signal, its effectiveness may decrease when the SNR is lower. Alternatively, the estimate obtained by the PE uses an independent set of RF-tones (see [9]) and is less affected by noise than the transmitted signal. When the SNR is higher, such as with VDSO>1.3 mV or fibre lengths of 75–100 km are used, the impact of FDPL on the EVM is not significant compared to the impact of NLD. For this motive, in these situations, the use of PE does not bring perceptible EVM improvement. Therefore, the EVM improvement achieved in those situations is attributed to NLD compensation.

5. Conclusion

A new full-band approach to mitigate nonlinear distortion using memory polynomials, instead of using a baseband-equivalent as previously considered, has been experimentally demonstrated and the performance has been assessed both in BtB and with fibre transmission in a WDM LR-PON.

The results have shown that, if the system is invertible, the better-performing MPs selected for DPD and DPostD are approximately the same. It has also been shown that DPostD is capable of compensating for the NLD caused by saturation/clipping, while DPD is not. However, in order to do so, the number of coefficients used by the MP in DPostD increases substantially. The results have shown that, in optical BtB, the maximum EVM improvement achieved with DPD and DPostD is 4.1 dB and 11.8 dB, respectively. However, when comparing the optimum EVMs, the EVM improvements are reduced to 1.5 dB and 3.0 dB with DPD and DPostD, respectively. With 125 km of fibre transmission, the maximum EVM improvements are reduced to 1 dB and 3 dB with DPD and DPostD, respectively. Therefore, the EVM improvements attained by NLD mitigation decrease significantly for the longest fibre lengths. The reason for this is that the dominant impairment in those cases is not the NLD, but the saturation of the SNR.

Acknowledgments

This work was supported by Fundação para a Ciência e a Tecnologia (FCT) from Portugal under contract SFRH/BD/66028/2009, by the European project FIVER-FP7-IST-4-249142 and partially supported by the MORFEUS-PTDC/EEITEL/2573/2012 project of FCT Portugal.

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4. Institute of Electrical and Electronics Engineers, “Standard IEEE 802.16 for local and metropolitan area networks Part 16: Air Interface for Fixed Broadband Wireless Access Systems,” 2009.

5. European Computer Manufacturers Association, “Standard ECMA-368: High Rate Ultra Wideband PHY and MAC Standard,” 2009.

6. F. Carvalho and A. Cartaxo, “Optimal electrical power distribution among coexisting OFDM-based signals in LR-PONs: theoretical and experimental analyses,” J. Opt. Commun. Netw. 6(6), 559–570 (2014). [CrossRef]  

7. J. Morgado, D. Fonseca, and A. Cartaxo, “Experimental study of coexistence of multi-band OFDM-UWB and OFDM-baseband signals in long-reach PONs using directly modulated lasers,” Opt. Express 19(23), 23601–23612 (2010). [CrossRef]  

8. M. Morant, T. Alves, A. Cartaxo, and R. Llorente, “Transmission impairment compensation using broadband channel sounding in multi-format OFDM-based long-reach PONs,” in Optical Fiber Communication Conference, pp. 1–3, paper OW3B.2, Los Angeles, CA, USA, March 2012.

9. F. Carvalho and A. Cartaxo, “Study on electrical power distribution among coexisting OFDM-based wired-wireless signals along long-reach passive optical networks,” J. Opt. Commun. Netw. 5(7), 813–824 (2013). [CrossRef]  

10. M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems, (Wiley, 1980).

11. Y. Shen, B. Hraimel, X. Zhang, G. Cowan, K. Wu, and T. Liu, “A novel analog broadband RF predistortion circuit to linearize electro-absorption modulators in multiband OFDM radio-over-fiber systems,” IEEE Trans. Microw. Theory Tech. 58(11), 3327–3335 (2010). [CrossRef]  

12. A. D’Andrea, V. Lottici, and R. Reggiannini, “Nonlinear predistortion of OFDM signals over frequency-selective fading channels,” IEEE Trans. Commun. 49(5), 837–843 (2001). [CrossRef]  

13. J. Cavers, “Amplifier linearization using a digital predistorter with fast adaptation and low memory requirements,” IEEE Trans. Veh. Technol. 39(4), 374–382 (1990). [CrossRef]  

14. D. Morgan, Z. Ma, J. Kim, M. Zierdt, and J. Pastalan, “A generalized memory polynomial model for digital predistortion of RF power amplifiers,” IEEE Trans. Signal Proces. 54(10), 3852–3860 (2006). [CrossRef]  

15. L. Ding, G. Zhou, D. Morgan, M. Zhengxiang, J. Kenney, J. Kim, and C. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004). [CrossRef]  

16. Z. Liu, M. Violas, and N. Carvalho, “Digital predistortion for RSOAs as external modulators in radio over fiber systems,” Opt. Express 19(18), 17641–17646 (2011). [CrossRef]   [PubMed]  

17. Y. Pei, K. Xu, J. Li, A. Zhang, Y. Dai, Y. Ji, and J. Lin, “Complexity-reduced digital predistortion for subcarrier multiplexed radio over fiber systems transmitting sparse multi-band RF signals,” Opt. Express 21(3), 3708–3714 (2013). [CrossRef]   [PubMed]  

18. Y. Liu, J. Zhou, W. Chen, and B. Zhou, “A robust augmented complexity-reduced generalized memory polynomial for wideband RF power amplifiers,” IEEE Trans. Ind. Electron. 61(5), 2389–2401 (2014). [CrossRef]  

19. T. Alves, J. Morgado, and A. Cartaxo, “Linearity improvement of directly modulated PONs by digital pre-distortion of coexisting OFDM-based signals,” in Advanced Photonics Congress, pp. 1–2, paper AW4A.2, Colorado Springs, CO, USA, 2012.

