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Abstract
Coherent superposition has been proposed to synthesize high-order quadrature amplitude modulation (QAM) by coherently superposing low-order QAMs in the optical domain. These approaches could effectively relax the digital-to-analog converter resolution and reduce the complexity of the driving electronics. However, in the superposition process, imperfect phase rotations (IPRs) in low-order QAMs will be transferred to the resultant high-order QAM. Importantly, the induced IPR cannot be compensated for by conventional linear equalizers and carrier recovery methods. To combat the induced IPR, herein, we propose a hierarchical blind phase search (HBPS) algorithm to compensate for the IPRs in synthesized high-order QAMs. The proposed HBPS can match the generation mechanism of the IPRs in coherent superposition, by tracing back and estimating the IPR in the QPSK-like constellation of each hierarchy and finally correcting the induced IPRs. Simulation and experimental results verify that this algorithm could effectively compensate for the IPR in the resultant 16-QAMs synthesized using coherent superposition approaches. The proposed HBPS shows significant optical signal-to-noise ratio (OSNR) gains compared to the conventional blind phase search (BPS) method for high-order QAMs coherently superposed using optical signal processing (OSP) and tandem modulators (TMs). Specifically, at the BER of 2.4e-2, the HBPS achieves a 1.5-dB OSNR sensitivity enhancement over the BPS in either OSP or TMs-based schemes, even with an imperfection rotation of up to $20^\circ$.
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1. Introduction
With the increasing demands for high throughput, multi-level modulation formats together with digital signal processing (DSP) are promising technologies for ensuring high spectral efficiency and transmission capacity [1]. However, in order to generate high-order modulation signals, higher resolutions of the effective number of bits (ENOB) in digital-to-analog converters (DACs) are required to generate multilevel driving electronics [2]. To release the requirement for the DAC’s resolution in the transmitter and reduce the complexity of driving electronics, one of the effective solutions is to generate high-order quadrature amplitude modulation (QAM) signal through coherent superposition in the optical domain. High-order QAMs could be synthesized by coherently superposing two low-order QAMs in the optical domain by either nonlinear or linear approaches [3–22]. In the nonlinear schemes [3–10], independent low-order QAMs are coherently superposed through optical signal processing (OSP) based on nonlinear optical effects such as four-wave mixing (FWM) to synthesize high-order QAMs. Whereas in the linear schemes, coherent superposition is achieved by tandem [11–16] or parallel [17–22] modulators, where binary or low-order driving electronics are employed as high-order QAM transmitters. However, during this process, the imperfect phase rotation (IPR) in low-order QAM is transferred through a nonlinear relationship to all quadrants in the complex plane, which leads to an IPR in the resultant constellation. In the high-order QAM synthesis scheme based on coherent superposition, the high-order optical QAM is considered to be decomposed into hierarchical low-order optical QAMs, and then coherently superimposed in the optical domain. However, in this synthesis process, if there is IPR in the low-order QAM of each hierarchy, it will introduce phase noise in the finally-synthesized high-order QAM signal. Unlike conventional high-order QAM synthesized by single-stage electro-optic modulation schemes, these phase noises induced in each hierarchy, overall, do not exhibit consistent phase rotation in the same direction. Therefore, it is challenging to compensate for them using conventional equalization algorithms. Especially due to the higher sensitivity of high-order QAM (64-QAM or beyond) to phase noise, IPR will have a more significant impact on the system performance. In [14], a phase-folded decision-directed (PF-DD) method was proposed to mitigate the IPR. However, since it is decision directed, the PF-DD may introduce extra symbol errors when partitioning the constellation in the phase folding process.
