formulation

MULTIPLE SOURCE DIRECTION FINDING WITH AN ARRAY OF M SENSORS USINGTWO RECEIVERSE.Fishler and H.MesserDepartment of Electrical Engineering–Systems,Tel Aviv University,Tel Aviv69978,Israel,E-mail::{eranf,messer}@eng.tau.ac.il.ABSTRACTMultiple source directionfinding algorithms(e.g.,MUSIC)are applied on simultaneous measurements collected by Msensors.However,practical considerations may dictate us-ing less receivers than sensors,such that the measurementscannot be collected simultaneously.In such cases,data is collected sequentially from the different array elements in a process which is referred to as”time varying preprocessing”[3],or”switching”.In this paper we study multiple source directionfinding(DF)with an array of M>2elements, where only two receivers are available.1.INTRODUCTION AND PROBLEMFORMULATIONDirectionfinding with fewer receivers then sensors via time varying processing is a very important issue[3].In may practical scenarios the number of receivers is considerably less then the number of sensors.Moreover,the tendency is to use the minimum number of receivers possible,that is only two receivers.Reducing the number of receivers results in a cheaper and much more simple designs.We in-vestigate the question of who many sources can be localized with only two receivers and how to perform this task with a computationally efficient algorithm.Assume q far-field narrow band sources impinge on the array,with p>q sensors,from directions{θ1,...,θq}. Using complex signal representation,the vector of received signals can be written as:x(t)=A(θ)s(t)+n(t)(1) where s(t)is the signals slowly varying complex envelope,n(t)is the additive noise,θis the vector of DOA’s,andA(θ)=[a(θ1),...,a(θq]where a(θ)is the array steering vector in directionθ.We denote by(x(t))i the i-th element of vector x(t).We assume that the source signals,s(t),are stationary and ergodic complex-valued,white,Gaussian random pro-cess with zero mean,uncorrelated with the noise process and with correlation matrix R S.The noise is assumed to be stationary and ergodic complex-valued,white,Gaussian random process with varianceσ2.It is well known that the correlation matrix of x(t),denoted by R x(θ),is given by:R x(θ)=A(θ)R S A(θ)H+σ2I(2)where(·)H denote the complex conjugate transpose opera-tion.The problem of estimatingθfrom a set of N snapshots of the array,x(t1),...,x(t N),is usually refereed to as the localization problem.The case of samples which are time dependent linear transformation of the array output is dis-cussed in[3].The resulted model for the measurements is y(t i)=G(t i)x(t i),where G(t i)is the time dependent linear transformation.Note that G(t i)is actually a matrix where the number of rows is the number of receivers in the system at time t i.We are interested in the special case where G(t i)is a 2×p matrix such that each row is a vector with all elements equal to zero and one element equal to1.For the sake of simplicity,we assume that we sample each combination of two elements N times.The total number of samples isp2·N samples,we denote this number by L.This assumption will be eased later.Note,y(t i)=[x(t i)k x(t i)l]T for some k=l∈{1,...,p}2.KNOWN RESULTS AND RELATION TO OTHERPROBLEMSIn[3]the ML estimate for general G(t i)was presented. This procedure involves maximization over all unknown pa-rameters,that isθ,σ2,R S.The maximization problem be-come extremely difficult even for as little as 2sources.The authors presented an ad-hoc approach,the GLS ,which re-duce the complexity of the estimator to a search over q pa-rameters.This approach is still very complicated and the computation load is huge.By noting that the problem can be modeled as the prob-lem of direction finding with time-varying array,one can apply all the results from [4]which includes,among others,the Cramer Rao lower bound (CRLB ).In this paper con-jectures about the complexity of the ML