20. D. Hsu, C. Wei, H. Chen, Y. Lu, C. Song, C. Yang, and J. Chen, “SSII cancellation in an EAM-based OFDM-IMDD transmission system employing a novel dynamic chirp model,” Opt. Express 21(1), 533–543 (2013). [CrossRef]   [PubMed]  

21. A. Zhu, P. Draxler, J. Yan, T. Brazil, D. Kimball, and P. Asbeck, “Open-loop digital predistorter for RF power amplifiers using dynamic deviation reduction-based Volterra series,” IEEE Trans. Microw. Theory Techn. 56(7), 1524–1534 (2008). [CrossRef]  

22. H. Cao, H. Nemati, A. Tehrani, T. Eriksson, J. Grahn, and C. Fager, “Linearization of efficiency-optimized dynamic load modulation transmitter architectures,” IEEE Trans. Microw. Theory Techn. 58(4), 873–881 (2010). [CrossRef]  

23. R. Llorente, M. Morant, E. Pellicer, M. Herman, Z. Nagy, T. Alves, A. Cartaxo, J. Herrera, J. Correcher, T. Quinlan, S. Walker, C. Rodrigues, P. Cluzeaud, A. Schmidt, R. Piesiewicz, and R. Sambaraju, “On-the-field demonstration of quintuple-play service provision in long-reach OFDM-based WDM-PON access networks,” European Conference and Exhibition on Optical Communication, paper We.4.F.1, London, UK, September 2013.

References

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  1. A. Cartaxo, J. Morgado, and D. Fonseca, “A perspective on optical-wireless converged NG-FTTH networks using directly modulated lasers,” in International Conference on Transparent Optical Networks, pp. 1–4, paper Mo.B4.3, Stockholm, Sweden, June 2011.
  2. J. Ulm and B. Weeks, “Next play evolution: beyond triple play and quad play,” in IEEE International Symposium on Consumer Electronics, pp. 1–6, Dallas, TX, USA, June 2007.
  3. 3rd Generation Partnership Project, “Evolved Universal Terrestrial Radio Access (E-UTRA), User Equipment (UE) radio transmission and reception (Release 8),” 2009.
  4. Institute of Electrical and Electronics Engineers, “Standard IEEE 802.16 for local and metropolitan area networks Part 16: Air Interface for Fixed Broadband Wireless Access Systems,” 2009.
  5. European Computer Manufacturers Association, “Standard ECMA-368: High Rate Ultra Wideband PHY and MAC Standard,” 2009.
  6. F. Carvalho and A. Cartaxo, “Optimal electrical power distribution among coexisting OFDM-based signals in LR-PONs: theoretical and experimental analyses,” J. Opt. Commun. Netw. 6(6), 559–570 (2014).
    [Crossref]
  7. J. Morgado, D. Fonseca, and A. Cartaxo, “Experimental study of coexistence of multi-band OFDM-UWB and OFDM-baseband signals in long-reach PONs using directly modulated lasers,” Opt. Express 19(23), 23601–23612 (2010).
    [Crossref]
  8. M. Morant, T. Alves, A. Cartaxo, and R. Llorente, “Transmission impairment compensation using broadband channel sounding in multi-format OFDM-based long-reach PONs,” in Optical Fiber Communication Conference, pp. 1–3, paper OW3B.2, Los Angeles, CA, USA, March 2012.
  9. F. Carvalho and A. Cartaxo, “Study on electrical power distribution among coexisting OFDM-based wired-wireless signals along long-reach passive optical networks,” J. Opt. Commun. Netw. 5(7), 813–824 (2013).
    [Crossref]
  10. M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems, (Wiley, 1980).
  11. Y. Shen, B. Hraimel, X. Zhang, G. Cowan, K. Wu, and T. Liu, “A novel analog broadband RF predistortion circuit to linearize electro-absorption modulators in multiband OFDM radio-over-fiber systems,” IEEE Trans. Microw. Theory Tech. 58(11), 3327–3335 (2010).
    [Crossref]
  12. A. D’Andrea, V. Lottici, and R. Reggiannini, “Nonlinear predistortion of OFDM signals over frequency-selective fading channels,” IEEE Trans. Commun. 49(5), 837–843 (2001).
    [Crossref]
  13. J. Cavers, “Amplifier linearization using a digital predistorter with fast adaptation and low memory requirements,” IEEE Trans. Veh. Technol. 39(4), 374–382 (1990).
    [Crossref]
  14. D. Morgan, Z. Ma, J. Kim, M. Zierdt, and J. Pastalan, “A generalized memory polynomial model for digital predistortion of RF power amplifiers,” IEEE Trans. Signal Proces. 54(10), 3852–3860 (2006).
    [Crossref]
  15. L. Ding, G. Zhou, D. Morgan, M. Zhengxiang, J. Kenney, J. Kim, and C. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
    [Crossref]
  16. Z. Liu, M. Violas, and N. Carvalho, “Digital predistortion for RSOAs as external modulators in radio over fiber systems,” Opt. Express 19(18), 17641–17646 (2011).
    [Crossref] [PubMed]
  17. Y. Pei, K. Xu, J. Li, A. Zhang, Y. Dai, Y. Ji, and J. Lin, “Complexity-reduced digital predistortion for subcarrier multiplexed radio over fiber systems transmitting sparse multi-band RF signals,” Opt. Express 21(3), 3708–3714 (2013).
    [Crossref] [PubMed]
  18. Y. Liu, J. Zhou, W. Chen, and B. Zhou, “A robust augmented complexity-reduced generalized memory polynomial for wideband RF power amplifiers,” IEEE Trans. Ind. Electron. 61(5), 2389–2401 (2014).
    [Crossref]
  19. T. Alves, J. Morgado, and A. Cartaxo, “Linearity improvement of directly modulated PONs by digital pre-distortion of coexisting OFDM-based signals,” in Advanced Photonics Congress, pp. 1–2, paper AW4A.2, Colorado Springs, CO, USA, 2012.
  20. D. Hsu, C. Wei, H. Chen, Y. Lu, C. Song, C. Yang, and J. Chen, “SSII cancellation in an EAM-based OFDM-IMDD transmission system employing a novel dynamic chirp model,” Opt. Express 21(1), 533–543 (2013).
    [Crossref] [PubMed]
  21. A. Zhu, P. Draxler, J. Yan, T. Brazil, D. Kimball, and P. Asbeck, “Open-loop digital predistorter for RF power amplifiers using dynamic deviation reduction-based Volterra series,” IEEE Trans. Microw. Theory Techn. 56(7), 1524–1534 (2008).
    [Crossref]
  22. H. Cao, H. Nemati, A. Tehrani, T. Eriksson, J. Grahn, and C. Fager, “Linearization of efficiency-optimized dynamic load modulation transmitter architectures,” IEEE Trans. Microw. Theory Techn. 58(4), 873–881 (2010).
    [Crossref]
  23. R. Llorente, M. Morant, E. Pellicer, M. Herman, Z. Nagy, T. Alves, A. Cartaxo, J. Herrera, J. Correcher, T. Quinlan, S. Walker, C. Rodrigues, P. Cluzeaud, A. Schmidt, R. Piesiewicz, and R. Sambaraju, “On-the-field demonstration of quintuple-play service provision in long-reach OFDM-based WDM-PON access networks,” European Conference and Exhibition on Optical Communication, paper We.4.F.1, London, UK, September 2013.

