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Abstract—In this paper, we propose an extension of Self-adaptive Differential Evolution algorithm (SaDE) to solve optimization problems with constraints. In comparison with the original SaDE algorithm, the replacement criterion was modified for handling constraints. The performance of the proposed method is reported on the set of 24 benchmark problems provided by CEC2006 special session on constrained real parameter optimization.
2006 IEEE Congress on Evolutionary Computation Sheraton Vancouver Wall Centre Hotel, Vancouver, BC, Canada July 16-21, 2006
Self-adaptive Differential Evolution Algorithm for Constrained Real-Parameter Optimization
“DE/rand/1”: Vi ,G = Xr ,G + F ⋅ Xr ,G − Xr G
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“DE/best/1”: Vi ,G = Xbest ,G + F ⋅ Xr ,G − X r G 1 2,

《2024年高考英语新课标卷真题深度解析与考后提升》专题05阅读理解D篇（新课标I卷）原卷版(专家评价+全文翻译+三年真题+词汇变式+满分策略+话题变式)目录一、原题呈现P2二、答案解析P3三、专家评价P3四、全文翻译P3五、词汇变式P4(一)考纲词汇词形转换P4(二)考纲词汇识词知意P4(三)高频短语积少成多P5(四)阅读理解单句填空变式P5(五)长难句分析P6六、三年真题P7(一)2023年新课标I卷阅读理解D篇P7(二)2022年新课标I卷阅读理解D篇P8(三)2021年新课标I卷阅读理解D篇P9七、满分策略（阅读理解说明文）P10八、阅读理解变式P12 变式一：生物多样性研究、发现、进展6篇P12变式二：阅读理解D篇35题变式（科普研究建议类）6篇P20一原题呈现阅读理解D篇关键词: 说明文;人与社会;社会科学研究方法研究;生物多样性; 科学探究精神;科学素养In the race to document the species on Earth before they go extinct, researchers and citizen scientists have collected billions of records. Today, most records of biodiversity are often in the form of photos, videos, and other digital records. Though they are useful for detecting shifts in the number and variety of species in an area, a new Stanford study has found that this type of record is not perfect.“With the rise of technology it is easy for people to make observation s of different species with the aid of a mobile application,” said Barnabas Daru, who is lead author of the study and assistant professor of biology in the Stanford School of Humanities and Sciences. “These observations now outnumber the primary data that comes from physical specimens(标本), and since we are increasingly using observational data to investigate how species are responding to global change, I wanted to know: Are they usable?”Using a global dataset of 1.9 billion records of plants, insects, birds, and animals, Daru and his team tested how well these data represent actual global biodiversity patterns.“We were particularly interested in exploring the aspects of sampling that tend to bias (使有偏差) data, like the greater likelihood of a citizen scientist to take a picture of a flowering plant instead of the grass right next to it,” said Daru.Their study revealed that the large number of observation-only records did not lead to better global coverage. Moreover, these data are biased and favor certain regions, time periods, and species. This makes sense because the people who get observational biodiversity data on mobile devices are often citizen scientists recording their encounters with species in areas nearby. These data are also biased toward certain species with attractive or eye-catching features.What can we do with the imperfect datasets of biodiversity?“Quite a lot,” Daru explained. “Biodiversity apps can use our study results to inform users of oversampled areas and lead them to places – and even species – that are not w ell-sampled. To improve the quality of observational data, biodiversity apps can also encourage users to have an expert confirm the identification of their uploaded image.”32. What do we know about the records of species collected now?A. They are becoming outdated.B. They are mostly in electronic form.C. They are limited in number.D. They are used for public exhibition.33. What does Daru’s study focus on?A. Threatened species.B. Physical specimens.C. Observational data.D. Mobile applications.34. What has led to the biases according to the study?A. Mistakes in data analysis.B. Poor quality of uploaded pictures.C. Improper way of sampling.D. Unreliable data collection devices.35. What is Daru’s suggestion for biodiversity apps?A. Review data from certain areas.B. Hire experts to check the records.C. Confirm the identity of the users.D. Give guidance to citizen scientists.二答案解析三专家评价考查关键能力，促进思维品质发展2024年高考英语全国卷继续加强内容和形式创新，优化试题设问角度和方式，增强试题的开放性和灵活性，引导学生进行独立思考和判断，培养逻辑思维能力、批判思维能力和创新思维能力。

explanatory factors for the observed input. A good representation is also one that is useful as input to a supervised predictor. Among the various ways of learning representations, this paper focuses on deep learning methods: those that are formed by the composition of multiple non-linear transformations, with the goal of yielding more abstract – and ultimately more useful – representations. Here we survey this rapidly developing area with special emphasis on recent progress. We consider some of the fundamental questions that have been driving research in this area. Specifically, what makes one representation better than another? Given an example, how should we compute its representation, i.e. perform feature extraction? Also, what are appropriate objectives for learning good representations?

