SymbolBased MultiLayer Iterative Successive Interference Cancellation for FasterThanNyquist Signaling
Abstract
A symbolbased multilayer iterative successive interference cancellation (MLISIC) algorithm is proposed to eliminate the intersymbol interference (ISI) for fasterthanNyquist (FTN) signaling. The computational complexity of the proposed MLISIC algorithm is much lower than most existing blockbased estimation algorithms. By comparison with existing symbolbased algorithms, the proposed MLISIC algorithm has higher estimation accuracy. Furthermore, by means of utilizing more accurate symbols and elaborate length of successive interference cancellation, an improved MLISIC (IMLISIC) algorithm is presented to further promote the estimation accuracy. Simulation results show in mild ISI cases, the proposed MLISIC and IMLISIC algorithms can approximate the theoretical performance even using 256amplitude phase shift keying (APSK) which is adopted in digital video broadcastingsatellitesecond generation extension (DVBS2X), i.e., bit error rate (BER) performance degradation is about 0.03 dB when the signaltonoise ratio is high. In other words, using the proposed MLISIC and IMLISIC algorithms in mild ISI cases, the spectrum efficiency can be improved with negligible BER performance degradation. In moderate ISI cases, BER performance degradation of our proposed IMLISIC algorithm is no more than 0.13 dB when adopting 64/128/256APSK, which is still satisfying. Besides, compared with existing symbolbased algorithms, the greater BER performance improvement can be achieved under severer ISI circumstances.
I Introduction
With the rapid development of the wireless communication technology in recent years, wireless devices grow exponentially, which results in more and more demand for communication services. In addition, the highthroughput satellite becomes a trend due to the scarcity of spectrum resources in satellite communications. Highorder modulations such as 256amplitude phase shift keying (APSK) and low rolling factors such as 0.05 are recommended in digital video broadcastingsatellitesecond generation extension (DVBS2X) [1], however, they are sensitive to the phase noise distortion and timing error respectively. As a nonorthogonal transmission scheme, fasterthanNyquist (FTN) signaling has attracted much more attention.
When designing the traditional communication system, the Nyquist criterion must be followed to avoid the intersymbol interference (ISI). Nevertheless, the orthogonality of Nyquist signaling is at the cost of sacrificing the spectrum efficiency. By deliberately introducing ISI, higher transmission rate and spectrum efficiency can be supported in the FTN system. Besides, FTN transmission can moderate the need for highorder modulations and low rolling factors. Thus, the adverse effect of the phase noise distortion and the timing error can be mitigated.
FTN signaling was first proposed by Mazo in 1970s. It is demonstrated in [2] that as long as the time acceleration parameter is no less than 0.802, i.e., , the minimum distance of uncoded binary sinc pulse transmission is not reduced in FTN signaling. Compared with the traditional Nyquist system, the FTN system can transmit about 25% more symbols in the same bandwidth without bit error rate (BER) performance degradation. The calculation of the minimum distance for FTN signaling was then given in [3]. Due to the fact that the pulse response of sinc pulse is infinite in time domain, raisedcosine (RC) pulses are introduced for FTN signaling in [4]. Since then, FTN signaling has been widely investigated in different aspects [5], such as FTN signaling using nonbinary or highorder modulations [6, 7]. The constrained capacity for FTN signaling is also clarified in [8], which shows that the FTN system can achieve a higher capacity than the Nyquist system unless sinc pulse is adopted. The advantages of frequencydomain FTN systems can be found in [9, 10]. In addition, FTN signaling has been applied to more scenarios, such as multicarrier [11, 12, 13] and multiinputmultioutput (MIMO) [14, 15] systems. Especially for satellite communications, FTN signaling is investigated as an innovative method to improve spectrum efficiency in [16, 17]. A receiver architecture for FTN signaling in the digital video broadcastingsatellitesecond generation (DVBS2) standard [18] is also presented [19].