In this paper, we propose an improved blind phase search (BPS) algorithm, named hierarchical BPS (HBPS), to correct IPR when synthesizing high-order QAMs through coherent superposition. The proposed HBPS can match the generation mechanism of the IPRs in coherent superposition, by tracing back and estimating the IPR in the QPSK-like constellation of each hierarchy and finally correcting the induced IPRs. To validate the performance of the proposed algorithm, numerical simulations and experiments are conducted. Simulation and experimental results verify that this algorithm could effectively compensate for the IPR in the resultant 16-QAMs and 64-QAMs synthesized using coherent superposition approaches. Comparing the conventional blind phase search (BPS) to the HBPS technique for coherent superposition via OSP and tandem modulators (TMs), notable optical signal-to-noise ratio (OSNR) gains are achieved. Specifically, the HBPS approach demonstrates a 1.5 dB OSNR sensitivity improvement compared to BPS when 16-QAM is synthesized via OSP or TMs. Additionally, the proposed algorithm can be extended to deal with IPR in synthesized high-order QAMs.
2. Coherent superposition for synthesizing high-order QAMs
2.1 Operation principle of coherent superposition
Figure 1 illustrates the operation principle of the high-order QAM synthesis based on coherent superposition. In this approach, high-order QAM is obtained by coherently superposing low-order QAMs in the optical domain. Since only low-order QAMs are required, it could effectively reduce the requirements for driving electronics in electro-optic modulators. Herein, the coherent superposition could be divided into coherent addition and coherent multiplication. Without loss of generality, unless otherwise specified, we use 16-QAM as an example for illustration. As shown in Fig. 1(a), two coherent optical carriers coded in QPSK with different power ratios could be coherently added in the optical domain to general high-order QAM, i.e., 16-QAM. The coherent addition could be realized through OSP based on nonlinear optical effects such as FWM, or linear approaches based on parallel modulators. On the other hand, the coherent multiplication could be implemented using TMs, shown in Fig. 1(b). In the coherent multiplication, an offset QPSK, which is squeezed to one of the quadrants, is cascaded with a standard QPSK, mapping the offset QPSK to all of the quadrants and finally generating 16-QAM in the complex plane. Note that the symbol mappings in these two approaches are different. The implementation details of the coherent superposition schemes will be discussed in Sec. 4.
Fig. 1. Coherent superposition approaches for synthesizing 16-QAM from QPSKs: (a) "coherent addition" by nonlinear OSP or parallel modulators; (b) "coherent multiplication" by TMs.
2.2 Imperfect phase rotation in high-order QAM synthesized by coherent superposition
For standard 16-QAM constellations with common phase noise, the constellation points in the distorted 16-QAM experience uniform phase angle rotations in the same direction. It is possible to estimate and correct these phase noise-induced rotations across the whole constellation in a single stage. In essence, conventional linear equalizers and carrier recovery algorithms can be employed to directly compensate for these rotations. However, in the 16-QAM synthesis schemes based on coherent superposition, as shown in Fig. 1, the synthesized 16-QAM $\vec {E_{LS}}$ could be considered to be composed of two basic QPSKs (QPSK1 $\vec {E_L}$ and QPSK2 $\vec {E_S}$). The IPR in each low-order QPSK would be transferred to all of the quadrants, thus distorting the synthesized 16-QAM. As an example, Fig. 2(a) illustrates an ideal 16-QAM synthesized by the OSP-aided nonlinear approach from perfect low-order QAMs. In contrast, Fig. 2(b) is a synthesized 16-QAM with IPRs of $\pi /10$ and $\pi /6$ for QPSK1 and QPSK2, respectively. The IPR cannot be effectively compensated for by a linear equalizer or traditional carrier recovery at the receiver. To mitigate such kind of IPR in coherent superposition schemes based on TMs, a phase-folded decision-directed (PF-DD) linear equalizer was proposed in [14], where constellation points in the in-phase/quadrature (IQ) plane were folded into one quadrant by using hard decisions along the coordinate axes for phase rotation correction. However, when the IPRs [$\varphi ^L, \varphi ^S$] increase to $\pi /6$ and $\pi /3$ for QPSK1 and QPSK2, respectively, as shown in Fig. 2(c), the excessive phase rotation leads to each small QPSK in the synthesized constellation diagram exceeding the predetermined quadrant, making it difficult to rectify the phase rotation even using the PF-DD.
