Iterative Symbol Recovery For Power Efficient DC Biased Optical OFDM Systems

Although orthogonal frequency division multiplexing (OFDM) has been known for its advantages in quite a while [1, 2, 3], only in recent years it has turned into a major field of interest in optical communications [4, 5]. Particularly attractive is the intensity-modulation direct-detection (IM/DD) scheme, which, owing to its simple implementation and low cost, constitutes one of the leading candidates for inter-data-center communications. For such systems, the data is modulated onto an electrical signal, $x(t)$, which can be either voltage or current, depending on the details of the electrical system, and an optical intensity modulator generates an optical signal with intensity $\alpha x(t)$, where $\alpha$ is a real-valued positive coefficient. We emphasize that the light’s intensity, and not its amplitude, is proportional to the electrical signal. Therefore, $x(t)$ must be a real-valued non-negative signal $x(t)\in\Re_{\text{\tiny{$+$}}}\ \forall t$, which generally does not hold for an unconstrained baseband OFDM signal. In order to generate a real-valued baseband OFDM signal, the vector of symbols of length $N$ (derived from a given constellation scheme), representing the bits, must meet the constraint of Hermitian symmetry, thereby sacrificing half of the allocated bandwidth. After generating the real-valued signal, further modification is needed so as to meet the non-negativity condition. To this end, two primary methods have been proposed: asymmetrically clipped OFDM (ACO-OFDM) [6, 7] and dc-biased optical OFDM (DCO-OFDM) [8, 9]. ACO-OFDM is a method suggested by Armstrong and Lowery [6] where the OFDM signal is generated such that only the odd frequency subcarriers carry the information (i.e. symbols from a given constellation), and the even frequency subcarriers are set to zero. After the IFFT is applied, the signal is clipped at the zero level. In this form, all of the clipping noise falls on the even subcarriers, and the data carrying odd subcarriers remain undistorted. This method has been shown to be efficient in terms of optical power and from an information theoretic perspective [10], however it uses only one quarter of the available bandwidth, hence it is wasteful in terms of spectral efficiency. DCO-OFDM is a method in which a DC bias is added to the real-valued baseband OFDM signal in order to reduce the probability that it assumes negative values. Since, owing to the high peak to average power ratio (PAPR) characterizing OFDM, the positivity of the signal cannot be guaranteed with $100\%$ certainty, the signal is clipped (i.e. its value is replaced by zero) in the intervals in which it is negative. The clipping operation causes distortion (mostly referred to as clipping noise) which limits the performance (in terms of symbol error rate (SER))[7]. The DC bias is set so as to ensure that the probability of a clipping event, and consequently the impact of clipping noise, remain acceptably low. The disadvantage of the DCO-OFDM is that the DC bias significantly increases the transmitted optical power. Several methods were proposed with the aim of addressing the power efficiency problem and coping with the implications of clipping noise. Most notable are the unipolar OFDM (U-OFDM) [11], which uses different time sample states and a rearrangement of the OFDM frame; asymmetrically companded DCO-OFDM (ADO-OFDM) [12], which uses a linear companding function in order to compress the negative part of the signal, thereby reducing the clipping noise, the Hartley Transform is proposed in [13] for optical IM/DD OFDM, and more. In this paper, we propose a new approach for meeting the non-negativity condition, which is based on replacing the negative parts of the DC-biased OFDM signal with their absolute value. In this case the traditional clipping noise is replaced with a stronger noise contribution, but one that is correlated with the signal. By applying an iterative signs estimation algorithm (ISEA) which will be presented as well, our approach enables a significant reduction in the DC bias level, and a corresponding improvement in power efficiency. In section II we review the mathematical model of an IM/DD channel and describe the proposed alternative to the clipping method. In section III we show simulation results where we compare the new suggested method to traditional DCO-OFDM and to a theoretical lower bound on performance. Section IV is devoted to conclusions.

We denote by $s(t)$ the data-carrying intensity waveform that is produced by the transmitter. After being photo-detected and electronically amplified, the electrical signal that impinges upon the analog to digital conversion unit (ADC) is given by

where $v(t)$ is zero mean additive white Gaussian noise (AWGN) with variance $\sigma_{v}^{2}$. Equation (1) contains a simplified description of the optical channel, as it assumes that optical propagation effects, most prominently loss and chromatic dispersion, are compensated for prior to photo-detection. In addition, noise following from possible optical amplification is assumed to be negligible. It is assumed that the ADC resolution is sufficiently high in order to justify the neglect of quantization noise. Finally, Since a host of standard synchronization methods for optical OFDM communication systems is available [14, 15], we shall follow common practice by assuming perfect synchronization.

