Clipping Noise Mitigation in Optical OFDM Using Decision-Directed Signal Reconstruction

In this paper, the performance of a new decision-directed signal reconstruction (DDSR) algorithm with optimal reconstruction thresholds for use with optical orthogonal frequency division multiplexing (OFDM) modulation is investigated. Clipping noise remains one of the main drawbacks in many practical optical OFDM transmission systems. It has been shown in previous studies that a time-domain based DDSR algorithm can be used to effectively reduce the clipping noise. However, in all existing works studying this DDSR algorithm, the decision threshold used to direct the time-domain signal reconstruction is fixed and equal to the level at which the transmitted signal is clipped. In this paper, it is shown that, due to the influences of the unavoidable noise at the receiver, fixing the DDSR threshold at the clipping level is not an optimal solution and can severely limit the advantages of using this algorithm. A detailed theoretical analysis is performed to investigate the key factors that influence the choices of the optimal DDSR thresholds. The presented simulation results are used to verify the accuracy of the theoretical analysis and also show that the performance of the DDSR algorithm is significantly improved by using the obtained optimal thresholds.


I. INTRODUCTION
Orthogonal frequency division multiplexing (OFDM) is increasingly being used as the modulation method for the next generation intensity modulated/direct detection (IM/DD) based optical wireless communication (OWC) systems [1]- [3]. Because the intensity of light can only be real and positive, a number of unipolar forms of optical OFDM have been developed [4]- [15]. The best known of these are asymmetrically clipped optical OFDM (ACO-OFDM) [4] and DC-biased OFDM (DCO-OFDM) [5]. In ACO-OFDM, only odd subcarriers are loaded with data and therefore the time-domain signal is antisymmetric. When the transmitted signal is clipped at a zero level, the clipping noise only presents on even subcarriers and consequently does not cause signal distortion. However, as only the odd subcarriers are used for data carrying, ACO-OFDM is very inefficient in terms of bandwidth [12]- [16]. In DCO-OFDM, a bipolar OFDM signal is first generated and then a DC bias is added. Next, any remaining negative peaks are clipped to generate a non-negative signal. Because the added DC bias does not carry data, adding a high DC bias is very power inefficient [5]. However, when the added DC bias level is too The associate editor coordinating the review of this manuscript and approving it for publication was Mohamed M. A. Moustafa . low, the non-linear clipping operation introduces significant signal distortion which results in data detection errors [5], [17], [18].
One approach to improve the performance of optical OFDM is to design spectral efficient ACO-OFDM where the even subcarriers are used to carry additional data. In [5], a modulation technique named asymmetrically clipped DC-biased optical OFDM (ADO-OFDM) is presented in which the odd subcarriers are used to transmit an ACO-OFDM signal and the even subcarriers are modulated with a DCO-OFDM signal. A similar method has been studied in [6] where the even subcarriers are used to transmit pulse-amplitude-modulated discrete multitone modulation (PAM-DMT) signals [7]. To further enhance the overall performance, Layered ACO-OFDM (LACO-OFDM) is introduced in [12]- [15]. In LACO-OFDM, multiple 'layers' of ACO-OFDM are configured to exploit the use of the even subcarriers. However, the main drawback of this layered or hybrid modulation approach is that the data decoding procedure is a successive interference cancellation process and the computational complexity increases dramatically when the number of layer increases. Moreover, to prevent error propagation, complex coding or signal processing methods [12], [19], [20] are usually required.
Another approach to improve the performance of optical OFDM is to use a decision-directed signal reconstruction (DDSR) algorithm which mitigates the clipping noise. In this algorithm, the receiver makes preliminary decisions about the transmitted data and from these constructive information of the original unclipped signal is extracted and then processed together with the received signal to perform more accurate data detections. This method was originally developed for radio frequency (RF) wireless applications and implemented in both the time domain [21] and the frequency domain [22]. It was later adapted for applications to optical OFDM in [23] and [24]. Both the time-domain and the frequency-domain based DDSR methods can be used as an iterative process to improve the estimation accuracy. In [25], it shows that a single shrinkage correction step can be used in place of the iterative process without any reduction in performance.