Fig. 2. Synthesized 16-QAM constellations: (a) standard 16-QAM without IPR; (b) distorted 16-QAM with $\pi /10$ IPR in QSPK1 and $\pi /6$ in QPSK2; (c) distorted 16-QAM with $\pi /4$ IPR in QSPK1 and $\pi /3$ in QPSK2.
3. Proposed hierarchical blind phase search algorithm
The minimum distance BPS algorithm is a carrier recovery technique used in coherent optical systems to estimate and compensate for phase offsets, ensuring accurate symbol demodulation in square $M$-QAM [23]. To combat the aforementioned IPR induced in the coherent superposition, we propose the hierarchical BPS (HBPS) algorithm. Considering the synthesis mechanism, as shown in Fig. 3(a), each symbol in the resultant 16-QAM constellation may suffer from IPR in different directions and angles, making it challenging to directly compensate for using conventional equalization and carrier recovery methods. Since 16-QAM is synthesized by superposing two fundamental QPSKs ($\vec {E_L}$ and $\vec {E_S}$ in Fig. 3(a)), in our proposed HBPS algorithm, two cascaded BPS stages are introduced to correct the IPR in each fundamental QPSK. In the first BPS stage (Stage 1), the constellation points in each quadrant of the received 16-QAM are treated collectively, resembling a QPSK-like constellation. As shown in Fig. 3, a standard QPSK constellation $s_i$ is provided and rotated by $N$ different test phases $\varphi ^L_n$. Then the MED between the received constellation $r_k$ and the rotated standard constellation is calculated individually. By averaging the MEDs in each quadrant, it is helpful to enhance the algorithm robustness. Then the global minimum of MED over $N$ different test phases could be obtained by Eq. (1).
Notably, another option is to use a constellation aggregation module in the symmetry BPS algorithm [24] as a replacement for the Euclidean distance calculation. Once the global MED has been calculated, the selector can use it to determine the estimated phase. The obtained phase is then applied to correct the IPR of the large QPSK, resulting in an output 16-QAM constellation $r_k^\prime$ as shown in Fig. 3(b).
Fig. 3. Algorithm architecture of the proposed hierarchical blind phase search: (a) $\textit {r}_k$: the received constellation of synthesized 16-QAM with IPRs; (b) $\textit {r}_k^\prime$: the output constellation of stage 1 in HBPS; (c) $\textit {s}_k$: the output constellation of Stage 2 in HBPS; $s_i$: standard QPSK constellation; MED: minimum Euclidean distance; mean: mean value; min: minimum value; slc: selector; $t_i$: standard 16-QAM constellation; QP: quadrant partition.
In Stage 2, a similar phase correction is performed. Different from Stage 1, the BPS is conducted on each small QPSK in each quadrant, and the mean of the small reference QPSKs in each quadrant is used as the rotation center for phase correction. After completing similar phase search and correction for each quadrant, the corrected constellation $s_k$ shown in Fig. 3(c) can be obtained.
4. Algorithm validation
Numerical simulations and experiments are conducted to validate the proposed HBPS algorithm in two coherent superposition schemes based on OSP and TMs. In addition, conventional BPS and PF-DD algorithm are performed as benchmarks.