Improved Algorithm for Estimating Pulse Repetition IntervalsKEN’ICHI NISHIGUCHI,Member,IEEEMASAAKI KOBAYASHI,Member,IEEEMitsubishi Electric CorporationThis paper presents an improved algorithm for estimating pulse repetition intervals(PRIs)of an interleaved pulse train which consists of several independent radar signals with different PRIs.The original version of this algorithm is a complex-valued autocorrelation-like integral,which leads to a kind of PRI spectrum wherein the locations of the spectral peaks indicatethe PRI values.The original algorithm,however,has a serious drawback in that it is vulnerable to timing jitter(PRI jitter).We analyze the cause of this vulnerability and propose an improved algorithm using overlapped PRI bins which have shifting time origins.The improved algorithm has proven to be quite effective in obtaining the PRI spectrum for jittered pulse trains,which enables detection of mean PRIs by thresholding.Manuscript received March4,1998;revised May28and November 29,1999;released for publication January3,2000.IEEE Log No.T-AES/36/2/05217.Refereeing of this contribution was handled by J.P.Y.Lee. Authors’addresses:K.Nishiguchi,Advanced TechnologyR&D Center,Mitsubishi Electric Corporation,8-1-1Tsukaguchi-Honmachi,Amagasaki-shi,Hyogo,Japan;M.Kobayashi,Communication Systems Center,Mitsubishi Electric Corporation,8-1-1Tsukaguchi-Honmachi,Amagasaki-shi,Hyogo, Japan.0018-9251/00/$10.00c°2000IEEE I.INTRODUCTIONWe deal here with the problem of estimating pulse repetition intervals(PRIs)of an interleaved pulse train,which is a superimposition of several independent radar signals with different PRIs.This problem arises in such areas as radar and electronic support measures(ESM)signal processing,wherethe estimated PRIs as well as the instantaneous pulse parameters,e.g.RF and direction of arrival(DOA), constitute important parameters for deinterleaving pulse trains that are interleaved[1—10].Various algorithms have been developed to estimate the PRIs of an interleaved pulse train.A comprehensive review of these algorithms is givenin[8].The common base of these algorithms is the autocorrelation function of the pulse train,whichis called the delta-¿histogram or time of arrival (TOA)difference histogram.In the autocorrelation function peaks are yielded at the locations of the PRIs contained in the original pulse train;however, many peaks are also yielded at the locations of integer multiples of the fundamental PRIs,i.e.,“subharmonics.”To detect fundamental PRIs,such algorithms as the cumulative difference(CDIF) histogram[9]and the sequential difference(SDIF) histogram[10]have been proposed.These algorithms intend to avoid the subharmonics by calculating the autocorrelation function partially and sequentially.On the other hand,there is an algorithm that can suppress the subharmonics in the autocorrelation function almost completely.One of the present authors[11,12]proposed a complex-valued autocorrelation-like integral,which yields a kindof spectrum whose peak locations indicate the fundamental PRIs.Nelson[13]also proposed the same integral formula independently.Our original algorithm for subharmonic suppression works well for detecting PRIs from an interleaved pulse train with constant PRIs[12].Ina practical situation,however,the original algorithm has a serious drawback in that it is vulnerable to PRI jitter due to measurement noise,quantization erroror intentional variation[8],even when it is not very large.We analyze the cause of the vulnerability of the original algorithm and propose an improved algorithm using overlapped