Modeling and compensation of transmitter nonlinearity in coherent optical OFDM

Siamak Amiralizadeh; An T. Nguyen; Leslie A. Rusch

doi:10.1364/OE.23.026192

1. Introduction

Orthogonal frequency-division multiplexing (OFDM) is a flexible modulation technique that has had wide adoption in wireline (telephone and cable) and wireless communications standards [1–3]. As optical communications move into a new era where spectral efficiency is of increased importance, OFDM is undergoing extensive examination as a key enabler. Robustness to channel dispersion and its flexible structure make optical OFDM attractive for both long-haul and short-reach applications [4–9].

In OFDM systems, an inverse fast Fourier transform (IFFT) is used to modulate the data onto orthogonal subcarriers for transmission. Some data sequences lead to a large number of subcarriers adding together with coherent phase, and this leads to high peaks in the OFDM signal. This is known as high peak-to-average power ratio (PAPR) and is a major challenge in OFDM systems. PAPR is especially problematic at the transmitter where OFDM signals are more susceptible to nonlinear distortions arising in the digital-to-analog converter (DAC), the electrical power amplifier (PA) and the optical modulator [10, 11].

Various schemes have been proposed and demonstrated to resolve the PAPR problem in OFDM systems including clipping, nonlinear companding, selective mapping (SLM), trellis shaping, predistortion and many other techniques with different efficiency and computational complexity [12, 13]. Recently, some of these approaches originally developed for wireless and wireline have been applied in optical OFDM systems. In [14], a Zadoff–Chu sequence is utilized to build a precoding matrix applied before the IFFT at the transmitter to reduce PAPR. Trellis shaping is used to reduce PAPR in [15]. With the SLM technique, 1 dB Q factor improvement is obtained in a quadrature phase-shift keying (QPSK) coherent optical OFDM (COOFDM) system when optical fiber is the only nonlinearity source [16].

Predistortion is deployed extensively to compensate for transmitter nonlinearities. In radio frequency (RF) OFDM, the main focus is usually on electrical PA nonlinearity [17–20]. In optical OFDM, on the other hand, predistortion is applied to mitigate nonlinearity of electrical-to-optical conversion block [21–25]. In [21–23], predistortion is based on inverting the sinusoidal transfer function of the MZM. An adaptive digital predistorter is utilized in [24] to compensate for optical modulator nonlinearity. Polynomials are used to estimate the nonlinear transfer function of optical modulator. A frequency-domain predistorter is used to compensate for nonlinearity of direct-detection optical OFDM systems in [25].

While the previous PAPR studies have focused on one or two sources of nonlinearity, we address the distortions as they appear along the transmitter RF front-end, from DAC to amplifier to modulator [26]. In this paper, we use combination of clipping and predistortion to overcome transmitter nonlinearity aggravated by high PAPR. We propose a simple strategy to characterize the nonlinear transfer function of the CO-OFDM transmitter; the nonlinearity estimate is used to calculate the correct predistortion to apply. We show that transmitter nonlinear response approximated by piecewise linear interpolation (PLI) leads to effective predistortion. We derive a semi-analytical solution for bit error rate (BER) that validates (via Monte Carlo techniques) the PLI approximation accurately captures transmitter nonlinearity. Note that although we use predistortion to suppress transmitter nonlinearities, in the characterization step, the estimated transfer function captures nonlinear effects of the whole back-to-back (B2B) system. Therefore, once system nonlinear response is accurately estimated, predistortion mitigates distortions regardless of the source of nonlinearity. Although fiber nonlinearity is exacerbated by high PAPR, compensation of fiber nonlinear effect requires complex methods and can be implemented independent of transmitter compensation [27]. We neglect fiber nonlinearity assuming a low launched power scenario in this work.

The paper is organized as follows. In section 2, we give a brief review of PAPR in OFDM. We describe the models used in our study and introduce parameters varied in simulations in section 3. The impact of transmitter nonlinearity on performance of optical OFDM is investigated in section 4, elaborating on our results reported in [26]. In section 5, we analyze the distribution of OFDM symbol errors with respect to PAPR. Based on this information, we establish our strategy to tackle transmitter nonlinear distortion in the presence of high PAPR. We provide a semi-analytical solution to estimate CO-OFDM system performance in the presence of nonlinearity. We also explain our proposed transmitter characterization method based on training sequences and the PLI approximation. In section 6, we examine the performance of the proposed method via Monte Carlo simulations considering different transmitter nonlinear characteristics and compare the results with theoretical predictions. Finally, we draw conclusions in section 7.

2. PAPR in OFDM

The baseband OFDM signal after IFFT at the transmitter is given by

x (t) = \frac{1}{\sqrt{N}} \sum_{k = 0}^{N - 1} X_{k} exp (j 2 π f_{k} t), 0 \leq t < T_{s},

where N is the FFT size, X_k are the complex data modulating subcarriers, f_k = k/T_s is the center frequency for k^th subcarrier and T_s is the period of the OFDM symbol.