On account of the violation of the Nyquist criterion in FTN system, it is inevitable to introduce undesired ISI. Hence, lots of research literature focused on the ISI cancellation algorithms. As known, the optimal ISI cancellation algorithm is the maximum likelihood sequence estimation (MLSE), which is not practical due to its nondeterministic polynomial (NP)hard computational complexity. Based on the Malgorithm and the BahlCockeJelinekRaviv (BCJR) algorithm, J. B. Anderson proposed the MBCJR algorithm for FTN signaling in [20]. Since the MBCJR algorithm only keeps several maximum states as possible states at each time, it can maintain the good performance when reducing the computational complexity of the BCJR algorithm. But the computational complexity of the MBCJR algorithm is still prohibitive for the practical implementation. In [9, 21], the truncated Viterbi algorithm and Malgorithm are also investigated for the uncoded singlecarrier FTN system. In 2013, a low complexity and effective frequencydomain equalizer (FDE) for the FTN system is proposed [22]. To a certain extent, however, its spectrum efficiency is decreased because of the insertion of guard interval, e.g., cyclic prefix. As a commendable method to eliminate ISI, the performance of precoding in the FTN system can be found in [23, 24, 25]. An iterative FTN detector in the presence of phase noise and carrier frequency offset is also developed in a factor graph framework [26]. Two successive symbolbysymbol sequence estimators are proposed to estimate binary and 4quadrature amplitude modulation (QAM) FTN signaling in [27]. It is proved that one of these two sequence estimators, which is named after successive symbolbysymbol with gobackK sequence estimator (SSSgbKSE), performs well in mild ISI cases and can significantly increase the transmission rate and spectrum efficiency. The semidefinite relaxationbased sequence estimation presented in [28] shows pretty good performance, but its computational complexity is disappointing when the block length is not that small.
Inspired by the successive symbolbysymbol sequence estimators in [27], we present a multilayer iterative successive interference cancellation (MLISIC) algorithm in this paper. And then, the improved MLISIC (IMLISIC) algorithm is proposed to further promote the estimation accuracy. The main contributions of this paper are summarized as follows:

Firstly, in consideration of the promising prospect of FTN signaling in satellite communications, FTN signaling is extended to highorder APSK modulations, which are adopted in DVBS2X, in this paper. As far as we know, this is the first effort in the literature reported. In addition, all the modulation types adopted in DVBS2X are evaluated in FTN systems.

Secondly, we propose a new MLISIC algorithm. The proposed MLISIC algorithm is based on the reception of one single symbol, instead of a block. Hence, in contrast to most of the blockbased sequence estimation algorithms mentioned above, lower processing delay can be achieved. Since the MLISIC algorithm is realized in time domain, it is not necessary to insert guard interval in FTN signaling, which ensures the high spectrum efficiency of FTN signaling. Besides, the computational complexity of our proposed MLISIC algorithm is quite lower than those of most blockbased sequence estimation algorithms. Even compared with the SSSgbKSE algorithm in [27], the proposed MLISIC algorithm has lower computational complexity in most cases.

Thirdly, based on the MLISIC algorithm, the IMLISIC algorithm is also proposed by using more accurate symbols and elaborate length of successive interference cancellation. The IMLISIC algorithm has higher estimation accuracy than the MLISIC algorithm. Hence, the proposed IMLISIC algorithm can achieve better BER performance than the proposed MLISIC algorithm even when their computational complexities are equal.

Fourthly but not the last, simulation results show that our proposed algorithms are characteristic by higher estimation accuracy than the FDE algorithm in [22] and the SSSgbKSE algorithm in [27] in mild and moderate ISI cases. For all the modulation types adopted in DVBS2 and DVBS2X, our proposed algorithms can approximate the theoretical performance in mild ISI cases. Even under moderate ISI circumstances, for which the SSSgbKSE algorithm is not suitable, the proposed algorithms perform quite well.
The rest of this paper is organized as follows: Section II introduces the system model of FTN signaling. The constrained capacity for FTN signaling is also covered in Section II. Section III discusses the proposed MLISIC and IMLISIC algorithms in detail. Simulation results are demonstrated in Section IV. Section V makes a brief conclusion.
Ii FTN SYSTEM MODEL AND The Constrained Capacity
In order to better illustrate our proposed algorithms, the FTN system model is first introduced in this section. After that, the relation among Shannon capacity, the capacity of FTN system and the capacity of Nyquist system is briefly explained.