4.1 Performance of HBPS in 16-QAM synthesis via OSP-aided nonlinear coherent superposition
To realize the coherent superposition illustrated in Fig. 1(a), it is crucial to ensure the coherence between two optical carriers. To satisfy this requirement, as discussed in [5] and [8], two independent QPSKs are carried in two optical sub-carriers to form twin single-sideband Nyquist QPSK (twin-SSB-Nyquist-QPSK) as the input of the FWM-based OSP unit for coherent superposition. Figure 4(a) illustrates the experimental setup of the OSP-aided coherent superposition scheme, where 16-QAM sub-carrier is synthesized by aggregating sub-carriers (sidebands) in a twin-SSB-Nyquist-QPSK. An IQ modulator (IQM) is first used to synthesize a twin-SSB signal with independent 28 GBaud QPSKs. OSP-aided data aggregation is then used to coherently superpose two sub-carrier QPSKs, resulting in generating a Nyquist 16-QAM data carrier with an increased modulation level. Figures 4(b)–(d) shows the principle of OSP-aided synthesis approach using FWM in a piece of high non-linear fiber (HNLF). In Fig. 4(b), the twin-SSB-Nyquist-QPSK signal consisting of upper (L at $f_1$) and lower (S at $f_2$) subbands is combined with two comb sources (at $f'_1$ and $f'_2$) and then launched into a piece of HNLF. The frequency spacing between the two-tone comb is $\Delta f = f'_2 - f'_1= f_2 - f_1$. As shown in Fig. 4(c), with the aid of FWM, replicas of upper (L1) and lower (L2) sub-carriers of original twin-SSB-Nyquist-QPSK would be generated at frequency $f_2=f_c -\Delta f/2$ and $f_1=f_c + \Delta f/2$, aligning with the original upper (L) and lower (S) sidebands, respectively. Subsequently, owing to the coherence between them, the newly-generated sub-carriers (at L1 and S2) and original sub-carries in the twin-SSB-Nyquist-QPSK signals (QPSK1@S and QPSK2@L) are coherently superposed with a power ratio of 1:2. Finally, a 16-QAM sub-carrier is generated. For example, as shown in Figs. 4(b)–(d), the QPSK2 at L and QPSK$1'$ at S2 are coherently superposed to generate 16-QAM. Note that the power ratio between the original and newly-generated sub-carriers should be adjusted to ensure the uniform distribution of the synthesized 16-QAM levels. The optical spectra before and after the FWM are illustrated in Fig. 4(e) and (f), respectively. The key parameters are depicted in Table 1.
Fig. 4. 16-QAM synthesis via OSP: (a) experimental setup; the principle of OSP-aided synthesis approach using FWM in HNLF: (b) input spectrum before HNLF; (c) newly-generated replicas after FWM around the input twin-SSB; (d) synthesized 16-QAM sub-carrier; (e) optical spectra before the FWM; (f) optical spectra after the FWM.
Table 1. Key parameters
At the receiver, an optical coherent receiver is used to demodulate the synthesized signal. In the DSP flow shown in Fig. 5, the received signal is initially downsampled to 2 samples per symbol. Subsequently, the signals undergo a direct current block for the removal of direct current, and an automatic gain control is applied to achieve amplitude normalization. Following this, the Gram-Schmidt process is employed for IQ orthogonalization of the signals. After frequency offset estimation and compensation, a linear channel equalizer based on the least mean square is used for channel equalization. To compensate for residual frequency offset and laser linewidth, a second-order phase-locked loop is employed after the linear equalizer for carrier frequency recovery. Following the carrier recovery, the proposed HBPS is used to mitigate residual phase noise and the IPR introduced in the coherent superposition synthesis. Additionally, the previously reported BPS and PF-DD methods are used for comparison. Finally, symbol synchronization, symbol demapping, BER, and EVM calculations are performed to evaluate the performance of the system.
Fig. 5. The receiver DSP with three different schemes: BPS, PF-DD, and our proposed HBPS; DC block & AGC: Direct Current block & Automatic Gain Control; Freq. Offset Comp.: Frequency Offset Compensation; Symbol Synch.: Symbol Synchronization; Symbol Demap.: Symbol Demapping.
Figure 6 presents the experimentally measured constellation diagrams when around $10^\circ$ and $20^\circ$ IPRs are induced in large and small QPSKs, respectively. The constellation after linear equalization is depicted in Fig. 6(a) in red color. A second-order carrier recovery loop is then deployed to estimate and compensate for the residual frequency offset. The corresponding constellation is shown in Fig. 6(b). After the carrier recovery loop, the proposed HBPS is applied for phase rotation corrections. In Fig. 6(c) and (d), the IPRs in large and small QPSKs have been corrected in Stage 1 and Stage 2, respectively.