Assuming that the input data are statistically independent and identically distributed (i.i.d.), the real and imaginary part of x (t) are orthogonal and uncorrelated. The real part ℜ{x(t)} and imaginary part ℑ{x(t)} each has a Gaussian distribution with zero mean and variance of $\frac{1}{2} E [{| X_{k} |}^{2}]$ for large N (N > 64) based on the central limit theorem. Accordingly, the instantaneous signal power has a central chi-squared distribution with two degrees of freedom; its probability density function (PDF) is strictly decreasing with a maximum at zero. We will exploit this fact to justify our suggested mitigation approach in section 5.

PAPR of an OFDM symbol is defined as the ratio between maximum instantaneous power and average power, given by

PAPR = \frac{max_{0 \leq t < T_{s}} [{| x (t) |}^{2}]}{\frac{1}{T_{s}} \int_{0}^{T_{s}} {| x (t) |}^{2} d t} .

Assuming that OFDM symbols have constant average power, PAPR is essentially determined by the instantaneous power which varies with the data sequence transmitted. For example, if subcarriers add together coherently, large peaks will be observed in the OFDM signal envelope leading to high PAPR. Furthermore, PAPR increases with increasing N. The number of symbols in the constellation used to transmit multiple bits per subcarrier has negligible effect on PAPR statistics [28]. Throughout this paper, we use FFT size of 256 and investigate 16-ary quadrature amplitude modulation (16-QAM) signaling in simulations.

3. System model

In this section, we explain the system model employed in simulations and analysis. We particularly focus on nonlinearity sources at the transmitter and introduce three different parameters capturing the distortion contribution of each component. We vary these parameters to quantify the performance degradation in section 4.

Figure 1 shows the block diagram for an M-QAM CO-OFDM system. Pseudo random binary sequences (PRBSs) with length of 2²¹ − 1 are used to generate 16-QAM symbols. IFFT and cyclic prefix addition are two key blocks in OFDM systems. An IFFT is used to modulate data onto orthogonal subcarriers. A cyclic prefix is appended to the time domain OFDM signal to facilitate channel equalization at the receiver. In simulations, the IFFT size is 256 of which 196 data-bearing subcarriers are available in each OFDM symbol. The remaining 60 subcarriers are left empty to achieve an oversampling factor of R_os = 1.3. Oversampling reduces the impact of aliasing at the DAC [6]. Eight samples are added to each OFDM symbol as a cyclic prefix, leading to 9.43 ns symbol duration with 28 GS/s sampling rate and a bit rate of 83.1 Gb/s.

Fig. 1 M-QAM CO-OFDM system block diagram. SMF: single-mode fiber, S/P: serial-to-parallel, P/S: parallel-to-serial.

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The OFDM signal is clipped at the transmitter to avoid large peaks and decrease quantization noise induced by the DAC. Normalized clipping level, the so-called clipping ratio (CR) defined as the ratio between clipping level and rms power of the OFDM signal, is used to quantify clipping [29]. A predistortion block is also included in the OFDM transmitter shown in Fig. 1 which will be explained in section 5. After digital-to-analog conversion, the in-phase/quadrature (I/Q) signals are amplified with an RF amplifier and drive an I/Q optical modulator biased at the null point and modulating an ideal continuous-wave (CW) laser. Amplified spontaneous emission (ASE) noise is added to the signal in the channel before detection. The signal is coherently detected by a local oscillator (LO). After analog-to-digital-conversion, the signals are processed at the OFDM receiver to recover the data bits and calculate BER. At least 200 errors are detected for each point to accurately estimate BER. The DAC, electrical PA and optical modulator each play an important role in the nonlinear behaviour of the transmitter and they are explained in more detail.

3.1. DAC

When OFDM signals are converted from digital to analog domain, they experience quantization noise due to finite DAC resolution. In a DAC with k bits resolution, the closest of 2^k levels is assigned to each input sample. As the maximum and minimum levels of the DAC are determined by the input signal amplitude range, OFDM signals with high PAPR have wider excursions which must span the same DAC levels, and thus suffer from excessive quantization noise. In our simulations, we find the maximum (minimum) amplitude of an OFDM frame consisting of 1000 OFDM symbols and assign it to the highest (lowest) level of quantization. The remaining 2^k − 2 levels are uniformly distributed between the minimum and maximum levels. After quantization we upsample the signal by a factor of 4 and apply a fourth-order super-Gaussian low-pass filter (LPF). This model has enough accuracy to simulate most DAC effects [29]. We find it sufficient in our study, without need of a more comprehensive model based on effective number of bits (ENOB) that includes integral nonlinearity (INL) and differential nonlinearity (DNL) of the DAC [30].

