Xisheng Wu,Yanbo Wu,Min Zhu,Dong Li

Abstract:To increase the spectral efficiency of the underwater acoustic (UWA) communication system,the high order quadrature amplitude modulations (QAM) are deployed.Recently,the probabilistic constellation shaping (PCS) has been a novel technology to improve the spectral efficiency.The PCS with high-order QAM is introduced into the UWA communication system.A turbo equalization scheme with PCS was proposed to cancel the severe inter-symbol interference (ISI).The non-zero a priori information is available for the equalizer and decoder before turbo iteration.A priori hard decision approach is proposed to improve the detection performance and the equalizer convergence speed.At the initial turbo iteration,the relation between the a priori information and the probability of the amplitude of 16QAM symbols in one dimension is given.The simulation results verified the efficiency of the proposed method,and compared to the uniform distribution (UD),the PCS-16QAM had a significant improvement of the bit error rate (BER) performance with PCS-adaptive turbo equalization (PCS-ATEQ).The UWA communication experiments further verified the performance superiority of the proposed method.

Keywords:high order quadrature amplitude modulation;probabilistic constellation shaping;UWA multipath channel;turbo equalization

1 Introduction

In the additive white Gaussian noise (AWGN)communication system,the uniform MQAM formats are usually deployed,leading to a gap of about 1.53 dB from the Shannon limit.Actually,the gap can be reduced by optimizing the distribution of the constellation mapping [1].With the help of the PCS,the constellation points within a lower power layer are transmitted with a higher probability.

In the optical communication system,the constellation shaping has attracted significant attention in recent years.The constellation shaping includes the geometric constellation shaping(GCS) and the probabilistic constellation shaping (PCS) [2].In the GCS,the location of the constellation points will be changed.But there is no simple solution to find locations of the GCS constellation point for arbitrary channel conditions and the irregular constellation points increase the complexity.In contrast to the GCS,the PCS is more convenient for the practical application.The PCS is simple to be realized for different probabilities and the constellation points are regular,which are placed on the grid of a square QAM template.It only changes the different constellation points’ probabilities.Thus the PCS algorithm is more popular than the GCS algorithm and it is adopted in this article.

In the underwater acoustic (UWA) communication system,the literatures [3−5]analyzed the constellations-constrained capacity of a nonlinear parametric channel by optimizing the input distribution.The optimal algorithm is extremely complex and the channel model of simulation is simple,where the long time delay and the multipath of UWA channel were not considered.The communication system capacity with UD/PCS-16QAM and UD/PCS-64QAM were calculated in[6],respectively.

The UWA channel is multipath.The received signal will suffer from the ISI.To mitigate the ISI,the channel equalization technology is applied.In addition to some traditional equalization methods,artificial intelligence (AI) in underwater acoustic communication for a better performance [7−10].But there is no equalization scheme specially designed for PCS UWA communication system.Turbo equalization is a scheme where the channel equalizer and the channel decoder are jointly processing in an iterative way.The soft information is exchanged in a process between the equalizer and the decoder.In the practical UWA communication system,the long distance communication induces a low signal to noise ratio (SNR).The minimum mean square error (MMSE) turbo equalization has been proven to be effective to removing the ISI in low SNR situations [11]and adopted in UWA communication in [12].Therefore,in this paper,the MMSE turbo equalization was chosen as the baseline scheme.

We combine the PCS with the high-order QAM for the single carrier UWA communication.A PCS-ATEQ scheme is proposed to cancel the ISI introduced by the multipath channel.A nonequal probability decision way for the received symbols is given.Finally,the simulated and experimental results have verified the superiority of the proposed method.

The rest of this paper is structured as follows.In Section 1,we introduce the coding and modulation scheme of probabilistic shaping.In Section 2,the UWA communication system model is described.The equalization algorithm with the probabilistic constellation shaping and the decoding algorithm is presented.A PCS-ATEQ scheme with non-equal a priori symbol probability information is proposed.Several simulation experimental results and undersea experimental results are shown in Section 3 and Section 4,respectively.Section 5 concludes the paper.

