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Abstract
Spectral efficient frequency division multiplexing (SEFDM) can greatly increase the spectral efficiency for intensity modulation/direct detection (IM/DD) optical communication systems. The sphere detection algorithm (SD) is the most efficient way to get the maximum likelihood (ML) solution to solve the inter-carrier interference (ICI) induced by the bandwidth compression of SEFDM system. However, SD algorithm is restricted by the numbers of subcarriers for SEFDM system, especially for larger numbers. Therefore, a sorted Gram-Schmidt (SGS) orthogonal decomposition algorithm, which can be applied to any subcarrier numbers, is proposed to overcome this restriction. To the best of our knowledge, the searched paths of FSD are researched for the first time to balance performance and complexity. Based on the analysis, a soft-tree-width sphere detection algorithm (SSD) is proposed and demonstrated by simulation and experiment. The results show that the proposed algorithm can greatly reduce the computational complexity (at least 40%) with the same system performance. The proposed algorithms are a promising candidate for flexible and efficient SEFDM systems. The SEFDM with the proposed detector is significant for the IM/DD optical systems.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Driven by the emerging network application such as big data, cloud computing, 5G, etc., the increasing demand for high speed and large capacity data transmission requires communication system to increase the spectral efficiency (SE). Intensity modulation/direct detection optical transmission system is considered as an appropriate solution to expand spectral efficiency due to its low cost and its simple configurations in cost-sensitive and short-range scenarios [1–3]. IM/DD systems have been widely studied to increase the SE with different advanced modulation formats, such as quadrature amplitude modulation (QAM), pulse amplitude modulation (PAM), and discrete multi-tone (DMT) [4–9]. In order to further increase the SE, direct detection faster-than-Nyquist technology (FTN) has been proposed to compress the signal bandwidth for IM/DD systems [10,11].
FTN is one of the most popular technologies to improve SE through compressing sub-carriers or carriers closer than the Nyquist signaling [12]. With the advantages of the compressing subcarriers, FTN can transmit higher speed signal with lower optical and electronic bandwidth. Therefore, FTN can realize higher-speed communication with limited bandwidth compared with Nyquist signal. With the development of integrated circuit chips and the decoding algorithms, the FTN signal has been widely used in bandwidth-limited scenarios such as fiber-optic communications, visible light communications, satellite broadcasting systems, mobile communication systems and so on [13–17]. However, the key challenge of FTN signaling system is the self-created interference which gives rise to the inter-carriers interference (ICI) because of the loss of the orthogonality between the subcarriers [12–14,18]. For single-carrier FTN system, many research methods have been reported to deal with the ICI of FTN signaling at a proper performance including maximum likelihood sequence estimation (MLSE), the decision feedback equalization (DFE), Tomlinson-Harashima precoding (THP) and multi-input multi-output detector (MIMO) [19,20]. Multi-subcarriers FTN systems, which are also called spectral efficient frequency division multiplexing (SEFDM), are widely researched in intensity modulation and direct detection (IM/DD) optical communication systems. However, SEFDM suffers the complicated ICI due to the multi-subcarriers. Therefore, many efforts have also been made to balance the trade-offs between performance and complexity [18]. Linear algorithms, such as zero forcing (ZF), truncated singular value decomposition (TSVD), minimum mean squared error (MMSE) and iterative detection (ID) have been proposed to compensate the ICI of SEFDM with poor performance and high complexity [20–25]. The maximum likelihood (ML) algorithm can be seen as the optimum detector to deal with the ICI. However, the highest complexity makes it not being implemented in the real communication [26–28]. Sphere detector (SD) is proposed to provide the ML performance with a greatly reduced complexity. In subsequent research, many efforts have been done to optimize the SD algorithm in consideration of system’s the performance and complexity [29–33]. The proposed SD algorithm is with the fixed tree width (Tw), named fixed SD (FSD). ZF/MMSE linear algorithms as the preposition algorithm of FSD are proposed to decrease the search radius with proper complexity for real experiment [24]. TSVD-FSD is employed for good complexity/performance trade-off to optimal ML and SD detection [30]. Then, ID algorithm and its optimal algorithm- cascaded BPSK ID (CBID) are also proposed to further modify the pre-algorithm of FSD for getting the best search radius [31–33]. However, these algorithms mainly focus on modifying the pre-algorithms and the SD algorithms which provide the ML performance have not been analyzed except the search radius (provided by the pre-algorithms). Besides, the SEFDM systems in these papers only work at a small certain number of subcarriers (e.g. 12, 15 and 16), which cannot be suitable for flexible optical networks and lose the advantages of multi-subcarriers communication system.