3.2. Electrical PA

Modeling the nonlinear response of PAs can take many forms. In this work, we use a simple SSPA model in which the amplifier is viewed as a memoryless nonlinearity [28]. In the SSPA model, AM/PM conversion is assumed to be negligible and AM/AM conversion is given by

g_{SSPA} (A (t)) = \frac{g_{0} A (t)}{{[1 + {(\frac{A (t)}{A_{sat}})}^{2 p}]}^{\frac{1}{2 p}}},

where A(t) is the input signal amplitude, g₀ is the amplifier gain, A_sat is the input saturation amplitude and p adjusts AM/AM saturation sharpness. We take p = 2 as a good approximation of available commercial amplifiers [3]. For p = 2, the input signal experiences 1.5 dB gain compression when it is at the saturation level. The gain g_SSPA (A(t)) can also be cast from Eq. (3) as

g_{SSPA} (A (t)) = \frac{g_{0} \sqrt{P_{sat}}}{{[1 + {(\frac{P_{sat}}{P_{in} (t)})}^{p}]}^{\frac{1}{2 p}}},

where P_in (t) is input power and

P_{sat} = A_{sat}^{2}

. In order to quantify nonlinear distortion induced by PA, we define input backoff (IBO) as the ratio between input saturation and average power, i.e., IBO = P_sat/P_avg. While higher IBO gives a more linear response, it leads to an inefficient PA. High bandwidth communications systems require significant trade-off of efficiency and distortion, as wideband linear amplifiers significantly increase system cost.

3.3. Optical modulator

Depending on the structure of subcarriers and the OFDM transmitter, an I/Q modulator or a single external modulator can be deployed for electrical-to-optical conversion in CO-OFDM. Simulations presented in this paper assume an I/Q modulator with symmetric nonlinear response for I and Q branches, i.e., no I/Q imbalance. We expect similar behaviour in terms of transmitter nonlinearity for CO-OFDM systems using either I/Q or a single modulator.

In this study, a Mach-Zehnder modulator (MZM) is assumed in both simulations and analysis. The transfer function of an MZM is illustrated in Fig. 2. In CO-OFDM systems, the modulator is biased at the null point and the driving signal amplitude is kept between ±V_π where V_π is half-wave voltage of MZM. We adjust maximum (minimum) of the RF OFDM signal to V_π (−V_π) to ensure clipping is excluded from the modulator induced nonlinearities. To sweep the MZM nonlinear effect, we multiply the driving signal amplitude, V_d (t), by α_d (α_d ≤ 1) considering some backoff in the MZM operating range. In this case, the output optical field of the MZM, E_out (t), can be expressed as

E_{out} (t) = E_{in} (t) sin (\frac{π α_{d} V_{d} (t)}{2 V_{π}}),

where E_in (t) is the input optical field to be modulated. Smaller α_d leads to better nonlinear performance, but comes with the cost of optical power inefficiency. In other words, as α_d decreases, the impact of optical modulator nonlinearity on the signal decreases; however, the output signal will be more attenuated due to higher insertion loss of the optical modulator.

Fig. 2 Transfer function of MZM biased at null point for CO-OFDM.

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Considering combination of the nonlinearity from amplifier and optical modulator, the overall normalized nonlinear function of the transmitter is given by

g_{TX} (A) = sin (\frac{π α_{d}}{2 V_{π}} \frac{g_{SSPA} (A)}{g_{SSPA} (A_{\max})}),

where A_max is the maximum input amplitude. Notice that the transmitter nonlinear function has a symmetry to the origin, i.e., it is an odd function. Therefore, for convenience we assume that g_TX (A) acts on magnitude alone.

4. Impact of transmitter nonlinearity on OFDM performance

To investigate the impact of transmitter nonlinear distortions on performance of optical OFDM, we vary the parameters introduced in section 3 for DAC, electrical PA and MZM for a 16-QAM CO-OFDM system—similar to our previous study for QPSK modulation format [26]. One thousand OFDM symbols (called one OFDM frame in the remainder of the paper) are transmitted in Monte Carlo simulations presented in this section.

Figures 3(a)–3(c) show error vector magnitude (EVM) versus IBO for a DAC with respectively 4, 5 and 6 bits resolution. Intentionally clipping the signal at the transmitter can suppress quantization noise. CR is swept in Monte Carlo simulations to find the clipping level offering lowest EVM for each DAC resolution. The CRs are adjusted at 1.9, 2.1 and 2.25 for 4-, 5- and 6-bit DAC, respectively. The performance improvement obtained is significant when a lower resolution DAC is employed; however, the improvement gradually disappears as higher resolution DACs are used, compare Fig. 3(a) with Fig. 3(c). In addition, clipping is not effective in mitigating the nonlinearity from electrical PA and MZM. This can be explained by the difference in nature of DAC quantization noise and the nonlinearity induced by electrical PA and MZM. High peaks in the OFDM signal lead to larger separation of DAC quantization levels, which exacerbates quantization noise for all samples regardless of their amplitude. Therefore, limiting PAPR of the OFDM signal improves overall performance while sacrificing accuracy of high amplitude samples. On the other hand, performance degradation in electrical PA and MZM is due solely to compression of relatively high-amplitude samples; low-amplitude samples remain largely unaffected. As a result, while clipping is effective to reduce DAC quantization noise, other mitigation techniques should be utilized to overcome nonlinearity induced by electrical PA and MZM (the same applies to other DAC nonlinearities such as INL and DNL).