estimator were also presented.This conjectures state that in the general case the ML estimator is not separable.In [2]it was shown that in cases where the number of sensors is smaller then the number or sources the CRLB for the DOA ’s,θ1,...,θq ,does not approach to zero as the SNR approach to infinity.It is well known that when the number of sources is smaller than the number of sensors the CRLB for the DOA ’s approach to zero when the SNR approach to infinity or the number of snapshots approach to infinity.Eigenvector based method for the case of time-varying arrays had been proposed in [1].In this paper two possible Eigenvector based method had been proposed.One is based on interpolating matrix and the other is based on focus ma-trix.Both this methods can not be used in our case due to the big difference in the steering vectors between different time instances.3.THE IDENTAFABILITY PROBLEMIt is well known that under the model (1)and some very weak conditions on the array,one can localize up to p −1sources.The following theorem establish the maximal number of sources that can be localized when using only two receivers.Theorem 1The maximal number of possible sources which can be localized is p −1,that is the problem is identifiable up to p −1parameters.Proof 1Proving that up to p −1sources could be local-ized is equivalent to proving that f (y (t 1),...,y (t L )|θ)=f y (t 1),...,y (t L )|θfor every θ=θ ,wheref (y (t 1),...,y (t L )|θ)is the joint p.d.f.of y (t 1),...y (t L )given the vector of parameters θ.Let ˜y (t i )be a column vector with 2 p2elements,given by:˜y(t i )=[y (t i )Ty (t i +N )T···y (t i +L −N )T ]T(3)It is clear that {˜y (t i )}N i =1contain all the available samplesand thus it is an equivalent representation of the data.It can be easily verified that {˜y (t i )}N i =1are -plex gaussian vectors with block diagonal correlation ma-trix given by the following:(R ˜y )j j (R ˜y )j j +1(R ˜y )j j +1(R ˜y )j +1j +1 = (R x )kk (R x )kl (R x )lk (R x )ll(4)were k,l are the sensors sampled at the j -th switching.It is clear from the structure of R ˜y that a simple one to one mapping between R x and R ˜y exists.Since both y and ˜yare gaussian random vectors it is suffice to prove that for every θ=θR ˜y |θ=R ˜y |θ.Assumethat exsist a pair θ=θ such that R ˜y |θ=R ˜y |θ.Since,as can easily seen from (4),there exists a one to one mappingbetween R ˜y and R x also R x |θ=R x |θ.We found a pair θ=θ such that R x |θ=R x |θ which contradict the factthat we can localize up to p −1sources.Thus R ˜y |θ=R ˜y |θThis theorem provides a very important result,at each time instant we are sampling a sub array of size two which in turn enable us to localize only one source.but coher-ently combining all the results from the sub arrays enables to localize p −1sources,the same number as if we were sampling the all array with p receivers.4.EIGENVECTORS BASED METHODS The ML estimator for θrequires q dimensional search at lest.The eigenvector based method like the MUSIC offers a way to reduce this complexity to one dimensional search.This reduction in complexity might be crucial since still to-day with the most advanced DSP,searching in more then two dimensional can not be performed in real time.We next describe a new eigenvector based procedure which can be used in our problem.We start with the follow-ing equivalent description of the data.Let y (t i )be a column vector with p elements.Let all the elements be equal to zero except,say the k,l elements,which are equal to x (t i )k and x (t i )l respectively,that is k,l are the two array elements which are sampled at time t i .It is clear that defining y (t i )as a p element vector is an equivalent way to describe theavailable data.Denote by ˆR Y =1NL i =1y (t i )y (t i )H the empirical correlation matrix.The expected value of ˆRY is given by the following:R Y =A (θ)R S A (θ)H +σ2I +Λ(5)where Λis a diagonal matrix