Although the time-domain and the frequency-domain based DDSR methods are fundamentally very similar, they process the clipped signal in slightly different ways and thus have different performance which has been studied and compared in [26]. In the frequency-domain based method, a time-domain signal is first regenerated after performing preliminary decisions. This regenerated signal is then clipped at the same level as the transmitted signal to estimate the clipping noise in the frequency domain which is next linearly subtracted from the original received signal before performing the final data detections. The potential drawback of this approach is that all constellations of the estimated clipping noise are subjected to the influences of the errors made in the preliminary decisions. When they are linearly removed from the received signal, all resulting signal constellations are influenced by both the detection errors and the receiver noise. In the time-domain based DDSR method, the signal component removed by the clipping operation is first estimated based on the preliminary decisions about the transmitted data. Then it is combined with the original received signal in the time domain which consequently reduces the influence of the clipping noise. However, unlike the frequency-domain based approach in which the estimated clipping noise is linearly subtracted from the received signal, the time-domain signal reconstruction step is a nonlinear process and based on a DDSR threshold. In this process, the signal samples are scanned one by one and compared to the DDSR threshold to be decided for updating or not. Using this approach, the updated signal samples are only affected by the decision errors and the signal samples which are not updated/replaced are only influenced by the receiver noise. To make the best of this method, all clipped signal samples are required to be updated.
However, in all the existing works studying this DDSR algorithm [21], [23], [25]- [28], particularly the time-domain method, the threshold used to direct/guide the time-domain signal reconstruction is fixed at a level at which the signal was clipped at the transmitter. This is arguably the optimal approach when the clipping distortion is the only impairment. However, due to the unavoidable noise at the receiver, a large number of clipped signal samples are above this signal reconstruction threshold and these signal samples cannot be successfully processed or updated. Consequently, in any practical transmission systems, the approach considered in [21], [23], and [25]- [28] is not an optimal solution and can severely degrade the advantages of using this algorithm for mitigating the clipping distortion.
In this paper, a new approach of using the DDSR algorithm is studied. We first analyze the properties of the DCO-OFDM signal and show how the optical power and the clipping distortion of the transmitted signal depend on the clipping levels. Then, we study the symbol error rate (SER) performance of the conventional DCO-OFDM receiver in which the detected signal is affected by both the clipping operation at the transmitter and the noise generated at the receiver. Furthermore, we show how the decision errors introduced in the frequency domain based on the maximum likelihood (ML) criterion affect the regenerated time-domain signal sequence. The optimal threshold for the DDSR algorithm is then determined by analyzing the theoretical signal-to-noise ratio (SNR) of the reconstructed signal. Moreover, each stage of the theoretical analysis is verified using Monte Carlo simulations. The results show that the SNR of the processed signal significantly depends on the choice of the DDSR threshold. The optimal threshold which maximizes the SNR is shown to be dependent on a range of parameters including the noise power, the constellation size and the clipping level. More importantly, we show that the BERs decrease dramatically by using the obtained optimal threshold. This paper shows many new viewpoints of using the DDSR algorithm for mitigating the clipping noise in optical OFDM systems. It makes a number of important new contributions which include: 1) A detailed analysis which studies how the clipping distortion and the transmitted optical power are affected by both the top clipping and the bottom clipping. Moreover, by calculating the optical power, it is shown that the top of the signal is usually clipped much less than its bottom. 2) Derivation of the detection error probability when the OFDM signal is affected by both the nonlinear clipping operation and the receiver noise.
3) The study of a new type of time-domain noise which is caused by the detection errors in the frequency-domain using ML decoders. 4) Both theoretical SNR analysis and BER simulation results showing how the overall system performance is related to a range of parameters and what parameters can support the optimal performance. The rest of the paper is structured as follows. Section-II summarizes DCO-OFDM modulation and also studies the related clipping distortion. Section-III introduces the DDSR algorithm for clipping noise mitigation with a flexible signal reconstruction threshold. The noise in the regenerated time-domain signal sequence is then modelled and analyzed in Section-IV. Section-V shows how the theoretical SNR is related to the choice of the signal reconstruction threshold. The simulated BERs in Section-VI are presented to verify the theoretical analysis and show the advantages of using the optimal signal reconstruction threshold. Finally, Section-VII concludes the paper.