Fig. 3 EVM versus electrical PA IBO for 16-QAM CO-OFDM system with (a) 4-, (b) 5-and (c) 6-bit DAC; dashed red line: without clipping, solid blue line: with clipping. BER of 3.8 × 10⁻³ is considered as forward error correction (FEC) limit.

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Figure 3 also indicates that when IBO is higher than a certain value, the amplifier response is very linear and no improvement in EVM can be achieved by increasing IBO. Driving the modulator across the full range considerably deteriorates OFDM performance. For example, when IBO is equal to 4 dB, α_d = 0.6 and α_d = 1 give respectively 2% and 5.4% EVM degradation with respect to the case where an ideal optical modulator is used. This suggests the importance of adjusting MZM driving voltage; by slightly reducing driving OFDM signal amplitude range, performance can be enhanced significantly. We emphasize that use of power backoff for electrical PA and modulator reduces power efficiency. Therefore, although higher backoff leads to linear transmitter response, it decreases the output optical power which makes it more susceptible to noise and can lead to performance degradation. This effect is not considered in Fig. 3. The impact of α_d on modulation efficiency of optical modulator will be discussed in section 6.

5. Transmitter nonlinearity characterization

The PAPR of an OFDM signal directly affects its performance in the presence of nonlinear distortions. In [16], Goebel et al. showed there is a correlation between OFDM symbol PAPR and Q factor; OFDM symbols with higher PAPR have lower Q factor. As discussed in section 2, the PDF of instantaneous signal power is strictly decreasing. Thus, OFDM symbols with high PAPR are less likely to occur. To motivate our proposed predistortion strategy we will examine the distribution of the bit errors as a function of PAPR. This distribution will be indexed by the IBO to assess the impact of the PA.

5.1. Bit error distribution

Figure 4(a) displays error distribution versus PAPR after clipping for one OFDM frame with different clipping levels when DAC resolution is 4 bits, IBO is equal to 6 dB and the MZM is driven across the full range, i.e., α_d = 1. Random binary data are used to generate 16-QAM symbols. Errors are densely aggregated in PAPR range between 6 and 10 dB when no clipping is applied. PAPR higher than 10 dB leads to poor performance, but PAPR this high appears so rarely that this PAPR range causes few errors.

Fig. 4 (a) Bit error distribution for IBO = 6 dB, α_d = 1 and 4-bit DAC as a function of PAPR for one OFDM frame (1000 OFDM symbols) with different clipping levels. (b) Normalized gain versus normalized input amplitude of the electrical PA. Rectangles show the range within one standard deviation of the mean of PAPR for different clipping levels. Star represents the PA input voltage amplitude equal to the average amplitude when IBO = 0 dB. A video is available for this plot when sweeping IBO ( Visualization 1).

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Clipping decreases PAPR and moves the error distribution toward lower PAPR ranges. By reducing CR, the number of errors decreases while the distribution moves further to left. Note that clipping the signal decreases quantization noise for all samples. Therefore, the reduction in number of errors occurs for all PAPR ranges. As clipping itself induces distortion, despite lower PAPR of the OFDM signal, total number of errors increases by further decreasing CR from the optimum value of 1.9. We verified that the distribution of the number of OFDM symbols versus PAPR is similar to the error distribution, suggesting that the performance of the OFDM system is mainly determined by PAPRs with greater probability, i.e., those in the modal region.

Figure 4(b) shows normalized gain versus the input amplitude normalized by the saturation amplitude of the PA. The operating point of the amplifier is determined by the IBO; with high IBO, the average of input amplitudes is in the linear region. A star indicates average amplitude level when IBO is equal to 0 dB in Fig. 4(b). The gray shading denotes the region where gain compression is more than 3 dB. The rectangles represent the modal region of the PAPR probability density—the range within one standard deviation of the mean value—for different CRs. This range depends on both PAPR and IBO as

\frac{A_{\max}}{A_{sat}} = \sqrt{\frac{PAPR}{IBO}} .

In order to avoid gain compression more than 3 dB at the electrical PA, the ratio between maximum input amplitude and saturation amplitude should be less than 1.3. Equivalently, the difference between PAPR and IBO should be less than 2.3 dB. With IBO as low as 0 dB, it is not possible to avoid distortion at the PA by utilizing common PAPR reduction methods. For example, by using the SLM technique with reasonable complexity and overhead, the PAPR would still be more than 6 dB (see Fig. 1 in [12]). Therefore, after initial PAPR reduction, other methods must be employed to mitigate the distortions from the electrical PA and MZM.

Figure 5 depicts the distribution of errors as a function of PAPR for various IBOs and CR equal to 1.9. We observed less than 0.1 dB variation in the mean of PAPR. The distributions shown in Fig. 4 are relatively similar for different amplifier IBOs. Simulations also showed that with a higher DAC bit resolution less improvement is observed by clipping; however, the OFDM signal experiences roughly the same nonlinear distortion at the electrical PA and MZM as the 4-bit DAC case.

