CHE Hui ,PENG Dingxiang ,GUO Fachang ,and BAI Yong

1.School of Information and Communication Engineering,Beijing University of Posts and Telecommunications,Beijing 100876,China;2.Ruijie Networks Co.,Ltd,Fuzhou 350002,China;3.School of Information and Communication Engineering,Hainan University,Haikou 570228,China

Abstract:Multi-carrier faster-than-Nyquist (MFTN) can improve the spectrum efficiency (SE).In this paper,we first analyze the benefit of time frequency packing MFTN (TFP-MFTN).Then,we propose an efficient digital implementation for TFP-MFTN based on filter bank multicarrier modulation.The time frequency packing ratio pair in our proposed implementation scheme is optimized with the SE criterion.Next,the joint optimization for the coded modulation MFTN based on extrinsic information transfer(EXIT) chart is performed.The Monte-Carlo simulations are carried out to verify performance gain of the joint inner and outer code optimization.Simulation results demonstrate that the TFPMFTN has a 0.8 dB and 0.9 dB gain comparing to time packing MFTN (TP-MFTN) and higher order Nyquist at same SE,respectively;the TFP-MFTN with optimized low density parity check(LDPC) code has a 2.9 dB gain comparing to that with digital video broadcasting (DVB) LDPC.Compared with previous work on TFP-MFTN (SE=1.55 bit/s/Hz),the SE of our work is improved by 29% and our work has a 4.1 dB gain at BER=1×10-5.

Keywords:faster-than-Nyquist,filter bank multi-carrier,timefrequency packing (TFP),spectrum efficiency (SE),joint optimization.

1.Introduction

Multi-carrier faster-than-Nyquist (MFTN) can improve the spectrum efficiency (SE) by packing time interval between shaping pulses or frequency spacing between adjacent subcarriers.MFTN is a potential key technology for the mm-Wave and THz communication [1] based on the beamforming [2].The MFTN introduces inter-symbol interference (ISI) or inter-carrier interference (ICI)due to the violation of orthogonality condition.Rusek and Anderson introduced the MFTN in [3] and further explored it in [4].The TFP (TFP)-MFTN improves the SE by packing both time interval and frequency spacing.We take the following approach: Firstly,we study the signaling limit for MFTN with the Gaussian input,i.e.,MFTN limit,and check TFP gain for MFTN by the numerical calculation method.If there is indeed a TFP gain for MFTN in term of the MFTN limit,we design the efficient digital implementation (EDI) of MFTN.Then we calculate the information rates (IR) for MFTN based on the EDI with the finite alphabet input.If there is indeed a TFP gain for MFTN in term of the IR (or SE),we design the coded modulation (CM) MFTN.Finally,we do the BER simulation for the CM-MFTN and check a TFP gain for MFTN in term of bit error rate (BER).The maximum TFP gain derives from the joint optimizations for MFTN,such as time packing ratio (TPR) τ,frequency packing ratio (FPR) ν and channel code.

The MFTN limit is achieved by numerical calculation method with the Gaussian input,which has the low complexity.Rusek and Anderson in [5] only considered the time packing (TP) for faster-than-Nyquist (FTN) system with Gaussian input.The authors in [6,7] only considered the frequency packing (FP) for MFTN with the Gaussian input.However,the benefit of MFTN using TFP (TFP-MFTN) with the Gaussian input has not been fully investigated and understood.We use the matrix analysis method [8,9] to study the benefit of MFTN using TFP with the Gaussian input.Our study concludes that TFP-MFTN can achieve a better performance than TPMFTN system,and reduce the gap between TP-MFTN and Shannon limit.The Shannon limit employs the sinc pulse and Gaussian input,and is optimal in theory.