In this paper, we analyze the SD algorithm including the search radius and search path in detail for the first time, to the best of our knowledge. Based on this analysis, we find that the Cholesky decomposition (CHOD) used in SD algorithm cannot be always suitable for any subcarrier numbers, especially for larger numbers. Therefore, a sorted-Gram-Schmidt decomposition (SGS) is proposed to solve this problem and suitable for any subcarrier numbers. Based on the analysis of search path, we propose a soft-tree-width-SD (SSD) algorithm instead of FSD to reduce the complexity with the same performance or a little transmission performance improvement. At last, an experiment is demonstrated and the results show that the SEFDM with the proposed detector can reach the forward error correction (FEC) limit with lower bandwidth compared with Nyquist signal. The proposed detector with SEFDM is significant for the IM/DD optical commination systems.
2. SEFDM system and ICI
SEFDM is a multi-subcarriers FTN technology, and the frequency distance between the sub-carriers in SEFDM is defined by
where, $\mathrm{\alpha}$ is the bandwidth compression factor (BCF), and T is the duration of one SEFDM symbol. As for OFDM system, BCF=1, while BCF<1, the normalized SEFDM system signal can be given by
Fig. 1. (a) Block diagram of SEFDM, (b) illustration of OFDM, (c) illustration of SEFDM under 0.8 BCF
At the receiver, the received SEFDM signal $r(t)$ with additive white Gaussian noise (AWGN)$w(t)$ can be expressed as
And this equation can be expressed in matrix by
where, R is an N-dimensional vector of distorted symbols after demodulating the received symbol using FrFT operation. C is an N×N-dimensional matrix describing the ICI. This matrix is derived by correlating the sub-carriers F and its conjugate transpose matrix F* F is a N×N-dimensional matrix, and the elements in the matrix can be expressed as [28–31]:It is defined as C = F×F*.
3. Principle of SGS and SSD
3.1 FSD algorithm
The FSD algorithm comes from the ML estimation, and the estimation ${S_{ML}}$ of the originally sent symbols are given by the least square problem as [29]:
where, $||{\ast } ||$ denotes the Euclidean norm and m is the constellation cardinality. Because of the high complexity of ML solution, FSD algorithm is proposed. The FSD algorithm solves the ICI by examining only the points that exist within an N dimensional radius g such that Eq. (6) can be calculated as:Then, substituting by P = C-1R, leads Eq. (7) to:
The transformation of the problem in Eq. (8) into sphere-wise representation is achieved by Cholesky decomposition (CHOD) as chol(C*C)=L*L [25,27,28], where L is an N×N upper triangular matrix. Consequently, Eq. (8) can be expressed as
In order to reduce the complexity of FSD, the previous proposed papers use unconstrained pre-algorithms (TSVD, ID and so on) to the received signal R and optimal the search radius g by getting R’ and g’ instead of R, g, respectively [25–29]. Due to the upper triangular structure of L, successive interference cancellation is used to eliminate interference layer by layer. The survival nodes in each layer are the same. Therefore the algorithm is called fixed SD. An example of FSD algorithm with Tw=2 is shown in Fig. 2. After searching the limited sub-space of the problem, the correct transmitted symbols can be demodulated.
3.2 SGS decomposition
In conventional FSD detector, Eq. (8) applies Cholesky decomposition to form the upper triangular matrix L to reduce the complexity. As we know, the conditions of C*C being CHOD should be a positive definite matrix. However, for any number of sub-carriers, the C*C cannot be CHOD, especially for larger number, example for 32, 64 and so on. In previous papers, SEFDM system works at a certain number, e.g. 12, 15 and 16 [28,20–32]. In this case, the SEFDM system may not be used for flexible optical network and loses the advantages of communication system.