2 The Design of the Probabilistic Shaping Modulation

2.1 PCS-MQAM UWA Communication System

A single-carrier UWA communication system with PCS is considered.The system model for probabilistic constellation shaped MQAM (PCSMQAM) signal is displayed in Fig.1,whereMis the order of the QAM constellation.The design of PCS modulation system is as follows.

2.2 PCS Encoder for MQAM

The square MQAM can be expressed as the Cartesian product of twom-amplitude shift key constellation [13],wherem=log2M.For sake of clarity,we describe the PCS-MQAM scheme for a real mASK constellation,which can be mapped to the phase and quadrature of the MQAM symbols.The distribution matching (DM) is a scheme that is used to transformkuniform data bitsu=[u0,u1,u2,...,uk−1]tonamplitudesA=[A0,A1,...,An−1]with a desired distribution,whereAi ∈{1,3,...,m −1}is the amplitude absolute value of the mASK.ThenamplitudesAcan also be transformed into thekuniform bits.That is to say,the DM is invertible.A symbollevel constant composition distribution matcher(CCDM) was proposed in [14].It is the common DM scheme for the PCS scheme.The output of CCDM is an amplitude sequence.Each amplitude will be labeled into a set of fixed bits.We can obtain the binary sequenceudmfrom thebinary labelingfunction,which will be introduce in the following section.

2.3 The Forward Error Correction Coding and Mapping

After the binary labeling,the binary forward error correction (FEC) is employed to encode the binary sequenceb.To keep the same probability distribution with the amplitude sequenceA,the binary FEC encoder must be systematic.The systematic binary FEC (SB-FEC) can satisfy the uniform check bit assumption [15].If the redundancy bits are the modulo-two sum of a large enough number of data bits,the distribution of them is close to uniform.A systematic generator matrix binary code has the formatGgen=[I P].The PCS encoder and systematic binary FEC encoder is displayed in Fig.2.

The output sequence of the SB-FEC coder can be divided into two parts,the uniform bit sequence and the non-uniform bit sequence.The former is the sign bits and the latter is the amplitude bits.The non-uniform data bits have the probabilistic information.Thereafter,we can reassign the encoded bits by Fig.3.Then we can map the data bits to symbols.The probability of each two-dimensional QAM constellation is the product of corresponding one-dimensional ASK probabilities.The 2-D symbols probability can be quantified by the 1-D amplitudes distribution.The stringd=d0d1···dm−1denotes the label of the mASK constellation point.For instance,the 16QAM modulation can be generated by two 4ASK.The mapping scheme of 4ASK modulation is described in Fig.4.The constellation point of 4ASK can be defined asB,whereB ∈{−3,−1,+1,+3}.TheP(|B|=1) andP(|B|=3)are respectively defined as the probability of the amplitude±1 and±3,which are the modulus of the 4ASK amplitudes.They satisfy the equation:P(|B|=1)+P(|B|=3)=1.For the 4ASK constellation points,we use the sign label functionβsign(·) and the amplitude label functionβamp(·) to label the amplitudes of 4ASK,as

Fig.3 Diagram of the bit-level assignment

Fig.4 Mapping for 4ASK modulation

Apparently,the sign label functionβsign(·) and the amplitude label functionβamp(·) is invertible.

3 System Model and Improved Equalizer for PCS

3.1 System Model Description

A single-carrier transmission with PCS method is considered.The PCS encoder method mentioned above will be briefly reviewed in this subsection.At the transmitter,it is convenient to define two paralleled uniform data bit sequencesuanduγ,which are used for PCS code rate self-adaption.When the data bit sequenceuis fed to the PCS encoder (DM and binary labeling),a data bit sequenceudmwith a desired probability information (a priori information) can be obtained.Then the bit sequenceb=[udmuγ]will be fed to SBFEC encoder.The encoded bit sequence iscand it include the non-uniform amplitude bits and the uniform sign bits.The non-uniform amplitude bits are the labeling bits.The uniform sign bits include the uniform check bits and the uniform data bitsuγ.After the bit-level assignment,cis divided into a matrix withmrows,and theith column coded bits setci=[ci,1,ci,2,...,ci,m]is mapped to ASK symbols by the binary reflected gray code (BRGC).The transmitted symbols are formed by the formula that maps two mASK symbols to a MQAM symbolx.The symbolxis taken from aM-ary constellationχ={α0,...,αM−1}.To satisfy the power constraint,we scale symbolxby the constellation scaling ∆>0.The scaling∆ will be merged into the constellation symbolxfor convenience writing.

