quantum systems substantially differ from classical ( i.e. , non - quantum ) systems in that state variables are represented by noncommutative operators acting on a hilbert space ; see e.g. , .such noncommutativity imposes some critical constraints on the structure of a quantum controller .this makes it difficult to analyze / synthesize feedback control systems for quantum systems .however , _ quantum filtering theory _ has clarified that a number of quantum control problems can be formulated and solved within the framework of standard classical stochastic control theory .a brief description of the filter - based approach to quantum control is as follows .the plant dynamics are given by a _quantum stochastic differential equation _, where the state is a noncommutative random variable .the dynamics are partially monitored by means of a continuous measurement that allows us to construct an estimator of the plant variables .the resulting filter is a classical stochastic differential equation called the _ belavkin equation _ or _stochastic master equation_. our objective is to synthesize an effective controller such that the filter shows a desirable behavior .  for this problem ,two types of control law have been proposed .the first one is a simple proportional feedback of the output signal .the second one is a feedback of the estimate of the plant variables , which we call the _ filter - based controller_. a more detailed description of these two controllers will be given in the next section , but we here note that , for the implementation of the filter - based controller , a non - negligible computation time is required to process the estimation .therefore , from a practical point of view , a filter - based feedback controller should be considered taking the feedback delay into explicit account .for example , steck et al .have numerically studied the issue of delay in the case of feedback stabilization of atomic motion . however , to the authors best knowledge , there have been no theoretical means to perform such investigations in the quantum case .  in this paperwe study the effect of the delay in quantum systems with the full use of several techniques for analyzing stability of stochastic delay differential systems ; see e.g. , and references therein .in particular , we focus on the control problem of a quantum spin system , which has also been studied in .this system is very important , since it is one of the most basic components in quantum information processing .this paper is organized as follows . section ii reviews quantum filtering and control . in particular , we discuss delay in this feedback control scheme .section iii is the main part of this paper .theorem 1 gives a delay - dependent stability criterion for a class of nonlinear stochastic systems including some quantum spin systems .the effectiveness of the result is then verified by deriving a stabilizing controller for the spin- particle case .* notation * for and , .the subscript is omitted when is the identity matrix .a function is said to be _ negative _ ( resp . _ positive _ ) in if ( resp . ) for any . a subset in said to be _ semi - algebraic _ if with polynomials .let be the set of -valued uniformly continuous functions on  ] .given a probability measure , the probability and expectation are denoted by and .we say an event occurs almost surely if .if it exists , the _ infinitesimal generator _ of a function along a markov process is denoted by i.e. , - \tilde x}{t} ] and =0~(\forall s\leq t) ] . fortunately ,in many important cases , especially in quantum optics , we can build such systems .the filter is then given by surprisingly , this is the same form as the classical filtering equation except the symmetrized terms .we now introduce a _ density matrix _ ; in a finite - dimensional case , it belongs to the convex set where is determined from the system .the statistics of the measurement results of an observable is completely characterized by .for example , the -th moment of the outcomes is given by .thus the conditional expectation should also be represented in terms of a time - dependent density matrix as , which together with leads to the time - evolution of . in particular ,when the homodyne detection scheme is used , the most simple form of it is given by the following _ stochastic master equation _: +l\rho l^*-{\frac{1}{2}}l^*l\rho-{\frac{1}{2}}\rho l^*l.\end{aligned}\ ] ] here , is an observable called _ hamiltonian _ , representing the energy of the system .the measurement operator determines how the system interacts with the measurement apparatus ( e.g. a laser field ; see figure 1 ) .   in a typical situation ,the hamiltonian term is a function of the control input ; .our goal is to design such that the filter of eq .has a desirable behavior .note that , as in the classical case , the last term is a classical wiener increment .this implies that eq .is a classical stochastic differential equation to which several techniques developed in control theory can be applied .the proportional output feedback controller ( is the gain ) is often considered and was implemented in the experimental setup of spin - squeezing control . on the other hand , note that we can compute by using the past output sequence by eq .