An Orthogonal Wavelet Transform Fractionally Spaced Blind Equalization Algorithm Based on the Optimization of Genetic Algorithm

LIAO Juan(廖娟)，GUO Ye-cai(郭业才)，2，JI Tong-ying(季童莹)

(1.School of Electrical and Information Engineering，Anhui University of Science and Technology，Huainan 232001，Auhui，China;2.College of Electronic and Information Engineering，Nanjing University of Information Science and Technology，Nanjing 210044，Jiangsu，China)

Introduction

In recent years，the blind equalization has become a major research direction in equalization technique.It does not require the training sequences，only using the statistical properties of the received signal to balance signal.In the blind equalization algorithm，a fractionally spaced blind equalizer(FSE)［1］oversamples the received signal.Its sampling rate is greater than symbol interval T.FSE has the slow convergence and the large steady-state error，without consideration of the input signal autocorrelation.Ref.［2 － 4］suggest that the wavelet transform has decorrelation capability.The orthogonal wavelet transform fractionally spaced blind equalization algorithm introduces wavelet transform into the fractionally spaced blind equalization algotirithm.It has fast convergence rate.However，the method of updating the weight vector is based on the idea of constant modulus algorithm in these algorithms.The updating method is a local convergent method，and the error function of algorithm must be guided.The genetic algorithm［5－7］is an adaptive probability search algorithm.It has strong robustness and global random search ability.It can quickly and efficiently find the global optimum solution in a complex，multi-peak and nondifferentiable vector space.It reduces the possibility of local convergence.Therefore，the performance of the equalizer can be improved by using it to optimize the equalizer weight vector.

In this paper，an orthogonal wavelet transform fractionally spaced blind equalization algorithm based on the optimization of genetic algorithm is proposed by introducing the genetic algorithm and the wavelet transform theory into fractionally spaced blind equalization algorithm(WT-FSE-GA).Compared with the fractionally spaced blind equalization algorithm based on orthogonal wavelet transform(WT-FSE)and the fractionally spaced blind equalization algorithm(FSE)，the proposed algorithm is more effective in eliminating inter-symbol interference and improving the performance of underwater communication.

青辰在远处观战，越看越是心惊。他知道师父会功夫，却从未想到，师父的功夫竟然如此高超。这个平日看起来弱不禁风老者，此时仿佛换了个人，在天葬台上忽起忽落，纵跃走转，一口天葬刀更是风雨不透，任唐飞霄七足连攻，却也攻之不下。

1WT-FSE

The fractionally spaced blind equalization algorithm based on orthogonal wavelet transform(WTFSE)is to introduce the orthogonal wavelet transform into the fractionally spaced blind equalization algorithm，as shown in Fig.1.The algorithm uses the same wavelet to transform each sub-channel input signals and normalize the signal energy［8］.It reduces the input signal autocorrelation.Then，in the transform domain，the algorithm uses constant modulus algorithm(CMA)to adjust the weights.It speeds up the convergence rate.

Structural design of the high level complex transfer Building

Fig.1 Structure of factionally spaced blind equalizer based on orthogonal wavelet transform

Suppose that{a(k)}is the period T of the transmitted signal sequence，and the lst(l=1，2，…，D)sub-channel impulse response is

where Ncis the length of channel impulse response;n(l)(k)is a Gaussian white noise of the lst sub-channel vector.

In terms of the wavelet transform theory，when the equalizer w(k)is a finite impulse response，w(k)can be expressed by a family of orthogonal wavelet functions and scale functions.Assuming that the length of each sub-channel equalizer is Mf=2J，and in the case of finite length，w(k)can be expressed as:

做好实习学生思想政治工作必须按照规范化的要求，健全工作机制，加强学生思政教育队伍建设。重视对专兼职学生管理工作者和辅导员的专题培训，提高他们的政治素质和管理水平。加强师德师风教育、建立健全相应的工作激励机制，提高他们的工作积极性。另外要加强党、团组织建设，实现党团建设阵地“战略转移”，做到哪里有党员、团员就把党小组和团小组建到哪里。还要建立一批稳定的校外实习基地并与实习单位建立起长期合作、双赢的关系，根据不同企业的特点对实习学生开展有针对性的思想政治教育。

where k=0，1，…，Mf－1，J，J is the maximum size of wavelet decomposition;kj=Mf/2j－1(j=1，2…，J)is the maximum shift of wavelet function;djmand vJmare the equalizer weights;φjk(k)and φjk(k)are the wavelet function and scale function.