The received time-domain signal at the timekis given by

wherehl,kdenotes thelth tap of the lengthLequivalent channel between the transmitter and receiver at time instantk,which produces ISI.xkis the symbol at time instantk.wkis the complex AWGN noise.

3.2 PCS-Adaptive Turbo Equalization

The receiver consists of two parts:an adaptive equalizer and an SB-FEC decoder.In the turbo equalization scheme,the equalization and the channel decoding are jointly optimized in an iterative way.The UWA channel has the characteristics of severe multipath delay spread and channel fading,leading to the severe ISI.To eliminate the ISI,the direct-adaptive equalizer (DAE)is employed at the receiver.

The output of the equalizer can be given as

The relations used to update the vectors of the coefficients can be obtained from the equations:

whereµis a very small step-size that controls the convergence properties of the equalizer.We can chooseµ=5×10−4in the following simulation experiments.

At the 0thturbo iteration,there is no a priori information for the traditional method.For PCS scheme,the a priori symbol probability information is not zero.To improve the symbol detection performance,the a priori hard decisions(HDs)is denoted on the equalizer output:

where the variance and mean of the symbols are obtained respectively as

The transmitted symbolsxiare not uniform.The a priori symbols probabilityPk,pri(xk=xi) is available to estimate the equalizer outputat the zeroth iteration.The log likelihood ratio(LLR) value ofci,jis defined as

The a priori probability informationis obtained from the CCDM empirical distribution at the 0thturbo iteration andLais form the decoder.The a priori symbol probability with the group of bitsmapped to the symbolαiis

For the PCS-16QAM,due to the symmetry of the four quadrants,we can calculate the a priori information at the 0thturbo iteration.It can be given by

Obviously,when the constellation symbols are uniform,the a priori information

As demonstrated in Fig.5,the a priori information available due to the PCS at the 0thturbo iteration.For space saving,the diagram leaves out the demapping and the mapping process.The a priori bit probabilityP(cj=ci,j) can be obtained from the CCDM.The a priori HDs are not enough accurate due to the background noise and the inaccurate a priori information from the CCDM.When the turbo iteration is performing,the decoder is made aware of the a priori probability informationLa(ci,j).The bitlevel probability is calculated by using the a priori LLRLa(cj),as

Fig.5 Block diagram of the adaptive Turbo equalizer scheme for PCS

At the following iterations,the a priori information from decoder will be updated for the LLR.The performance of the decoder will be improved when the a priori information is available.

4 Performance Analysis via Simulation

4.1 Parameters of Simulation Experiment

probability distributions for PCS-16QAM are given in Fig.6.The distribution is chosen from Maxwell-Boltzmann distributions.The system transmission rateRTxin theory can be calculated by

Fig.6 Graphical illustration of the employed probability distributions for PCS-16QAM.(a) P(1)=0.68;(b) P(1)=0.86

We simulate the UWA communication system with the uniform MQAM and probabilistically shaped MQAM.The value ofMis 16.The DM adopts CCDM,which is one of the most common DMs in [15].The FEC code is systematic binary low-density parity-check (LDPC) code.The code rates for 16QAM are 0.75 and 0.5,respectively.The FEC code is binary LDPC and its column weight is 3.The parameters of the MQAM signals evaluated are listed in Tab.1.Graphical illustration of the three employed

Tab.1 Fixed probability distribution (PD) of 16QAM

where H(P) is the entropy in bits,which depends on the probabilitiesPof the constellation symbols.The system transmission rateRTxcan be adjusted by changing the symbols distribution and the code rate.

To verify the performance in the UWA channel,the channel estimation of the practical lake trial channel is obtained.The channel impulse response (CIR) of the lake channel isshown in Fig.7.There are two extra paths,except the major path.In this paper,we use the estimated UWA channel as the simulation channel.The received signals will suffer from the ISI with the multipath effect.Under the aforementioned simulation conditions,the BER performance for the practical lake channel is given.For the practical lake channel,the shaped signal with PCSATEQ has a better performance,which coincides with the result in Fig.8.