( [ atomicsme ] ) .if it is possible to perform this computation on - line , we can implement controller of the state feedback form , i.e. , the filter - based controller . with this control the target stateis limited to the _ eigenstates _ of the measurement operator unlike the proportional feedback case where the target can be to some extent changed flexibly , but we can instead take much wider variety of designing methods of the filter - based controller .in fact , it has been proven that the lyapunov theory was successfully employed to show the global stability of the filter for some systems .moreover , it is known that the optimal controller for a general type of quantum optimal control problem is given by a filter - based controller .this is known as the _ separation theorem _ .however , in general , the time required to compute is not negligible compared to the time - constants associated with the dynamics of a nano - mechanical system . in other words , from a practical point of view , can not be used to determine .in view of this we should consider the delayed feedback control input , where denotes the delay length . note that this formulation is able to handle further delays , for example input delays .such input delays occur because the control input must be physically implemented by means of actuators .the purpose of this paper is to propose a rigorous methodology for analyzing the behavior of quantum control systems in the face of feedback delay .in this section we consider a cold atomic ensemble trapped in an optical cavity , as depicted in figure 1 .the total angular momentum operator of the atom around the -axis ( ) is given by ,\\ c_m&:=&\sqrt{(n - m)m},\quad m=1,2,\cdots , n-1,\\ f_z&:=&\frac{1}{2 } { \mathop{\rm diag}\nolimits}\ { n-1,n-3 , \\   & & \hspace{2cm}\cdots , -(n-3 ) , -(n-1)\ } , \end{aligned}\ ] ] where represents the number of atoms .the system interacts with a laser field oriented along the -axis at a homodyne - type photo detector , which implies .the system also interacts with an external magnetic field , which is oriented along the -axis , , where the control input corresponds to the magnetic field strength , which can be modified in time . as a result ,the controlled filter equation ( [ general - sme ] ) becomes {{\rm d}}t-{\frac{1}{2}}[f_z , [ f_z , \rho_t]]{{\rm d}}t \nonumber \\ & & \hspace*{3em }   \mbox { }     + \sqrt{\eta}\big(f_z\rho_t       + \rho_tf_z-2{\mathop{\rm tr}\nolimits}(f_z\rho_t)\rho_t\big){{\rm d}}w_t,\end{aligned}\ ] ] where  ] , , and . combining these inequalities and we obtain with }{^{\mbox { \sf\scriptsize t}}}\right\|_{\tau x+s { t}^{-1}s{^{\mbox { \sf\scriptsize t } } } } ^2\\   g_1({\tilde x } ) & : = & \int _{ -\tau } ^0 \left\| { \left [   \begin{array}{cc }    e{^{\mbox { \sf\scriptsize t}}}&\underline f(s){^{\mbox { \sf\scriptsize t}}}\end{array}\right]}{^{\mbox { \sf\scriptsize t}}}\right\|_\xi^2    { { \rm d}}s \ge 0\\   g_2({\tilde x } ) & : = & \int _ { -\tau } ^0 \|g({\tilde x}(s))\|_t^2 { { \rm d}}s \\   & & - \left\| { \tilde x}(0 ) - { \tilde x}(-\tau ) -\int _ { -\tau } ^0 \underline f(s ) { { \rm d}}s \right\|_{t}^2,\\   \xi & : = & { \left [   \begin{array}{cc }    x&s\\    s{^{\mbox { \sf\scriptsize t}}}&r   \end{array}\right ] } \ge 0.\end{aligned}\ ] ]  let us take the expectation after substituting .we can show } =   0\ ] ] by using the it isometry .we thus have } \le  { { \mathbb e}\left [ { \tilde \upsilon(x_t , x_{t-\tau } ) } \right ] } .\ ] ] finally , by the assumption on and defining } s{^{\mbox { \sf\scriptsize t}}}{\left [   \begin{array}{c }    x_t\\#2   \end{array}\right]}\in { { \mathbb r}}^{2n},\ ] ] we obtain therefore eq . ( [ lyap2 ] ) follows .this completes the proof .this subsection focuses on a _spin-1/2 _ model such that the system is composed of only a single particle .in this case , the density matrix is in . the filter equation ( [ atomicsme ] ) without the input ( i.e. , ) shows the following probabilistic convergence : ~~   \mbox{or}~~   \rho_t\rightarrow\rho_{\downarrow }   : = \left [ \begin{array}{cc }      0 & 0 \\      0 & 1 \\    \end{array } \right].\ ] ] this phenomenon is known as _ quantum state reduction _ . here ( resp . ) denotes the eigenstate ( of ) for which the monitored spin state of the atom is deterministically up ( resp . down ) .note that when , these two matrices are the only equilibrium points of eq .( [ atomicsme ] ) .our goal is to design a feedback control law that achieves the deterministic convergence of to the prescribed target , which is either or , as we choose .it is shown in that the control input with \rho{_{\mbox{\scriptsize f}}})\ ] ] achieves the control objective when both and are chosen appropriately ) _ locally _ stabilizes . unfortunately ,both and are equilibria of the closed - loop system .hence , when is close to the eigenstate that is not the regulation point , must be prevented from converging to it .this is done by the first term .see for a discussion on the effect of delays when a switching control law is employed instead . ] .in this subsection , we derive a sufficient condition for this control law to globally stabilize the spin-1/2 system in the face of feedback delay .let us rewrite eq .( [ atomicsme ] ) in terms of the regulation error } : =   \left\ {   \begin{array}{cl }    \rho{_{\mbox{\scriptsizef}}}- \rho_t , & { \rm if\ } \rho{_{\mbox{\scriptsize f}}}= \rho_\uparrow\\    \rho_t - \rho{_{\mbox{\scriptsize f } } } , & { \rm if\ }\rho{_{\mbox{\scriptsize f}}}= \rho_\downarrow .\end{array }  \right.\ ] ] it can easily be verified that is equivalent to }{^{\mbox { \sf\scriptsize t}}}\rightarrow 0 $ ] .when we apply the control input with given by eq .( [ eqn : u ] ) , the dynamics of are independent of and are given by eq .( [ sdde ] ) with ,\\  g(x ) & : = & \sqrt{\eta }  { \left [   \begin{array}{c }    2x { ^{\mbox{\tiny ( 1)}}}(x { ^{\mbox{\tiny ( 1)}}}- 1)\\#2   \end{array}\right ] } ,   \label{sdde4 }\\  k & : = & { \left [   \begin{array}{cc }    k_1&k_2   \end{array}\right]}.\end{aligned}\ ] ] note that means is in the circular domain  } \in { { \mathbb r}}^2 :    \psi(x):=x{^{\mbox{\tiny ( 1)}}}(x{^{\mbox{\tiny ( 1)}}}-1)+{x{^{\mbox{\tiny ( 2)}}}}^2 \le 0 \right\}.\ ] ] it can be verified that , independently of , the solution of eq .( [ sdde ] ) does not exit almost surely . in summary , according to theorem 1 , if the following sos decomposition problem has a solution , then the control objective is achieved :  [ prob : sos ] with the definitions above , let .then , find , positive - definite matrices , and polynomials , such that are the sum of squares of polynomials in and .  we provide a numerical example to illustrate the effectiveness of theorem [ maintheorem ] .decision polynomials are restricted to quadratic functions .let and which gives the control law whose stabilizing effect for the delay - free case was examined in ( * ? ? ?