PRI bins which have shifting time origins.The improved algorithm has proven to be quite effective in obtaining the PRI spectrum for jittered pulse trains,which enables detection of mean PRIs by thresholding.The organization of this paper is as follows.In Section II,the basic algorithm for subharmonic suppression is reviewed.In Section III,the performance of the basic algorithm and the degradation due to PRI jitter are analyzed.In Section IV,an improved algorithm is proposed.Section V discusses methods of automatically detecting PRIsfrom the spectrum of the improved algorithm as well as the detection performance of the algorithm.Finally, Section VI draws some conclusions.II.PRI TRANSFORMIn this section we review the basic algorithm for subharmonic suppression,which we refer to as the PRI transform[12].A.Definition and PrincipleLet t n,n=0,:::,N¡1be pulse arrival times, where N is the number of pulses.If we consider the TOA as the only parameter of each pulse,the pulse train can be modeled as a sum of unit impulses,g(t)=N¡1Xn=0±(t¡t n)(1)where±(¢)is the Dirac delta function.We consider the following integral transformation of g(t)[8,11—13], D(¿)=Z1¡1g(t)g(t+¿)exp(2¼it=¿)dt(2)where the domain of¿is¿>0.This integral is referred to as the harmonics rejecting correlation function in[8]or as the Nelson TDOA histogram in [14].However we give it the brief name of the PRI transform since its absolute value gives a kind of PRI spectrum wherein the locations of the spectral peaks indicate the PRI values[11,12].The PRI transformis similar in its form to the autocorrelation function defined byC(¿)=Z1¡1g(t)g(t+¿)dt(3)and also similar to the Fourier transform F[g](¡1=¿) =R11g(t)exp(2¼it=¿)dt.Substituting(1)into(2)and (3)yieldsD(¿)=N¡1Xn=1n¡1Xm=0±(¿¡t n+t m)exp[2¼it n=(t n¡t m)](4)C(¿)=N¡1Xn=1n¡1Xm=0±(¿¡t n+t m):(5)The difference between the PRI transform andthe autocorrelation function is that the former has the phase factor exp(2¼it=¿)or exp[2¼it n=(t n¡t m)], and this factor plays an important role in suppressing the subharmonics which appear in the autocorrelation function.To explain the effect of the phase factor of the PRI transform,let us define the phase of a pulse train. The pulse arrival times of a pulse train with a single PRI,which we refer to as a single pulse train,can be written ast n=(n+´)p,n=0,1,2,:::(6) where p is the PRI and´is a constant.We define the phase of the pulse train byµ=2¼´mod2¼:(7) Two phases,µ1andµ2,are equivalent if they satisfyµ1=µ2mod2¼,or exp(iµ1)=exp(iµ2).In symbols we writeµ1´µ2.The phase of a single pulse train with the PRI p can also be obtained byµ´2¼t n=p=2¼t n=(t n¡t n¡1)(8) for all t n,n=1,2,:::.Therefore,the phase is calculated in terms of every two adjacent pulses.Next,we consider the autocorrelation function of a single pulse train.Substituting(6)into(5),we obtainC(¿)=N¡1Xl=1(N¡l)±(¿¡lp):(9)Although the impulses located at¿=lp,l=2,3,::: are the subharmonics of the PRI p,from another viewpoint these impulses can be considered indicators of the pulse trains with PRI lp.Actually,the single pulse train with pulse TOAs given by(6)can be decomposed to l single pulse trains with the samePRI lp as shown in Fig.1(a).By definition,the phases of these l pulse trains becomeµ1=µ=l,µ2= (µ+2¼)=l,:::,µl=(µ+2¼(l¡1))=l,whereµ´2¼´, 0·µ<2¼.If we represent these phases by points on the unit circle as in Fig.1(b),the vector sum of these points become zero except when l=1.The phase of the pulse train that includes the pulse pair(t m,t n)as adjacent pulses is given by2¼t n=(t n¡t m).This implies that if we multiply each term