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

LIAO Juan(廖娟)，GUO Ye-cai(郭業才)，2，JI Tong-ying(季童瑩)

(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.

青辰在遠處觀戰，越看越是心驚。他知道師父會功夫，卻從未想到，師父的功夫竟然如此高超。這個平日看起來弱不禁風老者，此時仿佛換了個人，在天葬臺上忽起忽落，縱躍走轉，一口天葬刀更是風雨不透，任唐飛霄七足連攻，卻也攻之不下。

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:

做好實習學生思想政治工作必須按照規范化的要求，健全工作機制，加強學生思政教育隊伍建設。重視對專兼職學生管理工作者和輔導員的專題培訓，提高他們的政治素質和管理水平。加強師德師風教育、建立健全相應的工作激勵機制，提高他們的工作積極性。另外要加強黨、團組織建設，實現黨團建設陣地“戰略轉移”，做到哪里有黨員、團員就把黨小組和團小組建到哪里。還要建立一批穩定的校外實習基地并與實習單位建立起長期合作、雙贏的關系，根據不同企業的特點對實習學生開展有針對性的思想政治教育。

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.

由于中東歐16國都相對較小，每個國家又都有自己的國語，且均屬于小語種。而這些語種目前在國內主流外語培訓市場上幾乎難覓蹤跡。因此，在外語培訓市場上，政府可以政策性鼓勵有一定資質的培訓機構開設中東歐國家的小語種培訓。

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］

由圖3可知，隨著黃精浸提液添加量的不斷增加，黃精酸奶的酸度一直提高，由80.4°T增加到91.8°T，基本符合國標規定的最佳酸度要求，因此僅僅從酸奶的酸度無法確定黃精浸提液的最佳添加量；當黃精浸提液由0.3%添加到0.7%時，感官評分呈現先上升后下降的趨勢，黃精浸提液添加0.5%時，感官評分最高為90分，因此通過對黃精酸奶酸度和感官評分的分析，確定黃精浸提液的最佳添加量為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］

● 聯網共享。不僅是單一系統內的攝像機頭數量激增，而且多個系統之間的圖像信息也需要互通互用。特別是隨著監控系統規模的擴大，用戶業務需求日趨復雜，典型的如平安城市項目，要實現城市級的信息聯網共享，將治安、卡口、電警和社會面監控等視頻點統一接入管理，其聯網需求、數據存儲、業務應用都對管理平臺提出了更高要求。

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 觀察指標 分析兩組患者術中出血量、手術時間、術后排氣時間、術后拔管時間、術后腹腔引流量、進食時間、住院時間。于手術前后抽取兩組患者5 ml的空腹靜脈血，以轉速3 000 r/min離心15 min。分離血清后，提取上清液，使用酶法檢測血清膽紅素(total bilirubin，TBIL)、間接膽紅素(indirect bilirubin，IBIL)水平。采取干化學法檢測堿性磷酸酶(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.

通過以上例子不難發現，《報告》在時間軸上是以特朗普總統正式上臺執政的時間為指示中心（When I came into office），其他時間狀語大多都圍繞這一時間點展開，時間距離大多不超過一年。這就意味著，說話者有意強調或者縮小句中所涉及實體在時間上與指示中心的距離。時間軸上距離較近的句子大都提到了外部威脅和挑戰，也就是《報告》所塑造的敵對實體（rogue regimes，nuclear weapons，serious challenges，the rise of China），并且在心理空間上將這些外部實體表征為不斷迫近的威脅，從而喚起公眾的恐懼情緒。

護理教育旨在培養護理專業人才，護理教育包括基礎理論教育及實踐實訓教育。護理理論教育是為實訓教學提供理論支持，實訓教學意在將護理理論知識應用于臨床，將知識轉化為能力，直接為人們提供健康服務。所以，護理教育可以說是“知信行”理論的具體應用。護理實訓教學效果決定了護士臨床工作能力。為適應人們不斷增加的健康衛生需求，護理實訓教學越來越受到重視，而基礎護理學課程則顯得尤為重要。當前在護理學教學改革中，以學生為主體，注重學生能力培養已成為新的教學理念[1]。然而，大部分醫學院校基礎護理學實訓教學仍采用傳統教學模式，即實訓教師在理論課基礎上進行流程講授—示教—學生模仿練習。

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

說起天窗，我們并不陌生。那些最初被我們稱為“囪”“通孔”的口子，就是最早期的“天窗”雛形。1941年，世界上第一座真正意義上的天窗在丹麥出現。在人們對建筑光環境的迫切需求下，建筑天窗應運而生，并深得許多設計師的青睞，開始出現在一些公共建筑及住宅項目中。進入21世紀，隨著科學技術的進步，天窗的設計與功能也變得越來越豐富，甚至一度成為現代建筑設計的潮流。既是窗戶也是天花，天窗的混合性使其能夠成為建筑空間中的關鍵元素。不得不說，利用天窗的形狀和角度加以靈活地布置，任何空間皆可變得獨特而美觀，僅僅是和天空近距離的接觸，就能改變空間給人的第一印象，這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) 分別計算5個數據集上Spectral 聚類算法、AP算法和IOCAP算法的運行時間，其中Posture數據集選取兩個不同規模的子集，實驗結果數據取多次運行的平均值，具體如表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.

張仲平心里暗笑，敢情人家把你當司機了，他不想搭理她，用手示意曾真不要說話，然后假裝打著電話：“好好……那不行，行……好好。行，那不行。不是，我的意思是說……好好好，你說你說你先說……”

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.

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