with diagonal equal to (p −1)·diag (A (θ)R S A (θ)H ).Note that if Λwas equal to zeroone could use the MUSIC or any other eigenvector basedmethod directly on ˆRY for obtaining an estimate of θ.It is quite easy to see that if all the elements of Λare equal.An eigenvector based method of obtaining an esti-mate for θcan still be used.This is since R Y can be divided to a signal subspace and a noise subspace.Two conditions insure that all the elements of Λare equal.1.The sources are uncorrelated,that is R S is a diagonal matrix.2.All the array elements has a constant gain in every direction θ,that is |a (θ)i |are equal for every i .Although this scenario is very important it can be rarely found in practice.In all other cases an eigenvector based method can not be used directly with R Y .We suggest to use the following ad-hoc solution for the problem.By not-ing the R Y (θ)differ from R X (θ)only by multiplication of the diagonal elements by p −1we suggest to divide the di-agonal elements of ˆRY by p −1.Denote by ˆˆR Y the resulted matrix,it can be easily verified that E {ˆˆRY }=R X (θ)and thus ˆˆRY can serve as approximation to R X and thus we can use ˆˆRY with all the eigenvector based methods,e.g MUSIC .We refer to the MUSIC used with our prepro-cessing as MMUSICExample:Consider a uniform linear array with 4omni-directional elements.Assume two equi-power,partially cor-related (ρ=0.25)sources at bearings 0◦,15◦and N =100.In Figure 1a typical spectrum of the MMUSIC is shown,together with a typical spectrum of the MUSIC (when ap-plied on ˆRY without preprocessing).Direction [Deg]C o s t F u n c t i o nFigure 1:Typical MUSIC and MMUSIC cost functionsIt is obvious that this method can be extended to cases when the number of samples taken from each sensor is not equal.Let n i be number of samples taken at the i -th switch-ing.Let ˆRY = y (t i )y (t i )H .It is easily verified that the expectation of ˆRY is given by the following formula:E {ˆR Y }=(A (θ)R S A (θ)H +σ2I )Ψ(6)where (Ψ)ij is the total number of snapshots taken from thei,j sensors simultaneously,also (Ψ)i,i is the total numberof snapshots taken from i -th sensor,anddenote elementby element ing similar preprocessing,ˆRY can be divided element by element with Ψand the resultedmatrix,denoted again by ˆˆR Y can be used with any eigen-value based method.5.SIMULATION STUDYAssume a uniform linear array with 4omni-directional ele-ments.Assume two equi-power sources at bearings 0◦,15◦,and N =100.The following two graphs depicts the MSE of the first source and the probability of detecting two sources for various correlation coefficients as a function of the SNR.SNRP c Figure 2:Probability of detecting two sources as a function of the SNRThe following two graphs depicts the MSE of the first source and the probability of detecting two sources as the a function of the number of snapshots taken and for various correlation coefficient.The SNR was kept fix at 10dB .6.REFERENCES[1]B.Friedlander and A.Zeira,“Eigenstructure-based al-gorithms for direction finding with time-varying ar-SNRM S E [d e g ]Figure 3:The MSE of the First source as a function of theSNRNP cFigure 4:Probability of detecting two sources as a function of the number ofsnapshotsSNRM S E [d e g ]Figure 5:The MSE of the First source as a function of the number of snapshotsrays,”IEEE Trans.on AES.V ol.32,pp.689-701,April 1996.[2]J.Sheinvald,“On Detection and Localization of Mul-tiple Signals by Sensor Arrays,”Ph.D.Dissertaion,Tel Aviv University,Israel.[3]J.Sheinvald and M.Wax,“Direction Finding withFewer Receivers via Time-Varying Preprocessing”,IEEE Trans.on SP .V ol.47,pp.2-10,January 1999.[4]A.Zeira and B.Friedlander,“Direction Finding withTime Varying Arrays”,IEEE Trans.on SP .V ol.43,pp.927-938,1995.[5]M.A.Doron,A.J.Weiss and H.Messer,”MaximumLikelihood direction finding of wide band sources”,IEEE Trans.on SP .V ol.41,pp.411-414,1993.。