on the right-hand side (RHS)of(5)by the phase factor exp[2¼it n=(t n¡t m)] and take the summation as in(4),the subharmonics appearing in the autocorrelation function would be suppressed.B.Discrete PRI TransformThe PRI transform defined by(2)or(4)has the form of the sum of the impulses,and hence it is inappropriate to calculate it numerically.We must obtain a discrete version of the PRI transform,which takes some finite values at discrete points on the¿-axis.Let[¿min,¿max]be the range of the PRI to be investigated.We separate this range into K small intervals,which we refer to as PRI bins(see Fig.2). The width of a PRI bin is b=(¿max¡¿min)=K,and its center is¿k=(k¡1=2)b+¿min,k=1,2,:::,K:(10)Fig.1.Subharmonic components of pulse train.(a)Decomposition.Single pulse train with PRI =p can be decomposed into l subharmonic components with PRI =lp .(b)Phases of l subharmonic components.µ1=µ=l ,µ2=(µ+2¼)=l ,:::,µl =(µ+(l ¡1)¼)=l.Fig.2.PRI bins.We define the discrete PRI transform as follows:D k =Z ¿k +b=2¿k ¡b=2D (¿)d¿=Xf (m ,n );¿k ¡b=2<t n ¡t m ·¿k +b=2gexp ·2¼it nn m¸:(11)Further we define a PRI spectrum by j D k j .We note that if b !0,then D k =b !D (¿)in the sense of distribution.For the sake of comparison,we calculate a discrete version of the autocorrelation function (4)as follows:C k =Z¼k +b=2¿k ¡b=2C (¿)d¿=number of pairs (t m ,t n )that satisfy ¿k ¡b=2<t n ¡t m ·¿k +b=2,k =1,2,:::,K:(12)Obviously,the following inequality holds for every kj D k j ·C k :(13)The discrete PRI transform is easily calculated by the following procedure.1)Initialization.Let D k =0for 1·k ·K and let n =1.2)Let m =n ¡1.3)Let ¿=t n ¡t m .If ¿·¿min go to 5.Else,if ¿>¿max go to 6.4)Processing for each pair (m ,n ).a)Choose k such that ¿k ¡b=2<¿·¿k +b=2.b)Update the PRI transform.D k =D k +exp(2¼it n =¿).5)Substitute m =m ¡1.If m <0go to 6.Else,go to 3.6)Substitute n =n +1.If n >N ¡1stop.Else,go to 2.III.PERFORMANCE OF ORIGINAL PRI TRANSFORM A.Application to Single Pulse Train with Constant PRILet us calculate the PRI transform of a single pulse train.Substituting (6)into (4)yields D (¿)=N ¡1X n =1n ¡1X m =0±(¿¡(n ¡m )p )exp2¼i (´+n )=N ¡1X l =1±(¿¡lp )exp 2¼i´N ¡l ¡1X n =0exp2¼in=(N ¡1)±(¿¡p )exp(2¼i´)+N ¡1X l =2±(¿¡lp )sin(N¼=l )exp ¼i (N ¡1+2´):(14)Similarly,the autocorrelation function of g (t )is obtained by substituting (6)into (5)as follows:C (¿)=(N ¡1)±(¿¡p )+N ¡1X l =2(N ¡l )±(¿¡lp ):(15)The first term on the RHS of (14)represents an impulse located at ¿=p ,and the absolute value of its coefficient is N ¡1,which is the same as that of the autocorrelation function.The second term on the RHS of (14)is the sum of impulses located at ¿=lp ,l =2,:::,N ¡1,and the absolute value of each coefficient is evaluated by¯¯¯¯sin(N¼=l )sin(¼=l )¯¯¯¯·1sin(¼=l )·l 2=¿2p :(16)We note that the RHS does not depend on N ,so that the ratio between the peak level at the locationsparison between PRI spectrum and autocorrelation function for pulse train with constant PRIs.(a)Input pulse train,which is superimposition of3single pulse trains with PRIs1,p and p(b)PRI spectrum.