II. SYSTEM DESCRIPTION
In this section, we describe the DCO-OFDM transmitter and calculate the expressions for the clipping distortion in the transmitted signal. These analyses include the calculations of the optical power, the shrinkage ratio and the variance of the clipping noise. Fig. 1 shows a typical DCO-OFDM transmitter. The vector, X = [X 0 , X 1 , . . . , X N −1 ], which is input to the IFFT is a bipolar complex vector representing the QAM modulated data which is to be transmitted, and N is the number of subcarriers. X is constrained to have Hermitian symmetry, i.e. X k = X * N −k for 0 < k < N /2. The zeroth and Nyquist subcarrier are also set to zero i.e. X 0 = X N /2 = 0. The output of the IFFT is the real vector, In the following, we assume that N is large, so that central limit theorem applies and x m is approximately Gaussian distributed with zero mean and variance of σ 2 . Using the specific form of the IFFT operation shown in (1) which has a coefficient of 1/ √ N , the discrete signals at the input and the output of the transform have the same average power [1]. This has many important advantages and also simplifies the analysis of many OFDM functions. In this case, which is also calculated in Appendix-A.

A. CLIPPING OPERATION
In most of the research works [5], [17], [29], [30], the time domain DCO-OFDM signal, x m , is clipped at its bottom at a level of −B DC and the clipped signal, x clip,m is given by However, the top of the signal may also be clipped to reduce the dynamic range of the signal in some systems. This may reduce the constraints on the design of the amplifier and/or the resolution of the digital-to-analog convertor (DAC) and/or the linearity of the LED or laser and/or the received signal resolution in photon counting based detection systems [31]- [34]. A more general double-sided clipping operation is given by where λ is the top level at which the signal is clipped. The PDF of the clipped signal becomes [5] dt is the Q-function. Next, a DC bias, B DC , is added onto x clip,m to generate a positive signal, s DCO,m , using Finally, a DAC converts the clipped discrete time domain signal, s DCO,m , into an analog signal, s DCO (t), to drive the transmitter.

B. CLIPPING DISTORTION
In IM/DD based optical transmission systems, the non-linear clipping operation discussed in Section-II-A usually has two functions. First, to generate a unipolar signal, the transmitted signal is typically clipped at its bottom to avoid adding a high DC bias which is very power inefficient. Second, the clipping operation also reduces the peak-to-average power ratio (PAPR) of the transmitted signal which is usually a problem in OFDM. However, at the same time, the nonlinear clipping causes both in-band signal distortion and out-of-band power emission [35]. In RF communications, the spectrum resources are carefully managed and the requirements on the out-of-band emission are usually very strict. However, in optical wireless communication, its spectrum is license free. More importantly, the out-of-band emission is generated in the electrical domain and does not cause interference between optical signals. Consequently, as discussed in [1], the out-of-band power emission is much less of a problem in OWC than in RF communications. In this paper, we also focus on studying methods which can be used to mitigate the in-band signal distortion. To analyze the signal distortion in a more general way, a bottom clipping ratio is set relative to σ by and a top clipping ratio is defined as According to Bussgang' theory [36], the non-linear clipping operation in (4) would both attenuate the signal and cause clipping noise. The clipped signal, x clip,m , can be rewritten as where α is the shrinkage ratio and c m is the clipping noise. When the signal is double-sided clipped, based on Bussgang's theory, the shrinkage ratio can be calculated using Using (11), α is plotted as a function of ρ and µ and shown in Fig. 2. We can see that increasing ρ or µ reduces the clipping distortion and thus results in a larger value of α. It also can be noticed that, when both ρ and µ are large (e.g. ρ = µ = 3), α is very close to one because the removed signal due to the clipping operation becomes very limited. When both ρ and µ are zeros, the transmitted signal is completely removed and α is therefore zero. When one side of the transmitted signal is completely clipped and another side is unclipped (e.g. ρ = 0 and µ = 3), α is 0.5.
Next, the variance of the double-sided clipping noise, σ 2 c , is studied. σ 2 c is calculated in Appendix-B and given by (12), as shown at the bottom of the page. It is shown in (12) that σ 2 c depends on σ 2 , ρ and µ. Based on this relationship, σ 2 c is plotted as a function of ρ and µ with σ 2 fixed at one in Fig. 3. It can be seen that σ 2 c equals zero when both ρ and µ are zeros or when both ρ and µ are very high (e.g. ρ = µ = 3). This is because when ρ and µ are very high, the removed signal due to clipping is negligible which results in an extremely small value of σ 2 c . When ρ and µ are zeros, the transmitted signal, x clip,m , is completely removed. In this case, since α is zero as shown in Fig. 2, the clipping noise part, c m , does not exist based on (10) and consequently σ 2 c is zero. It is interesting to notice that, unlike DCO-OFDM with single-sided clipping in which the variance of the clipping noise would become higher when the clipping ratio increases [25], increasing the clipping ratio does not mean σ 2 c would be increased in a double-sided clipping case.