For TFP-MFTN,we design the efficient digital implementation containing three parts: modulator,demodulator and equalizer.For the modulator and demodulator,there are four types implementation,i.e.,FTN mapper,overlap-add type implementation [10],direct implementation and EDI [11].The FTN mapper represents the MFTN shaping pulse using the orthogonal basis which is actually the transmitted pulses over the communication channel [12,13].The calculation of projection coefficient and the reconstruction in [12] would induce additional complexity [13].Wang et al.[13] proposed a new FTN mapper with low complexity for TP-MFTN but the cyclic prefix (CP) in [13] reduces the SE.There are serious outof-band (OOB) leak for the transmitted pulses for MFTN with FTN mapper [12,13] because of short time support.Peng et al.[14,15] employed overlap-add type implementation for the lattice staggered MFTN.The FPR in [14,15]is a rational number and not flexible for MFTN.The MFTN with overlap-add type implementation has high complexity when the denominator of the rational number is large.The overlap-add implementation is designed for low interference in [10].There are also serious OOB leak for the shaping pulses in [15] because of the short time support.The direct implementation for TFP-MFTN was employed in [4,16-18].The complexity for the previous MFTN signaling without EDI is high.We propose an EDI of MFTN based on filter bank multicarrier modulation.The filter for the EDI is performed at rate 1/Ts instead ofL/Ts.Per subcarrier equalization (PSE) is needed in order to reduce the remaining interference.There are three type PSE,i.e.,symbol-by-symbol (SBS)equalizer (zero forcing equalizer),linear minimum mean square error (LMMSE) equalizer and maximum a posteriori probability (MAP) equalizer.The complexity is low for the SBS equalizer with successive interference cancellation (SIC) [14,19] and without SIC [12,15,16].However,the presences of ICI and ISI make IR un-achievable for an SBS equalizer [17].The LMMSE equalizer in [13]eliminates the ISI for TP-MFTN.Both zero forcing and MMSE are the suboptimal equalization criterion.We can reduce complexity of the receiver with a little performance loss.However,we cannot pursue a reduction in receiver complexity at the cost of a significant reduction in performance.The MAP detector works with a truncated version of the channel response for MFTN in [4,18]and it may yield poor performance.Colavolpe et al.[17]have considered a MAP equalizer enhanced by the use of the channel shortening (CS) [20] for MFTN.However,the MFTN in [17] employs direct implementation and does not eliminate the ICI.We employ the MAP equalizer based on the CS (CS-MAP) together with the parallel interference canceller (PIC) for MFTN with the EDI.The complexity of PIC in this paper is lower than SIC in[4] because of the EDI.The CS-MAP equalizer eliminates ISI and the PIC eliminates the ICI.The TFP-MFTN may have better performance (eliminating both ISI and ICI) and its complexity is reduced by the EDI.

The achievable IR or SE can be practically approached by CM-MFTN with the joint optimization [18].There are two parts for the joint optimization,TFP system parameters (TPR and FPR) and the proper LDPC code.The TPR and FPR optimizations do not need to consider the specific forms of LDPC code but the LDPC code optimization need to consider the specific forms of MFTN.For the TPR and FPR optimizations,Rusek and Anderson had guided the selection of TPR and FPR in [4] by maximizing minimum Euclidean distance (MED).However,the MED is not a robust and reliable metric [21].The SBS equalizer is employed for the TPR and FPR optimization by maximizing IR in [16] and signal-to-interference ratio(SIR) in [19].However,the presence of ICI and ISI makes IR un-achievable by a simple SBS equalizer which is not works on a trellis [17].The MAP equalizers without CS in [18] and with CS in [17,22] are used for the TPR and FPR optimization by maximizing IR.However,the complexity of the TPR and FPR optimizations in[17,18,22] is high because of the direct implementation and extensive search over all possible TPR and FPR.We perform the TPR and FPR optimization based on EDI of MFTN and the initial FPR which is from the numerical calculation with the Gaussian input can speed up search.If TFP-MFTN with the optimal TPR and FPR has the gains compared to TP-MFTN and Nyquist systems at same SE,we will do channel code optimization for the optimal TFP-MFTN.Many standards adopt LDPC codes to achieve better performance and this paper carries out LDPC code optimization for the optimal TFP-MFTN with the EDI.Degree distribution or degree matric of LDPC is a key parameter affecting the entire CM-MFTN system.Bocharova et al.[23] and Yu et al.[24] optimized the degree matrices by minimizing the frame error rate (FER) for the single carrier (SC) FTN.However,the complexity of exhaustive search in [24] is very high.Secondini et al.[18] optimized the degree distribution by a curve fitting on extrinsic information transfer chart(EXIT) for MFTN,which is an exhaustive search over all possible degree distribution.We propose a particle swarm optimization (PSO) to replace the exhaustive search method for the degree distribution optimization.PSO is one of most famous artificial intelligence (AI)-based optimization algorithms [25,26].The bare-bones PSO (BBPSO) algorithm is the simplest of all PSOs.With the fitness function in [27],the BB-PSO algorithm [28,29] can search the most suitable degree distribution based on EXIT charts.Secondini et al.[18] built the parity check matrix of an LDPC code with the optimal degree distributions by the progressive edge growth (PEG) algorithm.However,neither exploiting girth only nor together with the extrinsic message degree yields an optimal performance of LDPC codes [23].In addition,the doping method [30] can remove the error floor and need a higher pinch-off threshold.The doping rate also is optimized by EXIT chart.

The Monte-Carlo simulations are carried out to verify performance gain of the joint optimization proposed for the CM-MFTN.Parts of our work were presented in[31].In this paper,we conduct further study and give more results on the benefit analysis of TFP-MFTN and the joint optimization for our proposed TFP-MFTN with the EDI.Our main contributions on the study of TFPMFTN can be summarized as follows:

(i) We analyze the benefit of the TFP-MFTN by the method of numerical calculations with Gaussian source and conclude that the TFP-MFTN system can achieve a better performance than TP-MFTN system.

(ii) We propose CS-MAP and PIC for the MFTN with the EDI.