Fig. 6. Experimentally measured constellations when around IPRs of $10^\circ$ in large QPSK and $20^\circ$ in small QPSK are induced in the synthesis: (a) the received signal after the linear equalization; (b) the output after the carrier recovery; (c) the output after Stage 1 of HBPS; (d) the output after Stage 2 of HBPS.
To comprehensively assess the performance of the proposed HBPS, the conventional BPS and previously-reported PF-DD are also investigated via numerical simulation when different IPRs are introduced in the synthesis. When the IPR in the synthesis is negligible, as shown in Figs. 7(a)–(d), the performance of the three algorithms is comparable. However, when the first stage IPR changes to $20^\circ$, as shown in Figs. 7(e)–(h), HBPS surpasses the other two schemes in EVM performance, which is 2.28 dB higher than PF-DD and 7.78 dB higher than BPS. Specifically, Fig. 7(e) shows the constellation after carrier recovery. The constellation after applying the BPS is shown in Fig. 7(f), indicating that the conventional BPS could not compensate for the excessive phase rotation. Figure 7(g) shows the equalized constellation after the PF-DD. Clearly, the constellation folding process introduces numerous misjudged points. As shown in Fig. 7(h), the proposed two-stage BPS effectively addresses the IPR induced in the synthesis process. These results confirm the effectiveness of the proposed algorithm.
Fig. 7. Constellations after carrier recovery [(a), (e)] and applying the BPS [(b), (f)], PF-DD [(c), (g)] and HBPS [(d), (h)] with IPRs of [$0^\circ$, $0^\circ$] [(a)-(d)] and [$20^\circ$, $0^\circ$] [(e)-(h)]: (a) the constellation after carrier recovery; (b) the corrected constellation using BPS: EVM = -23.74 dB; (c) the corrected constellations using PF-DD: EVM = -23.83 dB; (d) the corrected constellation using HBPS: EVM = -23.92 dB; (e) the constellation after the carrier recovery; (f) the corrected constellation using BPS: EVM = 16.44 dB; (g) the corrected constellations using PF-DD: EVM = 21.94 dB; (h) the corrected constellation using HBPS: EVM = 24.22 dB.
To systematically evaluate the effectiveness of the proposed HBPS in correcting IPR in high-order QAMs synthesized by nonlinear coherent superposition, we measured the bit-error rate (BER) of 112 Gb/s 16-QAM signals using HBPS with varying IPRs. As a comparison, we also applied the conventional BPS algorithm to compensate for IPR in the synthesized 16-QAM. Figure 8(a) and (b) show the BER performance of the proposed HBPS and conventional BPS under different IPRs. Our results show that compared to the conventional BPS, the proposed HBPS algorithm delivers superior performance in compensating for IPR in 16-QAM coherent superposition synthesis schemes, particularly when the IPR exceeds $10^\circ$. Furthermore, at the BER of 2.4e-2, the HBPS demonstrates an OSNR gain of up to 1.5 dB over the BPS scheme when the IPRs are [$0^\circ$, $20^\circ$].
Fig. 8. BER versus OSNR performance comparison of synthesized 112 Gb/s 16-QAM using HBPS and traditional BPS algorithms under different IPRs.
4.2 Performance of HBPS in high-order QAMs synthesis through TMs based linear coherent superposition
4.2.1 16-QAM
In contrast to the aforementioned nonlinear approach, coherent superposition also could be realized by a linear approach based on TMs (Fig. 1(b)), which is depicted in Fig. 9. Two binary PRBS data streams are used to drive the first IQM for offset QPSK signal generation in the first quadrant [16]. Subsequently, the modulated offset-QPSK is mapped to other quadrants by the followed standard QPSK modulation in the second IQM driven by the other two electrical binary PRBS data. Finally, the desired 16-QAM signal is generated. Note that the symbol mapping in the resultant 16-QAM is different from the nonlinear approach illustrated in Sec. 4.1.