C. OPTICAL POWER
In IM/DD based OWC systems, the optical power is a very important parameter when the overall performance of the system is investigated. Because intensity modulation is used, the optical power and hence the optical energy per bit depends on the mean of the transmitted signal not its variance. Based on the statistical properties of the DCO-OFDM signal, the transmitted optical power is calculated in Appendix-B and given by  In Fig. 4, the optical power of the transmitted signal is plotted as a function of ρ and µ with σ fixed at one. It can seen that the transmitted power is directly related to the amount of clipping at the bottom of the signal. This is because increasing µ means that a large DC bias is required to make the transmitted signal unipolar. By contrast, clipping at the top of the signal does not affect the bias and therefore the average optical power very much. So, to design a more power efficient transmission system without introducing much signal distortion, there is usually an imbalance between the top clipping and the bottom clipping. Typically, clipping at the top of the signal is much less than clipping at the bottom and so has little effect on performance [25]. Consequently, without loss of generality, the cases with single-sided bottom clipping are considered in the following parts of this paper.
In the following analysis, when the single-sided clipping shown in (3) is considered, the PDF of x clip,m becomes where The shrinkage ratio, α, and the variance of the clipping noise, σ 2 c , shown in (11) and (12) can be simplified as (16) and (17), as shown at the bottom of the page, after removing the terms which are associated with ρ [17]. β in (17) is a function of the clipping ratio and used in later calculations.

III. DCO-OFDM RECEIVER WITH DDSR
In this section, we describe a generalized DDSR algorithm which is used to effectively mitigate the non-linear clipping distortion. In most of the OWC systems, the generated noise at the receiver is mainly the shot noise caused by the ambient light and/or the thermal noise due to the electronics. These two types of noise can be well modelled as additive white Gaussian noise (AWGN) 1 [37], [38]. In this paper, an AWGN channel is also considered. Although this model does not consider fading, frequency selectivity, interference, it is still extremely useful for gaining insight into the underlying behaviour of the overall system. Moreover, many of the conclusions and analyses can be easily adapted to a system when other factors are considered. In the following, the received signal is given by where n(t) is AWGN. The variance of the noise is σ 2 n and its PDF is given by The related OFDM parameters considered in this paper in the following analysis are summarized in Table 1.

A. CONVENTIONAL DCO-OFDM RECEIVER
The performance of a conventional DCO-OFDM receiver depends on the details of the receiver design and in particular whether the receiver corrects for the signal attenuation due to clipping. In most papers on DCO-OFDM, no compensation is made for the constellation shrinkage that results from this attenuation and the clipping noise is treated as independent additive noise. However, in practice, channel estimation is based on transmitted pilot signals or known preambles. In this 1 Note that, in photon counting based OWC systems (e.g. when single photon avalanche diodes (SPADs) are used in the detector), the noise due to the ambient light or the signal itself is Poisson rather than Gaussian. However, in this paper, we consider the more commonly used silicon positiveintrinsic-negative (PIN) photodiode and thus the noise is signal-independent and AWGN is a more suitable noise model.  case, the equalizer automatically corrects for constellation shrinkage. Note that the shrinkage may significantly affect signal detection especially in situations when the clipping level and/or the QAM modulation order are high. According to Bussgang's theorem, although the transmitted signal and the clipping noise are not correlated, this does not mean that they are independent. We have shown in [25] that useful information about the transmitted signal can be extracted from the clipping noise using a DDSR algorithm.