2014 (2)

F. Carvalho and A. Cartaxo, “Optimal electrical power distribution among coexisting OFDM-based signals in LR-PONs: theoretical and experimental analyses,” J. Opt. Commun. Netw. 6(6), 559–570 (2014).
[Crossref]

Y. Liu, J. Zhou, W. Chen, and B. Zhou, “A robust augmented complexity-reduced generalized memory polynomial for wideband RF power amplifiers,” IEEE Trans. Ind. Electron. 61(5), 2389–2401 (2014).
[Crossref]

2013 (3)

2011 (1)

2010 (3)

H. Cao, H. Nemati, A. Tehrani, T. Eriksson, J. Grahn, and C. Fager, “Linearization of efficiency-optimized dynamic load modulation transmitter architectures,” IEEE Trans. Microw. Theory Techn. 58(4), 873–881 (2010).
[Crossref]

Y. Shen, B. Hraimel, X. Zhang, G. Cowan, K. Wu, and T. Liu, “A novel analog broadband RF predistortion circuit to linearize electro-absorption modulators in multiband OFDM radio-over-fiber systems,” IEEE Trans. Microw. Theory Tech. 58(11), 3327–3335 (2010).
[Crossref]

J. Morgado, D. Fonseca, and A. Cartaxo, “Experimental study of coexistence of multi-band OFDM-UWB and OFDM-baseband signals in long-reach PONs using directly modulated lasers,” Opt. Express 19(23), 23601–23612 (2010).
[Crossref]

2008 (1)

A. Zhu, P. Draxler, J. Yan, T. Brazil, D. Kimball, and P. Asbeck, “Open-loop digital predistorter for RF power amplifiers using dynamic deviation reduction-based Volterra series,” IEEE Trans. Microw. Theory Techn. 56(7), 1524–1534 (2008).
[Crossref]

2006 (1)

D. Morgan, Z. Ma, J. Kim, M. Zierdt, and J. Pastalan, “A generalized memory polynomial model for digital predistortion of RF power amplifiers,” IEEE Trans. Signal Proces. 54(10), 3852–3860 (2006).
[Crossref]

2004 (1)

L. Ding, G. Zhou, D. Morgan, M. Zhengxiang, J. Kenney, J. Kim, and C. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
[Crossref]

2001 (1)

A. D’Andrea, V. Lottici, and R. Reggiannini, “Nonlinear predistortion of OFDM signals over frequency-selective fading channels,” IEEE Trans. Commun. 49(5), 837–843 (2001).
[Crossref]

1990 (1)

J. Cavers, “Amplifier linearization using a digital predistorter with fast adaptation and low memory requirements,” IEEE Trans. Veh. Technol. 39(4), 374–382 (1990).
[Crossref]

Alves, T.

M. Morant, T. Alves, A. Cartaxo, and R. Llorente, “Transmission impairment compensation using broadband channel sounding in multi-format OFDM-based long-reach PONs,” in Optical Fiber Communication Conference, pp. 1–3, paper OW3B.2, Los Angeles, CA, USA, March 2012.

T. Alves, J. Morgado, and A. Cartaxo, “Linearity improvement of directly modulated PONs by digital pre-distortion of coexisting OFDM-based signals,” in Advanced Photonics Congress, pp. 1–2, paper AW4A.2, Colorado Springs, CO, USA, 2012.

R. Llorente, M. Morant, E. Pellicer, M. Herman, Z. Nagy, T. Alves, A. Cartaxo, J. Herrera, J. Correcher, T. Quinlan, S. Walker, C. Rodrigues, P. Cluzeaud, A. Schmidt, R. Piesiewicz, and R. Sambaraju, “On-the-field demonstration of quintuple-play service provision in long-reach OFDM-based WDM-PON access networks,” European Conference and Exhibition on Optical Communication, paper We.4.F.1, London, UK, September 2013.

Asbeck, P.

A. Zhu, P. Draxler, J. Yan, T. Brazil, D. Kimball, and P. Asbeck, “Open-loop digital predistorter for RF power amplifiers using dynamic deviation reduction-based Volterra series,” IEEE Trans. Microw. Theory Techn. 56(7), 1524–1534 (2008).
[Crossref]

Brazil, T.

A. Zhu, P. Draxler, J. Yan, T. Brazil, D. Kimball, and P. Asbeck, “Open-loop digital predistorter for RF power amplifiers using dynamic deviation reduction-based Volterra series,” IEEE Trans. Microw. Theory Techn. 56(7), 1524–1534 (2008).
[Crossref]

Cao, H.

H. Cao, H. Nemati, A. Tehrani, T. Eriksson, J. Grahn, and C. Fager, “Linearization of efficiency-optimized dynamic load modulation transmitter architectures,” IEEE Trans. Microw. Theory Techn. 58(4), 873–881 (2010).
[Crossref]

Cartaxo, A.