Fig. 5 Error distribution of one OFDM frame (1000 OFDM symbols) versus PAPR for different electrical PA IBOs when α_d = 1 and 4-bit DAC is used. The number of errors is averaged over 200 OFDM frames.

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Various techniques have been introduced in the literature to decrease PAPR of the OFDM signal. However, these methods are not very effective when significant nonlinear distortion is imposed by the electrical PA and modulator in the OFDM transmitter. The block diagram in Fig. 6 shows the advantage of employing predistortion to mitigation strategies. After decreasing PAPR with common PAPR reduction techniques, high power backoff should be considered for the input OFDM signal to the optical modulator and electrical PA to avoid severe nonlinear distortion. Predistortion improves power efficiency of the OFDM transmitter significantly by alleviating the requirement for high power backoff. Our examination of error distributions as a function of IBO leads us to propose the second strategy with predistortion.

Fig. 6 Comparison of power backoff requirement for the transmitter with and without predistortion.

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In order to implement the predistorter, the nonlinear response of the transmitter should be characterized. In the next section, we describe a simple technique for characterizing the nonlinear response of the transmitter. We provide theoretical analysis that enables us to evaluate performance of the system in presence of nonlinearity and accuracy of the nonlinear function estimation.

5.2. Evaluation of BER

Development of a theoretical equation to assess system performance in the presence of nonlinearity not only allows us to see the impact of nonlinear distortions on the system, but also gives us a tool to investigate the accuracy of the estimated predistortion function. Figure 7 summarizes the models used for the theoretical analysis. The DAC quantization noise and clipping is modeled as additive Gaussian noise [29]. With the assumption of a Gaussian distribution for real and imaginary parts of the OFDM signal, and by exploiting Bussgangs theorem, it has been shown previously that the memoryless nonlinearities in OFDM can be modeled by a constant factor and an additive noise term with zero mean [31, 32]. We use the same approach in modeling the electrical PA and MZM nonlinearity. Finally, additive white Gaussian noise (AWGN) is added to the signal as ASE noise.

Fig. 7 Block diagram of the models used for theoretical analysis. Each noise source has impact on total SNR and degrades signal quality.

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Let z(t), x(t) and g(·) be the output signal, input signal and nonlinear function of the OFDM transmitter where z = g(x). The output of the nonlinear block can be written as [18, 31–33]

z (t) = κ x (t) + y (t),

where κ is a constant factor and y(t) is an additive zero-mean noise component uncorrelated with the input signal, i.e., E[x(t)y(t)] = 0. The attenuation factor κ and variance of y(t),

σ_{y}^{2}

, can be written as

κ = \frac{E [x (t) z (t)]}{σ_{x}^{2}} = \frac{1}{σ_{x}^{2}} \int_{- \infty}^{\infty} x g (x) \frac{1}{\sqrt{2 π σ_{x}^{2}}} exp (- \frac{x^{2}}{2 σ_{x}^{2}}) d x,

σ_{y}^{2} = E [z {(t)}^{2}] - κ^{2} σ_{x}^{2} = \int_{- \infty}^{\infty} g^{2} (x) \frac{1}{\sqrt{2 x σ_{x}^{2}}} exp (- \frac{x^{2}}{2 σ_{x}^{2}}) d x - κ^{2} σ_{x}^{2},

where

σ_{x}^{2}

is the input signal variance. Although y(t) is not a Gaussian process, after the FFT block at the receiver, the noise component can be approximated as AWGN with variance of

σ_{y}^{2}

for large N due to central limit theorem arguments [32].

In [29], Berger et al. studied the distortions induced by clipping and quantization effect of DAC and showed that the Gaussian approximation leads to accurate BER estimation for BER above 10⁻⁴. In this work, we employ the presented model for the DAC, and calculate the DAC noise variance, $σ_{DAC}^{2}$ , by Eq. (31) in [29]. Hence, the signal-to-noise ratio (SNR) at the receiver is given by

SNR = R_{os} \frac{κ^{2} σ_{x}^{2}}{κ^{2} σ_{DAC}^{2} + σ_{y}^{2} + σ_{AWGN}^{2}},

where

σ_{AWGN}^{2}

denotes variance of optical channel noise due to ASE and R_os is the oversampling factor.

The SNR at the receiver can also be written as a function of SNR after each noise source. The SNR at the output of each block is given in Fig. 7. Each noise source has an influence on the SNR and degrades signal quality. Using the SNR definition at the output of each noise source, the SNR at the receiver can be written as

SNR \approx R_{os} \frac{{SNR}_{DAC} {SNR}_{NL} {SNR}_{AWGN}}{{SNR}_{DAC} + {SNR}_{DAC} {SNR}_{NL} + {SNR}_{DAC} {SNR}_{AWGN} + {SNR}_{NL} {SNR}_{AWGN}} .