Fig. 9. 16-QAM synthesis by TMs: LD: laser diode; IQM: in-phase/quadrature modulator; MZM: Mach-Zehnder modulator.
The BER performance at different OSNRs is investigated when applying the proposed HBPS and the conventional BPS to compensate for the IPRs in the synthesized 112 Gb/s 16-QAM. As shown in Fig. 10(a) and (b), similar to the results obtained in the nonlinear coherent superposition scheme, the proposed HBPS delivers superior performance compared to the BPS scheme in combating the IPR induced during the synthesis process, particularly when the second stage IPR increases to around $20^\circ$. Our results reveal that, at the BER of 2.4e-2, the proposed HBPS algorithm exhibits an OSNR sensitivity gain of approximately 1.5 dB compared to the BPS when IPRs are [$0^\circ$, $20^\circ$].
Fig. 10. BER versus OSNR performance comparison of synthesized 112 Gb/s 16-QAM using HBPS and traditional BPS algorithms under different IPRs.
4.2.2 64-QAM
The configuration described above for the 16-QAM synthesis can be extended to generate 64-QAM by incorporating one more QPSK modulator in tandem, which is driven by another two binary data streams. Figure 11 illustrates the 64-QAM synthesis through coherent superposition based on TMs. Certainly, in contrast to 16-QAM, the generation of 64-QAM involves three consecutive coherent superpositions of QPSK. It may introduce three different levels of IPRs, as illustrated in Fig. 12(a) [$15^\circ$, $0^\circ$, $3^\circ$] and (e) [$15^\circ$, $5^\circ$, $3^\circ$]. Given this generation mechanism, it becomes evident that conventional methods, such as BPS and PF-DD, depicted in Fig. 12(b), (f), (c), and (g), are incapable of completely correcting IPRs in a single stage. Recognizing the inherent challenge posed by IPRs in the generation process, the proposed HBPS emerges as a suitable solution. This technique, as depicted in Fig. 12(d) and (h), is expressly designed to align with the impairment generation mechanism, enabling it to achieve a comprehensive and effective correction of the constellation. Quantitatively, HBPS outperforms conventional BPS by 4.58 dB and PF-DD by 6.70 dB in EVM performance when the IPRs are [$15^\circ$, $5^\circ$, $3^\circ$].
Fig. 11. 64-QAM synthesis by TMs: LD: laser diode; IQM: in-phase/quadrature modulator; MZM: Mach-Zehnder modulator.
Fig. 12. Constellations after carrier recovery [(a), (e)] and applying the BPS [(b), (f)], PF-DD [(c), (g)] and HBPS [(d), (h)] with IPRs of [$15^\circ$, $0^\circ$, $3^\circ$] [(a)–(d)] and [$15^\circ$, $5^\circ$, $3^\circ$] [(e)–(h)]: (a) the constellation after carrier recovery; (b) the corrected constellation using BPS: EVM = -21.95 dB; (c) the corrected constellations using PF-DD: EVM = -23.36 dB; (d) the corrected constellation using HBPS: EVM = -28.19 dB; (e) the constellation after the carrier recovery; (f) the corrected constellation using BPS: EVM = -21.47 dB; (g) the corrected constellations using PF-DD: EVM = -23.59 dB; (h) the corrected constellation using HBPS: EVM = -28.17 dB.
For further assessment, we assess BER performance at different OSNRs using both proposed HBPS and traditional BPS schemes to correct IPRs in a synthesized 168 Gb/s 64-QAM, as shown in Fig. 13. The first and third stage IPR is fixed at $15^\circ$ and $3^\circ$, while the second stage is set to $0^\circ$, $5^\circ$ or $10^\circ$. For 64-QAM with BER of 3.8e-3, the proposed HBPS can achieve about OSNR sensitivity gain of 2 dB compared to the conventional BPS when the second stage IPR is $5^\circ$. At the same time, it is worth noting that when the imperfection in the phase rotation of the second stage increases to $10^\circ$, BPS is no longer functional, whereas HBPS is still able to achieve excellent performance.