F. Carvalho and A. Cartaxo, “Optimal electrical power distribution among coexisting OFDM-based signals in LR-PONs: theoretical and experimental analyses,” J. Opt. Commun. Netw. 6(6), 559–570 (2014).
[Crossref]

F. Carvalho and A. Cartaxo, “Study on electrical power distribution among coexisting OFDM-based wired-wireless signals along long-reach passive optical networks,” J. Opt. Commun. Netw. 5(7), 813–824 (2013).
[Crossref]

J. Morgado, D. Fonseca, and A. Cartaxo, “Experimental study of coexistence of multi-band OFDM-UWB and OFDM-baseband signals in long-reach PONs using directly modulated lasers,” Opt. Express 19(23), 23601–23612 (2010).
[Crossref]

M. Morant, T. Alves, A. Cartaxo, and R. Llorente, “Transmission impairment compensation using broadband channel sounding in multi-format OFDM-based long-reach PONs,” in Optical Fiber Communication Conference, pp. 1–3, paper OW3B.2, Los Angeles, CA, USA, March 2012.

A. Cartaxo, J. Morgado, and D. Fonseca, “A perspective on optical-wireless converged NG-FTTH networks using directly modulated lasers,” in International Conference on Transparent Optical Networks, pp. 1–4, paper Mo.B4.3, Stockholm, Sweden, June 2011.

T. Alves, J. Morgado, and A. Cartaxo, “Linearity improvement of directly modulated PONs by digital pre-distortion of coexisting OFDM-based signals,” in Advanced Photonics Congress, pp. 1–2, paper AW4A.2, Colorado Springs, CO, USA, 2012.

R. Llorente, M. Morant, E. Pellicer, M. Herman, Z. Nagy, T. Alves, A. Cartaxo, J. Herrera, J. Correcher, T. Quinlan, S. Walker, C. Rodrigues, P. Cluzeaud, A. Schmidt, R. Piesiewicz, and R. Sambaraju, “On-the-field demonstration of quintuple-play service provision in long-reach OFDM-based WDM-PON access networks,” European Conference and Exhibition on Optical Communication, paper We.4.F.1, London, UK, September 2013.

Carvalho, F.

Carvalho, N.

Cavers, J.

J. Cavers, “Amplifier linearization using a digital predistorter with fast adaptation and low memory requirements,” IEEE Trans. Veh. Technol. 39(4), 374–382 (1990).
[Crossref]

Chen, H.

Chen, J.

Chen, W.

Y. Liu, J. Zhou, W. Chen, and B. Zhou, “A robust augmented complexity-reduced generalized memory polynomial for wideband RF power amplifiers,” IEEE Trans. Ind. Electron. 61(5), 2389–2401 (2014).
[Crossref]

Cluzeaud, P.

R. Llorente, M. Morant, E. Pellicer, M. Herman, Z. Nagy, T. Alves, A. Cartaxo, J. Herrera, J. Correcher, T. Quinlan, S. Walker, C. Rodrigues, P. Cluzeaud, A. Schmidt, R. Piesiewicz, and R. Sambaraju, “On-the-field demonstration of quintuple-play service provision in long-reach OFDM-based WDM-PON access networks,” European Conference and Exhibition on Optical Communication, paper We.4.F.1, London, UK, September 2013.

Correcher, J.

R. Llorente, M. Morant, E. Pellicer, M. Herman, Z. Nagy, T. Alves, A. Cartaxo, J. Herrera, J. Correcher, T. Quinlan, S. Walker, C. Rodrigues, P. Cluzeaud, A. Schmidt, R. Piesiewicz, and R. Sambaraju, “On-the-field demonstration of quintuple-play service provision in long-reach OFDM-based WDM-PON access networks,” European Conference and Exhibition on Optical Communication, paper We.4.F.1, London, UK, September 2013.

Cowan, G.

Y. Shen, B. Hraimel, X. Zhang, G. Cowan, K. Wu, and T. Liu, “A novel analog broadband RF predistortion circuit to linearize electro-absorption modulators in multiband OFDM radio-over-fiber systems,” IEEE Trans. Microw. Theory Tech. 58(11), 3327–3335 (2010).
[Crossref]

D’Andrea, A.

A. D’Andrea, V. Lottici, and R. Reggiannini, “Nonlinear predistortion of OFDM signals over frequency-selective fading channels,” IEEE Trans. Commun. 49(5), 837–843 (2001).
[Crossref]

Dai, Y.

Ding, L.

L. Ding, G. Zhou, D. Morgan, M. Zhengxiang, J. Kenney, J. Kim, and C. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
[Crossref]

Draxler, P.

A. Zhu, P. Draxler, J. Yan, T. Brazil, D. Kimball, and P. Asbeck, “Open-loop digital predistorter for RF power amplifiers using dynamic deviation reduction-based Volterra series,” IEEE Trans. Microw. Theory Techn. 56(7), 1524–1534 (2008).
[Crossref]

Eriksson, T.

H. Cao, H. Nemati, A. Tehrani, T. Eriksson, J. Grahn, and C. Fager, “Linearization of efficiency-optimized dynamic load modulation transmitter architectures,” IEEE Trans. Microw. Theory Techn. 58(4), 873–881 (2010).
[Crossref]

Fager, C.

H. Cao, H. Nemati, A. Tehrani, T. Eriksson, J. Grahn, and C. Fager, “Linearization of efficiency-optimized dynamic load modulation transmitter architectures,” IEEE Trans. Microw. Theory Techn. 58(4), 873–881 (2010).
[Crossref]

Fonseca, D.

J. Morgado, D. Fonseca, and A. Cartaxo, “Experimental study of coexistence of multi-band OFDM-UWB and OFDM-baseband signals in long-reach PONs using directly modulated lasers,” Opt. Express 19(23), 23601–23612 (2010).
[Crossref]

A. Cartaxo, J. Morgado, and D. Fonseca, “A perspective on optical-wireless converged NG-FTTH networks using directly modulated lasers,” in International Conference on Transparent Optical Networks, pp. 1–4, paper Mo.B4.3, Stockholm, Sweden, June 2011.

Giardina, C.

L. Ding, G. Zhou, D. Morgan, M. Zhengxiang, J. Kenney, J. Kim, and C. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
[Crossref]

Grahn, J.

