1. Introduction

In recent years, the continuously and explosively increasing amounts of data due to rising applications such as cloud computing/storage, 4G/5G wireless communication, artificial intelligence and so on have been boosting the demand for high-capacity optical communications, particularly for short-reach optical interconnects between/within data centers and high-performance computers. A wide variety of methods, including advanced modulation [1–6], error correction coding [7–12], and digital signal processing (DSP) [13–15], have been studied in attempts to improve the system capacity. For short-reach communication scenario, the intensity-modulation and direct-detection (IM-DD) solution provides great advantages of low power consumption and high density. While for most practical IM-DD channels, frequency response is usually non-flat even fluctuated in the frequency domain. That is caused by all possible contributions which influences channel frequency response, like limited bandwidth of optical and electrical devices and fiber dispersion. Discrete multi-tone (DMT) modulation was proposed as a way to address this problem, by loading modulations with different bit numbers on individual subcarriers with reference to the channel’s signal-to-noise-ratio (SNR) response [16,17]. However, for conventional DMT, the constellations of individual subcarriers are all equip probability distributed.

In fact, a uniform distribution with equip probability demonstrates high achievable information rate (AIR) at a high SNR. In noisy conditions, like most practical optical links, the optimal distributions with the highest capacities are not always uniform [18]. Particularly for DMT, the non-uniform distributions with the highest AIRs would also differ for different modulations of different subcarriers, according to their corresponding SNRs. Moreover, for the bit-loading process of DMT with targeted spectral efficiency, the optimal loaded entropies are continuous decimals rather than pure integers based on Fischer’s theory [16]. As a result, during the bit-loading process of conventional DMT, the stepped bit numbers (integers) cannot fit the channel response accurately. It requires extra power allocation to alter the subcarriers’ SNRs in return.

At the same time, probabilistic shaping (PS) method is widely utilized in coherent systems to adjust the distributions of the constellations, so that corresponding AIR can be improved effectively [19–24]. Very recently, an entropy loading scheme of DMT was reported to fit channel SNR response based on PS [25–27]. For [25], entropy-loaded DMT was realized by using an IQ modulator assisted with Kramers-Kronig receiver. In [26], entropy loading is proposed after water-filling power adaptation, to get rid of extra power reallocation. Corresponding investigations are done on a coherent system with maximum subcarrier number of 16. In [27], entropy-loaded DMT without power allocation is proposed by using Maxwell-Boltzmann (MB) distribution in visible light communication system, with AIR of 204 Mb/s. For IM-DD optical interconnection applications, there is still a research blank about the PS for frequency-resolved entropy loading, especially for achieving 100-Gbps per lane.

Here we demonstrate a frequency-resolved adaptive probabilistic shaping method refer to channel frequency response, for the 112-Gbps DMT-modulated IM-DD optical interconnection system. The continuously bit (in terms of entropy) loading is realized by adapted probability distributions, allowing for better fitting to channel frequency response with simultaneous shaping gain. Proof-of-concept studies have been carried out via both simulations and experiments. Performances of PS-DMT at a raw rate of 112 Gbps was investigated theoretically based on Maxwell-Boltzmann and dyadic distributions. The corresponding optical signaling was realized experimentally by direct modulation of a vertical-cavity surface-emitting laser (VCSEL) at 112 Gbps with 100-m multimode fiber (MMF) transmission successfully achieved.

2. Principle

For a practical optical channel, frequency responses always exhibit fading characteristic at high frequencies due to fiber dispersion, constrained bandwidth of optical and electrical devices. As a result, corresponding frequency response may roll down even fluctuate which resulting in non-flat response, as illustrated by the orange curve shown in Fig. 1(a). In this case, conventional DMT modulation maximizes capacity by loading paralleled bits on the frequency-resolved subcarriers with reference to the SNR-to-frequency response. Therefore, DMT modulation intuitively produces a higher capacity for such channels, than other formats, such as pulse-amplitude modulation (PAM) and carrier-less amplitude phase modulation. For DMT, the individual subcarriers are loaded using quadrature amplitude modulation (QAM) signals with different entropies. Typical AIR curves with respect to the SNR are briefly plotted in Fig. 1(b) for a non-uniform and the uniform probability distributions [19]. The QAM signals with the non-uniform probability distribution have a higher AIR in the low SNR region. Moreover, the non-uniform distribution can be adapted refer to different SNR.