* subsection iv.g ) .other parameters are chosen to be and . in this case ,problem [ prob : sos ] has a solution ; that is , the target state in the controlled system is shown to be stable .it took 3.01 seconds to check the feasibility of problem 1 using a computer with a pentium 4 3.2ghz processor and 2 gb memory .  by setting the target state , we performed a numerical simulation .time responses of the function \ ] ] are shown in figure [ fig : time ] ( 30 sample paths and their average ) .this function gives the distance from the target state , i.e. , ( resp . ) if and only if ( resp . ) . the initial state is given by for . from figure[ fig : time ] it can be seen that stability is achieved .  in principle, the numerical approach introduced in this subsection is applicable to the stability analysis of the general multi - spin system despite time - delays .the computational complexity grows quickly with the dimension .very high dimensional problems are therefore computationally intractable . on the other hand ,there exist some _ analytical _results for the -dimensional delay - free case .the authors are currently investigating computational approaches which combine the aforementioned numerical and analytical methods , in order to overcome this computational issue .( 85,70 ) ( 6,5 ) ( 40,0)time ( 0,62) a practical point of view , filter - based quantum control problems should be formulated taking feedback delay into explicit account .a delay - dependent stability criterion was derived for a class of nonlinear stochastic systems including some quantum spin control systems .a semi - algebraic approach was shown to be useful for incorporating the structure of density matrices .theorem 1 was motivated by quantum spin control systems .theorem [ maintheorem ] can deal with any stochastic delay system having the three properties listed above it .many finite - dimensional quantum systems satisfy these properties .hence theorem [ maintheorem ] is applicable to a wide class of finite - dimensional quantum systems .this paper is a first attempt to analyze quantum systems which suffer from feedback delays .hence , many important and interesting problems are left unsolved .the research topic mentioned in remark 2 is one of them .d. yue and q .- l .han , delay - dependent exponential stability of stochastic systems with time - varying delay , nonlinearity , and markovian switching , _ ieee trans .217 - 222 , 2005 .p. a. parillo , structured semidefinite programs and semialgebraic geometry methods in robustness and optimization .phd thesis , california institute of technology , 2000 .p. a. parillo , semidefinite programming relaxations for semialgebraic problems , _ math. program ._ , ser .293 - 320 , 2003 .s. prajna , a. papachristodoulou and p. a. parrilo , sostools ver 2.02 : sum of squares optimization toolbox for matlab , 2004 . j. sturm , sedumi version 1.1 , 2006 .k. kashima and k. nishio , global stabilization of two - dimensional quantum spin systems despite estimation delay , _ proc .

Abstract: feedback control ( based on the quantum continuous measurement ) of quantum systems inevitably suffers from estimation delays . in this paper we give a delay - dependent stability criterion for a wide class of nonlinear stochastic systems including quantum spin systems . we utilize a semi - algebraic problem approach to incorporate the structure of _ density matrices_. to show the effectiveness of the result , we derive a globally stabilizing control law for a quantum spin-1/2 systems in the face of feedback delays .  quantum control , delay systems , sum of squares , filtering , spin systems


over the last few decades , the increasing demand for wireless communications has motivated the exploration for more efficient usage of spectral resources ( ) .in particular , it has been noticed that there are large portions of spectrum that are severely under - utilized .recently , cognitive radio technologies ( cr ) have been proposed as a means to intelligently use such spectrum opportunities by sensing the radio environment and exploiting available spectrum holes for secondary usage . in cr systems ,secondary users are allowed to `` borrow ( or lease ) '' the usage of spectrum from primary users ( licensed users ) , as long as they do not hinder in the proper operation of the primary users communications .unfortunately , as we move to make cr technologies commercial , which will allow secondary users to access spectrum owned by primary users , we will face the inevitable risk that adversaries will be tempted to use cr technology for illicit and selfish purposes .if we imagine an unauthorized user ( invader ) attempting to sneak usage of spectrum without obeying proper regulations or leasing the usage of the spectrum , the result will be that both legitimate secondary users and primary users will face unexpected interference , resulting in significant performance degradation across the system .the challenge of enforcing the proper usage of spectrum requires the notion of a `` spectrum policing agent '' , whose primary job is to ensure the proper usage of spectrum and identify anomalous activities occurring within the spectrum . as a starting point to being able to police the usage of spectrum , we must have the ability to scan spectrum and effectively identify anomalous activities . towards this objective ,there has been several research efforts in signal processing techniques that can be applied to the spectrum scanning problem .for example , in , the author s presented methods for detecting a desired signal contained within interference . similarly , detection of unknown signals in noise without prior knowledge of authorized users was studied in .as another example , in , the authors proposed a method to detect anomalous transmission by making use of radio propagation characteristics .in authors investigated what impact on spectrum scanning can have information about the over - arching application that a spectrum thief might try to run .however , these works tend to not examine the important `` interplay '' between the two participants inherent in the problem the invader , who is smart and will attempt to use the spectrum in a manner to minimize the chance