(iii) We perform the joint optimization for TFP-MFTN with the EDI.We employ a BB-PSO to replace the exhaustive search method for the degree optimization.Especially,we find that the average degree of LDPC for the FTP-MFTN system is usually smaller than that for the Nyquist system.

2.System model

The system model for coded modulation MFTN system is illustrated in Fig.1.

Fig.1 Coded modulation MFTN system model

In the transmitter,the information vectoruis encoded by a binary channel encoder to generate a codewordc,whereci∈{0,1}.The code rate of the channel encoder isr.The codewordcis permuted by a random bit-interleaver.The output of bit-interleaverdfeeds the doping encoder,which is a rate-1 recursive convolutional code,to remove the error floor.The output of doping encoder,v,is delivered to theM-ary signal mapper which employs a quadrature amplitude modulation (QAM) and Gray labelling.M-ary symbol sequencexis delivered to the MFTN transmitter.In multicarrier systems,information is transmitted over pulses which usually overlap in time and frequency.The transmitted signals(t)of a MFTN system[4,32,33] can be expressed in time domain as

whereLis the number of the subcarriers,Kis the number of the FTN symbols on every subcarrier andxl(k)denotes the transmitted symbol at subcarrier positionland time positionk,which is expressed as

andx(kL+l) belongs to a given zero-meanM-ary complex constellation.The transmitted basis pulsehl,k(t)[32,33] in (1) is defined as

whereTsdenotes the time spacing andFcis the subcarrier spacing.It is essentially a time and frequency shifted version of the prototype filterh(t).The shaping pulse (3)is different from [16] and an additional phase component leads to an efficient polyphase filter bank structure in (8).TheT-orthogonal root raised cosine (RRC) pulse with a roll-off factor β is employed as the prototype filter (or shaping pulse).

whereTis the orthogonal symbol time,Wis the orthogonal subcarrier spacing andW=(1+β)/T.In order to maintain almost the same power spectral density (PSD),the shaping pulse has a time-truncation to ±ζTaroundt=0 and ζ=15 in this paper.

When τ=1,orthogonality between the neighboring shaping pulses is ensured by the Nyquist zero-ISI theorem.When τ ＜1,MFTN packs the transmission interval of the neighboring shaping pulses and introduces ISI,as shown in Fig.2.When ν=1,orthogonality between subcarriers is ensured by using non-overlapping spectral characteristics as compared with the overlappingsinc(f)type spectrum employed in OFDM.When ν ＜1,MFTN packs the subcarrier spacing and introduces ICI,as shown in Fig.3.The MFTN with τ ＜1 and ν=1 is denoted as TP-MFTN.The MFTN with τ=1 and ν ＜1 is denoted as FP-MFTN.The MFTN with τ ＜1 and ν ＜1 is denoted as TFP-MFTN.

Fig.2 Time domain compression of MFTN

Fig.3 Frequency domain compression of MFTN

The MFTN signals(t) is transmitted over the additive white Gaussian noise (AWGN) channel and the received signal is given byr(t)=s(t)+w(t),where the white Gaussian noisew(t) has two sides PSDN0/2.The MFTN receiver projects the received signalr(t) onto the basis pulseshl,k(t),that is,

where * means convolution operation,and the superscript (·)?indicates the conjugate operation.The output of the MFTN receiver feeds the PIC.We evaluate the performance of this communication system where the PSE is employed at the receiver.The MFTN equalizer employs the max-log-MAP equalizer for each subcarrier.The FTN equalizer uses with CS method [20,34] based on the ungerboeck observation model [35].

As shown in Fig.1,the channel encoder serves as the outer code;the cascade of doping encoder,signal mapper and MFTN transmitter serves as the inner code.The log likelihood ratio (LLR) values are passed between the FTN equalizer and the LDPC decoder.After several iterations,a sequencecan be found,which is regarded as the estimate ofu.

We emphasize two benchmarks which are important for evaluating the BER performance of the CM-MFTN with the joint optimization and the two benchmarks are ignored by most of the papers.The benchmark A is the Shannon limit which has the same SE as the CM-FTN (or CM-Nyquist).The benchmark A shows us the gap between performance of the CM-FTN (or CM-Nyquist)with joint optimization and the optimal performance in theory.This gap analysis is suitable for both Nyquist and FTN system.If this gap is too large,the CM system is poor.The MFTN with the convolutional code can achieve full gain [4].However,this MFTN with the convolutional code is poor because its upper performance limit(the Nyquist with convolutional code) has 3.5 dB loss compared to the benchmark A at BER=3 × 10-5.Therefore,we replace convolutional codes with LDPC codes in this paper.The benchmark B is the CM-Nyquist( τ=1,ν=1) which has the same SE as CM-MFTN and executes the joint optimization.The optimized modulation and channel code for CM-Nyquist can be found in many communication standards.The benchmark B shows us the gap between performance of the CM-MFTN with joint optimization and the CM-Nyquist with joint optimization in practice.If the CM-MFTN with joint optimization has no gain compared to the benchmark B,we do not need MFTN because of its high equalization complexity.MFTN with convolutional codes has a 1.5 dB loss compared to the benchmark B at BER=3 × 10-5.Thus,we need the joint optimization to improve the MFTN performance.