The Gram-Schmidt decomposition (GS) that can realize the QR (QL) decomposition, is used to channel estimation for wireless communication. For conventional GS decomposition, the diagonal element of the upper triangular L is lower from ${l_{1,1}}$ to ${l_{n,n}}$. A sorted- Gram-Schmidt decomposition (SGS) is proposed in this paper which can realize the decomposition after reordering the column to maximize the diagonal elements from ${l_{n,n}}$ to ${l_{1,1}}$. The SGS can suitable for any FFT size. The detailed algorithm can be seen in Ref. [26]. SGS in Ref. [26] is proposed for space-time codes.
In this paper, we use the SGS decomposition instead of CHOD. In previous papers, Chol(C*C)=L*L, and in this paper, SGS(CU)=QL, where, U is the reorder matrix, L is the upper triangular matrix with reordered diagonal elements, and Q*Q = I unit matrix. Therefore, the Eq. (8) can be expressed as:
The Eq. (10) is similar with the Eq. (9). Therefore, similar to CHOD, the influence of ICI can be successfully reduced. However, SGS can be suitable for any matrix C compared with CHOD. The CHOD only be suitable for positive definite matrix C*C. When the matrix is positive defined matrix, the decomposition is unique and SGS is same as CHOD. Table 1 is the processing flow and description of SGS and CHOD.
Table 1. SGS decomposition, CHO decomposition and their descriptions
3.3 SSD
The received signal after pre-algorithms is sent into FSD algorithm for detection. The successive interference cancellation is used to eliminate interference layer by layer according to Eq. (10). Figure 2 illustrates the FSD algorithm with a 3 carrier in OOK system. It is obvious that, for FSD algorithm, the retained nodes, which is named tree width (Tw) at every layer are the same. There is also the trade-off between the performance and the Tw. The larger the Tw is, the better the performance of the SEFDM system is. However, the larger Tw will make complexity drastically increase. For large number of subcarriers, the Tw should be increased as well in order to obtain acceptable performance.
The retained nodes are selected from Eq. (10). Therefore, the retrained nodes and discarded nodes can be analyzed from Eq. (10). For SGS algorithm, U in Eq. (10) is the just reordered matrix, which is used to reorder the recovered signal. For simplicity, we neglect the U in Eq. (10). Due to the upper triangular matrix L, Eq. (9) can be written as:
The decoding process can be regarded as descending down in a tree in which each node has M branches. If a TCF exceeds g, the entire branch and all its descendants are pruned. In order to decrease the complexity of SD, FSD algorithm is proposed, which selects Tw survival nodes (& PED < g) at each layer. Figure 3 shows the relationship between the survival nodes and TCF. For traditional algorithms, such as MMSE-FSD, TSVD-FSD, ID-FSD and other FSDs, the survival nodes in each layer are fixed, so the larger Tw can improve the performance, while deeply increases the complexity. The trade-off between the performance and the complexity is decided by the Tw, so the optimal Tw need to be calculated and analyzed.
We analyze the reason that causes the performance degradation firstly. Assume that we have two candidate symbols (nodes) s3 and s4 at K-th layer, and TCF are
${T_{k,3}}({s_3}) = \sum\limits_{i = N}^K {{e_{i,3}}({s_3})}$ and ${T_{k,4}}({s_4}) = \sum\limits_{i = N}^K {{e_{k,4}}({s_4})}$, respectively. Suppose s3 is the ML solution at the layer of K, then, ${T_{k,3}}({s_3}) < {T_{k,4}}({s_4})$, and FSD algorithm selects the s3 as candidate instead of s4. If we also consider these two nodes in the next layer of K-1, then the TCF are
Although in the layer K, ${T_{k,3}}({s_3}) < {T_{k,4}}({s_4})$, the added PED can also have deeply influence on the TCF, which may make ${T_{k\textrm{ - }1,5}}({s_5}) > {T_{k\textrm{ - }1,7}}({s_7})$ in the layer K-1. In other words, the s4 is the best ML solution branch at the final decision. However, in the upper layer K, the ML solution s4 has been discarded. Even though we select the best TCF at early layers, the excluded PEDs are still possible to make the discarded nodes become the best ML solution. Thus, the error at early layer will propagate and make us miss the ML solution. The description can be seen from the nodes 3, 4 of 1-layer and 5, 7 of 0-layer in the Fig. 3. For conventional FSD algorithm, increasing the Tw can retain more nodes, which reduces the probability of missing ML solution, however, improves the computational complexity.