 

Fig. 1 AIR (in terms of GMI) gain by PS-DMT. (a) Frequency responses for ideal (green line) and practical channels (orange curve); (b) AIR with respect to SNR for the uniform distribution and a non-uniform distribution, along with the Shannon limit; (c) Corresponding AIR-frequency responses for the ideal (green) and practical (orange) channels. The solid curves indicate the Shannon limit. The dotted and dashed curves represent the AIR curves for the uniform distribution and adaptive non-uniform distribution, respectively.

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Consequently, for an ideal channel with a flat and high SNR response, AIR can be improved by loading with more bits per symbol. However, this is always a limited option during practical implementation. For DMT-modulated system, AIR-to-frequency curve can be obtained by known SNR and distribution. The AIR curves with respect to frequency can then be plotted as shown in Fig. 1(c). The green solid straight line represents Shannon limit for such a channel. The limited number of loaded bits leads to the AIR gap to Shannon limit, as indicated by the green dotted straight line. If using PS with the non-uniform probability distribution optimized for a low SNR, the AIR may be reduced because of such mismatched distribution for a high SNR, as indicated by the green dashed straight line. However, AIR performance is totally different for practical channels with constrained and fluctuating SNR responses, as illustrated by the orange curves shown in Fig. 1(c). When SNR is constraint, PS can offer considerable AIR gain, which more approaching to Shannon limit. The orange solid curve depicts the corresponding Shannon limit. Significant AIR gain can be obtained because of the shaping gain of the PS, particularly in the low SNR region, as indicated by the orange dotted and dashed curves.

In addition, for conventional DMT, the loaded bit numbers must be step integers to perform bit-to-constellation mapping. This is the reason why rounding-off is required during the conventional bit loading process. As a result, it requires extra power allocation of subcarriers to alter the corresponding SNRs, for adapting the channel response, as plotted in Fig. 2(a). When taking the digital subcarriers into the whole-link analysis, the adapted SNR-frequency response after ideal power reallocation should be flat for those subcarriers with same loaded bit numbers. However, such adaptation is always not precise enough for perfect channel fitting. The proposed PS-DMT method can get rid of power reallocation because continuous bit loading (in terms of entropy) can be realized without any reduction of net rate. Therefore, the curve of the effectively loaded bit can perfectly fit the channel response, with all subcarriers’ power unchanged, as shown in Fig. 2(b).

 

Fig. 2 (a) Conventional DMT using discrete bit loading and power allocation; (b) PS-DMT using entropy loading.

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For conventional DMT, to achieve a balanced signal quality for all subcarriers of a DMT symbol with target spectrum efficiency, more bits are loaded on the subcarrier with the higher SNR, and fewer bits are loaded on those subcarriers with lower SNRs. The bit-loading process can be performed numerically based on Chow and Fischer algorithm [16]. The main reason why chose Fischer’s algorithm in this work, is to obtain the decimal entropies to be loaded on every subcarriers. The optimal loaded bit number B_i (which is always decimal) of the i^th subcarrier can be calculated based on a known noise power for a DMT symbol, as described by Eq. (1) in [16], where B_total is the total bit numbers to be loaded, M is the total number of subcarriers, and N_i is the noise power for the i^th subcarrier assuming normalized signal power. Consequently, with known the SNR for a typical subcarrier, N_i can be directly obtained by calculated the ratio of adapted subcarrier’s power and corresponding SNR. For conventional DMT, the bit loading process includes channel SNR estimation, rounding-off function of B_i and power reallocation of individual subcarriers, as shown in Fig. 3.

 

Fig. 3 Bit loading metrologies of conventional DMT and proposed PS-DMT schemes.

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While for PS-DMT, the loaded entropies on the subcarriers can be arbitrarily decimal through careful selection of the non-uniform distributions. An adaptive approach to the optimal loaded bit number B_i of Eq. (1) can thus be achieved. In this way, entropies can be loaded very accurately with reference to the channel response. Additionally, the AIR gain produced by PS is significant when the SNR is limited to a low value. Benefitting from these two advantages, the proposed PS-DMT technique is expected to be an effective enabling method to approach the Shannon limit for SNR-constrained and fluctuating channels.