According to the signal theory，the output of the equalizer z(k)is

where rjm(k)，sJm(k)are the wavelet and scale transform coefficients，respectively.the lst sub-channel output y(l)(k)is transformed by the orthogonal wavelet，and the result is

The unknown weights of the lst sub-channel equalizer is denoted as

where H represents a conjugate transposition.

由于中东欧16国都相对较小，每个国家又都有自己的国语，且均属于小语种。而这些语种目前在国内主流外语培训市场上几乎难觅踪迹。因此，在外语培训市场上，政府可以政策性鼓励有一定资质的培训机构开设中东欧国家的小语种培训。

The iteration formula of lst sub-channel equalizer weight vector is

where，RCMis module value of signal and RCM=E{|a(k)|4}/E{|a(k)|2};e(k)is error and e(k)=RCM－|z(k)|2;［^R(l)(k)］－1is energy normalized matrix of the lst sub-channel and［^R(l)(k)］－1=

The selection operation is to use the fitness probability of each individual to determine the possibility of its entry into the next generation.In this paper，the selection operation is to use the ergodic random sampling method to replace the roulette selection method.The roulette selection method uses a single pointer，but the ergodic random sampling method uses K equidistant pointers.The number of the chosen weight vector is denoted as K.1/K is defined as the distance of the selective pointer.The position of the first pointer is determined by the uniform random number of［0，1/K］.So K individuals are selected by K pointers.The cumulative probability of selected individual is close to the pointer position and is calculated by an individual selective probability.If the number of the individual is K and f(Wi)is defined as the fitness value of the ith weight vector Wi，the selective probability of Wican be written as

where β is a forgetting factor.Eq.(3)－ (11)constitute the fractionally spaced equalization algorithm based on orthogonal wavelet transform(WT-FSE).

2 WT-FSE Based on the Optimization of Genetic Algorithm

The traditional CMA searches the equalizer weight vector by the fast gradient descent search method［9］and determines the iterative equations.However，this search method only takes a local region into consideration，and its cost function must be derivative.And GA is a random search algorithm，which uses the group search strategy and exchanges information between individuals.It doesn’t depend on the gradient information，and its cost function needn’t to be derivative.It has strong robustness and global search ability.Therefore，the genetic algorithm can be introduced into the orthogonal wavelet transform fractionally spaced blind equalization algorithm，the global search feature is used to find the best equalizer weight vector，and the equalizer weight vector is adjusted without the guidance of gradient information.The basic idea is that the weight vector of each sub-channel equalizer is taken as a decisive variable of genetic algorithm，the input signal of each sub-channel equalizer is transformed by wavelet transform，and the transformed signal is the input sequence of genetic algorithm.Combined with the cost function of CMA，the objective function or the fitness function of genetic algorithm is determined.And the best equalizer weight vector is searched by the genetic algorithm.

The cost function of equalizer is represented by the average time of error function.Suppose that the length of the received signal sequence is N，and the cost function can be calculated by

where z(k)is the equalizer output，RCMis equalizer modulus.Therefore，each evolution generation receives N input signals.In each generation，the N input signals are firstly balanced by WT-FSE.Then they are evoluted into new species by the genetic algorithm and are taken as the initial population of the next generation.The flow chart of one sub-channel is shown in Fig.2.

Fig.2 Flow chart of genetic optimized algorithm

1)Generation of initial population

The mutation operation is a complementary search operation in the genetic operation.It approaches the optimal solution in view of a local.Here the real-mutation is used.Wi(m)is defined as the mth tap value of the ith less differentiated weight vector individual，and the mth tap value of the ith mutated weight vector individual is given by W'i(m):

In the genetic algorithm，the operational objects are the groups.Before the operation，the initial population needs to be prepared.For simplicity，［－ 1，1］is initially identified as the solution space of genetic algorithm，and a certain number of individuals are randomly generated.These individuals construct an initial group.Each individual corresponds to a equalizer weight vector，and the length of individual corresponds to the length of the equalizer weight vector.An individual encoding method is a real-coded method.The initial population is generated by stochastic method.W is defined as the initial population and given W=［W1，W2，…，WM］，where Wi(0 ＜ i≤M)corresponds with a weight vector of the equalizer.