Based on the analysis of PED and TCFs, we can easily find that the discarded nodes are decided by the TCF (TCF is a sum of PEDs). The reason why the ML solution miss is that the added PED in later layer changes the order of TCF, which means the discarded nodes at early layers may be retain at the later layers. The TCF at K layer is a sum of K PEDs from Eq. (10). The PED can be calculated by the Eq. (12). Thanks to the triangle decomposition L, we can easily reduce the complexity at a good performance. One reason is that according to upper triangle matrix L of the SGS (or CHOD), the value of diagonal elements is larger than the other elements of the same rows, therefore, the PEDs at each layer are mainly depending on the diagonal elements. Thus, the Eq. (11) can be expressed as ${e_{i,n}}({s_n}) \approx {|{{l_{i,i}}({p_i} - {s_{i,n}})} |^2}$. The other reason is that the PEDs at each layer nearly equal to each other because the diagonal elements have almost the same value except at early layers. Based on these two reasons, assuming the PEDs are the same at each layer, then the Eq. (13) can be modified as ${T_k}(s) \approx k\ast e(s)$. It is easy to reduce from the equation that the added PED at K layer just accounts a little for the TCF, that is, in this layer the PED can be hard to change the order the TCF at the later layer. Therefore, the Tw is decreased as the layer is increased. In the same way, the Tw in early layer should be properly larger to reduce the possibility of missing the ML solution. The Tw can be set as descending order, such as [12,8,4,2] for 4 layers decoding. In this way, the performance will be increased with the lower complexity compared with 8 Tw FSD.
In the formal deduction, this paper assumes that the diagonal elements are almost the same for each layer. In fact, if the elements at early layers can be enlarged, then ${e_{k - 1}} > {e_k}$. The PEDs are so small to change the TCF. Therefore, the performance will further improve. In Section 3.2, we apply SGS decomposition to reorder the column of the diagonal elements, except for getting the upper triangle decomposition. For the SGS algorithm, the lager diagonal elements can be put at the early layers while the small diagonal elements can be put at the later layers. Therefore, the ML solution can be put for the smallest TCF at each layer. It can be concluded that the SSD algorithm is more suitable for SEFDM decoding.
The algorithms of SGS-SSD and CHOD-FSD are descripted in Fig. 4. The pre-algorithms can be ZF, TSVD and ID. $S_1^{pre}$ and $S_1^{\textrm{un} - pre}$ are the constrained matrix. The unconstrained matrix of R from pre-algorithms can be seen in Ref. [22–26] for detail.
4. Simulation and results
Based on the discussion of the decomposition algorithm SGS and path search algorithm SSD, we build a SEFDM transmission system via MATLAB and VPI TransmissionMaker 9.0 environment, as is shown in Fig. 5. In the simulation, the symbol rates are set to be 8 Gbaud (for BCF=0.8), 8.5 Gbaud (BCF=0.85), and 7.5 Gbaud (BCF=0.75). The symbol rate is 10 Gb/s. The transmit data stream is a pseudo random bit sequence of length 215-1, and mapped to 4/16 QAM. After IFrFT, adding CP, and serial to parallel transforming, the SEFDM signal is generated by MATLAB at the transmitted side. Then, two symbols are added into the head of the SEFDM symbols to be synchronization and channel estimation. The baseband SEFDM signal is subsequently modulated by a 10 GHz RF signal for directed modulation and then sent into modulator. The sampling rate is 40 GSa/s. The generated SEFDM signal modulated by an intensity modulator is transmitted into single mode fiber, with an optical amplifier of 5 dB noise figure. The optical carrier is generated by a laser with linewidth of 100 kHz and power of 0 dBm. A variable optical attenuator (VOA) is utilized to modify the receiver optical power (ROP) and optical signal noise ratio (OSNR). At the receiver side, the signal is converted to electronic signal by a photo-diode. Then, the training sequence is extracted for synchronization and channel estimation. After FrFT, the SEFDM symbols are decoded by the FSD and SSD algorithms. In this paper, ID algorithm is used as the pre-algorithm with 4 iterations for the search radius of SD.