The proposed bit-loading methodology for PS-DMT is illustrated in the flowchart as shown in Fig. 3. A rounding-up function is performed for B_i to obtain standard constellations for QAM-N noted as B_i*, and N denotes the constellation number. The corresponding bit-to-symbol mapping is performed as $\text{N}=2^{B_i^{*}}$. Finally, the optimization problem must be solved to search for the distribution of the 1D PAM signals. The optimization object is to obtain the 1D probability distribution with entropy that is most close to $\frac{B_i}{2}$. These distributions are subject to the Maxwell-Boltzmann (MB) equation P = e^−μ|x| with a variable μ. Therefore, the optimization problem can be simplified as:

Because in the case of Bi* = 3, two orthogonal components of QAM-8 are different: one is OOK, another is PAM-4. In this case, Eq. (2) with assuming the same modulation for two orthogonal parts, is no longer valid. As a result, Eq. (3) is stated specifically for the case of Bi* = 3, aiming to find the distribution P of the PAM-4 component:

For quantitatively evaluating performance of proposed PS-DMT, there are several metrics like bit-error rate (BER) before FEC, Q factor, symbol-error rate (SER) and error vector magnitude (EVM). Very recently, AIR is proved as a powerful metric for coded-modulation systems, because it can indicate how much information can be reliably transmitted per symbol, and also exhibits inherent relationship to FEC [28]. In the case of bit-wise decoding, generalized mutual information (GMI) can represent AIR. Under this circumstance, normalized GMI can be used for predicting the code rate of optimal FEC for achieving error free [29,30]. Consequently, we use GMI and NGMI as the performance metrics in this work.

In fact, for IM-DD optical links, the inter-symbol interference (ISI) caused by limited bandwidth and fiber dispersion will induce non-Gaussian distortion. However, DMT separates its spectrum by several subcarriers, which can get rid of ISI. In this case, noise for every subcarrier can be approximated as Gaussian type. As a result, GMI value of the i^th subcarrier can be calculated by Eq. (4) [31], in which, m is the bit number (equal to entropy) of a QAM symbol, i is the index of subcarrier, $B_k^i$ is the k^th bit of QAM symbol for the i^th subcarrier, $L_k^i$ is estimated log-likelihood ratio (LLR) of the k^th bits for the i^th subcarrier.

To evaluate the AIR performance of the proposed PS-DMT scheme, the optical signaling is modeled based on the additive white Gaussian noise (AWGN) channel with a constrained bandwidth. The attenuation at frequency domain is characterized using a Gaussian low-pass filter as described by Eq. (5), where SNR_{f = 0} is defined as the SNR value at DC, and BW_3dB is the 3-dB bandwidth of the channel.

Based on known SNR_{f = 0} and BW_3dB, the bit loading of the proposed PS-DMT can then be realized through solution of the optimization problem above. Typically, BW_3dB is set at 25 GHz, with SNR_{f = 0} of 20 dB. With the aim of providing the raw rate of 112 Gbps, the bit loading results of both conventional DMT and the proposed PS-DMT scheme are investigated, as plotted in Fig. 4(a). The red circles are the loaded bit numbers for conventional DMT, which are all stepped integers. Moreover, we take the digital subcarriers into consideration for analyzing the whole-link SNR response. The reason why calculating the whole-link response with extra consideration of power-altered subcarriers, is for evaluating the channel adaptation of conventional DMT and proposed PS-DMT. Because loaded bit numbers for conventional DMT are integer only, power reallocation has to be performed to alter the subcarriers’ power (raw-tooth shape) to maintain same SNRs for those subcarriers with same loaded bit numbers, aiming to assure equal and optimal performance margin over all subcarriers according to the statement in Column 18, Page 2 of [17]. As a result of which, SNR response (including digital subcarriers) after power reallocation should be in the same shape to loaded bit numbers of subcarriers, which is flat for those subcarriers loaded by same bit numbers. The adapted channel response (including digital subcarriers) after power reallocation is shown as red line in Fig. 4(a). For PS-DMT, bit loading is performed by solving the optimization problem stated in the article. The entropies adapted to the channel response are obtained for all 127 subcarriers and plotted as the blue triangles in Fig. 4(a). The adapted distribution (MB) of the 60th subcarrier is shown in Fig. 4(b), with entropy equal to 4.975.

 

Fig. 4 (a) Simulated channel SNR response (blue line), adapted channel response using power allocation (red line), and entropy loading results by PS-DMT; (b) The probability distribution of 60^th subcarrier for PS-DMT.