Fig. 13. BER versus OSNR performance comparison of synthesized 168 Gb/s 64-QAM using HBPS and traditional BPS algorithms under different IPRs.
5. Discussion
5.1 Laser linewidth tolerance
Figure 14 shows the EVM performance as a function of laser linewidths for synthesized 112 Gb/s 16-QAM using HBPS and traditional BPS through numerical simulation. To ensure fairness, consistent parameter settings were employed for both algorithms. It is observed that as the linewidth increases from 0.1 to 1 MHz, the EVM performances of both methods slightly degrade. Notably, there is no significant discernible difference in performance between the two algorithms, indicating that both have a similar linewidth tolerance.
Fig. 14. EVM performance comparison of synthesized 112 Gb/s 16-QAM using HBPS and traditional BPS algorithms under different linewidths.
5.2 Compensation range
Figure 15 presents the simulated EVM performance comparison of synthesized 112 Gb/s 16-QAM using HBPS and traditional BPS algorithms under different IPRs. The $\varphi ^L$ and $\varphi ^S$ are the IPRs induced in the first and second stage of 16-QAM synthesis. We fixed $\varphi ^L$ and examined the system’s EVM performance under different $\varphi ^S$. The figure indicates that the conventional BPS attains comparable performance to HBPS only when the same IPR ($\varphi ^L$ = $\varphi ^S$) is introduced in each stage. This is due to the fact that, under such circumstances, the accumulated IPRs introduced in the coherent superposition process result in an overall rotation of the constellation. In contrast, the QPSK-like constellation is employed in our proposed HBPS to estimate and compensate for the IPR introduced in each stage. Herein, as long as the IPRs introduced at each stage remain within the range of avoidable phase ambiguity of $\pm \pi /4$, HBPS can operate stably.
Fig. 15. EVM performance comparison of synthesized 112 Gb/s 16-QAM using HBPS and traditional BPS algorithms under different IPRs.
5.3 Algorithmic complexity
To better illustrate the computational complexity of BPS and HBPS in a quantitative manner, Table 2 provides a summary of the operation count required for both methods when considering $N$ test phase angles within a $M$-QAM symbol. $2N$ real multiplications and $3N$ real additions are required for conventional BPS. As described in Sec. 3, the proposed HBPS relies on straightforward calculations within the real plane. These computations are employed to determine the suitable test phase angle for compensation. In contrast to BPS, HBPS obviates the need for actual multiplication by capitalizing on the benefits of employing Manhattan distance. Additionally, the HBPS requires only a slightly higher number of real additions. However, with the increase in modulation order $M$, the computational complexity of real additions in HBPS will correspondingly increase. HBPS incurs some computational complexity, but at the same time, it significantly enhances system performance. Therefore, this cost is deemed acceptable.
Table 2. Computational complexity of BPS and HBPS
6. Conclusions
Coherent superposition has been proposed for synthesizing high-order QAMs. However, compensating for the IPR in the synthesized high-order QAMs based on coherent superposition approaches is difficult using conventional BPS or previously reported PF-DD algorithms. To overcome this challenge, we present an HBPS algorithm that utilizes stage-wise BPS to correct IPR. The effectiveness of HBPS has been confirmed through both the simulation and experimental results for synthesized 112 Gb/s 16-QAM and 168 Gb/s 64-QAM. Notably, when the IPR reaches up to $20^\circ$, HBPS for the synthesized 16-QAM via OSP or TMs can attain a 1.5 dB OSNR sensitivity improvement over BPS. Besides, the proposed scheme is also applicable to higher-order QAM.
Funding
Science and Technology Commission of Shanghai Municipality (SKLSFO2019-06); National Institute of Information and Communications Technology (02101); Japan Society for the Promotion of Science (22H01490); 111 Project (D20031); Sichuan Science and Technology Program (2021YFG0149).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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