B. DDSR ALGORITHM
Next, we generalize the existing DDSR algorithms described in [21] and [25]. In the DDSR algorithm, preliminary decisions are made about the transmitted data and from these an estimate is made of the signal removed in the clipping operation. These estimated signals are then combined together with the original received signal to reconstruct/estimate the signal before clipping and thus to mitigate the clipping noise. This DDSR algorithm is shown in Fig. 5 and explained as follows: 1) The signal received by the photodetector, y(t), is first filtered and passed through an analog to digital converter (ADC), and then sent into an FFT block to obtain the frequency domain signals, Y = 2) The shrinking effect is eliminated by implementing an equalization step to Y k byȲ k = α −1 Y k .
3) The equalized frequency domain signal,Ȳ k , is sent into an ML detector, and the output is where Z M −QAM is the M-QAM constellation space, M is the QAM constellation size. 4) The detected frequency domain signal, Z k , are converted into a time domain sequence, z m , using IFFT. Note that the samples in z = [z 0 , z 1 , . . . , z N −1 ] would have both positive and negative peak values. 5) A new time domain sequence,x m , is generated from y m and z m using DDSR algorithm bỹ where τ is the threshold used to direct the time-domain signal reconstruction. Using (21), the clipped signal samples in y m are assumed to be replaced with z m which would mitigate the effect of clipping distortion. In the DDSR algorithms described in [21] and [25], τ is fixed at the clipping level, −B DC . This would just provide the optimal performance when the transmitted signal is only subjected to the non-linear clipping distortion. However, when noise at receiver is also considered, the assumption that all the clipped signal samples in y m will be replaced with z m is not correct. For example, approximately half of the clipped samples in y m would have values greater than −B DC and would not be updated with z m . This significantly limits the advantages of using the DDSR algorithm. Fig. 6 shows an example of a time-domain signal sequence to explain the influence of the noise on the signal reconstruction process. In this specific example, four signal samples are clipped at the transmitter. At the receiver, with the influence of the noise, two clipped signal samples are below the clipping level and another two clipped signal samples are above the clipping level. Using the existing DDSR algorithm, only signal samples below −B DC (within the green circles) are updated with z m . Two clipped signal samples which are above −B DC (within the red circles) would not be successfully updated.
In this paper, we generalize the description of the DDSR algorithm by not limiting its threshold at the clipping level. The optimal value of τ is studied in later sections using both theoretical analysis and Monte Carlo simulations. 6) Finally,x m is converted into frequency domain using FFT, and then sent into an ML detector to recover the transmitted data.

IV. DECISION ERROR NOISE IN DCO-OFDM
In the above described DDSR algorithm, the ML decision process in step 3 and IFFT process in step 4 are intended to restore most of the negative peaks of the signal which have been removed by clipping. Thus, replacing clipped signal samples will usually result in a more accurate estimate of the signal before clipping and the data recovered from this will have a lower BER. However, as explained in the previous section, the ML decisions in (20) are made with the detected signal subjected to both the noise generated at the receiver and the clipping distortion introduced at the transmitter and decision errors would normally occur. Therefore, the regenerated time-domain signal sequence in step 4, z m , is not equal to the transmitted signal before clipping, x m . We term the difference between x m and z m as decision error noise (DEN). The DEN is given by In this section, we analyze the properties of the DEN noise which depend on a range of parameters including the signal power, the noise power, the clipping level, the QAM constellation size, M , and the number of OFDM subcarriers, N . We show that, when N is sufficiently large the DEN noise is approximately Gaussian. A theoretical analysis of its variance is presented which closely matches the simulation results. In the following, the SER performance of the conventional DCO-OFDM receiver is first determined and then used to calculate/estimate the number of OFDM subcarriers with detection errors. Based on these calculations, an expression for the variance of the DEN noise is obtained.