of being detected and fined , while also striving to maximize the benefit he / she receives from illicit usage of this spectrum ; and the scanner , who must be smart and employ a strategy that strategically maximizes the chance of detecting and fining the smart invader , with minimal cost .this challenge is made more difficult by the complexity of the underlying scanning problem itself : there will be large swaths of bandwidth to scan , and the system costs ( e.g. analog - to - digital conversion , and the computation associated with running signal classifiers ) associated with scanning very wide bandwidth makes it impossible to scan the full range of spectrum in a single instance .consequently , it is important to understand the strategic dynamics that exist between the scanner and the invader , while also taking into account the underlying costs and benefits that exist for each participant. this paper examines the interactions between the scanner and invader by formulating the problem using game theory .we find the optimal scanning strategy by selecting the scanning ( and , similarly , the invading ) bandwidth that should be employed in spectrum scanning .the organization of this paper is as follows : in section [ sec : problem ] , we first define the problem by formulating a two - step game in terms of payoff and cost functions : in the first step , each player chooses their bandwidths ; while in the second step each player uses the first step results to arrive at equilibrium strategies .we supply saddle point strategies for the game s first step .to gain insight into the problem , for the second step , in section [ sec : linear ] , we outline a linearized model of detection probability and arrive at the corresponding best strategies for each player in section [ sec : best response ] .we then explicitly obtain the equilibrium strategies in sections [ sec : nash equilibrium ] and [ sec : nash equilibrium_unknown ] for cases involving complete and incomplete knowledge of the invader s reward type . in section [ sec : numerical illustrations ] numerical illustrationsare supplied .finally , in sections [ sec : discussion ] and [ sec : appendix ] discussions , conclusions and the proofs of the announced results are offered to close the paper .in this section we set up our problem formulation .our formulation of the spectrum scanning problem involves two players : the scanner and the invader .the scanner , who is always present in the system , scans a part of the band of frequencies that are to be monitored , in order to prevent illegal usage by a potential invader of the primary ( scanner ) network s ownership of this band .we assume that the amount of bandwidth that needs to be scanned is much larger than is possible using a single scan by the scanner , and hence the scanner faces a dilemma : the more bandwidth that is scanned , the higher the probability of detecting the invader , but at the expense of increasing the cost of the rf scanning system .we assume that if the scanner scans a particular frequency band and the invader uses the band then the invasion will be detected with certainty if , and it will not be detected otherwise . without loss of generalitywe can assume that the size of the protected frequency band is normalized to 1 .the invader wants to use spectrum undetected for some illicit purpose .we consider two scenarios : ( a ) the reward for the invader is related to the width of the frequency band he uses if he is undetected .if he is detected he will be fined .so , the invader faces a dilemma : the more bandwidth he tries to use yields a larger payoff if he is not detected but also it increases the probability of being detected and thus to be fined , ( b ) the award for the invader is unknown to the scanner : he only knows whether it is related to the width of the frequency band the invader uses , or not .we formulate this problem as a two step game in the following two subsections .   in the first step of the game the scanner selects the band \subseteq [ 0,1] ] with a fixed upper bound frequency width to intrude , i.e. .so , and are pure strategies for the scanner and the invader .the scanner s payoff is 1 if the invader is detected ( i.e. \cap [ t_i , t_i+y]\not=\emptyset ] , and a strategy for the invader is to employ a width of frequency of size  ] is approximated by a linear function as follows : we assume that the scanning and intrusion cost as well as the invader s and scanner s utilities are linear in the bandwidth involved , i.e. , , , where . then the payoffs to the invader and the scanner , if they use strategies  ] respectively , become :   note that linearized payoffs have found extensive usage for a wide array of problems in wireless networks .of course , such approach simplifies the original problem and only gives an approximated solution. meanwhile it can also be very useful : sometimes it allows one to obtain solution explicitly , and allows one to look inside of the structure of the solution as well as the correlation between parameters of the system .in this section we give best response strategies for the scanner and the invader , i.e. such strategies that and .[ th_br ] in the second step of the game the scanner and the invader have the best response strategies and are given as follows :   , &  y=(c_s - f)/v,\\ b , &  y>(c_s - f)/v , \end{cases}\ ] ]    with    the equilibrium in the second step of the problem exists since the payoff to the scanner is linear on and the payoff to the invader is concave on . the equilibrium can be found by ( [ e_ne ] ) as a couple of strategies which are the best response to each other , i.e. and , i.e. the intersection of the best response curves ( figure [ fig_0 ] ) .such intersection always exists and is unique as shown in the following theorem .   