H. Cao, H. Nemati, A. Tehrani, T. Eriksson, J. Grahn, and C. Fager, “Linearization of efficiency-optimized dynamic load modulation transmitter architectures,” IEEE Trans. Microw. Theory Techn. 58(4), 873–881 (2010).
[Crossref]

Herman, M.

R. Llorente, M. Morant, E. Pellicer, M. Herman, Z. Nagy, T. Alves, A. Cartaxo, J. Herrera, J. Correcher, T. Quinlan, S. Walker, C. Rodrigues, P. Cluzeaud, A. Schmidt, R. Piesiewicz, and R. Sambaraju, “On-the-field demonstration of quintuple-play service provision in long-reach OFDM-based WDM-PON access networks,” European Conference and Exhibition on Optical Communication, paper We.4.F.1, London, UK, September 2013.

Herrera, J.

R. Llorente, M. Morant, E. Pellicer, M. Herman, Z. Nagy, T. Alves, A. Cartaxo, J. Herrera, J. Correcher, T. Quinlan, S. Walker, C. Rodrigues, P. Cluzeaud, A. Schmidt, R. Piesiewicz, and R. Sambaraju, “On-the-field demonstration of quintuple-play service provision in long-reach OFDM-based WDM-PON access networks,” European Conference and Exhibition on Optical Communication, paper We.4.F.1, London, UK, September 2013.

Hraimel, B.

Y. Shen, B. Hraimel, X. Zhang, G. Cowan, K. Wu, and T. Liu, “A novel analog broadband RF predistortion circuit to linearize electro-absorption modulators in multiband OFDM radio-over-fiber systems,” IEEE Trans. Microw. Theory Tech. 58(11), 3327–3335 (2010).
[Crossref]

Hsu, D.

Ji, Y.

Kenney, J.

L. Ding, G. Zhou, D. Morgan, M. Zhengxiang, J. Kenney, J. Kim, and C. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
[Crossref]

Kim, J.

D. Morgan, Z. Ma, J. Kim, M. Zierdt, and J. Pastalan, “A generalized memory polynomial model for digital predistortion of RF power amplifiers,” IEEE Trans. Signal Proces. 54(10), 3852–3860 (2006).
[Crossref]

L. Ding, G. Zhou, D. Morgan, M. Zhengxiang, J. Kenney, J. Kim, and C. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
[Crossref]

Kimball, D.

A. Zhu, P. Draxler, J. Yan, T. Brazil, D. Kimball, and P. Asbeck, “Open-loop digital predistorter for RF power amplifiers using dynamic deviation reduction-based Volterra series,” IEEE Trans. Microw. Theory Techn. 56(7), 1524–1534 (2008).
[Crossref]

Li, J.

Lin, J.

Liu, T.

Y. Shen, B. Hraimel, X. Zhang, G. Cowan, K. Wu, and T. Liu, “A novel analog broadband RF predistortion circuit to linearize electro-absorption modulators in multiband OFDM radio-over-fiber systems,” IEEE Trans. Microw. Theory Tech. 58(11), 3327–3335 (2010).
[Crossref]

Liu, Y.

Y. Liu, J. Zhou, W. Chen, and B. Zhou, “A robust augmented complexity-reduced generalized memory polynomial for wideband RF power amplifiers,” IEEE Trans. Ind. Electron. 61(5), 2389–2401 (2014).
[Crossref]

Liu, Z.

Llorente, R.

M. Morant, T. Alves, A. Cartaxo, and R. Llorente, “Transmission impairment compensation using broadband channel sounding in multi-format OFDM-based long-reach PONs,” in Optical Fiber Communication Conference, pp. 1–3, paper OW3B.2, Los Angeles, CA, USA, March 2012.

R. Llorente, M. Morant, E. Pellicer, M. Herman, Z. Nagy, T. Alves, A. Cartaxo, J. Herrera, J. Correcher, T. Quinlan, S. Walker, C. Rodrigues, P. Cluzeaud, A. Schmidt, R. Piesiewicz, and R. Sambaraju, “On-the-field demonstration of quintuple-play service provision in long-reach OFDM-based WDM-PON access networks,” European Conference and Exhibition on Optical Communication, paper We.4.F.1, London, UK, September 2013.

Lottici, V.

A. D’Andrea, V. Lottici, and R. Reggiannini, “Nonlinear predistortion of OFDM signals over frequency-selective fading channels,” IEEE Trans. Commun. 49(5), 837–843 (2001).
[Crossref]

Lu, Y.

Ma, Z.

D. Morgan, Z. Ma, J. Kim, M. Zierdt, and J. Pastalan, “A generalized memory polynomial model for digital predistortion of RF power amplifiers,” IEEE Trans. Signal Proces. 54(10), 3852–3860 (2006).
[Crossref]

Morant, M.

M. Morant, T. Alves, A. Cartaxo, and R. Llorente, “Transmission impairment compensation using broadband channel sounding in multi-format OFDM-based long-reach PONs,” in Optical Fiber Communication Conference, pp. 1–3, paper OW3B.2, Los Angeles, CA, USA, March 2012.

R. Llorente, M. Morant, E. Pellicer, M. Herman, Z. Nagy, T. Alves, A. Cartaxo, J. Herrera, J. Correcher, T. Quinlan, S. Walker, C. Rodrigues, P. Cluzeaud, A. Schmidt, R. Piesiewicz, and R. Sambaraju, “On-the-field demonstration of quintuple-play service provision in long-reach OFDM-based WDM-PON access networks,” European Conference and Exhibition on Optical Communication, paper We.4.F.1, London, UK, September 2013.

Morgado, J.

J. Morgado, D. Fonseca, and A. Cartaxo, “Experimental study of coexistence of multi-band OFDM-UWB and OFDM-baseband signals in long-reach PONs using directly modulated lasers,” Opt. Express 19(23), 23601–23612 (2010).
[Crossref]

A. Cartaxo, J. Morgado, and D. Fonseca, “A perspective on optical-wireless converged NG-FTTH networks using directly modulated lasers,” in International Conference on Transparent Optical Networks, pp. 1–4, paper Mo.B4.3, Stockholm, Sweden, June 2011.