Iia FTN System Model
The FTN system model used in this paper is illustrated in Fig. 1, where AWGN indicates the additive white Gaussian noise. The FTN signaling shaping is composed of a upsampler and a square root raised cosine (SRRC) shaping filter. P and Q represent the times of upsampling or downsampling used for the corresponding modules. As described in [29], the corresponding time acceleration parameter can be formulated as , where indicates the division operation. It is worth mentioning that if P equals to Q, the system is the traditional Nyquist system.
Symbols after the upsampler can be expressed as
(1) 
where N represents the total number of transmitted symbols. is the independent and identically distributed data symbols.
The transmitted signal of the FTN system shown in Fig. 1 can be written as
(2) 
where is the average energy of the transmitted symbols. is a unit energy pulse. is the sampling rate after the upsampler. and indicate the symbol rates of the Nyquist and FTN systems, respectively.
The received signal after the SRRC matching filter can be written as
(3) 
where . is the zeromean AWGN with variance .
Assuming perfect timing synchronization between the transmitter and the receiver, symbols after the downsampler can be denoted as [28]
(4) 
where is the desired symbol. The latter item represents the ISI caused by adjacent symbols.
When transmitted symbols are and sinc pulse is adopted, the received symbols in the noiseless Nyquist and FTN () systems are shown in Fig. 2, where numbers such as 0.61 and 0.32 represent values of ISI.
IiB Constrained Capacity of the FTN System
As described in [30], Shannon defined the capacity of the AWGN channel as
(5) 
where and represent the onesided bandwidth and the average power of transmitted signals, respectively. is the power spectrum density (PSD) of AWGN. is the Fourier transform of the shaping pulse .
According to [8], the capacity of FTN system under the constraint can be denoted as
(6) 
Correspondingly, the capacity of Nyquist system is
(7) 
Hence, it can be concluded that the constrained capacity of FTN system equals to the AWGN capacity defined by Shannon. Besides, the capacity of FTN system is no less than that of the corresponding Nyquist system, i.e., . And only when is the sinc pulse.
(14) 
Iii Proposed Multilayer Iterative Successive Interference Cancellation
Given that our proposed algorithms are inspired by the successive symbolbysymbol sequence estimators in [27], the SSSgbKSE algorithm is first introduced in this section. And then, our proposed algorithms are explicated in detail. The comparison of computational complexity is presented at the end of this section.
Iiia SSSgbKSE in [27]
Based on the FTN system model in Section II, symbols after the downsampler can be rewritten as
(8) 
where .
Therefore, in order to estimate the symbol , the interference of symbols before the receiving symbol should be eliminated. Based on this idea, the estimation of the successive symbolbysymbol sequence estimator (SSSSE) in [27] can be expressed as
(9) 
where rounds to the nearest symbol in the constellation.
The computational complexity of the SSSSE algorithm is quite low, however, it is sensitive to the error propagation, which means the estimation accuracy of the latter symbols will be seriously affected by the incorrectly estimated symbol. In [27], the SSSgbKSE algorithm is presented to reduce the error propagation effect and improve the estimation accuracy. The first step of the SSSgbKSE algorithm is consistent with the SSSSE algorithm. Once is estimated for the first time, the previous K estimated symbols, i.e., , can be reestimated by using the current estimated symbol . With the reestimated symbols , can be reestimated as
(10) 
where represents the reestimated symbol. It can be concluded that for the SSSgbKSE algorithm, each symbol is estimated up to times.
(17) 
The formula used for reestimating can be expressed as
(11) 
IiiB Proposed MLISIC
As depicted in (8), received symbols can be estimated correctly as long as the ISI caused by adjacent symbols is completely eliminated. Therefore, once is received, the desired symbol can be estimated for the first time as
(12) 
which is the first step of the proposed MLISIC algorithm. It is worth mentioning that all the symbols used on the right side of (12) are received symbols.
Once is obtained, the symbols after the first estimation can be used to reestimate as
(13) 
Similarly, when the symbol is received, the th () estimation is given in (14).