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Under AWGN condition, theoretical GMI values are calculated for both conventional DMT and PS-DMT. For DMT signals, GMI calculation is performed over all the subcarriers. The GMI of a single subcarrier can be obtained by Eq. (4). The GMI values for all 127 subcarriers are plotted in Fig. 5(a), with BW_3dB of 25 GHz and SNR_{f = 0} of 20 dB. It indicates that the AIR values of PS-DMT (MB distribution) are much higher than those of conventional DMT, at lower-order subcarriers (1^st to 80^th). While for higher-order subcarriers, shaped QAM signals has lower AIRs than uniform ones. In addition, the MB distributions require large-length block to perform PS coding. Here, we use Geometric Huffman coding (GHC) to match dyadic distributions to MB ones. GHC aims to obtain the optimized dyadic distribution, whose Kullback-Leibler distance to desired distribution is minimized. The detailed descriptions about GHC is given in [32]. To obtain dyadic distribution, a Huffman tree requires to be constructed with updating rule as Eq. (6), with assumption of $p_1\ge p_2\ge p_3\ge \dots \ge p_m$, and $p^{\text{'}}$ is the updated root during every interaction.

 

Fig. 5 (a) Loaded entropy of conventional DMT (red circles), PS-DMT (MB) (blue triangles) and PS-DMT (dyadic) (black circles), and corresponding GMI; (b) The GMI-to-SNR curve for the 60^th subcarrier.

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Loaded entropies of PS-DMT (dyadic) are plotted as light gray dots in Fig. 5(a). Although the entropies of PS-DMT (dyadic) cannot fit channel response perfectly, positive AIR gain still can be obtained for lower-order subcarriers. Moreover, a phenomenon is observed that dyadic distribution performs better at low frequencies rather than MB distribution. We think that is because that the optimal entropies calculated by Eq. (1) exceed the Shannon capacity for those subcarriers. While, dyadic distribution has more probabilities gathering to the center of constellation. As a consequent, dyadic distribution has higher AIR than MB for those subcarriers. To investigate the reason of shaping gain more clearly, the GMI-to-SNR curve for the 60^th subcarrier (QAM32) is plotted in Fig. 5(b). The dark rectangular marks the zone with positive shaping gain of PS. Without power reallocation, SNR at the 60^th subcarrier is 17.46 dB. Adapted SNR is 17.06 dB after power reallocation. Thus the GMI gain of PS-DMT is 0.4 bit/symbol for the 60^th subcarrier.

To further investigate the performance improvement of PS-DMT, total GMI value for a DMT symbol (contain 127 subcarriers) can be obtained by summing up GMI values of all subcarriers. Thus, the AIR improvement contours for PS-DMT (MB) and PS-DMT (dyadic) are plotted versus both SNR and 3-dB bandwidth in Figs. 6(a) and 6(b), respectively.

 

Fig. 6 AIR gain versus varying SNR and bandwidth: (a), PS-DMT (MB); (b), PS-DMT (dyadic).

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Figure 6(a) indicates that the maximum AIR improvement over 189 bits/symbol can be obtained by using PS-DMT (MB) for bandwidths below 20 GHz, when SNR is lower than 15 dB. When bandwidth is over 20 GHz, the AIR improvement over 25 bits/symbol can be obtained due to the shaping gain of MB distributions. The improvement of data rate is not strictly monotone increasing with SNR. It can be explained that the positive shaping gain is achieved at certain SNRs, typically shown in Fig. 5(b). For dyadic distributions, the data rate improvement of 29.1 bits/symbol also can be achieved at a broad region with bandwidth over 25 GHz.

3. Experimental results and discussion

The experiment investigation of a 112-Gbps optical interconnection is carried out using a VCSEL-MMF link. The experimental setup is illustrated in Fig. 7 below. The gray box of Fig. 7 encloses the channel SNR estimation process, which is based on sending a prior-known multi-tone signal (QPSKs) to enable calculation of the SNR for every subcarrier. In the hardware part, the arbitrary waveform generator (AWG) converts the digital sequences into analogue signal to drive the VCSEL chip, with a DC source provided for tuning bias. The 3D alignment platform is for both electrical and optical coupling of the VCSEL. At the receiver, a photo detector (PD) is used for signal detecting, and a digital storage oscilloscope (DSO), is used to sample the detected signal for processing. The bit-loading process is realized using the Fischer algorithm (for the conventional DMT) and the proposed continuous bit-loading algorithm.

 

Fig. 7 Experimental setup of VCSEL-MMF optical link.