(c)Autocorrelation function.corresponding to the PRI and the noise level decreasesas N becomes larger.B.Application to Interleaved Pulse Train with ConstantPRIsIn order to analyze the PRI transform of aninterleaved pulse train,we represent the interleavedpulse train as follows:g(t)=MX¹=1g¹(t)(17)whereg¹(t)=N¹¡1Xn¹=1±(t¡t n¹),¹=1,:::,M(18)are single pulse trains with PRI p¹,¹=1,:::,M.The PRI transform of(17)becomesD(¿)=MX¹=1D¹¹(¿)+MX¹=1MXº=1(º=¹)D¹º(¿)(19)whereD¹º(¿)=Z1¡1g¹(t)gº(t+¿)exp(2¼it=¿)dt:(20)The first term of the RHS of(19)is the sum of the PRI transforms of single pulse trains with constant PRIs.Hence,if the PRIs are different,this term has notable peaks only at the locations corresponding to the PRIs of the single pulse trains as in the case of a single pulse train described in the previous section.The second term on the RHS of(19)is causedby the mutual interference between different single pulse trains.This interference gives rise to a noise-like spectral shape in the PRI transform and its level can be evaluated probabilistically using the Poisson arrivalTABLE IParameters of PRI Transform(Figs.3and5)model,shown in the Appendix.As a result,the value of the discrete PRI transform at the PRI bins thatdo not correspond to any true PRIs or their integer multiples are evaluated byq hj D k j2i<p(21) where h¢i means the sample average,and½denotes the pulse density.Fig.3shows an example of the PRI spectrum of an interleaved pulse train,where the PRIs of the input pulse train are1,p2,and p5.The parameters used are shown in Table I.As is apparent from the figure, the subharmonics that appeared in the autocorrelation function are suppressed almost completely by the PRI transform.C.Application to Jittered Pulse TrainsWe assume that the input pulse train is a single pulse train and the TOAs are represented ast0=0(22)t n=t n¡1+p(1+²n),n=1,2,:::,N¡1(23) where p is the average PRI and²n is the relative deviation of the adjacent pulse interval fromthe average PRI.We further assume that²n s are independently and identically distributed random variables with mean0and the standard deviation¾=p n Under these assumptions,the phase of the pairs of adjacent pulses is given byµn =2¼t n =(t n ¡t n ¡1)=2¼n +²1+¢¢¢+²n1+²n´2¼²1+¢¢¢+²n ¡n²nn+2¼(²1+¢¢¢+²n ¡n²n ):(24)When n is large ²1+¢¢¢+²n is the order of pn while n²n is the order of n ,so that the latter is significant and the phase is approximated byµn +¡2¼n²n(25)which means that the phase error increases in proportion to n .We suppose that all the pulse pairs (t n ¡1,t n ),n =1,2,:::,N ¡1are gathered into the k th PRI bin.Then the PRI transform at the bin is given byD k =N ¡1X n =1eiµn+N ¡1X n =1e ¡2¼in²n :(26)Let q (²)be the probability density function of ²n ,so the expectation of D k can be written ash D k i +N Xn =2Z 1¡1e ¡2¼in²q (²)d²:(27)We calculate this expectation for the following twocases.1)Uniform distribution.If q (²)is given byq (²)=½1=2a ,¡a ·²·a ,0,otherwise,(28)where 0<a <1=2,then h D k i =N ¡1X n =1Z a ¡acos2¼n²¢12a d²=N ¡1X n =112¼nasin2¼na !1¡1,as N !1(29)(see Fig.4(a)).When a is sufficiently small h D k i can be approximated byh D k i +1Z 2¼aN 0sin µdµ(30)which takes the maximum value (1=2¼a )R ¼0(1=µ)£sin µdµ=0:295=a at N =1=2a .2)Gaussian distribution.If q (²)is given byq (²)=p 2¼¾2¡²2=2¾2(31)Fig.4.h D k i versus N .(a)In the case of uniform distribution.(b)In the case of Gaussian distribution.thenh D k i =N ¡1X n =1Z a ¡acos2p ¡²2=2¾2d²=N ¡1X n =1e ¡(2¼n¾)2=2(32)which is a monotonically increasing function of N(see Fig.4(b)).When ¾is sufficiently small h D k i can be approximated byh D k i +1Z 2¼¾N 0e ¡µ2=2dµ(33)which approaches 1=2p2¼¾=0:1995=¾as N !1.As is seen by the above examples,though h D k i increases in proportion to N when N is small,it does not exceed an upper bound.The upper bound decreases as the PRI jitter becomes larger.On the other hand the noise level of the PRI transform increases in proportion to p N ,so that the ratio between the peak level at the locationscorresponding to the PRI and the noise level decreases as N becomes larger.In Fig.5the results of the PRI transform applied to an interleaved pulse train which includes 3single pulse trains with PRI jitter are shown.The parameters used are shown in Table I.As is shown in the figure,the spectral peaks corresponding to true PRIs were submerged in the noise even in the case of a =0:01(Fig.5(b)).Fig.5.PRI spectrum by PRI transform.Input data is an interleaved pulse train with PRI jitter.Mean PRIs are 1,p and pandjitter follows uniform distribution with width 2a as in(28).