农药常用专业词汇（剂型，表面活性剂，设备仪器，作物等Pesticide Formulation农药剂型1. Dust formulation 粉剂2. Wettable Powder (WP) 可湿粉剂3. Emulsifiable Cocentration (EC) 乳油4. Control release formulation 控制释放剂5. Slow release formulation 缓释剂型6. Capsule release formulation 胶囊释放剂7. Granule formulation 颗粒剂Pelletised granule formulation8. Smoke generator formulation 烟雾剂Aerosol formulation 气雾剂9. Soluble powder formulation 可溶性粉剂10. Fumigant formulation 熏蒸剂11. Multipurpose formulation 混合制剂12. Water flowable formulation 水悬剂13. Suspension concentrate formulation (SC) 悬浮剂14. Emulsion in Water formulation (EW) 水乳剂15. Suspend emulsion formulation (SE) 悬乳剂16. Micro emulsion formulation (ME) 微乳剂17. water dispersing granule (WDG) 水分散粒剂Dry flowable formulation 干胶悬剂18. water soluble granule (WSG) 水溶性粒剂③设备与仪器Instrument Ⅰ仪器Ⅰ（基础分析）1. Gas chromatography 气相色谱2. Liquid chromatography 液相色谱3. Electro microscope (scanning) 电子显微镜4. Electro probe 电极5. Heat weight balance 压力天平6. Hg extru meter 汞压力计7. Electronic balance 电子天平8. Surface tension meter (surface tensiometer) 表面张力仪9. Viscosimeter黏度计10. Particle size counting analyzer 粒度仪Particle distribution analyzer粒度分布仪11. Contact angle meter （忘了，呵呵！）12. Moisture content analyzer 水分测定仪Equipment and apparatus设备与仪器④稀释剂与填料Diluent carrier and filling (filler) 稀释剂与填料Solvent (aromatic compound or aliphatic compound) 溶剂1.Xylene 二甲苯2.Toluene 甲苯3.acetone丙酮4.甲醇Methanol5.ethyl alcohol 乙醇农药专业英语⑤蔬菜与谷物Vegetable and Grain (cereal)蔬菜与谷物1. Soybean 大豆2. Bean sprout 豆芽3. Bean 菜豆4. Pea豌豆5. Onion 洋葱6. Garlic 大蒜7. Ginger 生姜8. Tomato 西红柿9. Potato 土豆sweet potato 红薯10. Cucumber 黄瓜11. Cabbage 包菜12. Rape (colza) 油菜13. Celery 芹菜14. Cauliflower花椰菜15. Spinach 菠菜16. Mushroom 蘑菇17. Sugar beet 甜菜18. Pea nut (groundnuts) 花生19. Carrot 胡萝卜20. Radish 萝卜21. Green pepper 青椒22. Lettuce 莴苣, 生菜23. Chilies 红辣椒24. Eggplant 茄子25. corn 玉米26. Wax gourd (whit gourd) 冬瓜27. Watermelon 西瓜28. Pumpkin 南瓜29. Kaoliang (sorghum) 高粱30. Corn (maize) 玉米31. Rice (paddy) 水稻32. Wheat 小麦33. Barley 大麦34. Oat 燕麦35. Rye 黑麦36. Millet 小米，稷, 粟农药专业英语⑥经济作物Economic Plant经济作物1. Tobacco 烟草2. Cotton 棉花3. Tea 茶树4. Sugar cane 甘蔗5. Cacao 可可豆6. Coffee 咖啡7. Panax 人参Ginseng 高丽参gen-seng 西洋参8. Jute 黄麻9. Flax 亚麻10. Hemp 大麻11. Sesame 芝麻12. Rose 玫瑰13. Palm 棕榈树14. Ornament 观赏植物15. Peppermint 薄荷16. Mulberry 桑树17. Sugar sweet 甜菜18. Wheat 小麦19. Rice (paddy) 水稻20. Rape 油菜21. Maize (corn) 玉米农药外贸专业英语⑦质检报告Analysis Certificate 分析报告Test Report检验报告单Sample Name 样品名称Sample Source样品来源或产地Sample Quantity取样量Standard样品名称Batch No. 批号Testing Date (Examine Date) 测试时间Analytical Results 分析结果Item 项目Result Note 结果记录A.I. Content (%) 有效成分质量分数Water content (%) 含水量pH Value (25℃) pH值Viscosity (25℃mpas) 粘度.Stability 稳定性Gravity比重Wettable time (Sec) 润湿时间Suspensibility (%)悬浮率Mean Particle Size (?m) 平均粒度Accelerated storage stability 低温稳定性Qualification (qualified) 合格Conclusion 结论Signature 签名Operator (Analyst) 操作者，检验者Checker复核者Assessor审核者。