2)Calculation of objective function

Every generation of GA receives a certain input signal.These signals are provided with the sub-channel signals and are transformed by wavelet transform.They are balanced by CMA，and their cost function can be calculated by Eq.(12).The cost function acts as the objective function of genetic algorithm.

3)Determination of fitness function［10］

由图3可知，随着黄精浸提液添加量的不断增加，黄精酸奶的酸度一直提高，由80.4°T增加到91.8°T，基本符合国标规定的最佳酸度要求，因此仅仅从酸奶的酸度无法确定黄精浸提液的最佳添加量；当黄精浸提液由0.3%添加到0.7%时，感官评分呈现先上升后下降的趋势，黄精浸提液添加0.5%时，感官评分最高为90分，因此通过对黄精酸奶酸度和感官评分的分析，确定黄精浸提液的最佳添加量为0.5%。

The purpose of blind equalization algorithm is to iterate the cost function to the minimum and gets the best equalizer weight value.The goal of genetic algorithm is to get the individual.The individual has the maximum fitness function value.To resolve this conflict，the fitness function of GA is defined as

where α is a scale factor，which is a random number belonging to［0，1］.

The combined output of all sub-channel equalizers is

4)Design of genetic operators［11］

● 联网共享。不仅是单一系统内的摄像机头数量激增，而且多个系统之间的图像信息也需要互通互用。特别是随着监控系统规模的扩大，用户业务需求日趋复杂，典型的如平安城市项目，要实现城市级的信息联网共享，将治安、卡口、电警和社会面监控等视频点统一接入管理，其联网需求、数据存储、业务应用都对管理平台提出了更高要求。

Genetic operation includes selection operation，crossover operation and mutation operation.

The crossover operation plays a central role in the genetic operation.It is the main method to produce a new individual.Taking account of the real-coded method，the crossover operation uses the combination of crossover and linear to generate new cross-parameter and opens up a new search space.In the crossover operation，the two intersections are randomly set in the two encoded string，and new individuals are generated by linearly combining the coding values of cross-bit.If Wiand Wi+1are two parent individuals，the linearly combined individuals W'iand W'i+1are as follows:

1.3 观察指标 分析两组患者术中出血量、手术时间、术后排气时间、术后拔管时间、术后腹腔引流量、进食时间、住院时间。于手术前后抽取两组患者5 ml的空腹静脉血，以转速3 000 r/min离心15 min。分离血清后，提取上清液，使用酶法检测血清胆红素(total bilirubin，TBIL)、间接胆红素(indirect bilirubin，IBIL)水平。采取干化学法检测碱性磷酸酶(alkaline phosphatase，ALP)、淀粉酶(amylase，AMY)水平。

where J(Wi)is the cost function of equalizer;Wiis the equalizer weight vector individual and generated by GA.

通过以上例子不难发现，《报告》在时间轴上是以特朗普总统正式上台执政的时间为指示中心（When I came into office），其他时间状语大多都围绕这一时间点展开，时间距离大多不超过一年。这就意味着，说话者有意强调或者缩小句中所涉及实体在时间上与指示中心的距离。时间轴上距离较近的句子大都提到了外部威胁和挑战，也就是《报告》所塑造的敌对实体（rogue regimes，nuclear weapons，serious challenges，the rise of China），并且在心理空间上将这些外部实体表征为不断迫近的威胁，从而唤起公众的恐惧情绪。

护理教育旨在培养护理专业人才，护理教育包括基础理论教育及实践实训教育。护理理论教育是为实训教学提供理论支持，实训教学意在将护理理论知识应用于临床，将知识转化为能力，直接为人们提供健康服务。所以，护理教育可以说是“知信行”理论的具体应用。护理实训教学效果决定了护士临床工作能力。为适应人们不断增加的健康卫生需求，护理实训教学越来越受到重视，而基础护理学课程则显得尤为重要。当前在护理学教学改革中，以学生为主体，注重学生能力培养已成为新的教学理念[1]。然而，大部分医学院校基础护理学实训教学仍采用传统教学模式，即实训教师在理论课基础上进行流程讲授—示教—学生模仿练习。

According to Fig.1，the lst(l=1，2，…，D)subchannel output is

5)Judgment of termination

Termination of GA is determined by the maximum evolution generation.When the evolution generation doesn’t exceed the maximum evolution generation，the operation returns to the second step and repeats Steps(2)－(5).When the evolution generation exceeds the maximum evolution generation，the operation carries out Step(6).