(a)a =0:001.(b)a =0:01.(c)a =0:1.IV .MODIFICATION OF PRI TRANSFORMIn the previous section we saw that the peaks of the PRI spectrum derived from the original PRI transform are reduced in the case of jittered pulse trains.There are two factors that cause this reduction.One is that the phase error of the phase factor of the PRI transform is enlarged as the TOAs grow apart from the time origin.The other is that the pulse pairs,concentrated in a PRI bin when the PRI is constant,are distributed in several bins around the average PRI.In this section we describe a modified algorithm of the PRI transform to overcome these drawbacks.A.Shifting Time OriginsTo avoid the enlargement of the phase error caused by the TOAs becoming large,we may change thetimeFig.6.Subharmonic components of PRI.origin.The role of the phase factor exp[2¼it n =(t n ¡t m )]in the PRI transform (4)is to suppress thesubharmonics which appeared in the autocorrelation function,while keeping the peak levels at the PRIs.To do so,it is not necessary that the phases of all pulse pairs are determined using a common time origin.For example,in the case of Fig.6,it is sufficient that the phases of 3pairs (t 1,t 4),(t 2,t 5),(t 3,t 6)differ from each other by 2¼=3.Consequently,we may update the time origin in the period 3£PRI.In general to eliminate the k £PRI components,we may update the time origin in the period k £PRI.On the other hand to keep the peak levels at the PRIs we may update the time origin in the period of a PRI.Hence,in all cases we may update the time origin in the period ¿k in the k th PRI bin;therefore,a different time origin is necessary for each PRI bin.If there are no missing pulses,the condition of the time shift is that t m of the pair (t m ,t n )agrees with the previous time origin.However,we have to consider the missing pulses which occur in practical situations.It is also necessary to shift the time origin whent m ¡O k takes a value near some integer multiple of ¿k .Considering the above,we shift the time origins as follows.First we calculate a preliminary phase by´0=t n ¡O kk(34)where O k denotes the previous time origin of the k th PRI bin.Here we use ¿k instead of t n ¡t m toaccommodate the influence of the PRI jitter.Then we decompose the phase as´0=º(1+³)(35)where ºis an integer and ³is a real number such that ¡1=2<³<1=2.Finally we decide whether to shift the time origin or not according to the following conditions:1)when º=0,do not shift the time origin,2)when º=1,if t m =O k ,then let t n be the new time origin,3)when º¸2,if j ³j ·³0,then let t n be the new time origin,where ³0is a positive parameter that determines the mobility of the time origins.In Fig.7examples of the shift of time origins are shown.Fig.7.Shift of time origins.(a)When PRI bin includes PRI component.