The approximation is made by assuming the total signal power remains the same after clipping and DAC quantization, i.e.,

σ_{s}^{2} \approx σ_{x}^{2}

. Consequently, the bit error probability for M-QAM with Gray coding is given by

BER = \frac{2}{{log}_{2} M} \frac{\sqrt{M} - 1}{\sqrt{M}} erfc (\sqrt{\frac{3}{2 (M - 1)} SNR}) .

In order to calculate κ and $σ_{y}^{2}$ for the optical OFDM transmitter model, the integrals should be evaluated for the nonlinear function given by Eq. (6). Derivation of a closed-form equation for κ and $σ_{y}^{2}$ is usually a tedious task and not always possible. Therefore, we approximate the nonlinear function by PLI to obtain a simplified solution.

5.3. PLI for transmitter nonlinearity characterization and predistortion

Although different models are available to study nonlinear effects in optical OFDM transmitters, it is challenging to find the nonlinear response of a real system. The nonlinear function of the transmitter can be approximated with desired accuracy in small intervals by using polynomials [33]. This simplifies calculation of the integrals in Eq. (9) and Eq. (10). In this section, we take the same approach and approximate the nonlinear response of the transmitter by PLI (first-order polynomial) and show that linear interpolation is sufficient for accurate BER estimation. We also describe a training symbol-based method to extract the PLI approximation of the transmitter nonlinear function.

Let ĝ(x) be PLI of g(x) with m + 1 intervals given by

\hat{g} (x) = {\begin{array}{l} a_{i} x + b_{i}, & x_{i} \leq x < x_{i + 1} \\ g (x_{m + 1}), & x_{m + 1} \leq x, \end{array}

i = 1,...,m, where a_i is slope of the line in the i^th interval from x_i to x_i+1, b_i = g(x_i) − a_ix_i, x₁ = 0 and x_m+1 is the maximum amplitude of the signal at the output of the clipping block. By using ĝ(x) in Eq. (9) and Eq. (10), it is straightforward to obtain κ and

σ_{y}^{2}

as

κ = \frac{2}{σ_{x}^{2}} \sum_{i = 1}^{m} [a_{i} ψ_{2} (x_{i}, x_{i + 1}) + b_{i} ψ_{1} (x_{i}, x_{i + 1})] + \frac{2}{σ_{x}^{2}} g (x_{m + 1}) ψ_{1} (x_{m + 1}, \infty),

\begin{matrix} σ_{y}^{2} = 2 \sum_{i = 1}^{m} [a_{i}^{2} ψ_{2} (x_{i}, x_{i + 1}) + 2 a_{i} b_{i} ψ_{1} (x_{i}, x_{i + 1}) + b_{i}^{2} ψ_{0} (x_{i}, x_{i + 1})] \\ + 2 g^{2} (x_{m + 1}) ψ_{0} (x_{m + 1}, \infty) - κ^{2} σ_{x}^{2}, \end{matrix}

where ψ_k(v, w) is given in the appendix. The derived equations for κ and

σ_{y}^{2}

yields exact values as m goes to infinity. We found that m = 15 is accurate enough for estimation of BERs above 10⁻⁵.

The nonlinear response of the optical OFDM transmitter can be extracted with high accuracy if its behaviour is approximated by straight lines in sufficiently small input intervals. In order to characterize transmitter nonlinearity, we transmit training sequences and then compare the received instantaneous samples to the transmitted ones. For each amplitude level of the I and Q signals, a unique value can be found as the amplitude compression. If m intervals are considered between 0 and the maximum amplitude of transmitted samples, m points can be obtained by comparing the received and transmitted signals.

Let x_I [n] and x_Q [n] be the real and imaginary part of the OFDM signal after clipping, respectively. Assuming that the nonlinear distortion is the only impairment in the system, the OFDM signal after analog-to-digital converter (ADC) at the receiver is given by

r [n] = g_{I} (x_{I} [n]) + j g_{Q} (x_{Q} [n]),

where g_I (·) and g_Q (·) are the AM/AM characteristics of the channel for the I and Q arm, respectively. Let I_i,I and I_i,Q be

I_{i, I} = {x_{I} [n] | \frac{(i - \frac{1}{2}) A_{I, \max}}{m} \leq x_{I} [n] < \frac{(i + \frac{1}{2}) A_{I, \max}}{m}},

I_{i, Q} = {x_{Q} [n] | \frac{(i - \frac{1}{2}) A_{Q, \max}}{m} \leq x_{Q} [n] < \frac{(i + \frac{1}{2}) A_{Q, \max}}{m}},

i = 1,...,m, where A_I,max and A_Q,max are the maximum amplitude of the I and Q signal, respectively. The definition of these sets assures that all samples map into one of the prescribed intervals. Each sample experiences a certain amount of compression due to the nonlinear response of the transmitter. The compression factor is equal to the ratio between output and input signal amplitude. For each of the defined m intervals in I_i,I and I_i,Q, the average compression factors of the i^th interval, c_I(i) and c_Q(i), can be obtained respectively by