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The sampling rate of the AWG is set at a fixed value of 54 Gsa/s. To achieve the target raw rate of 112 Gbps using 127 subcarriers, the total loaded bit number required for a DMT symbol is 112/54*(128*2 + 10) = 552. The term “128*2” represents the Hermitian transformation and “10” is the length of the cyclic prefix. Based on the measured SNR responses, bit-loading processes are performed on 127 subcarriers for both conventional DMT and PS-DMT. In the optical B2B link case, the bit-loading results for conventional DMT are shown as the red points in Fig. 8, while the bit-loading results for the proposed PS-DMT (MB) scheme are plotted as blue points. Entropies of dyadic shaped PS-DMT are marked by blue lines. The adapted SNR response by power reallocation is re-measured by sending multi-tone QPSK signals with reallocated power, plotted as pink line in Fig. 8. In fact, the adapted response is not precisely matched to loaded bit numbers for conventional DMT. Fortunately, PS-DMT can get rid of this deviation, by directly adapting entropies to channel response. The corresponding optimized distributions for 12^th and 66^th subcarriers are shown in Fig. 8(a), respectively. For MB distribution, a distribution matcher (DM) is required for coding uniform-distributed binary sequence to the sequence whose output symbol is non-uniformly distributed [33]. While for dyadic distributions, corresponding non-uniform distribution can be obtained directly by binary mapping. For an example, in order to obtain a symbol with probability of 2^-L, L-length mapping has to be performed on binary sequence for this symbol. In practical case, such a variance-length coding may result in severe synchronization problem in the receiver. Insertion of ambiguity bits can ideal with this issue through maintaining same bit number for every symbol. The dyadic distributions are also inserted in Fig. 8(b). In optical B2B case, the entropies of MB and distribution for the 12^th subcarrier are 6.613 and 6.5, respectively. For the 66th subcarrier, the entropies of MB and distribution are 5.3755 and 5.5. After 100-m OM3 fiber transmission, the bit-loading results and the optimized distributions are plotted as shown in Fig. 9. The entropies of MB and distribution for the 12^th subcarrier are 6.9939 and 7, respectively. For the 66th subcarrier, the entropies of MB and distribution are 5.316 and 5.5. The channel fading becomes severe because of modal dispersion, and thus more bits are allocated on the lower-frequency subcarriers. These results show that continuous entropy loading provides a better fit to the SNR response when compared with the conventional DMT.

 

Fig. 8 Experimental bit loading results for DMT and PS-DMT in the optical B2B case: (a) Bit-loading results for conventional DMT (red dots), PS-DMT (blue dots) and PS-DMT (dyadic) (blue line). (b) Shaped probability distributions of two typical subcarriers (12^th and 66^th subcarriers) for PS-DMT.

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Fig. 9 Experimental bit loading results for DMT and PS-DMT after 100-m transmission: (a) Bit-loading results for conventional DMT (red dots), PS-DMT (blue dots) and PS-DMT (dyadic) (blue line). (b) Shaped probability distributions of two typical subcarriers (12^th and 66^th subcarriers) for PS-DMT.

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With an optical power of 3.5 dBm for the optical B2B case, the demodulated constellations for both DMT and PS-DMT at the receiver are plotted as shown in Fig. 10. In fact, the demodulated constellations will rotate if no synchronization is performed. Thus, we estimated the phase-to-frequency response during training process, for correcting the rotated constellations. Corrected constellations are drawn for two selected typical subcarriers (12^nd and 66^th). In addition, right side of Fig. 10 shows the constellations of these typical subcarriers after 100-m OM3 fiber transmission with a received optical power of 3.3 dBm. The improvement in the signal quality related to the PS shaping gain can be observed visually by the clearer constellations that appear after PS when compared with those obtained before PS. It can also be seen that the constellations become clearer after PS. This occurs because the Euclidean distances between the symbols are broadened, with more symbols being gathered at the center, when the average power is fixed. After 100-m OM3 fiber transmission, however, the channel-efficient bandwidth is reduced, and more bits are loaded on the lower-frequency subcarriers. As a result, the constellations are much noisier after the 100-m transmission. Additionally, the shaped constellations are obviously clearer, with more bits being allocated at their centers. Moreover, one may observe that the shaped constellations are not ideally circular, that is because constellations become distorted after modulation and fiber propagation.

 

Fig. 10 Constellations of 12^th and 66^th subcarriers for optical B2B case (3.5-dBm received power) and after 100-m OM3 fiber transmission (3.3-dBm received power).