The complete estimation of the proposed MLISIC algorithm is shown in Fig. 3, where indicates the number of iteration. Each circle in Fig. 3 represents a received or estimated symbol. The subscript of the green circle in a layer is consistent with that of the symbol reestimated using symbols in the current layer. The orange and blue circles represent the symbols used for the estimation of the corresponding layer. In particular, the blue circle indicates the latest estimated symbol obtained by the current layer.
The proposed MLISIC algorithm is summarized in Algorithm 1.
IiiC Proposed IMLISIC
The MLISIC algorithm takes advantage of symbols in the previous layer to estimate symbols in the current layer, which means the estimation of each layer is independent. However, symbols that have been estimated in the current layer are not fully used in the estimation of the current layer. In addition, the latest estimated symbol in each layer is not used for the estimation that is performed after receiving the next symbol. Therefore, the IMLISIC algorithm is proposed to further improve the estimation accuracy. In order to be consistent with the MLISIC algorithm, assume that the number of iteration is and the singleside symbol length used for the ISI elimination in the th () iteration is .
After is received, the first iteration of the proposed IMLISIC algorithm can be denoted as
(15) 
It should be noted that this formula eliminates the ISI by using both symbols after the first estimation and received symbols.
And then, in the second layer and in the third layer can be used to estimate as
(16) 
Correspondingly, the th () estimation is denoted in (17).
Furthermore, the latest estimated symbol after the th () estimation can be used to update the estimated symbols with the corresponding subscript in the previous layers. Hence, () must be set elaborately. And should be set carefully to fully eliminate the ISI. The latest estimated symbol in the th () layer can contribute to the next estimation of the th () layer if
(18) 
where and indicate subscripts of the latest estimated symbols in the th and th layers, respectively.
The optimal () should make sure that the latest estimated symbol in the th () layer can be used for the next estimation of all the previous layers. In this case, the relation between and can be defined as
(19) 
where the optimal length increases exponentially with the number of iteration . Therefore, this formula is applicable to cases with small (generally, the satisfactory performance can be achieved when ranges from 2 to 6). In addition, it is not necessary to meet the optimal length, and the estimation accuracy can be sacrificed to simplify the complexity. One of these simplification methods can be represented as
(20) 
with which the latest estimated symbol in the current layer only contributes to the next estimation of the previous layer.
Similarly, the estimation diagram of the proposed IMLISIC algorithm is shown in Fig. 4. Circles in the first layer indicate the received symbols. Orange and blue circles represent the right symbols used for the estimation of the next layer. Yellow, green, and orange circles in the current layer indicate the left symbols used for the estimation of the current layer.
The proposed IMLISIC algorithm is summarized in Algorithm 2.
IiiD Comparison of the Computational Complexity
In order to eliminate the ISI on both sides of the desired symbol, the MLISIC algorithm needs to carry out additions/subtractions and multiplications in every estimation. Hence, given the estimation parameter and , additions/subtractions and multiplications are needed. As for the IMLISIC algorithm, additions/subtractions and multiplications are required in the th () estimation. The total computational complexity includes additions/subtractions and multiplications. Comparison with the algorithms in [27] is shown in Table I.
Algorithm 



SSSSE in [27]  
SSSgbKSE in [27] 



Proposed MLISIC  
Proposed IMLISIC 
With the U random samples of the Gaussian randomization and the block length , the algorithms presented in [28] require the computational complexity of and , respectively. These algorithms show satisfactory performance, whereas their computational complexities are too high for the practical implementation.
Besides, the multiplication numbers of the SSSgbKSE algorithm and the proposed algorithms are shown in Fig. 5, where . In this case, it can be seen that the multiplication numbers of the proposed algorithms are lower than that of the SSSgbKSE algorithm when K or L is small (the most common cases), otherwise, our proposed algorithms require more complexity. Furthermore, the computational complexity of the IMLISIC algorithm is higher than that of the proposed MLISIC algorithm if . In particular, at least additions/subtractions and multiplications are necessary for the IMLISIC algorithm when meets the optimal length, however, the computation complexity is still acceptable as long as and are not that large (common scenarios). The higher estimation accuracy can also be ensured even if is decreased to reduce the complexity. In addition, we will show that our proposed MLISIC and IMLISIC algorithms can achieve better BER performance with lower computational complexity than the SSSgbKSE algorithm in the next section.