6)Selection of the best weight vector individual

说起天窗，我们并不陌生。那些最初被我们称为“囱”“通孔”的口子，就是最早期的“天窗”雏形。1941年，世界上第一座真正意义上的天窗在丹麦出现。在人们对建筑光环境的迫切需求下，建筑天窗应运而生，并深得许多设计师的青睐，开始出现在一些公共建筑及住宅项目中。进入21世纪，随着科学技术的进步，天窗的设计与功能也变得越来越丰富，甚至一度成为现代建筑设计的潮流。既是窗户也是天花，天窗的混合性使其能够成为建筑空间中的关键元素。不得不说，利用天窗的形状和角度加以灵活地布置，任何空间皆可变得独特而美观，仅仅是和天空近距离的接触，就能改变空间给人的第一印象，这大概就是天窗的魅力所在。

In the each generation，the weight vector of the largest objective function is selected.Considering the real-time of algorithm and the zero-forcing condition of blind equalization algorithm，the best weight vector individual of the generation is regarded as the best weight vector individual of the next generation when the best weight vector individual is extracted.

This paper introduces the genetic algorithm into the ideas，methods and process of WT-FSE，and uses the genetic algorithm to optimize the equalizer weight vector.The algorithm is called orthogonal wavelet transform fractionally spaced blind equalization algorithm based on the optimization of genetic algorithm(WT-FSE-GA).

3 Simulation

The proposed algorithm was simulated to prove its effectiveness，and compared with FSE and WTFSE.

In the simulation test，D=2，the population size is 100，the crossover and mutation probabilities of GA are 0.7 and 1/16，respectively，and the maximum evolution generation is 100，as shown in Fig.1.

Simulation test 18 PSK signals are transmitted to a mixed-phase water acoustic channel，the impulse response of channel is given by h=［0.313 2－ 0.104 0 0.890 8 0.313 4］［12］;the weight length of equalizer is set to 32;SNR is set to 25 dB;the power initialization value is 4;the 16th taps of the weight vectors of WT-FSE and FSE are initialized to one;in three algorithms， the step sizes are μT/2-FSE-CMA=0.002，μWT-T/2-FSE-CMA=0.001 and μWT-T/2-FSE-GA-CMA=0.004;the DB2 wavelet is used to decompose the output signal of each subchannel，and the decomposition level is 3，the power initialization value is set to 4，and the forgetting factor β is set to 0.999 9.The results of 500 Monte-Carlo simulations are shown in Fig.3.

It can be seen from Fig.3(a)that the convergence rate of WT-T/2-FSE-GA is 500 steps and 1 000 steps more than WT-T/2-FSE and T/2-FSE，respectively.The MSE(mean square error)of WT-T/2-FSE-GA is essentially the same as WT-T/2-FSE and has a drop of about 1.5 dB compared with the T/2-FSE.Fig.3(b，c，d)shows that the constellation map of output signal in WT-T/2-FSE-GA is the clearest.

1) 分别计算5个数据集上Spectral 聚类算法、AP算法和IOCAP算法的运行时间，其中Posture数据集选取两个不同规模的子集，实验结果数据取多次运行的平均值，具体如表1所示.

Simulation test 216 QAM signals are transmitted to the minimum phase water acoustic channel;the impulse response of channel is given by h=［0.965 6－ 0.090 6 0.057 8 0.236 8］［12］，the weight length of equalizer is set to 32;SNR is set to 25 dB;the power initialization value is 4;the 2th taps of the weight vectors of WT-FSE and FSE are initialized to one.In three algorithms，the step sizes are μT/2-FSE-CMA=0.000 014， μWT-T/2-FSE-CMA= 0.000 014， and μWT-T/2-FSE-GA-CMA=0.000 002.The DB2 wavelet is used to decompose the output signal of each subchannel，and the decomposition level is 3，the power initialization value is set to 4，and the forgetting factor β is set to 0.999 9.The results of 2 000 Monte-Carlo simula-tions are shown in Fig.4.