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Experimental GMI values for signals on individual subcarriers are calculated by Eq. (4). In detail, with assuming noise on every subcarrier as Gaussian type, LLR for the k^th bits for the i^th subcarrier can be estimated by Eq. (7), in which $X_k^1$ means the real value of symbol whose k^th bit is ‘1’, $Y_k^1$ is the imaginary value of symbol whose k^th bit is ‘1’

The GMI curve of the compensated signal has a similar shape to that of the bit-loading numbers, but with a reduced value because of noise. In addition, after PS, the GMI curve becomes much smoother, as shown in Fig. 11(a), and has a similar shape to the loaded entropies. The proposed continuous bit-loading algorithm can adapt the loaded entropies to fit the channel’s SNR response well, and thus the calculated GMI curve is much smoother for PS-DMT. Moreover, the theoretical GMI values of individual subcarriers are also plotted as dashed dots for comparison. The experimental results are in the similar shape to theoretical ones but smoother. After 100-m OM3 fiber transmission, the GMI curves for both conventional DMT and PS-DMT are plotted as shown in Fig. 11(b). Moreover, total GMI values for all subcarrier are summed up for calculating GMI of DMT symbol. In optical B2B case, GMI values for conventional DMT, PS-DMT (MB) and PS-DMT (dyadic) are 476.95 bits/symbol, 495.25 bits/symbol and 486.74 bits/symbol. Consequently, PS-DMT by using MB and dyadic distributions can offer 18.3 bits/symbol and 9.78 bits/symbol AIR gain, respectively.

 

Fig. 11 Generalized Mutual information (GMI) values: (a) Conventional DMT and PS-DMT for the optical B2B case, with received optical power of 3.5 dBm; (b) Conventional DMT and PS-DMT after 100-m OM3 MMF transmission, with received optical power of 3.3 dBm.

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As stated before, NGMI can be used for predicting the post-FEC BER. Moreover, information can be reliably recovered only when $R_c\ge \text{NGMI}$, in which R_c is the FEC code rate, according to [29,30]. As a result, the NGMI limit of 25%-overhead soft-decision FEC (25% SD-FEC) is 0.8 [29,30]. Consequently, in order to investigate the power sensitivity gain induced by PS-DMT, NGMI values are then calculated with varying received optical power values, as shown in Fig. 12. If there is no noise and signal distortion, NGMI value equals to 1. As the optical power decreases, NGMI also decreases rapidly because of the low SNR. In the optical B2B case, with an optical power of more than 1 dBm, the SNR is sufficient to keep the NGMI value stable. However, for optical powers below 1 dBm, any reduction in the SNR also reduces NGMI values of high-density constellations, which results in sharply reduced NGMI values. After 100-m OM3 fiber transmission, more bits are allocated to the lower-frequency subcarriers because the bandwidth is more constrained. In this case, when the optical power decreases, the higher-density constellations with larger bit-loading numbers suffer greater NGMI reduction than in the optical B2B case. To enable error-free signaling at 112 Gbps assuming 25% SD-FEC, power sensitivity gains of 0.7 dB and 0.64 dB can be obtained for the optical B2B case and the 100-m OM3 fiber transmission case, assisted by PS-DMT (MB). For PS-DMT (dyadic), power sensitivity gains are 0.44 dB and 0.29 dB for optical B2B and 100-m transmission.

 

Fig. 12 Experimental NGMI values and data rate of reliable communication under various received optical powers.

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4. Conclusion

In this paper, an optical signaling method to improve the data rate of the DMT-modulated IM-DD optical interconnection links is proposed based on frequency-resolved adaptive probabilistic shaping. This method can significantly improve the signaling capacity since two significant benefits are simultaneously utilized: 1) the shaping gain of PS for carriers with limited signal-to-noise-ratio (SNR) and 2) the frequency-resolved continuous entropy loading for better fitting to the channel frequency response. Proof-of-concept investigations have been carried out via both simulations and experiments, for MB and dyadic distributions. AIR improvements of MB and dyadic distribution versus varying SNR and bandwidth were investigated theoretically. In addition, optical signaling was realized experimentally using a commercial VCSEL at 112 Gbps data rate with 100-m OM3 fiber transmission achieved. 0.64-dB and 0.29-dB power gains have been achieved for 100-m OM3 fiber transmissions, by using MB and dyadic shaping of DMT, respectively.