Iv Simulation Results
Based on the FTN system model in Section II, the performance of the proposed MLISIC and IMLISIC algorithms is evaluated in this section. Quadrature phase shift keying (QPSK), 8phase shift keying (8PSK) and 16/32/64/128/256APSK are considered. Assuming that the order of SRRC filters is known and equals to 201, rolling factors of 0.3, 0.4 and 0.5 are applied in the FTN system. Besides, we only consider the time acceleration factors of 0.8 and 0.9. Unless otherwise mentioned, the length of cyclic prefix for the FDE algorithm in [22] is 10 and the block length equals to 1024, which means the spectrum efficiency is decreased by 0.98%.
ISI type 


0.8  25%  0.3,0.4  relatively severe ISI  QPSK, 8PSK, 16APSK  
0.5  moderate ISI  16/32/64/128/256APSK  
0.9  11%  0.3  mild ISI 

For the convenience of readers, all the simulated cases are summarized in Table II, where represents the improvement ratio of spectrum efficiency compared with the corresponding Nyquist system.
Iva Simulations in Mild ISI Cases
BER performance in mild ISI cases with is shown in Fig. 6. Other simulation parameters can be seen in Table III, where is equivalent to for the IMLISIC algorithm.
Modulation  Algorithm  

SSSgbKSE  6  3  
MLISIC  6  2  
IMLISIC  [7,6]  2  
32APSK  SSSgbKSE  8  3  
MLISIC  8  2  
IMLISIC  [9,8]  2  
64APSK  SSSgbKSE  8  4  
MLISIC  8  3  
IMLISIC  [8,8,8]  3  
128APSK  SSSgbKSE  8  5  
MLISIC  8  4  
IMLISIC  [8,8,8,8]  4  
256APSK  SSSgbKSE  13  5  
MLISIC  13  4  
IMLISIC  [13,13,13,13]  4 
Conclusions drawn from Fig. 6 are summarized as follows. Firstly, the proposed algorithms are superior to the SSSgbKSE and FDE algorithms no matter which modulation is adopted. And it should be noted that according to Table I and Table III, the computational complexities of the simulated MLISIC and IMLISIC algorithms are lower than that of the SSSgbKSE algorithm. In other words, our proposed MLISIC and IMLISIC algorithms can achieve better BER performance with lower computational complexity than the SSSgbKSE and FDE algorithms in these cases. Secondly, it can be observed that the theoretical performance of all the simulated modulations (256APSK included) can be approached with our proposed algorithms. The BER performance degradation is no more than 0.03 dB for 256APSK when the theoretical BER is , and it could be lower when adopting lowerorder modulations. By contrast, the FDE algorithm can not approximate the theoretical performance in the face of mild ISI cases, and the SSSgbKSE algorithm can only achieve this performance when adopting loworder modulations, such as QPSK. Particularly, the performance degradation of the SSSgbKSE algorithm is almost 2 dB when adopting 256APSK. Compared with the SSSgbKSE and FDE algorithms, therefore, there is no doubt that our proposed algorithms are more suitable for higherorder modulations in mild ISI cases. Furthermore, the performance difference between the proposed MLISIC and IMLISIC algorithms is tiny due to the mild ISI.
IvB Simulations in Moderate ISI Cases
BER performance in moderate ISI cases with is shown in Fig. 7. Other simulation parameters are listed in Table IV.
Modulation  Algorithm  
16APSK  SSSgbKSE  8  4 
MLISIC  8  4  
IMLISIC  [13,7,6]  3  
32/64APSK  SSSgbKSE  8  5 
MLISIC  8  5  
IMLISIC  [10,9,8,7,6]  5  
128/256APSK  SSSgbKSE  8  6 
MLISIC  8  6  
IMLISIC  [13,12,11,10,9,8]  6 
In contrast with the curves Fig. 6, due to the severer ISI, the BER performance of most algorithms in Fig. 7 becomes worse even though their complexities may be higher. In particular, BER performance of the SSSgbKSE algorithm is much better when adopting 256APSK because of the increase of . As shown in Fig. 7, the FDE algorithm is quite disappointing in moderate ISI cases. Nevertheless, our proposed MLISIC and IMLISIC algorithms are still superior to the other two algorithms and achieve excellent performance in moderate cases. To be more specific, for the theoretical BER of , the performance degradation of the proposed MLISIC and IMLISIC algorithms is almost 0.06 dB when adopting 16/32APSK. Using 64/128/256APSK, the degradation of our proposed MLISIC and IMLISIC algorithms is about 0.13 dB and 0.19 dB, respectively. Certainly, the performance of our proposed algorithms can be further improved by increasing or . In addition, there is almost no difference in BER performance among the SSSgbKSE algorithm and our proposed algorithms when signal to noise ratio (SNR) is low.