Figure 6 shows the Tw selection based on the complexity reduction and the performance. The SEFDM system is modulated by 4 QAM, 16 subcarriers and BCF of 0.8. The reference OSNR is the FSD algorithm at Tw of 8. The results show that the complexity reduces rapidly and the performance is almost same at first, and then the trend reverses. Therefore, the inflection point is selected as the optimal Tw value.
Figure 7 shows the results of SGS-SSD algorithm and CH-FSD algorithm under different Tw. The modulation format is 4QAM. The Tw1 for SGS-SSD means that the Tw at each layer is [4,16,16,16,16,8,8,8,8,4,4,4,4,2,2,2], and Tw2 for SGS-SSD is [4,12,12,12,12,4,4,4,4,4,4,4,4,2,2,2], while Tw for CH-FSD algorithm is fixed 16, 8, 4 or 2 at each layer, respectively. When BCF=0.85, the SGS-SSD algorithm has almost the same performance with the CH-FSD at the Tw of 16 and 8. The complexity for SGS-SSD is much lower than the CH-FSD. However, the performance of SGS-SSD outperforms CH-FSD at the Tw of 4 and 2 with appropriate complexity. As for BCF=0.75, the difference between SGS-SSD and CH-FSD is almost the same. The performance of SEFDM is similar to that of OFDM. The faintly uneven BER curves are coming from imperfect channel estimation and algorithm error. These comparisons neglect the imperfect channel estimation. In the later simulation, the Tw2 and Tw=8 are utilized by SGS-SSD and CH-FSD algorithms, respectively.
Table 2 shows the complexity and performance of SGS-SSD algorithm and CH-FSD algorithm for comparison. The complexity (PEDs) is calculated by the number of PEDs $e(s)$. However, we should note that one PED is counted as 1 in the Table. The computation complexity is increased from high layers to low layers, according to Eq. (11). The computation complexity (CC) of one PED at the layer i can be expressed as:
where, CM is the complex multiplication operation and CA is the complex addition. From Table 2, the proposed algorithm decreases the Tw at the later layer. SGS-SSD with Tw2 can reduce 48% complexity of the CM and CA compared with CH-FSD with Tw=8 for BCF of 0.8 and 0.75, respectively. The performances have been proved almost the same, which are only 0.03 dB and 0.1 dB of OSNR penalties.Table 2. Performance and complexity comparison
In section 3, the CHOD decomposition is suitable for any subcarrier numbers, for example 32 carriers. Therefore, we use GS algorithm (QR decomposition) instead of the CHOD for 32 subcarriers. The performances of GS-FSD (CH-FSD) and SGS-SSD algorithms under 32 subcarriers (16 subcarriers) with different BCFs are demonstrated in Fig. 8. The modulation format is 4QAM. The Tw for FSD are 8 and the Tws for 16 and 32 subcarriers SSD are Tw2 and Tw3= [4,12,12,12,12,12,8,8,8,8,8,8,8,8,8,8,8,8,4,4,4,4,4,4,4,4,4,4,4,4,2,2,2,2], respectively. Table 3 shows the complexity of SSD and FSD algorithm under different subcarriers. The SSD can reduce 48% and 40% complexity of the CM compared with FSD for 16 and 32 subcarriers, respectively. For BCF=0.85, SGS-SSD and GS(CH)-FSD are almost the same. For BCF=0.8, the OSNR penalties of SGS-SSD are about 0.03 dB and 0.4 dB better than GS(CH)-FSD for 16 and 32 subcarriers, respectively. For BCF=0.75, the OSNR penalties of SGS-SSD are 0.12 dB and 1 dB better than GS(CH)-FSD for 16 and 32 subcarriers, respectively.
Table 3. Complexity of SSD and FSD for different subcarriers
Figure 9 shows the performance of SGS-SSD algorithm and CH-FSD algorithm for 16QAM under 16 subcarriers. The Tw is set as the same as 4QAM. As BCF=0.85, the performance of SGS-SSD is almost the same as that of CH-FSD. For BCF=0.8, only SGS-SSD algorithm can reach the FEC threshold. For this case, the SGS-SSD can reduce 48% complexity of the CM compared with CH-FSD.