图记单词：词根form-（形式，形成）含义：形态，形式；形成来源：拉丁语名词forma（形态，形式），及所衍⽣的动词formare（形成）form：[fɔrm]n.形式，形状；形态v.形成；构成；排列；塑造formal：['fɔrml]adj.正式的。

结构分析：formal=form（形式）+al（形容词后缀）→形式的→正式的informal：[ɪn'fɔrml]adj. ⾮正式的；不拘礼节的；随便的；⽇常使⽤的结构分析：informal=in（否定前缀）+formal（正式的）→⾮正式的formality：[fɔr'mæləti]n.礼节；拘谨；仪式；正式⼿续结构分析：formality=formal（正式的）+ity（名词后缀）→礼节，拘谨format：['fɔrmæt]n.格式；版式v.使格式化；规定…的格式；设计版式结构分析：format=form（形成）+at（过去分词后缀）→形成的结果→格式formation：[fɔr'meʃən]n.形成；构造；编队结构分析：formation=form（形成）+ation（动名词后缀）→形成，构造，编队formative：['fɔrmətɪv]adj.形成的；造型的；格式化的n.构词要素结构分析：formative=form（形成）+ative（形容词后缀）→形成的formula：['fɔrmjələ]n.公式，准则；配⽅结构分析：formula=form（形成，构成）+ula（指⼩名词后缀）→构造好的⼩东西，规定好的东西→公式，准则，配⽅formulate：['fɔrmjə'let]vt.⽤公式表⽰；精确地表达；构想，规划结构分析：formulate=formul(a)（公式）+ate（动词后缀）→⽤公式表⽰→精确地表达→构想，规划formulation：[,fɔrmjə'leʃən]n. 构想，规划；公式化；简洁陈述结构分析：formulation=formulat(e)（构想，规划）+ion（名词后缀）→构想，规划conform：[kən'fɔrm]v.（使）遵守；符合；使⼀致；（使）顺从；适应结构分析：conform=con（⼀起）+form（形成）→⼀起形成→使形态⼀致→遵守，使⼀致conformity：[kən'fɔrməti]n. ⼀致，适合；符合；相似结构分析：conformity=conform（使⼀致）+ity（名词后缀）→符合，⼀致conformation：[,kɑnfɔr'meʃən]n.⼀致，符合；构造；构象结构分析：conformation=conform（使⼀致）+ation（动名词后缀）→⼀致，符合deform：[dɪˈfɔrm]vt.使变形；使成畸形vi. 变形；变畸形结构分析：deform=de（向下，远离）+form（形态）→（使）远离（正常）形态→（使）变形，（使）变畸形deformity：[dɪ'fɔrməti]n.畸形；畸形的⼈或物；道德⽅⾯的缺陷结构分析：deformity=deform（变畸形）+ity（名词后缀）→畸形deformation：['difɔr'meʃən]n. 变形；畸变结构分析：deformation=deform（变畸形）+ation（动名词后缀）→变形，畸变uniform：[junəˌfɔrm]adj. 统⼀的；⼀致的；始终如⼀的n.制服vt.使⼀致，配以制服结构分析：uniform=uni（同⼀的）+form（形式）→形式统⼀的（⾐服）→⼀致的；制服uniformity：[,jʊnə'fɔrməti]n. ⼀致；同样结构分析：uniformity=uniform（⼀致的）+ity（名词后缀）→⼀致inform：[ɪn'fɔrm]vt.通知；告诉；报告vi.告发；告密结构分析：inform=in（使）+form（形成）→使形成（知识体系）→教导，使知道→通知，报告information：['ɪnfɚ'meʃən]n.信息，资料；知识；情报；通知结构分析：information=inform（教导，使知道）+ation（动名词后缀）→信息，资料informative：[ɪn'fɔrmətɪv]adj. 教育性的，有益的；情报的；见闻⼴博的结构分析：informative=inform（教导，使知道）+ative（形容词后缀）→教育性的，情报的transform：[træns'fɔrm]v.改变，使…变形；转换；转化结构分析：transform=trans（跨越，从⼀端到另⼀端）+form（形态）→换成另⼀种形态→变形，转化transformer：[træns'fɔrmɚ]n.变压器；促使转化的⼈结构分析：transformer=transform（转化）+er（者）→转化者transformation：[,trænsfɚ'meʃən]n.转化；转换；改⾰；变形结构分析：transformation=transform（转化）+ation（动名词后缀）→转化，转换reform：[rɪ'fɔrm]n.改⾰，改良；改正v.改⾰，⾰新；重组结构分析：reform=re（再次）+form（形成）→再次形成→改⾰reformation：[,rɛfɚ'meʃən]n. ⾰新；改善结构分析：reformation=reform（改⾰）+ation（动名词后缀）→⾰新reformative：[rɪ'fɔrmətɪv]adj. 改⾰的；改良的结构分析：reformative=reform（改⾰）+ative（形容词后缀）→改⾰的reformatory：[rɪ'fɔrmətɔri]adj.改⾰的；感化的；⾰新的n. 少年管教所；妓⼥教养所结构分析：reformatory=reform（改⾰，改良）+atory（形容词后缀）→改⾰的，改良的（场所）→管教所，教养所。