Fig.7 The estimated CIR of the practical lake trial channel

Fig.8 The BER performance comparison between different PDs of 16QAM with different iteration times of PCS-ATEQ:(a) P(1)=0.5,Rc=0.75,RTx=3;(b) P(1)=0.68,Rc=0.75,RTx=2.8088;(c) P(1)=0.5,Rc=0.5,RTx=2;(d) P(1)=0.86,Rc=0.75,RTx=2.1685

4.2 Performance Analysis of the Simulation Results

Fig.8 shows the BER performance with the proposed turbo equalizer with PCS algorithm for the estimated lake channel.We compare the three different symbol distributions and two different code rates under five iterations of the turbo equalization.Between the 0thand the first equalization iteration,the a priori probability information is obtained by the PCS-encoder scheme and be fed to the equalizer and decoder.The decoder will feed the equalizer with more accurate a priori information.There is a great BER performance gain between the 0thequalization iteration and the first equalization iteration.

When the FEC code rateRcis the same(Scheme 1,Scheme 2 and Scheme 4),the Scheme 4 has the best convergence performance,but its transmission rateRTxis lowest (seen in Tab.1).Clearly,the BER performance is better when the FEC code rate is 0.5 than the FEC code rate is 0.75 with the uniform symbol distribution(Scheme 1 and Scheme 3).But the BER performance of Scheme 4 with FEC code rateRc=0.75 and transmission rateRTx=2.1685 is better than Scheme 3 withRc=0.5 andRTx=2 because of the shaping gain.We consider the limited condition,when the number of the symbols with high power is major force,the distribution of PCS-16QAM is closer to the distribution of QPSK.Meanwhile,the information entropy of PCS-16QAM is closer to 2 when the entropy of UD-16QAM is 4.

Under the same conditions,the shaped signals can achieve a better performance than the unshaped signals.Specifically,to the speed,the equalizer utilizes the a priori hard decision way for received symbols and provides LLRs with non-zero a priori probability information at the 0thiteration.The probability mass function(PMF) of the shaped constellation is used for a priori probability computing at the initial iteration,namely.

5 Undersea Experimental Results

5.1 Experimental Setup

Considering the multipath in the horizontal channel of the shallow water,the single-input multiple-output (SIMO) was adopted to improve the communication performance.The receiving linear array was a 4-ary linear array.At the transmitter,the uniform 16QAM and the PCS-16QAM modulations were generated.The total number of the data frames for each modulation format was 8.The carrier frequency was 4 kHz and the bandwidth was 2 kHz.The parameters of the PCS 16QAM were the same of the Scheme 3 and the Scheme 4 in the last Section,which have almost the same transmission rateRTX.

5.2 Experimental Results

The channel impulse responses (CIRs) and the amplitude frequency responses (AFRs) of the four subchannels are estimated described in Fig.9.The multipath energy spread is long and the subchannels manifest severe frequency selectivity.Tab.2 lists data processing results of the uniform 16QAM and the PCS-16QAM with almost the same transmition rate.The number of error frames with PCS-16QAM is zero,meanwhile,the number of the error frames with uniform 16QAM is 8.The information rate of the PCS-16QAM can achieve 6 kbit/s over a 10 km commmunication distance.

Fig.9 The CIRs and AFRs of the four subchannels:(a) The CIRs of the four subchannels;(b) The AFRs of the four subchannels

Tab.2 The performance of the uniform 16QAM and PCS-16QAM

6 Conclusion

In this paper,the probabilistic constellation shaping technology is applied to the UWA system with multipath channel.An equalizer with PCS algorithm is proposed to cancel the ISI.Based on the non-zero a priori information at the initial turbo iteration,the PCS-ATEQ and the channel decoding are jointly optimized.It is verified by the simulation results that the PCS-16QAM has a better performance with PCSATEQ iteration.The practical undersea experimental results also show that the PCS-16QAM can achieve error-free frame at a distance of 10 km compared to the traditional uniform 16QAM.The PCS technology with the high order modulation is efficient for the UWA communication system and it achieves an excellent performance.