Figure 10 shows the transmission length versus required OSNR at the FEC of 3.8×10−3 for SGS-SSD and CH-FSD algorithms. The modulation format is 4QAM. The OSNR penalties are increased when the fiber lengths are increased with CD. The performance of OFDM increases faintly with the increase of fiber length. Without considering the chromatic dispersion, the performance is almost same. The OSNR penalties induced by the increased fiber length are from the imperfect channel estimation, which is used to compensate the CD in the fiber. The channel estimation algorithm in SEFDM system is the algorithm widely used the OFDM as usual. However, the self-created ICI of SEFDM system, which is absent in OFDM, has greatly influence on performance of the channel estimation algorithm. Therefore, the SEFDM system suffers from this penalty. When the fiber length is longer, larger OSNR is required to compensate the shortcoming of the channel estimation. That is the reason that the SEFDM systems are widely used in short-reach transmission systems.
5. Experiment and results
In order to verify the performance of the proposed algorithms, an experiment of direct detection optical SEFDM system is demonstrated, as is shown in Fig. 11. At the transmitter, SEFDM signal is generated with different BCF, 2 Gbaud for 0.8 BCF and 1.835 Gbaud for 0.75. Due to the bandwidth limitation of photo-detector, the bandwidth of our system is no more than 8 Gbaud. The SEFDM signal is set as 4 QAM with 16 or 32 subcarriers. Then the formed SEFDM is modulated on the radio frequency (RF) of 5 GHz for the optical direct modulation to avoid the second-order inter-modulation distortion due to the square law photo-detector (PD). The RF-SEFDM signal is generated in the off-line computer platform. Then, the RF-SEFDM signal is sent into the Agilent arbitrary waveform generator (Keysight M8195A) at the sampling rate of 64 GSa/s with 8-bit resolution. After amplified by a linear broadband electrical amplifier, the signals are used to driven the intensity modulator with 3 dB bandwidth of 20 Gband. A laser with a linewidth of 100 kHz is used to generate the optical carrier. With the launch power of 2 dBm, the optical signal is transmitted in to a 7 km single mode fiber (SMF). At the receiver, a variable optical attenuator are used to adjust the received optical power. The optical signal is then detected by PD with a bandwidth of 8 Gbaud. Then, the RF signal is fed into Tektronix real-time oscilloscope at the sampling rate of 80 GSa/s and processed off-line.
Figure 12 shows the performance comparison of SGS-SSD and CH-FSD algorithms under different BCF. The subcarriers of SEFDM are 16, and the signal is modulated as 4 QAM. The Tw of SGS-SSD is set as Tw2, and the Tw of CH-FSD is fixed at 8. From Fig. 12, when the BER is reaching the HD-FEC threshold for back to back (BTB) transmission, the performances of SGS-SSD algorithm achieve 0.15 dB and 0.16 dB ROP improvement than that of CH-FSD for the BCF of 0.8 and 0.75, respectively. As for 7 km SMF transmission, the ROP improvements of SGS-SSD are about 0.16 dB and 0.15 dB than that of CH-FSD, respectively. The results show that the SGS-SSD algorithm with Tw2 can reduce 48% complexity without performance deterioration.
For 32 subcarriers, the CHOD algorithm cannot work. Therefore, we use GS algorithm instead. The performances comparison for SGS-SSD and GS-FSD is demonstrated in the Fig. 13. The Tw of GS-FSD is set as 8 and that of SGS-SSD is set as Tw3. The ROP penalties of SGS are 0.4 and 0.6 dB better than that of GS-FSD for BTB and 7 km SMF transmission, respectively.
6. Conclusion
In conclusion, this paper proposes SGS decomposition instead of CHO decomposition, which can be suitable for any subcarrier numbers. The reason of performance degradation of FSD algorithm is analyzed in detail for the first time. Based on the analysis, a better search path algorithm is proposed to reduce the computational complexity without decreasing the transmission performance. These two algorithms have been demonstrated by simulation and experiment. For larger number of subcarriers system (32 subcarriers), the SGS decomposition can operate as well, while the CHO decomposition cannot work. The results also show that the proposed algorithms can reduce the complexity of FSD at least 40% without the performance deterioration. SEFDM systems with the proposed detector is a promising candidate for IM/DD communication systems.
Funding
National Natural Science Foundation of China (61471179, 62001181); China Postdoctoral Science Foundation (2018M642843).
Disclosures
The authors declare no conflicts of interest.
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