T. Alves, J. Morgado, and A. Cartaxo, “Linearity improvement of directly modulated PONs by digital pre-distortion of coexisting OFDM-based signals,” in Advanced Photonics Congress, pp. 1–2, paper AW4A.2, Colorado Springs, CO, USA, 2012.

Morgan, D.

D. Morgan, Z. Ma, J. Kim, M. Zierdt, and J. Pastalan, “A generalized memory polynomial model for digital predistortion of RF power amplifiers,” IEEE Trans. Signal Proces. 54(10), 3852–3860 (2006).
[Crossref]

L. Ding, G. Zhou, D. Morgan, M. Zhengxiang, J. Kenney, J. Kim, and C. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
[Crossref]

Nagy, Z.

R. Llorente, M. Morant, E. Pellicer, M. Herman, Z. Nagy, T. Alves, A. Cartaxo, J. Herrera, J. Correcher, T. Quinlan, S. Walker, C. Rodrigues, P. Cluzeaud, A. Schmidt, R. Piesiewicz, and R. Sambaraju, “On-the-field demonstration of quintuple-play service provision in long-reach OFDM-based WDM-PON access networks,” European Conference and Exhibition on Optical Communication, paper We.4.F.1, London, UK, September 2013.

Nemati, H.

H. Cao, H. Nemati, A. Tehrani, T. Eriksson, J. Grahn, and C. Fager, “Linearization of efficiency-optimized dynamic load modulation transmitter architectures,” IEEE Trans. Microw. Theory Techn. 58(4), 873–881 (2010).
[Crossref]

Pastalan, J.

D. Morgan, Z. Ma, J. Kim, M. Zierdt, and J. Pastalan, “A generalized memory polynomial model for digital predistortion of RF power amplifiers,” IEEE Trans. Signal Proces. 54(10), 3852–3860 (2006).
[Crossref]

Pei, Y.

Pellicer, E.

R. Llorente, M. Morant, E. Pellicer, M. Herman, Z. Nagy, T. Alves, A. Cartaxo, J. Herrera, J. Correcher, T. Quinlan, S. Walker, C. Rodrigues, P. Cluzeaud, A. Schmidt, R. Piesiewicz, and R. Sambaraju, “On-the-field demonstration of quintuple-play service provision in long-reach OFDM-based WDM-PON access networks,” European Conference and Exhibition on Optical Communication, paper We.4.F.1, London, UK, September 2013.

Piesiewicz, R.

R. Llorente, M. Morant, E. Pellicer, M. Herman, Z. Nagy, T. Alves, A. Cartaxo, J. Herrera, J. Correcher, T. Quinlan, S. Walker, C. Rodrigues, P. Cluzeaud, A. Schmidt, R. Piesiewicz, and R. Sambaraju, “On-the-field demonstration of quintuple-play service provision in long-reach OFDM-based WDM-PON access networks,” European Conference and Exhibition on Optical Communication, paper We.4.F.1, London, UK, September 2013.

Quinlan, T.

R. Llorente, M. Morant, E. Pellicer, M. Herman, Z. Nagy, T. Alves, A. Cartaxo, J. Herrera, J. Correcher, T. Quinlan, S. Walker, C. Rodrigues, P. Cluzeaud, A. Schmidt, R. Piesiewicz, and R. Sambaraju, “On-the-field demonstration of quintuple-play service provision in long-reach OFDM-based WDM-PON access networks,” European Conference and Exhibition on Optical Communication, paper We.4.F.1, London, UK, September 2013.

Reggiannini, R.

A. D’Andrea, V. Lottici, and R. Reggiannini, “Nonlinear predistortion of OFDM signals over frequency-selective fading channels,” IEEE Trans. Commun. 49(5), 837–843 (2001).
[Crossref]

Rodrigues, C.

R. Llorente, M. Morant, E. Pellicer, M. Herman, Z. Nagy, T. Alves, A. Cartaxo, J. Herrera, J. Correcher, T. Quinlan, S. Walker, C. Rodrigues, P. Cluzeaud, A. Schmidt, R. Piesiewicz, and R. Sambaraju, “On-the-field demonstration of quintuple-play service provision in long-reach OFDM-based WDM-PON access networks,” European Conference and Exhibition on Optical Communication, paper We.4.F.1, London, UK, September 2013.

Sambaraju, R.

R. Llorente, M. Morant, E. Pellicer, M. Herman, Z. Nagy, T. Alves, A. Cartaxo, J. Herrera, J. Correcher, T. Quinlan, S. Walker, C. Rodrigues, P. Cluzeaud, A. Schmidt, R. Piesiewicz, and R. Sambaraju, “On-the-field demonstration of quintuple-play service provision in long-reach OFDM-based WDM-PON access networks,” European Conference and Exhibition on Optical Communication, paper We.4.F.1, London, UK, September 2013.

Schetzen, M.

M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems, (Wiley, 1980).

Schmidt, A.

R. Llorente, M. Morant, E. Pellicer, M. Herman, Z. Nagy, T. Alves, A. Cartaxo, J. Herrera, J. Correcher, T. Quinlan, S. Walker, C. Rodrigues, P. Cluzeaud, A. Schmidt, R. Piesiewicz, and R. Sambaraju, “On-the-field demonstration of quintuple-play service provision in long-reach OFDM-based WDM-PON access networks,” European Conference and Exhibition on Optical Communication, paper We.4.F.1, London, UK, September 2013.

Shen, Y.

Y. Shen, B. Hraimel, X. Zhang, G. Cowan, K. Wu, and T. Liu, “A novel analog broadband RF predistortion circuit to linearize electro-absorption modulators in multiband OFDM radio-over-fiber systems,” IEEE Trans. Microw. Theory Tech. 58(11), 3327–3335 (2010).
[Crossref]

Song, C.

Tehrani, A.

H. Cao, H. Nemati, A. Tehrani, T. Eriksson, J. Grahn, and C. Fager, “Linearization of efficiency-optimized dynamic load modulation transmitter architectures,” IEEE Trans. Microw. Theory Techn. 58(4), 873–881 (2010).
[Crossref]

Ulm, J.