(b)When PRI bin includes subharmonic componentofPRI.Fig.8.PRI bins for modified PRI transform.B.Overlapped PRI BinsTo avoid the reduction of the peaks by the distribution of the pulse pairs,the width of the PRI bins must be greater than the width of the PRI jitter.However this causes the degradation of theresolution of the estimated PRIs and makes it difficult to deinterleave an interleaved pulse train.To resolve this dilemma we may use overlapped bins (see Fig.8).Let ²be the upper limit of the PRI jitter.Let K be the number of PRI bins.We determine the center of each PRI bin in the same way as before,i.e.,¿=k ¡1=2(¿max ¡¿min )+¿min ,k =1,2,:::,K(36)where [¿min ,¿max ]is the range of PRI to beinvestigated and K is the number of PRI bins.Then the width of the PRI bin may be set asb k =2²¿k :(37)C.Modified PRI TransformBy combining shifting time origins and overlapped PRI bins we obtain the following modified PRI transform algorithm.1)Initialization.Let D k =0for 1·k ·K and let n =2.2)Let m =n ¡1.3)Let ¿=t n ¡t m .If ¿·(1¡²)¿min go to 5.Else,if ¿>(1+²)¿max go to 6.4)Calculate the range of PRI bins:k 1=·³¿1+²¡¿min´Á¢¿¸+1,k 2=·µ¿1¡²¡¿min¶Á¢¿¸+1where ¢¿=(¿max ¡¿min )=K .5)Repeat the next 5steps (from 6to 10)for k =k 1,:::,k 2.6)Initialization of the time origin.If the k th PRI bin is used for the first time,then let O k =t n .7)Calculate the preliminary phase and decompose it:´0=(t n ¡O k )=¿k ,º=[´0+0:4999:::],³=´0=º¡1:8)Shift of the time origin.If either of the following conditions are satisfied then let O k =t n .a)º=1and t m =O k .b)º¸2and j ³j ·³0.9)Calculate the phase:´=(t n ¡O k )=¿k .10)Update the PRI transform.D k =D k +exp(2¼i´).11)Substitute m =m ¡1.If m <1go to 12.Else,go to 3.12)Substitute n =n +1.If n >N stop.Else,go to 2.In Fig.9the results of the modified PRI transform are applied to the same pulse train as in Fig.5.The parameters used are shown in Table II.As is apparent from the figure,the spectral peaks corresponding to the true PRIs are recovered.Although the number of all pulse pairs (t m ,t n )s is N (N ¡1),only those that satisfy ¿min ·t n ¡t m ·¿max are processed by the modified PRI transform,so that the processing time of the modified PRI transform is proportional to N½(¿max ¡¿min ),where ½is the pulse density.Fig.10shows the CPU time of the modified PRI transform on an Intel Pentium IIITABLE IIParameters of Modified PRI Transform (Figs.9—15)Fig.9.PRI spectrum by modified PRI transform.Input data is same as in Fig.5.(Values of mean PRIs are 1,p and pand jitter follows uniform distribution with width 2a as in (28).)(a)a =0:001.(b)a =0:01.(c)a =0:1.Fig.10.Processing time of modified PRI transform as measured on Intel Pentium III 550MHz processor (densities ½=1:0,2.150,and 2.695correspond to 1,3,and 5emitters,respectively).550MHz processor.The parameters are shown in Table II.Fig.10exhibits the linear dependence of the computational load on N ,which is common to a broad class of pulse deinterleaving algorithms [15].V .DETECTION OF PRIS USING MODIFIED PRITRANSFORM A.Threshold for Detection of PRIsTo detect PRIs from the result of the modified PRI transform,the PRI bins that correspond to the correct PRIs must be distinguished from the other PRI bins.This discrimination can be achieved by using three criteria:a criterion by observation time,a criterion for eliminating subharmonics,and a criterion for eliminating noise.The threshold used to detect PRIs can be established by these criteria.Criterion by Observation Time :If a single pulse train with a PRI ¿k exists in the entire observation time T ,then the number of pulses is T=¿k .On the other hand,j D k j denotes the number of pulses of the pulse train with a PRI ¿k ,and thus ideally it becomes j D k j =T=¿k .In actual situations,each pulse train doesnot always exist throughout the entire observation time and there are some missing pulses;accordingly, we make the following criterion with