A. SER IN DCO-OFDM
We first calculate the error probability. The SER of any QAM based transmission systems can be tightly bounded using the pairwise error probability (PEP) [39] which can be also adapted to the case of DCO-OFDM. We use PEP(i → j) to denote the probability that the ith QAM constellation point is transmitted and the jth constellation point is received after performing ML decisions. Taking the shrinkage factor, the clipping noise and the AWGN noise into consideration, PEP(i → j) can be calculated using [39] PEP where d ij is the Euclidean distance between the ith QAM constellation point and the jth QAM constellation point, σ o is the standard deviation of the sum of the clipping noise and AWGN. Since the clipping noise is independent of the AWGN, we can get The SER, P s , can be tightly upper bounded using [39] In this paper, in line with most of the practical transmission scenarios, we considered a system in which the variance of the signal is much greater than the variance of the noise. Therefore, a reasonable assumption is made that the QAM constellation detection failure only occurs in the situation that the ML decision is made to the neighbors of the transmitted QAM constellation point. The neighbors of a constellation point means its adjacent points to which it has the shortest Euclidean distance and this is illustrated in in Fig. 7 for both 4-QAM and 16-QAM. In this case, P s can be rewritten as where is the average number of the neighbors of a QAM constellation point [39] when a square QAM constellation set is considered. d is the shortest Euclidean distance between two adjacent constellation points as shown in Fig. 7. d can be expressed as a function of the variance of the QAM constellation points, E(|X k | 2 ), using [39] Finally, combining (17), (26), (27) and (28) gives and β is defined in (17). It can be seen from (29) that the SER is a function of the constellation size, the clipping ratio and the variance of the signal and the variance of AWGN. We then present the SER results using both the theoretical analysis derived in (29) and Monte Carlo simulations. Different DC bias levels and thus different clipping levels are considered for both 4-QAM and 16-QAM. In Fig. 8, the SER is plotted as a function of γ with µ fixed at 0.5, 1, 2 and 3. It shows that the simulation result matches very well with the theoretical results obtained using (29). When the bias level is high, the SER decreases when γ increases. However, when the bias is low, clipping noise dominates and consequently the SER plot plateaus.

B. THE VARIANCE OF DEN
Next, we analyze the variance of the DEN noise based the above detection error calculation. In an OFDM signal, when the overall number of subcarriers, N , is sufficiently large and the symbol error rate is P s , the number of subcarriers with detection errors is less random and approximately NP s . Consequently, the DEN noise, e m , which is defined in (22) can be modelled as a multi-subcarrier signal with NP s subcarriers which is added onto the transmitted signal before clipping. Moreover, since most of the incorrect decisions are made to the neighbours of the transmitted QAM constellation point, each subcarrier of e m is loaded with a 'point', χ, which has a Euclidean distance of d from its 'origin'. Therefore, z m is given by and the time-domain DEN noise can be expressed as As discussed above, since χ k has a distance of d from the origin of its Euclidean space, E(|χ k | 2 ) = d 2 . Furthermore, because only NP s subcarriers have errors, χ k is zero on N (1 − P s ) OFDM subcarriers. Since the input and the output of the FFT/IFFT have the same average power as shown in Appendix-A, the following relationship can be obtained.
Combining (28), (29), (33) gives We then perform simulations to verify the theoretical analysis of the variance of the DEN noise. In the previous analysis, we predicted that the DEN noise will be Gaussian when the number of subcarriers which have detection errors is sufficiently large within one OFDM signal. Therefore, the statistical properties of the DEN noise are related to the power of the signal, the power of the noise, the signal clipping level and the number of the OFDM subcarrier. In Fig. 9, the PDF of the DEN noise is obtained using both simulations and the theoretical model derived in (34). In the simulations, γ is fixed at 20 dB and the µ is set to be 0.5. 4-QAM is used on each OFDM subcarrier. In Fig. 9 (a)-(d), different numbers of OFDM subcarriers are considered. We can see that the DEN noise becomes approximately Gaussian when the number of subcarriers is 512. Moreover, when the DEN noise is Gaussian, our theoretical analysis matches very well with the simulation results. In the following analysis, the number of OFDM subcarriers is fixed at 512. This large number of OFDM subcarriers can potentially lead to a high PAPR. However, we also study a system in which the clipping distortion or clipping noise is the main impairment and the transmitted signal is severely clipped. The benefit of this severe clipping operation is that it also limits the PAPR.