Fig.4 Simulation results

It can be seen from Fig.4(a)that the convergence rate of WT-T/2-FSE-GA is essentially the same as WT-T/2-FSE，and is about 1 000 steps more than T/2-FSE.The MSE(mean square error)of WT-T/2-FSE-GA has a drop of about 1.3 dB compared with WT-T/2-FSE and T/2-FSE.Fig.2(b，c，d)shows that the constellation map of output signal in WT-T/2-FSE-GA is the clearest.

张仲平心里暗笑，敢情人家把你当司机了，他不想搭理她，用手示意曾真不要说话，然后假装打着电话：“好好……那不行，行……好好。行，那不行。不是，我的意思是说……好好好，你说你说你先说……”

Simulation results show that the genetic algorithm is used in WT-FSE to improve the steady-state error and convergence rate significantly.

4 Conclusions

In this paper，an orthogonal wavelet transform fractionally spaced blind equalization algorithm based on the optimization of genetic algorithm was proposed by introducing genetic algorithm and wavelet transform theory into a fractionally spaced blind equalization algorithm(WT-FSE-GA).The performance of the proposed algorithm reduces the possibility of local convergence by using the global search of genetic algorithm to optimize the weight vector.It has faster convergence rate and lower steady-state error by using the orthogonal wavelet transform to reduce the autocorrelation of the input signal and using the fractional space to oversample the signal.The simulated results from an underwater acoustic channel model show that WT-FSEGA has faster convergence rate and lower MSE over WT-FSE and FSE.So，WT-FSE-GA can more effectively improve the equalization performance.

［1］ZHANG Yan-ping，ZHAO Jun-wei.Underwater acoustic channel blind equalization algorithm based on fractionally spaced equalizer［J］.Acoustics and Electronics Engineering，2005，78(2):21 －23.(in Chinese)

［2］GUO Ye-cai.Blind equalization technology［M］.Hefei:Press of Hefei University of Technology，2007.(in Chinese)

［3］Cooklev T.An efficient architecture for orthogonal wavelet transforms［J］.IEEE SignalProcessing Letters(S1070-9980)，2006，13(2):77 －79.

［4］Tewfik A H，Kim M.Fast positive definite linear system solvers［J］.IEEE Trans on Signal Processing，1994，42(3):572－585.

［5］LI Yuan，ZHANG Li.A blind equalization algorithm based on neural network optimized by real-coded genetic algorithm［J］.Journal of Shanxi Datong University，2008，26(4):43 －48.(in Chinese)

［6］LI Yuan，ZHANG Li-yi.Blind equalization algorithm based on neural network and genetic algorithm［J］.Journal of Taiyuan University of Technology，2006，37(Special):69 －71.(in Chinese)

［7］ZHU Ting-ting，WANG Ying-min.The study of CMA based on genetic algorithm for underwater acoustic channel［J］.Technical Acoustics，2007，26(6):1274 －1278.

［8］HAN Ying-ge，GUO Ye-cai.Momentum term and orthogonal wavelet based CMA blind equalization［J］.Journal of System Simulation，2008，20(6):1559－1562.

［9］XIAO Ying，LIU Guo-zhi，LI Zhen-xing，et al.Blind equalization for underwater acoustic communication by genetic algorithm optimizing neural network［J］.Applied Acoustics，2006，25(6):340 －345.(in Chinese)

［10］LI Yuan.Blind equalization algorithm based on genetic algorithm and neural network［D］.Taiyuan:Taiyuan U-niversity of Technology，2006.(in Chinese)

［11］LEI Ying-jie，ZHANG Shan-wen，LI Xu，et al.MATLAB the toolbox of genetic algorithms and applications［M］.Xi’an:Xidian University Press，2004.(in Chinese)

［12］WANG Feng.Blind equalization algorithms using higherorder statistics for underwater acoustic channel［D］.Xi’an:Northwestern Polytechnical University，2003.(in Chinese)