the nash equilibrium as an intersection of the best response curves , scaledwidth=24.0% ]  [ th_2 ] let the invader s award be related to the width of band in use .this game has unique nash equilibrium in the second step , and it is given by table [ tbl1 ] with   [ cols="^,^,^,^,^,^",options="header " , ]   for marginal probabilities of the problem with unknown invader s type ( where indeed there is complete confidence in the invader s type ) we already have the solution .namely , if then the invaders reward depends on width in use .so , the equilibrium is unique and given by theorem [ th_2 ] . if then the invaders reward does not depend on width in use .thus , the equilibrium is unique again and it is . forthe inside probabilities the equilibrium is given by the following theorem  [ th_3 ] let it be unknown whether the invader s award is related to the bandwidth used .this game has unique nash equilibrium in the second step , and it is given by table [ tbl1 ] with and .as a numerical illustration of the scenario when the invader s reward depends on the bandwidth in use we consider , , , , .figures [ fig_1 ] and [ fig_2 ] demonstrate the equilibrium strategies as a function of the fine for the invader s characteristics .increasing fine makes the scanner and the invader employ larger and smaller bands correspondingly .the scanner would alter his strategy by a sudden jump at a threshold value ( for ) . in spite of the fact that the invader would vary his strategy continuously ,his payoff experiences a sudden drop .this also leads to an increase in detection probability by a jump .since the scanner already gets the upper band , while the invader still does not get to the lower band , further increasing of the fine leads to continuous decreasing of the detection probability due to the smaller bandwidth employed by the invader ( figure [ fig_3 ] ) .equilibrium strategy and when the invader s reward depends on the bandwidth in use , scaledwidth=24.0% ]   the equilibrium payoffs and when the invader s reward depends on the bandwidth in use , scaledwidth=24.0% ]   probability of detection when the invader s reward depends on the bandwidth in use , scaledwidth=24.0% ]  figure [ fig_4 ] and [ fig_5 ] demonstrate the equilibrium strategies as a function of probability ( i.e. uncertainty about the invader s reward type ) and fine .the result essentially depends on the invader s characteristics .for the equilibrium strategies do not depend on the probability .for they are constant for , while increasing of the scanner strategy by a jump at switching point drops also the invader s payoff by a jump .this `` jumping nature '' for the invader s payoff means that when assigning a fine one has to take into account the fact that sometimes a simple increase in fine might not change the invader s behavior , but that once a threshold value is hit , his behavior will change dramatically .   equilibrium strategy and with uncertainty about the invader s award , scaledwidth=24.0% ]   the equilibrium payoffs and with uncertainty about the invader s award , scaledwidth=24.0% ]   probability of detection with uncertainty about the invader s award , scaledwidth=24.0% ]in this paper we suggest a simple model of finding the optimal bandwidth to scan for detection of an invader and found the solution of this model explicitly .we have shown that the optimal width essentially depends on the scenario and under some conditions a small varying of network parameters and fine could lead to jump changes in the optimal strategies , as well as in the payoffs of the rivals .this mixture between continuous and discontinuous behavior of the invader under the influence of fine implies that the network provider has to carefully make a value judgement : some threshold values of fine could have a huge impact on the invader , while in the other situations a small increase will have a minimal impact on the strategies used .a goal for our future investigation , of course , is to investigate the non - linearized detection probability .also , since our investigation showed that the optimal scanning essentially depends on the invader s characteristics , we intend to extend our model to the case of incomplete information on the characteristics as well as to multi - scanner or multi - invader systems .30  adlakha , s. , johari , r. , and goldsmith , a. , _ competition in wireless systems via bayesian inference games . _ corr abs/0709.0516 , 2007 .akyildiz , i.f . ,lee , w.y . ,vuran , m.c . , and mohanty , s. , _ next generation /dynamic spectrum access/ cognitive radio wireless networks : a survey_. computer networks,50 , 2127 - 2159 , 2006 .brown , t.x . , sethi , a. , _ potential cognitive radio denial - of - service vulnerabilities and protection countermeasures : a multi - dimensional analysis and assessment_. mobile networks and applications , 13 , 516 - 532 , 2008 .digham , f.f . ,alouini , m.s . , and simon , m.k . ,_ on the energy detection of unknown signals over fading channels_. ieee transactions on communications , 55 , 21 - 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855 , 1999 .garnaev a. , trappe w. , _ the eavesdropping and jamming dilemma in multi - channel communications_. ieee icc 2013 .suppose that the invader uses a band with width and the scanner with equal probability employ a band from the set ( ) for ( for ) .the intervals composing and are separated from each other by at most .so , at least one band from for and from for intersects with .so , detection probability is greater or equal to for and it is is greater or equal to for .suppose that the scanner uses a band with width and the invader with equal probability employ a band from the set ( ) for ( for ) .the intervals composing and are separated from each other by more that .so , at most one band from for and from for intersects with .so , detection probability is less or equal to for and it is is less or equal to for and the result follows .     note that .so , for a fixed the payoff is a concave quadratic polynomial on getting its absolute maximum at .thus , the maximum of within  ] as it is given by ( [ e_th_br_2 ] ) .( a ) : :  let . then the equation   has the unique root within   ] .so , by  ( [ e_th_br_2 ] ) , for   ] .thus , by ( [ e_th_br_1 ] ) .so ,  ( [ e_equiv ] ) implies ( ) and ( ) .( c ) : :  let . then   for .thus , by ( [ e_th_br_2 ] ) ,   for  ] .so , by ( [ e_th_br_1 ] ) ,   , and , ( [ e_equiv ] ) implies  ( ) and ( ) .