3.Benefit analysis of TFP-MTN

The effect of TFP operation on spectral efficiency is studied by the numerical calculation method.Although ICI is presented due to FP,a single-carrier detector which treats ICI as noise [6] is considered at the receiver.We limit our investigation to the situation that one subcarrier is only interfered by adjacent subcarriers,i.e.,0.5 ≤ν ＜1.

3.1 SE for MFTN

The matrix analysis is an efficient numerical calculation method to obtain the SE of TFP-FTN system with Gaussian source.The formulas for capacity of multiple input multiple output (MIMO) system are derived by the matrix analysis method [8].The formulas for SE of TFP-MFTN system can also be formulated by using this method.

Insertings(t) into (5) yields the model

whereyl(k) is the received signal for thelth subcarrier without ICI,

ICIl+(k) and ICIl-(k) are adjacent ICIs for thelth subcarrier.ICIl+(k) and ICIl-(k) are from the (l+1)th and the(l-1)th subcarrier,respectively.

ηl(k) represents the complex color Gaussian noise for thelth subcarrier.

wherew(t) is the complex white Gaussian noise.The received signal for thelth subcarrier without ICI can be expressed as

where the operator ×represents Hadamard product and

whereg(l)=g(-l) andwg=2πl(wèi)FcTs.N=2ζ/τ+1,·represents the floor function.is the Hermitian-Toeplitz matrix,i.e.=.

The adjacent ICIs for thelth subcarrier can be expressed as

where

whereFcTs=τν(1+β) .We calculate the SE ηGby the FFT algorithm instead of singular value decomposition(SVD) [7] for the TFP-FTN system with the (complex)Gaussian input.The effect ofKvalue on spectral efficiency is shown in Fig.4.The calculation results converge very nicely forK≥105.This kind of good-natured convergence is encountered in all our numerical experiments.

Fig.4 Effect of K on spectral efficiency with β = 0.2, τ =0.83, v = 0.93

3.2 Benefit analysis of TFP-MFTN

The SE ηGfor the TFP-MFTN system with the different time and frequency packing ratios are evaluated by (28).ηGincreases with the decrease of TPR τ for bothν ≥0.5 and τ ≥1/(1+β) .ηGno longer increases with the decrease of TPR τ for both ν ≥0.5 and τ ＜1/(1+β).Hence,the optimal τopt=1/(1+β) for the different FPRs.TPFTN in [4] has the same conclusion and is a special case with ν=1.0 .Fig.5 shows a special case for ηGwith ν=0.93 .Then,the optimal FPR νoptfor TFP-MFTN with τoptalso can be found at a certain SE range.The optimized FPR for MFTN with Gaussian input employed as the initial FPR for MFTN with finite alphabet input.The optimal FPR for TFP-MFTN system with Gaussian input and β =0.2 is 0.93 when the SE ranges from 0 to 6 bit/s/Hz.Fig.6 shows the ηGversusEb/N0for TFP-MFTN with the optimal TPR ( τopt=0.83 for β=0.2).The TFPMFTN ( ν=0.93,τ=0.83) has a 0.06 dB gain comparing to the TFP-MFTN ( ν=0.85,τ=0.83) when the SE is 2 bit/s/Hz.It can be seen that the FPR is not as small as possible,and the FPR needs to be optimized.The TFPMFTN ( ν=0.93,τ=0.83) has a 0.25 dB gain comparing to TP-MFTN ( ν=1.0,τ=0.83) when SE is 2 bit/s/Hz.It can be seen that the TFP-MFTN can achieve a better performance than TP-MFTN at the same SE.The gap between the Shannon limit ( β=0,complex Gaussian input) and the TP-MFTN system ( ν=1.0,τ=0.83) is 0.41 dB when the SE is 2 bit/s/Hz.This gap is reduced to 0.16 dB by TFP-MFTN ( ν=0.93,τ=0.83).The reason that the FP can further reduce the gap is that the power of ICIintroduced by FP is negligible compared to the noise powerN0at low-to-moderate SNR and the reduction onFcresults in an increase on SE.TP-(M)FTN can reduce the gap between Shannon limit( β=0,complex Gaussian input) and Nyquist limit( β ≠0,complex Gaussian input),but it cannot eliminate the gap.TP-(M)FTN needs to further consider FP to further reduce this gap between Shannon limit and Nyquist limit.

Fig.5 ηG versus Eb/N0 for TFP-MFTN with the same FPR (0.93)and different TPRs (τ= 0.85, 0.83, 0.81), where β=0.2

Fig.6 ηG versus Eb/N0 for TFP-MFTN with the same TPR (0.83)and different FPRs (ν= 1.0, 0.93, 0.85), where β=0.2

4.Efficient digital implementation for MFTN

The MFTN signals(t) is sampled atTΔ=Ts/L,

where α=TsFc/L=τν(1+β)/L.