c_{I} (i) = \frac{\sum_{x_{I} [n] \in I_{i, I}} \frac{g_{I} (x_{I} [n])}{x_{I} [n]}}{| I_{i, I} |}, c_{Q} (i) = \frac{\sum_{x_{Q} [n] \in I_{i, Q}} \frac{g_{Q} (x_{Q} [n])}{x_{Q} [n]}}{| I_{i, Q} |},

where | · | denotes cardinality of the set. Averaging is done over numerous samples in each set to minimize impact of Gaussian noise on the estimated nonlinear response. Linear interpolation of the obtained m coordinates for each of I and Q arms gives the compression functions f_LI,I(x) and f_LI,Q(x) as

f_{LI, I} (x) = {\begin{array}{l} \frac{m [c_{I} (i) - c_{I} (i - 1)]}{A_{I, \max}} (x - \frac{i A_{I, \max}}{m}) + c_{I} (i), & \frac{(i - 1) A_{I, \max}}{m} \leq x < \frac{i A_{I, \max}}{m}, \\ c_{I} (m), & A_{I, \max} \leq x, \end{array}

f_{LI, Q} (x) = {\begin{array}{l} \frac{m [c_{Q} (i) - c_{Q} (i - 1)]}{A_{Q, \max}} (x - \frac{i A_{Q, \max}}{m}) + c_{Q} (i), & \frac{(i - 1) A_{Q, \max}}{m} \leq x < \frac{i A_{Q, \max}}{m}, \\ c_{Q} (m), & A_{Q, \max} \leq x, \end{array}

where c_I (0) = c_Q (0) = 0. The output of the nonlinear block is equal to the input amplitude multiplied by the compression factor. Therefore, the PLI approximation of the nonlinear response of the transmitter for I and Q signal is respectively given by

{\hat{g}}_{I} (x) = x f_{LI, I} (x) and {\hat{g}}_{Q} (x) = x f_{LI, Q} (x) .

The estimated transmitter nonlinear functions can be used to evaluate the system performance or pre-compensate for the induced nonlinear distortions in the system. In the latter case, inverse of the estimated functions, i.e.,

{\hat{g}}_{I}^{- 1} (x)

and

{\hat{g}}_{Q}^{- 1} (x)

, should be applied before DAC at the transmitter. While Eq. (6) assumes both I and Q arms experience the same nonlinearity g_TX, for the PLI we estimate the nonlinearity separately in the I and Q arms.

We note that inverse of the nonlinear response can also be applied at the receiver with more flexibility. However, the main problem with post-compensation is increased complexity of the receiver side signal processing as compared to pre-compensation where complexity increase is moderate. The envelope of the OFDM signal varies during transmission due to fiber dispersion. Therefore, the post-compensation block must be applied after frequency-domain equalizer imposing an extra FFT/IFFT operation.

6. Simulations

In this section we first validate the accuracy of the BER theoretical expression in Eq. (13). We next simulate the performance of our PLI-based predistortion when varying IBO and DAC resolution. We also show the impact of optical modulator input signal backoff on its power efficiency.

Monte Carlo simulations are run with the nonlinearity generated per Eq. (6) for various IBOs, and errors are counted to estimate BER. The theoretical expression for BER is evaluated under two scenarios: 1) with side information of the transmitter nonlinearity, Eq. (6), and using numerical integration of Eq. (9) and Eq. (10), and 2) with the PLI estimate of the nonlinear function, obtained from Monte Carlo simulations, using Eq. (15) and Eq. (16). This validation serves two purposes. Firstly they establish that the approximations made in developing the theoretical expression are valid, and secondly that the PLI estimate of the nonlinear function is a good predictor of BER performance. The better the PLI estimate, the better will be the performance of our pre-compensation based on this estimate.

In Fig. 8(a) the BER versus electrical PA IBO is plotted where solid lines refer to BER estimate with a Gaussian assumption and side information on nonlinearity, dashed lines with triangle markers refer to BER estimate with a Gaussian assumption and a PLI estimate of nonlinearity, and circle markers refer to BER estimate from Monte Carlo error counting. For PLI, we first transmit 40 OFDM symbols as training signals and estimate the transmitter nonlinear function by linear interpolation with m = 15. Afterwards, we extract a_i and b_i and calculate BER using Eq. (15) and Eq. (16). For an ideal DAC, the estimated nonlinear function is very accurate; both approaches give the same BER. However, when a 4-bit DAC is used PLI approach underestimates BER. This is expected since with a low-bit DAC, there is not enough resolution to accurately estimate the nonlinear function of the transmitter. We also note the good agreement between Monte Carlo simulations and theoretical BER calculation.

Fig. 8 (a) BER versus IBO for 16-QAM CO-OFDM. Solid lines refer to BER estimate with side information on nonlinearity, dashed lines with triangle markers refer to BER estimate with a PLI estimate of nonlinearity, and circle markers refer to BER estimate from Monte Carlo error counting. Black: ideal DAC, α_d = 0.7; blue: ideal DAC, α_d = 1; red: 4-bit DAC with CR = 1.9 and α_d = 1. The ASE noise power is assumed to be zero. The DAC noise variance is calculated by Eq. (31) in [29] and is equal to $0.013 σ_{x}^{2}$ . (b) Power attenuation at I/Q modulator due to input signal backoff. Solid lines: κ² obtained with side information on nonlinearity; markers: Monte Carlo simulations.