Abstract: in this paper we investigate the problem of designing a spectrum scanning strategy to detect an intelligent invader who wants to utilize spectrum undetected for his / her unapproved purposes . to deal with this problem we apply game - theoretical tools . we model the situation as a game between a scanner and an invader where the invader faces a dilemma : the more bandwidth the invader attempts to use leads to a larger payoff if he is not detected , but at the same time also increases the probability of being detected and thus fined . similarly , the scanner faces a dilemma : the wider the bandwidth scanned , the higher the probability of detecting the invader , but at the expense of increasing the cost of building the scanning system . the equilibrium strategies are found explicitly and reveal interesting properties . in particular , we have found a discontinuous dependence of the equilibrium strategies on the network parameters , fine and the type of the invader s award . this discontinuity on fine means that the network provider has to take into account a human factor since some threshold values of fine could be very sensible for the invader , while in other situations simply increasing the fine has minimal deterrence impact . also we show how different reward types for the invader ( e.g. motivated by using different type of application , say , video - streaming or downloading files ) can be incorporated into scanning strategy to increase its efficiency  .


data acquisition ( daq ) electronics in nuclear and particle physics have been shifting from analog to digital methods for many years . as part of this process , flash adc ( fadc ) chips and field programmable gate array ( fpga )modules are having ever increasing use .although these chips and modules provide a much greater range of data processing options along with the flexibility of firmware programming , considerably more data bits often must be passed through the electronics hardware , processed in computers , and stored for the long term .data analysis methods must also be able to handle the increased size and complexity of the raw data .in some situations , only a small fraction of the total data flow may contain meaningful information relevant to the physics being pursued .a common situation is that in which the signal from a detector component or a preamplifier is digitized in a fadc .the signal contains information about the energy deposited or collected as well as the time .it may have a shape that depends on the particle , or it may represent overlapping pulses . although the desired information ( e.g. , energy ) may need only a fraction of the data bits of the digitized pulse , sophisticated processing is often not possible on the fast time scale of the signals .so the digitized pulses are passed on and possibly stored for later analysis .an additional problem with such signals is that of distinguishing a real signal from a noise excursion , perhaps one that is many standard deviations from the average .precision experiments that push to the limits of sensitivity might falsely record such excursions while , due to noise fluctuations , miss good signals .to address such issues , we have been developing methods based on correlation coefficient ( cc ) algorithms .such cc methods can be used to pick out signals from a stream of input data , to scan through atomic or nuclear spectra to identify peaks , to obtain initial estimates of the yield and location of a peak for use in a detailed fitting program , for pedestal evaluations in the presence of noise , among other things .although the context for our work is the koto experiment at j - parc , benefits can also extend to many other experiments .our initial and primary interest in developing cc methods was for use in offline data analysis .we explore here the ability of cc algorithms to identify fadc signals and to distinguish them from noise .very good estimates of the energy and time of fadc signals can be obtained quickly and easily for use in event reconstruction .consideration is also given to ways in which the methods might be extended to online applications where only a small fraction of the data contains meaningful information , and/or the event rate through a daq system or the storage of data are limited .consider a set of data \{ } and a reference peak \{ } , each with points .a simple correlation function is that of the cosine similarity function  where each sum is over points .this normalized , -dimensional scalar product expresses how well two vectors are ` aligned ' in the space .more generally , the correlation coefficient is defined as  \ ,\left [ n\sum x_i^2 - ( \sum x_i)^2 \right ] \right\}^{1/2 } } \:.\ ] ] each sum is again over values .the value is actually more meaningful in that it expresses the fraction of the variance in the data \{ } which is accounted for by the hypothesis of the reference peak \{}. for example , if , about half of the variance in the data would need to be ascribed by something other than the reference peak .a reference peak can be shifted through a data spectrum to identify candidate peaks for which at some location exceeds a selection criterion .the sums , , and are all known and fixed .the sums involving alone can be adjusted at each step by dropping off the contributions from the first ( oldest ) data location and adding the contributions of the new location . however, the correlation sum must be recomputed at each step .to provide a context for the discussion , we consider the koto experiment which seeks to obtain the first observation of the  decay and a measurement of the cp - violating parameter in the standard model ( sm ) .a full description of the koto daq system , along with an overview of the experiment , is provided in ref .only some items relevant to the discussions in this paper are noted here .the heart of the detector system is a roughly circular array of 2716 small and large csi crystals , approximately 90 cm in radius , for detecting photons .the array has a hole of 20 cm by 20 cm , partly filled with some veto detectors , for the beams to pass through . in front of and surrounding the csi array are numerous detectors to veto events with charged particles and/or photons that miss the array.  signals from each csi crystal are sent to custom - made fadc modules , each with 16 inputs .the signals are passed through a low - pass filter that is designed to convert the pulse into a quasi - gaussian shape about 45 ns full width at half maximum , and with a long and relatively flat tail .this shape is digitized by a 14-bit , 125-mhz chip into 64 time samples ( initially 48 samples ) .the top of the peak in each channel is adjusted to be near the middle of the time samples , leaving room for samples at each end to estimate pedestals .an example of a fadc signal is shown in fig .