A change of notationi=nL+min (29) allows us to introduce the polyphase components,

where the shaping pulseh(t) is sampled atTΔ=Ts/L.In this paper,the default sampling period for MFTN is assumed to beTΔ.

From the perspective of a polyphase filter bank,orthogonal frequency division multiplexing (OFDM) has the same polyphase structure as (30).Considering ejk2π=1 for the integerk,α=1/Lfor OFDM,which leads to eA=1.The second summation part in (30) can be efficiently implemented with the inverse FFT (IFFT) for OFDM.When the shaping pulse of OFDM is rectangular pulse that only lasts a time ofTs,the first summation part in (30) disappears.Therefore,OFDM can be implemented with only an IFFT without filtering the signal in time domain.Unlike OFDM,MFTN filters the signal in time domain because of ζ ＞1.The filter operation introduces ISI,which may lead to increased complexity of the equalizer.

The part eAin (30) is related tolandnfor α ＜1/L,hence it needs to be recalculated at each subcarrier positionland time-positionn.Assume that α=φ/(κL) and φ ＜κ,φ,κ ∈N+.Assume thatLis a multiple of κ and defineP=L/κ .A change of notationl=κp+χ (0 ≤p＜P,0 ≤χ ＜κ,p,χ ∈N+) to (30) gives

The MFTN transmitter consists of three steps: fractional Fourier transform (FRFT) [37],filtering,and addition operation.Bailey [37] introduced two methods for FRFT.The increase of complexity is mainly due to the extension of input sequence.There is a trade-off between the flexibility of the TFP ratio and the complexity of the transceiver for TFP-MFTN.To avoid extending the input sequence,we focus on τν=0.5/(1+β),i.e.,α=0.5/L.τ and ν can be flexibly adjusted or optimized under the premise of α=0.5/L.Parts of the following subsections in this section have been presented in [31].

4.1 Transmitter based on IFFT and polyphase filter bank

Next,we design an EDI for MFTN with α=0.5/L.sm(n)can be split into two parts in this case,i.e.,κ=2 and defineP=L/2 .As shown in Fig.7,MFTN forα=0.5/Lcan be efficiently implemented by an IFFT together with a polyphase filter bank.The output of the IFFT is filtered by the polyphase component ofh(n) and the filtering operation is performed at rate 1/Tsand notL/Ts.For the direct implementation of the MFTN,each modulation symbol is filtered at a rateL/Ts.We show the number of operations to compute each output for both the direct and the EDI in Table 1 for τ=0.47.

Table 1 Number of complex multiplications per output of MFTN with 16, 32, 64, 128, and 256 subcarriers

Fig.7 MFTN transmitter with τν=0.5/(1+β)

4.2 Receiver based on FFT and polyphase filter bank

The EDI for MFTN receiver is shown in Fig.8,where the FFT operation is applied to the outputs of the polyphase filters.It can be seen that the implementation of the receiver in Fig.8 is mirrored (matched) to the implementation of the transmitter in Fig.7.The EDI of the MFTN receiver involves lower number of complex multiplications than the direct implementation.

Fig.8 MFTN receiver with τν=0.5/(1+β)

4.3 Receiver based on channel shortening and parallel interference canceller

As shown in Fig.9,the PSE was employed in the MFTN receiver to mitigate the ISI.MFTN is equivalent toLindependent ISI channels simultaneously transmitting signals in this situation.To limit the receiver complexity,we apply a CS technique to each subcarrier.In fact,excellent performance can be achieved by properly filtering the received signal before adopting a reduced-state detector.The main work of CS is calculating the front-end-filters (and) and the target responsefor the subcarrier.The front-end-filtersandexecute interference cancellation to make the energy of the channel impul se response (CIR) more concentrated.The front-end-filtersandin Fig.9 can be seen as the replacements ofandin Fig.8.The target responseapproximates the equivalent CIR of subcarrier.The mismatched channel law for thelth subcarrier base on CS can be referred to the channel law of the single carrier in [20],

Fig.9 Per subcarrier equalization based on channel shortening

If an outer code is concatenated to the MFTN system,the iterative detection based on PIC for the practical MFTN system is shown in Fig.10.The MFTN equalizer is defined in Fig.9.The ICI term for thelth subcarrier at thekth time-position is

Fig.10 PIC for MFTN

The ICI for all subcarrier can be calculated simultaneously as shown in Fig.11 and(k) are the soft estimates[38] ofxl(k).It can be seen that the EDI of MFTN reduces the complexity of PIC.