We shall present results for two systems with an FFT size of $N=1024$, normalized signal power $\sigma_{s}^{2}=1$, and constellations of type QPSK and 8-PSK. First, we consider the case of a signal-independent-noiseless channel and examine clipping noise versus AV noise. Figs. 3 and 4 show the constellation diagrams of the traditional and the proposed DCO-OFDM methods for QPSK and $8$-PSK respectively. As seen, a relatively low DC bias with $\kappa=1$ and $\kappa=1.3$ is applied for QPSK and $8$-PSK respectively, and while the proposed method’s estimators (before the slicer) are exactly the true values, the clipped signal suffers from a significant distortion, which makes it much more vulnerable to signal-independent noise. We remind that typical values for the DC bias used in traditional DCO-OFDM are at least two standard deviations ($\kappa\geq 2$) of the OFDM signal, and Figs. 3 and 4 show the symbols estimators for QPSK with $\kappa=1$ and for $8$-PSK with $\kappa=1.3$, only $50\%$ and $65\%$ respectively of the minimal conventional value (which reduces the transmitted optical power by $60\%$ and $46\%$ respectively). Clearly, the proposed method achieves satisfactory results with a significantly reduced DC bias. Fig. 5 shows the SER, defined as the number of incorrectly estimated symbols divided by the total number of symbols, as a function of the bias level with and without using the algorithm for QPSK and $8$-PSK. The numerically assessed value of $p_{a}$ is also shown in the figure, together with the analytical curve obtained from Eq. (26). The dotted vertical line indicates the bias level beyond which the ISEA algorithm converges to the exact value (SER$=0$). The value of $p_{th}$ corresponds to the intersection of this dotted vertical line with the curve describing $p_{a}$. Notice that the values of $p_{th}$ are $0.1355$ and $0.06654$ in the cases of QPSK and $8$-PSK, respectively. The corresponding respective values of $B_{DC}$ in Figs. 5(a) and 5(b) are $1.1$ and $1.5$, and they can be checked to agree well with estimates based on Eq. (32). Fig. 6 shows the average number of iterations needed for the ISEA algorithm to converge as a function of the bias for QPSK and $8$-PSK. Here too, the vertical dotted lines indicate the threshold bias level (as obtained from Figs. 5(a) and 5(b)). Evidently, when the algorithm converges to the correct solution (i.e. over the threshold), it does so within only few iterations.

We now move on to assessing the performance of the ISEA algorithm in the presence of AWGN (i.e. with $\sigma_{v}^{2}>0$). In this case we compare the ISEA performance not only with the that of the standard clipping method, but also with the performance of the selective mapping (SLM) method, which combats clipping noise by means of PAPR reduction [19], and was selected as a typical representative of the PAPR mitigation approach. The SLM method was implemented exactly as detailed in [19], using $128$ alternative transmit sequences.¹¹1Each transmit sequence is generated via an element by element multiplication by a phase vector, whose $N$ elements are constrained to Hermitian symmetry, so that $N/2-1$ of them are chosen randomly (and uniformly) from the set $\{-1,+1\}$. Furthermore, we assume that the phase vector of the chosen lowest PAPR sequence is known to the receiver. This number is significantly higher than the typical numbers used in SLM implementations, and hence the SLM results that we report can be viewed as an effective upper-bound for this method’s performance. The comparison of the ISEA approach with the SLM method will be relevant when considering the complexity implications in what follows. In addition, we also show the results for an AWGN channel that is not subject to the non-negativity constraint (where $y[n]=s[n]+v[n]$), and hence its SER constitutes a theoretical lower bound.

We now examine the dependence of the SER on the SNR in the cases of QPSK and $8$-PSK transmission, with the bias levels set slightly below and slightly above the corresponding thresholds (recall that the threshold $\kappa$ is $1.1$ in the case of QPSK and $1.4$ with $8$-PSK, as seen in Fig. 6). Figs. 7(a) and 7(b) show the SER versus the SNR [defined as $10\log_{10}(\sigma_{s}^{2}/\sigma_{v}^{2})$] slightly below the $\kappa$-threshold, whereas in Figs. 7(c)–7(d) $\kappa$ is slightly higher than the threshold value. When $\kappa$ is set slightly below the threshold (Figs. 7(a) and 7(b)), the performance of the ISEA and the two other clipping-based methods are comparable in the low SNR regime, but in the relevant range, with the SER set to $10^{-3}$ the ISEA scheme gains relative to the other methods. With QPSK the ISEA result is similar to that of SLM and it is $\sim 1$[dB] better in SNR than in the case of standard clipping. With 8-PSK the SNR gain with respect to standard clipping and SLM, is $8$[dB] and $4$[dB], respectively. When $\kappa$ is slightly above the threshold value (Figs. 7(c) and 7(d)), the performance of the ISEA method rapidly converges to the performance of the unconstrained AWGN channel, implying that the positivity requirement does not introduce any penalty. Figure 8 shows the SER versus $\kappa$ for a fixed SNR value, illustrating the difference between the various methods below and above the threshold value of $\kappa$. Note that unlike in the case of the other two methods, the performance of the ISEA scheme rapidly approaches the theoretical lower bound. Table I shows the threshold values of $\kappa$ and the gain in SNR with respect to standard bias and clipping with the SER set to $10^{-3}$ for a few common constellations. As seen, the proposed method outperforms clipping in SNR requirement while reducing the DC bias. It is noteworthy that the DC bias threshold was computed for a noiseless channel, hence the threshold in a noisy channel would most likely be higher for low SNRs and would approach the noiseless threshold as the SNR increases. This explains the degraded performance of the proposed method for low SNRs, expressed by the gap between the lower bound and ISEA’s curves in the low SNR region of Fig. 7(c) and 7(d).