a margin:j D k j¸®T¿k(38)where®is a tunable parameter.Criterion for Eliminating Subharmonics:If¿k is the PRI of a pulse train,then ideally j D k j=C k+number of pulses of the single pulse train. Otherwise,if¿k is the subharmonics of the PRI of some single pulse train,then jD k j¿C k.Therefore, we can judge whether¿k is a PRI or its subharmonics by the criterion:j D k j¸¯C k(39) where¯is a tunable parameter.This criterion is effective for jittered pulse trains,which has incomplete suppression of the subharmonics by the modified PRI transform.Criterion for Eliminating Noise:To detect PRIs from the result of the PRI transform,it is necessary that the levels of the PRI bins that correspond to the correct PRIs are much larger than the noise level,i.e., the level of the PRI bins other than those including the PRIs or their integer multiples.In the case of the modified PRI transform,however,it is not easy to estimate the noise level because of the shifting time origins.Therefore,we have devised a criterion that uses the estimate of the noise level of the original PRI transform.As is shown in the Appendix,if D k is a noise component of the original PRI transform,then the variance of j D k j is less than T½2b k,where½is the pulse density and b k is the width of the k th PRI bin. Using this variance,we can judge that the k th PRIbin includes some component other than noise by the following criterion:j D k j¸°q T½2b k(40)where°is a tunable parameter.If D k is the value of the original PRI transform,then°=3is adequate by the“three-¾criterion.”Since the noise level of the modified PRI transform is greater than that of the original PRI transform,it is necessary to choose a value of°not less than3.Combining the above three criteria we canestablish the threshold as follows:A k=max½®T¿k,¯C k,°q T½2b k ¾(41)where three tunable parameters are®,¯,and°.We tuned the values of these parameters through simulations under various conditions to increase the detection probabilities and to reduce the false alarm probability.All numerical examples in this paper were calculated by the following common values:®=0:3,¯=0:15,°=3:Fig.11.Determination of threshold and detection of PRIs. (a)Components of threshold(RHS of(41)with®=0:3,¯=0:15,°=3).(b)Detection of PRIs based on threshold.In Fig.11,an example of the components of the above threshold and the detection by the thresholdis shown.As the figure clearly shows,we can easily detect correct PRIs by finding the peaks that exceed the threshold.B.Detection PerformanceThere are mainly three factors that affect the detection performance:number of input pulses, number of emitters(single pulse trains),and jitter width.To investigate the influence of these factors, computer simulation was performed under various conditions.The input data was generated by the superimposition of all or part of five emitters with average PRIs of1,p p p and p All emitters obey the uniform jitter with the same peak-to-peak jitter width of10%,20%,or30%.Figs.12—14show the PRI spectrum and the detection results using the threshold described in the preceding section.The parameters of the modified PRI transform are shown in Table II.The detection results shown in Figs.12—14as well as others are summarized in Table III.When the number of input pulses is1000(Fig.12),up to5emitters with a10%PRI jitter can be detected.If the jitter width is expanded to30%, the number of detected emitters is reduced to threeor four.In some cases,there are false detections.It seems,however,that these false detections are caused by the nonoptimality of the current threshold.。