[ fig : data_peak ] .the signals in each module are placed in a 4- pipeline , sufficient to retain all data . upon receipt of a suitable trigger pulse, the fadcs then transmit the pipelined data for all time samples of the event to level 2 ( l2 ) modules .the l2 modules prepare the data for routing through 1-gb ethernet links to a computer array for event building , followed by ethernet transfer from j - parc to the nearby kek laboratory for long - term storage .the daq hardware is fully synchronized to an 8-ns clock .for each event , the goal of data analysis is to identify the ` hit ' crystals , their energy deposits , and their times so that clusters corresponding to the incident photons can be constructed .kinematic analyses are then applied to reconstruct the decay modes and their properties .it is assumed for the discussion here that suitable methods are applied to determine pedestals .our analyses typically involve an algorithm the utilizes the averages of the first 8 and the last 8 time samples , with checks on possible accidental peaks in those regions .the discussion is also focused on the csi crystals as they determine the kinematics of an event .   for this work , a model reference peak \{ }was constructed by averaging the shapes obtained from a large number of peaks in the data of an early koto test run . a representation of the shape is shown in fig .[ fig : data_peak ] .this parent reference peak consists of 44 samples of real ( floating point ) numbers with a maximum value of 1.0 . because the raw data are in the form of 14-bit integers , the analysis is made much faster by coding eq .( 2 ) with integer arithmetic and bit shifts . an auxiliary reference peak of 21 integers is thus used in the data analysis routines .the data for this peak includes all values of the parent reference peak down to the 2 - 3% level , where the tail becomes flat .the peak is then scaled from the parent reference peak so that the sum of the 21 values is .( see sec . 5 .)  to find hits , a pedestal - subtracted time spectrum is prepared for each crystal ( extended with samples of zero at each end to accommodate the samples before the maximum of the reference peak ) .the reference peak is then scanned through a time spectrum , a correlation coefficient is calculated at each step , and values that exceed the user s criterion are placed in a table along with a label of the time sample .in addition , the cc routine automatically calculates the area of the signal peak and the weighted time sum , and includes them in the candidate table .the user s code selects the highest cc value from the table , if any , to record a hit , along with the area and mean time .the area is converted to a value for the peak maximum by a fixed constant .the energy is obtained from this peak maximum by applying experimental calibration constants .it is to be noted that the information returned to the user is obtained directly from the data and is not the result of a fitting procedure .the cc value is simply a measure of the quality of the correlation with respect to a reference shape .statistical precision is improved by using a sum over the peak distribution rather than a single sample at the maximum .our standard analysis code contains some other useful features .for example , a time spectrum can be scanned prior to the cc analysis to see if it satisfies a tight criterion for having no energy , at which point it is removed from further processing . a simple method is to find the maximum and minimum values of all time samples and test if their difference is less than a specified value ( _ e.g. _ , 4 - 6 ) .also , because the peaks of all the csi signals are located near a fixed time sample by design , it is not necessary to scan over the full time distribution but only over a region around the nominal time sample .this option can eliminate crystals that have only out - of - time accidentals .two cc criterion levels are available for test and development purposes : one in the cc routine to fill its table of hit information for the user , and a second , higher one for final user selection .finally , an energy cut can be applied , commonly near the noise floor .real data are often not the best source for testing a correlation coefficient method .the energies and times have uncertainties from calibration methods , pedestal values that must be determined empirically , non - linearities , noise from known and unknown sources , and many other issues . in short , the true values are unknown and the purpose of the data analysis codes is to estimate the best values .the best tests are against monte carlo ( mc ) simulations , even with their own uncertainties .extensive mc studies were made to test the ability of the cc method to properly detect photon events in the csi array , and especially to explore its sensitivity to small signals down to the noise region . in these studies ,pseudo - fadc data were generated and then processed through the offline analysis code in the same way as real data .  in generating the pseudo - fadc data ,a pedestal level was generated randomly for each crystal .the values could be selected for the analysis , or an algorithm could be used to extract a pedestal from the first 8 and/or the last 8 time samples of a time spectrum . in the case of the algorithm ,the root - mean - square ( rms ) deviations of pedestals from their averages were about 0.7 adc counts .the pedestal fluctuations depend on noise in the system . from the data sheet for the fadc used in the experiment as well as tests with the fadc boards , a standard deviation of about counts was chosen .combined with a model energy conversion scale factor of 9 counts / mev , the noise floor is near 0.6 mev ( 2 - 3 ) .hence , the mc simulations of energy deposits in the crystals were converted to adc counts , digitized with respect to the parent reference peak , and added with gaussian distributed noise fluctuations at each time sample to the pedestal value of a crystal .the signal data were also given random time fluctuations within one time sample .apart from the effects of noise fluctuations , the conversion scale factor cancels in round - trip mc simulations .single photons for ten energies between 1 and 1000 mev were directed to the csi crystals .except for the lowest energies , the photon energy is deposited over several individual crystals .the individual deposits and the summed total energy were recorded by the mc code .the analysis code with a cc value was used to select hits .a window of about time samples about the expected location was used , and there was no energy cut .the results from the analysis were compared against the original mc energy and time values , crystal by crystal for each event , to find matched , missed , and false hits .full details are given in a separate tech note , with key features summarized here .even though the csi crystals have a radiation length of 27 , there is some inefficiency in collecting all of the energy of an electromagnetic shower .separate mc studies have shown that and from the showers ( as well as neutrons from photonuclear processes ) can be emitted and carry energy back upstream .in addition to the intrinsic loss of energy through physical processes , analysis codes may not be able to identify all of the deposited energy of a photon due to ambiguities arising from photo - statistics , noise , and/or limitations of the detection algorithm . at some point , no method that reports energy deposits below the noise floor can be considered to be reliable .  