5.Time-frequency packing ratio optimization based on efficient digital

We describe the criterion used to evaluate the ultimate performance limits of the EDI of the MFTN with finite alphabet input and to perform the TFP ratio pair optimization.As aforementioned,the PSE is used.We compute the IR and achievable SE when the channel inputs are finite alphabet (i.e.,i.i.d.random variables belonging to a given constellation).The computed IR represents an achievable lower bound on the IR of the actual channel,according to mismatched detection.A proper auxiliary channel [39,40] for the mismatched detection is provided by CS.

The SE for MFTN with finite alphabet input is represented as

whereI(xl,) represents the IR for thel-subcarrier.The IR for thel-subcarrier withM-ary input symbols in (35)can be calculated by the simulation-based method [39]which employs forward recursion of the BCJR algorithm and the auxiliary channel.

where the branch metric

andxlare known for calculatingP(,xl).

The aim of the MFTN optimization is to find the best values of τ and ν for a given SE at the minimumEb/N0,i.e.,

wheref(τ,ν)=min{Eb/N0:ηF(Eb/N0,τ,ν)=η0} andη0is the given SE.

The TPRs τ ranges from 0.42 to 0.49 in steps of 0.01 and the initial FPR for ηFis the optimized FPR for ηG.Fig.12 shows some typical results forτ=0.416 7,0.47,0.49,whereL=32,M=4,β=0.2 andLr=6.The Shannon limit is also shown in Fig.12.The optimal packing ratio pair (τ,ν) is (0.47,0.886 6) for TFP-MFTN at ηF=2 bit/s/Hz.When the SE is 2 bit/s/Hz,the optimal TFP-MFTN ( τ=0.47,ν=0.886 6) has a 0.34 dB gain comparing to the TP-MFTN ( τ=0.416 7,ν=1.0).When the SE is 2 bit/s/Hz,the gap between the optimal TFPMFTN with τ=0.47,ν=0.886 6 and benchmark A is less than 0.2 dB.

Fig.12 ηF versus Eb/N0 for TFP-MFTN with different FPRs ν and β=0.2 , subject to τν=0.5/(1+β)

As shown in Fig.13,we compare SE of our work based on the EDI and TFP system parameters optimization to SE of the previous works.The TFP-MFTN with the RRC pulse in [16] employed a direct implementation and SBS detector,and optimized the TFP values by maximizing the achievable IR.The TFP-MFTN with the Gaussian shaping pulse in [19] employed an SBS detector,and optimized the TFP values by maximizing the SIR.The TFP-MFTN with the optimal hexagonal lattice and Gaussian shaping pulse in [15] employed an overlap-add type implementation and SBS detector,and optimized the TFP values by maximizing the achievable SE.The TFPMFTN in [16] has 0.9 dB gain (Es/N0) comparing to TFPMFTN in [15] at SE=1.5 bit/s/Hz.The TFP-MFTN in[15] may not be very effective comparing to the TFPMFTN in [16].Because the overlap-add type implementation in [10] is suggested for low interference.The TFPMFTN in this paper has 4.6 dB gain (Es/N0) comparing to TFP-MFTN in [16] at SE=1.5 bit/s/Hz.The gain originates from that the MAP equalizer eliminates ISI instead of treating ISI as noise at the price of high complexity.

Fig.13 ηF versus Es/N0 for TFP-MFTN in this paper and TFPMFTN in [15], [16] and [19]

6.Coded modulation MFTN optimization based on EXIT chart

The parameters for the coded modulation MFTN are listed in Table 2.The packing ratio pair (τ,ν) is (0.47,0.886 6)for the MFTN hereafter,i.e.,τν(1+β)=0.5.The doping rate and degree distribution λ of LDPC in Table 2 are optimized respectively through the EXIT chart in this paper.

Table 2 Parameters for coded modulation MFTN

At low SNR,the dynamics of iterative decoding are more important and EXIT chart is a powerful semi-analytical tool for analyzing the convergence behavior of the iterative decoding [41].

6.1 Degree distribution optimization based on EXIT chart

We optimize the QC LDPC based on EXIT chart for TFPMFTN.The optimization of LDPC for TFP-MFTN is divided into three steps: (i) find a good degree distribution;(ii) construct a good base matrix;(iii) and search an appropriate degree matrix.In Subsection 6.1,the inner decoder in Fig.1 and the variable nodes of LDPC are regarded as decoder I,and the check nodes of LDPC are regarded as decoder II.With the same fitness function in[27],the BB-PSO algorithm basing on EXIT charts can search the most suitable degree distribution for the QC LDPC.

wheref(λ)=min{Eb/N0:Ψ(λ,Eb/N0,r,τ,ν)≥0} is the fitness function,λ={λi} denotes the variable node degree distribution vector,λiis the fraction of edges which connects to degree-ivariable nodes,f(λ) is the lowest SNR for the LDPC with λ such that the tunnel is open.Ψ(λ,Eb/N0,r,τ,ν)indicates the dependence on variable node degree distribution vector λ,

where N(a,b) represents the Gaussian distribution with meanaand standard deviationb.To find the optimal degree distribution vector,each particle moves in the direction to its previously best positionpiand local optimalgiin the swarm.