J. Ulm and B. Weeks, “Next play evolution: beyond triple play and quad play,” in IEEE International Symposium on Consumer Electronics, pp. 1–6, Dallas, TX, USA, June 2007.

Violas, M.

Walker, S.

R. Llorente, M. Morant, E. Pellicer, M. Herman, Z. Nagy, T. Alves, A. Cartaxo, J. Herrera, J. Correcher, T. Quinlan, S. Walker, C. Rodrigues, P. Cluzeaud, A. Schmidt, R. Piesiewicz, and R. Sambaraju, “On-the-field demonstration of quintuple-play service provision in long-reach OFDM-based WDM-PON access networks,” European Conference and Exhibition on Optical Communication, paper We.4.F.1, London, UK, September 2013.

Weeks, B.

J. Ulm and B. Weeks, “Next play evolution: beyond triple play and quad play,” in IEEE International Symposium on Consumer Electronics, pp. 1–6, Dallas, TX, USA, June 2007.

Wei, C.

Wu, K.

Y. Shen, B. Hraimel, X. Zhang, G. Cowan, K. Wu, and T. Liu, “A novel analog broadband RF predistortion circuit to linearize electro-absorption modulators in multiband OFDM radio-over-fiber systems,” IEEE Trans. Microw. Theory Tech. 58(11), 3327–3335 (2010).
[Crossref]

Xu, K.

Yan, J.

A. Zhu, P. Draxler, J. Yan, T. Brazil, D. Kimball, and P. Asbeck, “Open-loop digital predistorter for RF power amplifiers using dynamic deviation reduction-based Volterra series,” IEEE Trans. Microw. Theory Techn. 56(7), 1524–1534 (2008).
[Crossref]

Yang, C.

Zhang, A.

Zhang, X.

Y. Shen, B. Hraimel, X. Zhang, G. Cowan, K. Wu, and T. Liu, “A novel analog broadband RF predistortion circuit to linearize electro-absorption modulators in multiband OFDM radio-over-fiber systems,” IEEE Trans. Microw. Theory Tech. 58(11), 3327–3335 (2010).
[Crossref]

Zhengxiang, M.

L. Ding, G. Zhou, D. Morgan, M. Zhengxiang, J. Kenney, J. Kim, and C. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
[Crossref]

Zhou, B.

Y. Liu, J. Zhou, W. Chen, and B. Zhou, “A robust augmented complexity-reduced generalized memory polynomial for wideband RF power amplifiers,” IEEE Trans. Ind. Electron. 61(5), 2389–2401 (2014).
[Crossref]

Zhou, G.

L. Ding, G. Zhou, D. Morgan, M. Zhengxiang, J. Kenney, J. Kim, and C. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
[Crossref]

Zhou, J.

Y. Liu, J. Zhou, W. Chen, and B. Zhou, “A robust augmented complexity-reduced generalized memory polynomial for wideband RF power amplifiers,” IEEE Trans. Ind. Electron. 61(5), 2389–2401 (2014).
[Crossref]

Zhu, A.

A. Zhu, P. Draxler, J. Yan, T. Brazil, D. Kimball, and P. Asbeck, “Open-loop digital predistorter for RF power amplifiers using dynamic deviation reduction-based Volterra series,” IEEE Trans. Microw. Theory Techn. 56(7), 1524–1534 (2008).
[Crossref]

Zierdt, M.

D. Morgan, Z. Ma, J. Kim, M. Zierdt, and J. Pastalan, “A generalized memory polynomial model for digital predistortion of RF power amplifiers,” IEEE Trans. Signal Proces. 54(10), 3852–3860 (2006).
[Crossref]

IEEE Trans. Commun. (2)

A. D’Andrea, V. Lottici, and R. Reggiannini, “Nonlinear predistortion of OFDM signals over frequency-selective fading channels,” IEEE Trans. Commun. 49(5), 837–843 (2001).
[Crossref]

L. Ding, G. Zhou, D. Morgan, M. Zhengxiang, J. Kenney, J. Kim, and C. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
[Crossref]

IEEE Trans. Ind. Electron. (1)

Y. Liu, J. Zhou, W. Chen, and B. Zhou, “A robust augmented complexity-reduced generalized memory polynomial for wideband RF power amplifiers,” IEEE Trans. Ind. Electron. 61(5), 2389–2401 (2014).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

Y. Shen, B. Hraimel, X. Zhang, G. Cowan, K. Wu, and T. Liu, “A novel analog broadband RF predistortion circuit to linearize electro-absorption modulators in multiband OFDM radio-over-fiber systems,” IEEE Trans. Microw. Theory Tech. 58(11), 3327–3335 (2010).
[Crossref]

IEEE Trans. Microw. Theory Techn. (2)

A. Zhu, P. Draxler, J. Yan, T. Brazil, D. Kimball, and P. Asbeck, “Open-loop digital predistorter for RF power amplifiers using dynamic deviation reduction-based Volterra series,” IEEE Trans. Microw. Theory Techn. 56(7), 1524–1534 (2008).
[Crossref]

H. Cao, H. Nemati, A. Tehrani, T. Eriksson, J. Grahn, and C. Fager, “Linearization of efficiency-optimized dynamic load modulation transmitter architectures,” IEEE Trans. Microw. Theory Techn. 58(4), 873–881 (2010).
[Crossref]

IEEE Trans. Signal Proces. (1)

D. Morgan, Z. Ma, J. Kim, M. Zierdt, and J. Pastalan, “A generalized memory polynomial model for digital predistortion of RF power amplifiers,” IEEE Trans. Signal Proces. 54(10), 3852–3860 (2006).
[Crossref]

IEEE Trans. Veh. Technol. (1)

J. Cavers, “Amplifier linearization using a digital predistorter with fast adaptation and low memory requirements,” IEEE Trans. Veh. Technol. 39(4), 374–382 (1990).
[Crossref]

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Opt. Express (4)

Other (9)

R. Llorente, M. Morant, E. Pellicer, M. Herman, Z. Nagy, T. Alves, A. Cartaxo, J. Herrera, J. Correcher, T. Quinlan, S. Walker, C. Rodrigues, P. Cluzeaud, A. Schmidt, R. Piesiewicz, and R. Sambaraju, “On-the-field demonstration of quintuple-play service provision in long-reach OFDM-based WDM-PON access networks,” European Conference and Exhibition on Optical Communication, paper We.4.F.1, London, UK, September 2013.