V. SNR PERFORMANCE
In this section, we analyze the SNR performance of the system when the DDSR algorithm is used for clipping noise mitigation. As shown in (21), the processed signal,x m , is reconstructed using y m and z m and therefore encounters three types of noise. In particular, the signal samples reconstructed using y m experience the AWGN noise and/or the clipping noise and the signal samples which are replaced with z m are subject to the DEN noise. It can be noticed that the probability that a received signal sample is updated with y m or z m would affect the SNR of the processed signal and this probability depends on the choice of the DDSR threshold. In the following analysis, P z is used to denote the probability that y m is replaced with z m in the signal reconstruction procedure and consequently 1 − P z is the probability that y m remains not being updated. P z would depend on 1. the choice of the signal reconstruction threshold and 2. the statistical properties of the received signal. In order to obtain P z , we first need to find the PDF of x clip,m + n m , p cn (x). Since x clip,m is independent of n m , p cn (x) can be obtained using the convolution of p clip (x) shown in (14) and p n (x) shown in (19) and calculated using (35), as shown at the bottom of the page. Then, the theoretical calculation shown in (35) is verified using simulations. Both the theoretical and the simulated PDFs of the clipped OFDM signal with AWGN noise are shown in Fig. 10. We can see from Fig. 10 that the statistical properties of the signal are affected by the variance of the AWGN noise. Also, the PDF changes when different values of B DC or µ are considered. More importantly, it verifies that the PDFs obtained using the above theoretical analysis are very consistent with the simulated results. Note that, in DCO-OFDM, the bottom clipping ratio or B DC is usually defined in dB as 10 log 10 (µ 2 +1) dB [5], [17] which is also considered in this paper. Table-2 shows several examples of how µ is related to B DC (in dB) with different values.
In the DDSR process, in order to replace most of the clipped signal samples in y m with z m , the DDSR threshold should be chosen at the clipping of −B DC and with an offset. For example, in Fig. 6, all four clipped signal samples (rather than only two samples) can be successfully updated if the DDSR threshold is chosen to be slightly above −B DC . It can be noticed that the choice of this offset depends on the variance of the noise. To obtain more general conclusions, the offset coefficient is set relative to, σ n , using ησ n . Then, the threshold becomes −B DC + ησ n and signal reconstruction operation shown in (21) is rewritten as Note that, in a practical transmission system, in order to implement (36), the variance of the AWGN is required to be pre-estimated at the receiver side and the optimal value of η is obtained in the following analysis. Next, the probability that the received signal samples (including both the clipped signal samples and the unclipped signal samples) are below the signal reconstruction threshold, P z , can be obtained using (37), as shown at the bottom of the page. In (37), P z,1 is the probability that a transmitted signal sample is clipped and this clipped signal sample has a value below the DDSR threshold after the influence of the AWGN noise. This is because Q (B DC /σ ) is the probability that a transmitted signal sample is clipped and 1−Q (ησ n /σ n ) is the probability that the value of a clipped sample is below the DDSR threshold after AWGN noise is added. P z,2 is the probability that a transmitted signal sample is unclipped and this unclipped signal sample has a value below the DDSR threshold after the influence of the AWGN noise. Since only the signal samples which are updated using z m are affected by the DEN noise and the updating probability is P z , the variance of the DEN noise inx m is σ 2 e P z . Similarly, 1 − P z is the ratio/probability of the signal samples which are not updated and therefore remain being affected by AWGN. Consequently, the variance of the AWGN inx m is σ 2 n (1 − P z ). Furthermore, after implementing the signal reconstruction procedure shown in (36), the shrinkage ratio increases since the number of clipped signal samples are reduced. The probability that a clipped signal sample is above the DDSR threshold and therefore remaining being clipped is Q (ησ n /σ n ). Taking this into account, the new shrinkage factor, α new , becomes and the variance of the clipping noise becomes Q(η)σ 2 c . Taking all these factors into account, the SNR of the processed signal can be given by which is shown to be a function of η and therefore the DDSR threshold. Note that when the DDSR threshold is extremely high, the reconstructed signal,x m , would be exactly z m . Also, when the DDSR threshold is extremely low,x m would be y m . Both z m and y m would be decoded with the same performance. Therefore, we predict that there is a value of η which can provide the optimal performance. In Fig. 11, the theoretical SNR is plotted as a function of η with γ fixed at different values. In these results, in order to see the effectiveness of the studied algorithm on mitigating clipping noise, clipping noise rather than the AWGN noise is the dominant noise source. Consequently, the added DC biases are low so that the transmitted data cannot be successfully decoded using conventional receiver processing. In these plots, the DC bias is set by 1 dB for 4-QAM and 4 dB for 16-QAM. It can be seen from Fig. 11 that the SNR of the processed signal significantly depends on η. The SNR first increases and then decreases when η is increased from 0 to 10. Note that η = 0 is related to the previous DDSR method in which the signal reconstruction threshold is fixed at −B DC . The value of η which can maximize the SNR varies as γ changes. When γ is changed from 21 dB to 30 dB, the optimal η varies from 2 to 3.5.
This approach of using the theoretical SNR to locate the optimum DDSR thresholds can be implemented efficiently without a significant amount of simulations. However, the SNR in (39) is defined in the time domain while the ML decisions are made in the frequency domain. Unlike AWGN noise, both the clipping noise and the DEN noise after the time domain signal reconstruction no longer have a Gaussian-like form when the signal is transformed back to the frequency domain (see Fig. 3 (c) in [25]). Therefore, the variances of the clipping noise and the DEN noise cannot precisely indicate the their actual influence on the signal detection process. Also, because of this reason, the classical formulas used to show the relationship between BER and SNR in an AWGN channel do not apply in this situation.