In this paper,the topologies structure {Ti} is a ring,which is suggested by Kennedy [42].In the ring topology structure,a particle communicates with two of its neighbors using a ring topology.The main steps of BBPSO searching the degree distribution for MFTN system with LDPC are listed as follows [43].

After the degree distribution λ is determined,Algorithms 1 and 3 in [23] instead of PEG can help us find the good base matrix and degree matrix for QC LDPC.

QC LDPC is optimized by the above method and the parameters for TFP-MFTN are listed in Table 2.Eb/N0=2.0 dB for TFP-MFTN with SE=2.0 bit/s/Hz (τ=0.47,ν=0.886 6) in Fig.12.The minimalEb/N0=2.5 dB (i.e.,2 dB+0.5 dB) is a reasonable reference value for TFPMFTN in the degree distribution search process.The optimal vector (λ2,λ3,λ19)=(0.813 6,0.025 4,0.161 0)for TFP-MFTN and the average variable node degree is about 2.36.The parameters for the optimized QC LDPC are listed in Table 3.

Table 3 LDPC parameters

6.2 Doping rate optimization based on EXIT chart

As shown in Fig.1,the inner decoder’s input is comprised of the apriori LLRs denoted byLi,a(d).Ii,a(d) denotes the mutual information between the bit streamdandLi,a(d).Ii,e(d) denotes the mutual information between the bit streamdand the bit stream extrinsic LLR valuesLi,e(d).Io,a(c) is the mutual information between the coded bitscand the apriori LLRsLo,a(c);Io,e(c) is the mutual information between the coded bitscand the extrinsic LLR valuesLo,e(c).The doping code [30] which is a rate-1 recursive convolutional code can introduce the dependencies between various consecutive bits to bend up the inner decoder transfer characteristics at highIi,a(d).However,the doping method may pull down the curve for smallIi,a(d) as shown in Fig.14.The doping rate [30]Dpshould be optimized to make a trade-off between low pinch-off thresholds and low error floors.The doping rateDpwhich was optimized by comparing EXIT charts of different doping rates is 100 for coded modulation MFTN in this paper.The influences of doping on the EXIT charts of the inner decoder are shown in Fig.14 and the abbreviations w/o means without hereafter.The iterative decoding between the inner code without doping and LDPC (d2.36) gets stuck before reaching the point (1.0,1.0),which indicates an error floor that cannot be overcome by increasing the number of iterations.At higherEb/N0,the inner code transfer curves without doping is lifted up and the error floor is lowered,but not eliminated.

Fig.14 Doping effects on the EXIT chart of TFP-MFTN inner decoders and Eb/N0 = 3 dB

7.Theoretical and simulation results

7.1 Theoretical results

In order to verify the effectiveness of the joint optimization method proposed,we compare the theoretical performance (EXIT prediction performance) of the optimized TFP-MFTN with that of the TP-MFTN system.The two systems have the same spectral efficiency.The parameters for TFP-MFTN are listed in Table 2.The TP-MFTN has the same parameters with the TFP-MFTN expect(τ,ν)=(0.416 7,1.0).

Fig.15 shows the EXIT characteristics of the TFPMFTN-doping and TP-MFTN-doping.The EXIT characteristics of the optimized LDPC(d2.36) decoder and DVB LDPC [44] decoder is also shown in Fig.15.There is an open convergence tunnel between EXIT characteristic of LDPC(d2.36) and TFP-MFTN-doping.The EXIT characteristic of the LDPC(d2.36) decoder is fitted for the TFPMFTN-doping.It should be noted that the tunnel is narrow and long,hence more iterations are needed,such as 30 iterations in this paper.The TP-MFTN-doping system needs a higher SNR than TFP-MFTN-doping system to form an open tunnel with the LDPC(d2.36).

Fig.15 EXIT chart prediction for joint MFTN and LDPC optimization

There is an intersection between the TFP-MFTN-doping equalizer’s EXIT chart and DVB LDPC decoder’s chart.The EXIT characteristic of the DVB LDPC decoder is unfitted for the TFP-MFTN.The DVB LDPC whose average variable node degree is about 3.499 needs a higher SNR than LDPC(d2.36) to form a convergence tunnel with the TFP-MFTN.We found that the LDPC with a lower average variable node degree has a lower pinch-off threshold for the TFP-MFTN which experiences severe ISI or ICI.

7.2 Monte-Carlo simulation results

We further carry out Monte-Carlo simulations to verify performance gain of the joint optimization.We compare the BER performance of the optimized TFP-MFTN system with that of the TP-MFTN and the benchmark B.The benchmark B employs the rate-3/5 DVB LDPC (code length=648 00) [44],β=0.2,andM=16.Other simulation parameters are shown in Table 2 and Table 3.