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In Fig. 8(b), we present the power attenuation due to input signal backoff in optical modulator when DAC and electrical PA are ideal. Although decreasing α_d solves the optical modulator nonlinearity problem, it reduces the power efficiency of the modulator. Therefore, it is essential to use predistortion to suppress nonlinearity while driving the optical modulator in a power efficient regime. We also observe that modulation efficiency increases by decreasing CR; however, if the OFDM signal is over-clipped, the clipping noise will degrade performance. Simulations are in agreement with the theoretical predictions. We notice that the accuracy of theoretical calculations decrease when CR decreases. For low CRs, the assumption of Gaussian distribution for the OFDM signal becomes less accurate. Therefore, theoretical predictions have low accuracy in this regime.

Figures 9(a) and 9(b) show BER versus received SNR, for 4-bit and 5-bit DAC, respectively. Solid lines refer to BER estimate with side information on nonlinearity and markers show Monte Carlo simulation results. When 4-bit DAC is used, transmitter nonlinear function cannot be estimated with sufficient accuracy. In addition, applying predistortion increases PAPR of the DAC input signal which in turn leads to more quantization noise from DAC. Therefore, predistortion gives negligible performance improvement due to inadequate bit resolution. For 5-bit DAC, the nonlinearity characterization with PLI is very accurate and predistortion gives significant improvement. BER is always above FEC threshold of 3.8 × 10⁻³ without predistortion. When predistortion is employed, received SNR of 18 dB is sufficient to achieve BER below FEC.

Fig. 9 BER versus received SNR for 16-QAM CO-OFDM system with (a) 4-bit DAC and CR = 1.9; (b) 5-bit DAC and CR = 2.1. Circles: α_d = 1, squares: α_d = 0.7.

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7. Conclusions

We studied the impact of high PAPR in an optical OFDM transmitter and proposed a simple predistortion scheme to enhance system performance taking into the account the entire transmitter chain of DAC, electrical PA and optical modulator. We used the statistical properties of the PAPR and the distinct nature of the quantization noise from electrical PA and MZM nonlinearity to propose a new predistortion strategy. We apply clipping along with digital predistortion to mitigate performance degradation due to high PAPR without incurring power efficiency degradation. Our simulations show when predistortion is used, BER below FEC threshold of 3.8 × 10⁻³ can be achieved with a 5-bit DAC, IBO equal to 2 dB and α_d = 1 for optical modulator in 16-QAM CO-OFDM. Clipping is known as the simplest PAPR mitigation method and predistortion can be implemented via a lookup table with minimal hardware complexity. While fiber nonlinearity was neglected in this work (assuming low launched powers to the fiber), residual nonlinearity from transmitter can interact with fiber nonlinearity influencing performance of the predistortion scheme. We leave fiber nonlinearity impact on predistortion for future study.

A Appendix

In this section, we introduce ψ_k(v, w) function and give the required formula to calculate κ and $σ_{y}^{2}$ . ψ_k(v, w) is defined as

ψ_{k} (v, w) = \frac{1}{\sqrt{2 π σ_{x}^{2}}} \int_{v}^{w} x^{k} exp (- \frac{x^{2}}{2 σ_{x}^{2}}) d x .

For k = 0, 1 and 2, ψ_k(v, w) is given by

ψ_{0} (v, w) = Q (\frac{v}{σ_{x}}) - Q (\frac{w}{σ_{x}}),

ψ_{1} (v, w) = \frac{σ_{x}}{\sqrt{2 π}} [exp (- \frac{v^{2}}{2 σ_{x}^{2}}) - exp (- \frac{w^{2}}{2 σ_{x}^{2}})],

ψ_{2} (v, w) = σ_{x}^{2} ψ_{0} (v, w) + \frac{σ_{x}}{\sqrt{2 π}} [v exp (- \frac{v^{2}}{2 σ_{x}^{2}}) - w exp (- \frac{w^{2}}{2 σ_{x}^{2}})],

where Q(x) is the tail probability of the standard normal distribution.

Acknowledgments

This work was supported by TELUS and NSERC under CRD grant 437041.

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Modeling and compensation of transmitter nonlinearity in coherent optical OFDM

Abstract

1. Introduction

2. PAPR in OFDM

3. System model

3.1. DAC

3.2. Electrical PA

3.3. Optical modulator

4. Impact of transmitter nonlinearity on OFDM performance

5. Transmitter nonlinearity characterization

5.1. Bit error distribution

5.2. Evaluation of BER

5.3. PLI for transmitter nonlinearity characterization and predistortion

6. Simulations

7. Conclusions

A Appendix

Acknowledgments

References and links

Supplementary Material (1)

Cited By

Figures (9)

Equations (27)

Optics Express