T. Alves, J. Morgado, and A. Cartaxo, “Linearity improvement of directly modulated PONs by digital pre-distortion of coexisting OFDM-based signals,” in Advanced Photonics Congress, pp. 1–2, paper AW4A.2, Colorado Springs, CO, USA, 2012.

M. Morant, T. Alves, A. Cartaxo, and R. Llorente, “Transmission impairment compensation using broadband channel sounding in multi-format OFDM-based long-reach PONs,” in Optical Fiber Communication Conference, pp. 1–3, paper OW3B.2, Los Angeles, CA, USA, March 2012.

M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems, (Wiley, 1980).

A. Cartaxo, J. Morgado, and D. Fonseca, “A perspective on optical-wireless converged NG-FTTH networks using directly modulated lasers,” in International Conference on Transparent Optical Networks, pp. 1–4, paper Mo.B4.3, Stockholm, Sweden, June 2011.

J. Ulm and B. Weeks, “Next play evolution: beyond triple play and quad play,” in IEEE International Symposium on Consumer Electronics, pp. 1–6, Dallas, TX, USA, June 2007.

3rd Generation Partnership Project, “Evolved Universal Terrestrial Radio Access (E-UTRA), User Equipment (UE) radio transmission and reception (Release 8),” 2009.

Institute of Electrical and Electronics Engineers, “Standard IEEE 802.16 for local and metropolitan area networks Part 16: Air Interface for Fixed Broadband Wireless Access Systems,” 2009.

European Computer Manufacturers Association, “Standard ECMA-368: High Rate Ultra Wideband PHY and MAC Standard,” 2009.

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Figures (8)

Fig. 1.
Fig. 1. Implementation schemes of digital (a) pre-distortion and (b) post-distortion, using a digital signal processor (DSP) programmed with a memory polynomial (MP), adjusted by the coefficients w, to compensate for the distortion introduced by the transmission system S. Remark: the signals u and v in implementations (a) and (b) are different.
Fig. 2.
Fig. 2. Experimental setup employed to evaluate the performance improvement achieved by mitigating the NLD in the transmission system S using DPD and DPostD. The necessary MP, w coefficient estimator and signal (de)/modulator are implemented in the digital domain by MATLAB programs.
Fig. 3.
Fig. 3. Input-output characteristics of system S, of the experimental setup of Fig. 2 in optical back-to-back, for sp of (a) 10%, (b) 30% and (c) 100%.
Fig. 4.
Fig. 4. PSDs obtained at DSO output with the experimental setup in optical BtB with sp=10%, without BBNIM (red line) and with BBNIM (black line). In (a), DPD; in (b), DPostD.
Fig. 5.
Fig. 5. PSDs obtained at DSO output with the experimental setup in optical BtB with sp=25%, without BBNIM (red line) and with BBNIM (black line). In (a), DPD; in (b), DPostD.
Fig. 6.
Fig. 6. PSDs obtained at DSO output with the experimental setup in optical BtB with sp=50%, without BBNIM (red line) and with BBNIM (black line). In (a), DPD; in (b), DPostD.
Fig. 7.
Fig. 7. EVM of OFDM-GbE signal as a function of RMS voltage at DSO output, transmitted along the experimental setup of Fig. 2 in optical back-to-back when using DPD, DPostD or no BBNIM.
Fig. 8.
Fig. 8. EVM of OFDM-GbE signal (○) without BBNIM, (□) with DPD, (⋄) with DPostD and (◂) with PE, as a function of RMS voltage at DSO output, transmitted along the experimental setup of Fig. 2 with (a) 75 km, (b) 100 km and (c) 125 km of total fibre length.

Tables (3)

Tables Icon

Table 1. a-vectors, number of coefficients, C, and corresponding number of multiplications per sampling interval that are required to calculate the output samples of the best performing MP for DPD and DPostD in the experimental setup of Fig. 2 in optical BtB, as a function of peak and RMS amplitudes of the input signal (expressed as percentage of sM).

Tables Icon

Table 2. Optimum EVM and maximum EVM improvements, and corresponding RMS voltage at DSO input, with DPD and DPostD, for optical BtB.

Tables Icon

Table 3. a-vectors of the best performing MPs for DPD and DPostD in the experimental setup of Fig. 2 for total fibre lengths of 75, 100 and 125 km, as a function of sp.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

v [ n ] = p = 1 P q 1 = 0 a p q p = 0 a p h p ( q 1 , q 2 , , q p ) m = 1 p u [ n q m ]
u [ n ] = MP ( w , v [ n ] ) = p = 1 P q 1 = 0 a p q 2 = 0 q 1 q p = 0 q p 1 w p ( q 1 , q 2 , , q p ) m = 1 p v [ n q m ] .
C p = m p = 1 a p m p 1 = 1 m p m 1 = 1 m 2 1 , 1 p P
MP ( w , v [ n ] ) = q 1 = 0 37 w 1 ( q 1 ) v [ n q 1 ] + q 1 = 0 12 q 2 = 0 q 1 q 3 = 0 q 2 w 3 ( q 1 q 2 , q 3 ) v [ n q 1 ] v [ n q 2 ] v [ n q 3 ] .
V = [ V 1 C 1 V 2 C 2 V P C P ] } N s
V p = [ ( v [ 0 ] ) p ( v [ 1 ] ) ( v [ 0 ] ) p 1 ( v [ a p ] ) p 1 v [ 1 a p ] ( v [ a p ] ) p ( v [ 1 ] ) p ( v [ 0 ] ) ( v [ 1 ] ) p 1 ( v [ 1 a p ] ) p 1 v [ 2 a p ] ( v [ 1 a p ] ) p ( v [ N s 1 ] ) p ( v [ N s a p ] ) p ] .
w = ( V H V ) 1 V H u ,
w i + 1 = w i + μ ( V H V ) 1 V H ( u V w i ) ,

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