(i) Doping and PIC.Fig.16 shows the BER performance of TFP-MFTN system with or without doping.The TFP-MFTN has an error floor at BER=5×10-3and the TFP-MFTN-doping removes the error floor.The BER performances are consistent with the EXIT analysis in Fig.14.The TFP-MFTN-doping has an approximately 0.1 dB loss compared to the TFP-MFTN atBER=5×10-2.TFP-MFTN-doping with PIC achieves an approximate 0.3 dB gain comparing to TFP-MFTN-doping without PIC.In the following,TFP-MFTN uses PIC technology by default.

Fig.16 Impact of doping and PIC on BER

(ii) Optimized outer code.Fig.16 shows the BER performances of TFP-MFTN-doping with DVB LDPC and the optimized LDPC.TFP-MFTN-doping with LDPC(d2.36) has a 2.9 dB coding gain comparing to TFPMFTN-doping with DVB LDPC at BER=1×10-6.The optimized LDPC is more fitted for TFP-MFTN-doping than DVB LDPC,which confirms the theoretical EXIT chart prediction in Fig.15.However,Nyquist-4QAM with LDPC(d2.36) has a 3.1 dB coding loss comparing to Nyquist-4QAM with DVB LDPC at BER=1×10-6.It can be seen that the degree distributions of LDPC matching to FTN and Nyquist system are very different.The FTN system has severe ISI or ICI,and the average degree of QC LDPC for the FTN system is usually smaller than that for the Nyquist system.

(iii) TFP-MFTN,TP-MFTN and benchmarks.Fig.17 shows the BER performances of TFP-MFTN-doping,TPMFTN-doping and benchmark B with the same SE.TFPMFTN-doping has a 0.9 dB TFP gain comparing to the benchmark B.The result demonstrates that TFP-MFTN can achieve higher energy efficiency than the benchmark B at the same SE.TP-MFTN-doping has 0.1 dB TP gain comparing to the benchmark B.TFP-MFTN-doping has 0.8 dB gain comparing to TP-MFTN-doping,which confirms that TFP is better than TP in Fig.6,Fig.12 and Fig.15.The optimized packing ratio pair plays a key role in the joint optimization gain.These gains are summarized in Table 4.The BER performances are consistent with the SE and EXIT chart analysis.TFP-MFTN-4QAMdoping-d2.36 LDPC has a 1.3 dB loss comparing to benchmark A and this loss is less than 1.5 dB.

Fig.17 BER performance of MFTN and Nyquist system

Table 4 Gain for TP/TFP-MFTN dB

The MFTN in [4] has the same SE with our work,where (τ,ν)=(0.88,0.437 1),β=0.3,the convolutional code (7,5) serves as the outer code,and the SIC and MAP equalizer are employed.As shown in Fig.17,TFP-MFTNdoping with LDPC(d2.36) has a 2 dB gain comparing to[4] at BER=1×10-4.The optimized LDPC code plays a key role in the joint optimization gain for this comparison case.The SC-FTN in [24] has the approximately equal energy efficiency (Eb/N0) to our work atBER=2×10-6,where τ=0.5,β=0.4,the optimized rate-1/2 LDPC serves as the outer code and the MAP equalizer is employed.Compared with [24],the SE of our system is improved by 40%.The optimized packing ratio pair and CS play the key role in the joint optimization gain for this comparison case.TFP-MFTN with the optimal hexagonal lattice and Gaussian shaping pulse in [15] employed an overlap-add type implementation and SBS detector,and employed the rate-1/2 DVB LDPC.Compared with[15] (SE=1.55 bit/s/Hz),the SE of our work is improved by 29% and our work has a 4.1 dB gain (Eb/N0) at BER=1×10-5.Reference [15] employed the soft demapper,and we have provided the complexity comparison in [1].

8.Conclusions

In this paper,we show the benefit of TFP-MFTN compared with TP-MFTN and Shannon limit by the numerical calculation method.When the SE is 0-6 bit/s/Hz,TFP-MFTN system can achieve a better performance than TP-MFTN system.Then,we propose an EDI and MAP equalization based on CS and PIC for MFTN.We obtaine the optimal packing ratio pair with SE criterion for the proposed EDI.When the SE is 2 bit/s/Hz,the gap between the TFP-MFTN system with the optimal packing ratio pair and Shannon limit is less than 0.2 dB.Furthermore,we perform the inner and outer code joint optimization for TFP-MFTN.Finally,EXIT chart prediction and Monte-Carlo BER simulations are carried out to verify performance gain of the joint optimization for TFPMFTN.TFP-MFTN with doping has a 0.8 dB and 0.9 dB gain comparing to TP-MFTN with doping and higher order Nyquist,respectively.Compared with the previous work on TFP-MFTN in [15] (SE=1.55 bit/s/Hz),the SE of our work is improved by 29% and our work has a 4.1 dB gain (Eb/N0) at BER=1×10-5.However,there is still a 1.3 dB gap between the Shannon limit and the BER performance of the optimized TFP-MFTN.We will work on reducing the gap in our future study on this subject.