In Fig. 9 we compare the tolerance of the three methods to chromatic dispersion. It should be noted in this context that schemes based on direct laser modulation, such as the ones that we are considering, are not intended for highly dispersive links. Yet, comparing the three schemes with respect to this phenomenon can be viewed as a plausible indicator of their relative tolerance to inter-symbol-interference mechanisms in a more general sense. Since the absolute value operation underpinning the ISEA approach produces larger spectral broadening than standard-clipping or SLM, the ISEA method is expected to be less tolerant to dispersion. Indeed, this reality is seen in Fig. 9, where the rate at which the ISEA performance deteriorates with increasing dispersion is higher than it is with the other two approaches. Nonetheless, the vast advantage of the ISEA approach more than compensates for this sensitivity.

Finally, we wish to address the implications of the proposed ISEA method with respect to the complexity of the overall system. The receiver of the ISEA scheme involves one FFT and one IFFT operation per iteration and hence the complexity increase in comparison with the standard clipping-based DCO-OFDM approach, is determined by the number of iterations that are required for the algorithm to converge. As witnessed by the simulation results displayed in Fig. 6, the average of the number of iterations needed for convergence (above threshold) is practically equal to $3$ and its standard deviation is between one order of magnitude and two orders of magnitude smaller.²²2This was also confirmed in simulations with a broad variety of system set-ups. This implies that the effective complexity of the ISEA method is larger by a factor of 6 relative to standard clipping, where the receiver performs a single FFT operation. It is interesting to compare this cost, with the cost of a typical PAPR reduction method, such as SLM [19]. The SLM method also involves additional IFFT operations, whose number equals the number of alternative transmit sequences that it includes. Hence, an SLM method with the same complexity overhead as the ISEA would include 6 alternative transmit sequences, and its performance would be inferior to that of the SLM results that we have shown in Figs. 7–9, where the number of alternative transmit sequences was 128. Although we have only considered SLM explicitly, it must be noted that SLM is not considerably different from other PAPR reduction methods in terms of its complexity performance tradeoff.³³3Other relevant methods in this context are; partial transmit sequence, interleaving, tone reservation, and tone injection. As detailed in [19],[20] these methods also involve additional FFT/IFFT operations and/or solvers of optimization problems. All these methods aim at reducing the PAPR, as opposed to the ISEA method which copes with it. In this sense, the PAPR reduction methods are complimentary to the ISEA approach and in principle the two can be combined with one another.

We note in passing, that assuming synchronization, a scheme with the same receiver but with a different, significantly easier to implement transmitter will yield exactly the same results presented above. The simplified transmitter, proposed in Fig. 10, differs from the original by applying the DC bias and the absolute value operation digitally on the samples, followed by simple clipping after the DAC, to ensure non-negativity. In this manner, all the values at the samples instances are identical to those produced by the transmitter presented in Fig. 1, hence leading to identical results.

We proposed a new approach for meeting the non-negativity constraint in IM/DD. The new scheme, combines the absolute value operator and the ISEA algorithm, and shows a significant improvement over the standard DCO-OFDM method, both in terms of SER performance for a given DC bias and in terms of the bias needed for a given required SER. With a relatively moderate DC bias-index of $3.87$[dB] and $5.12$[dB] for QPSK and $8$-PSK respectively, the theoretical lower bound of a standard AWGN channel is achieved. Along with a rudimentary analysis of the AV noise, both analytically approximated lower bound on the DC bias and simulation results of the proposed method have been presented. Our analysis was performed for the case of a linear system, neglecting component imperfections and hence provides an upper bound for the system performance. The consideration of the tolerance of the scheme for system non-idealities is left for future study.