VI. MONTE CARLO SIMULATION RESULTS
In this section, Monte Carlo simulations are performed to locate the optimal DDSR thresholds which can provide the minimum BER. In Fig. 12, the BER is simulated as a function of η with γ fixed at different values. It can be seen that the BER first decreases and then increases when η increases. The optimal DDSR threshold which can minimize the BER varies slightly for different values of γ . More importantly, comparing Fig. 12 to Fig. 11, we can notice that the optimal DDSR thresholds obtained using Monte Carlo simulations have almost identical values with those found using theoretical SNR analysis especially when γ becomes large.  Next, we show the BER performance of overall transmission when the DDSR algorithm is used with optimal thresholds. Its performance is compared to both the conventional receiver and the previous clipping noise mitigation receiver in which the signal reconstruction thresholds are fixed at the clipping levels. In these simulations, the transmitted optical power is set to unity and the BER is simulated as a function of E b,opt /N 0 . E b,opt is the transmitted optical energy per bit and given by where b is the transmission bit rate and N 0 is the single-sided noise power spectral density. In these simulations, both 4-QAM and 16-QAM are considered. As shown in Fig. 13, compared to the conventional receiver, although the receiver using previous DDSR clipping noise mitigation can improve the performance of the system, the BER has a plateau level at which the BER would not decrease as E b,opt /N 0 increases. Using the receiver with optimal clipping noise mitigation can significantly improve the performance of the system for both 4-QAM and 16-QAM and also avoid the BER plateau level for 4-QAM.

VII. CONCLUSION
In this paper, we study the performance of an enhanced time-domain based DDSR algorithm which has a flexible signal reconstruction threshold for the mitigation of clipping noise in optical OFDM systems. We show that, although using the previous DDSR algorithm can mitigate the clipping noise in DCO-OFDM, the performance of the system remains being severely affected when the added DC bias is too low. Due to the impacts of the unavoidable noise at the receiver, fixing the signal reconstruction threshold at the clipping level is not an optimal solution and can limit the advantages of this DDSR algorithm. In this paper, we study how the ML detections performed in the frequency-domain affects the regenerated time-domain signal sequence by introducing the DEN noise. Furthermore, by analyzing how the variances of the clipping noise, the AWGN and the DEN noise are affected after the time-domain signal reconstruction, a theoretical SNR expression of the processed signal is calculated which can be used to efficiently locate the optimal DDSR threshold. Finally, Monte Carlo simulations are performed to show the advantages of using this method compared to both the conventional receiver and the previous DDSR algorithm.

APPENDIX A THE CALCULATION OF E (|X | 2 ) = σ 2
The IFFT in (1) is paired with the FFT given by Based on (41), E(|X k | 2 ) can be calculated as Since E(x m x * p ) = 0 when m = p, E(|X k | 2 ) is then further calculated as Consequently, using the specific form of the IFFT/FFT pair shown in (1) and (41), the discrete signals at the input and the output of the transform have the same average power.

APPENDIX B THE CALCULATIONS OF THE VARIANCE OF THE CLIPPING NOISE AND THE OPTICAL POWER FOR DOUBLE-SIDED CLIPPED OPTICAL OFDM SIGNALS
In order to obtain the variance of the clipping noise, σ 2 c , we first need to calculate the variance of x clip , σ 2 x clip . Then, based on the relationship shown in (10), σ 2 c can be determined using Because the mean of x clip,m is not zero due to the non-linear clipping operation, both the mean of x clip,m , m x clip and E(x 2 clip ) need to be calculated first to get σ 2 x clip . E(x 2 clip ) is calculated as where ζ is calculated in (46), as shown at the top of the next page. Combining (45) and (46) gives Next, the mean of x clip is calculated as Then, the variance of x clip,m is calculated using (49), as shown at the top of the page. Furthermore, based on (44), the variance of the clipping noise is obtained using The transmitted optical power is the mean of s DCO,m which can be obtained using