✧剂型（formulation;preparation）：剂型指方剂组成以后，根据病情与药物的特点制成一定的形态。

为适应治疗或预防的需要而制备的药物应用形式，称为药物剂型，简称剂型。

✧制剂(preparation):制剂是根据药典或药政管理部门批准的标准，为适应诊断、治疗或预防的需要而制成的药物应用形式的具体品种。

✧方剂（prescription）：凡按医师处方专为某一病人调剂的并确切指明具体用法、用量的药剂称为方剂。

✧调剂学（DispensingPharmacy）：方剂一般是在医院药方中制备的，研究方剂的调制理论、技术和应用的科学称为调剂学。

✧目：计量单位目粒度是指原料颗粒的尺寸，一般以颗粒的最大长度来表示。

网目是表示标准筛的筛孔尺寸的大小。

在泰勒标准筛中,所谓网目就是6.4516平方厘米即 1平方英寸（2.54厘米≈1英寸）面积中的筛孔数目,并简称为目。

✧崩解剂（Disintegrating agent）：崩解剂是指能使片剂在胃肠液中迅速裂碎成细小颗粒的物质，从而使功能成分迅速溶解吸收，发挥作用。

这类物质大都具有良好的吸水性和膨胀性，从而实现片剂的崩解。

除了缓控)释片以及某些特殊用途的片剂以外，一般的片剂中都应加有崩解剂。

✧增溶剂（increase solvent）：具有增溶能力的表面活性称为增溶剂。

✧助溶剂（Solubilizers 或称 Solubilizing agents）：助溶是指难溶性药物在水中，当加入第三种物质时能增加其溶解度；这种物质就称为助溶剂（助溶与增溶不同，其主要区别在于加入的第三种物质是低分子化合物，而不是胶体电解质或非离子表面活性剂。

如：苯甲酸钠、水杨酸钠、乙酰胺等。

）。

✧潜溶剂（cosolvent）：在混合溶剂中各溶剂在某一比例时，药物的溶解度比在各单纯溶剂中溶解度出现极大值，这种现象称为潜溶（cosolvency），这种溶剂称为潜溶剂。

✧絮凝剂（flocculant;flocculating agent）：可使液体中分散的细粒固体形成絮凝物的高分子聚合物。