IvC Simulations in relatively Severe ISI Cases
BER curves in relatively severe ISI cases with and are shown in Fig. 8 and Fig. 9, respectively. Other simulation parameters are listed in Table V.
Modulation  Algorithm  
QPSK  SSSgbKSE  6  3 
MLISIC  6  3  
IMLISIC  [8,7,6]  3  
8PSK  SSSgbKSE  6  4 
MLISIC  6  4  
IMLISIC  [9,8,7,6]  4  
16APSK  SSSgbKSE  8  4 
MLISIC  8  4  
IMLISIC  [25,13,7,6]  4 
Through the observation of Fig. 8(a) and Fig. 8(b), BER performance of our proposed algorithms is still acceptable and better than that of the SSSgbKSE and FDE algorithms. As shown in Fig. 8(a), the degradation of our proposed MLISIC and IMLISIC algorithms is almost 0.07 dB and 0.12 dB for QPSK, respectively. As for 8PSK, the corresponding degradation is approximately 0.1 dB and 0.22 dB. When adopting the SSSgbKSE algorithm and our proposed algorithms as shown in Fig. 8(c), error floor appears, which means BER can not decline as SNR increases. Even in this case, the error floor can be reduced with our proposed algorithms. There is no error floor with the FDE algorithm, which shows the advantage of this algorithm to some extent.
Similarly, it can be seen from Fig. 9 that our proposed algorithms remain the superiority to the other two algorithms. Furthermore, BER performance of the IMLISIC algorithm is better than that of the MLISIC algorithm. However, it can be concluded from the results that the performance of these algorithms still has a lot of room for improvement, especially for highorder modulations.
IvD Analysis of the Proposed MLISIC and IMLISIC Algorithms
To further illustrate the proposed MLISIC and IMLISIC algorithms, their BER performance for the 32APSK FTN system with is shown in Fig. 10. It’s worth mentioning that although only one situation is demonstrated here, the conclusions are universal. And the parameter is chosen just because the phenomenon is more obvious. Comparing the green and blue curves, the superiority of the IMLISIC algorithm to the MLISIC algorithm can be verified. Besides, it can be concluded that increasing the number of iteration improves BER performance by comparison between the red and magenta curves. The magenta, blue and cyan curves indicate that in terms of the IMLISIC algorithm, BER performance degradation caused by the simplified is negligible if ISI is not quite severe.
V Conclusion
As a nonorthogonal transmission scheme, FTN signaling is a quite promising technology to alleviate the pressure brought by limited spectrum resources. Inspired by existing successive symbolbysymbol sequence estimators, we propose a low complexity and high accuracy sequence estimator, i.e., MLISIC, which requires additions/subtractions and multiplications for one single estimation. Besides, based on the MLISIC algorithm, the IMLISIC algorithm, which utilizes more accurate symbols and elaborate length of successive interference cancellation, is proposed to further improve the estimation accuracy. Although its computational complexity is higher than that of the MLISIC algorithm in most cases, it is superior to the MLISIC algorithm even when their complexities are equal. Compared with most blockbased algorithms, the computational complexities of the proposed MLISIC and IMLISIC algorithms are much lower. In addition, the performance of the proposed algorithms is superior to that of the SSSgbKSE algorithm under all the simulation cases considered in this paper. Furthermore, for all the modulation types adopted in modern satellite communication standards such as DVBS2 and DVBS2X, our proposed algorithms can approximate the theoretical performance in mild ISI cases, which is far beyond the performance of the FDE algorithm. Even under moderate ISI circumstances, the proposed algorithms perform satisfactorily.
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