with no energy cut on any crystal ,the csi array collected an average of 98.0% to 99.4% of the incident mc photon energy , increasing slowly over the 11000 mev range .for the same range , the analysis code collected an average of 70.2% to 98.6% of the photon energy over the csi array .note that the 30% discrepancy between the mc and analyzed hits for 1-mev photons is only 0.3 mev ; the discrepancy is 0.8% ( 8 mev ) for 1000-mev photons . if crystals with energy deposits less than 1.0 mev are excluded from the sums , both the mc and analysis codes collect nearly equal average fractions of the photon energies .the losses range from about 22.5% or .7 mev for 3-mev photons to 2% or 20 mev for 1000-mev photons .hence , with such a cut , it would be feasible on average to correct the energy of a photon cluster obtained from data by using an energy - dependent factor determined from mc simulations .  on average, the cc method provides very good agreement with the energies and times of the hits .summing over the energy differences , positive and negative , between the individual mc deposits and the analyzed hit deposits , the net difference was about 0.01 mev for every photon energy .the rms spreads of the differences were about 0.13 mev .similarly , the summed time differences were about 0.04 time samples with an rms spread typically between 0.60 - 0.70 of the 8-ns time step .all of these quantities were obtained with the use of known pedestals .if a pedestal algorithm is used , the summed energy differences were about 0.07 mev with a rms spread of 0.23 mev , while the summed time differences and rms spreads were unaffected .the cc method is not intended to provide high precision for these quantities .yet it provides excellent estimates for the quantities as starting points if needed for detailed fitting routines .   in the previous section , events were considered matched " if they had the same crystal i d , although the energies were generally close as well .but the energy regions for the missing and false hits also need to be examined .  the fraction , in percent , vs. low - energy deposits within a cluster of matched , missed , and false hits between mc and analyzed pseudo - data distributions are shown in fig .[ fig : hit_distrib ] for four of the pure photon energies .the percent distributions are similar in each case , with the lines for the matched and missed hits crossing at 50% near 0.6 mev ( in our model ) .this value can be interpreted as having an equal probability for either choice : below this value , noise prevails ; above this value , there is an improving likelihood of having a real hit .the missing mc hits are nearly all gone by 1 mev , with a few reaching perhaps 1.5 mev .hence , one may define the hard noise region mev , and a gray region 0.6 mev mev .these values agree well with separate studies of the noise regions based on pseudo - fadc events with zero energy .the energy distributions here are based on assumptions in the monte carlo simulations of the overall energy calibration of fadc counts per mev .they may be different for koto or any other experiment .in addition , photo - statistics can affect the behavior at these low energies , and are not included in the model .such fluctuations would increase the energy values for the hard and gray regions .all of our mc calculations showed that very few csi crystals have energy deposits in an event . out of 2716 crystals , a 1-mev photon deposits energy on average in 1.4 crystals , while a 1000-mev photon would have hits in 48 crystals .even complicated multi - photon decays had deposits only up to 200 hits per event , with average values of about 100 - 150 hits per event . hence , 95% or more of the event datahave no direct physics content .if crystals with real energy deposits can be cleanly distinguished from those with only noise , substantial enhancements in daq event rates , along with savings in analysis time and data storage , can be implemented .large compression factors of 20 , 30 , or more might be possible .to accomplish such a task , the correlation coefficient would need to be recomputed every time sample ( 8 ns for koto ) on data in a fadc pipeline .although challenging , a rearrangement of eq .( 2 ) along with the use of integer arithmetic and bit shifts has provided a solution that can be encoded in the fadc firmware .it is not necessary to know the actual cc value to make a decision about whether to retain some data or not , but only that it exceeds a preset value . in that case, it will be convenient to square eq .( 2 ) and rearrange it in a slightly different form . to ensure a positive correlation ,require that the numerator of eq .( 2 ) before squaring be greater than zero .then , defining , ^ 2 = \left\ { \rho^2 \left [ \frac{1}{n}\sum x_i^2 -  \frac{1}{n^2}(\sum x_i)^2 \right ] \right\ }  \left [ n\sum y_i^2 - ( \sum y_i)^2 \right ] \:.\ ] ] as mentioned in sec .4.1 , in our model is normalized to 512 ( ) , so .this scaling is needed to obtain sufficient accuracy with integer arithmetic for small signals .the factor in curly braces can be precomputed once .the selection criterion is met if the left - hand side is greater than the right - hand side .it is not necessary to complete the calculations over the full dynamic range of the signals because those that exceed some level will certainly be passed .if the test fails , a compression bit for the corresponding adc input to the module can be set , and referenced by subsequent data stages to recognize missing data .the monte carlo studies have shown that a value of is generally successful .it has the ability to reach well into the noise region without picking up an excessive number of false hits .a more conservative value of 0.6 may be better for fadcs , to preserve marginal cases for further analysis . for the mc data , the extra hits with had energies below 1 mev .values below 0.6 tended to be excessively sensitive to noise .as with the offline case , a time window can be used for the cc scans to suppress accidentals .  because the correlation coefficient function is complex and can take up considerable space in a fpga , replacing it with the similarity function eq .( 1 ) may be satisfactory .in offline comparisons , very little difference has been found between the results of the two functions .many experiments need to be sensitive to signals that border on or are embedded in the noise region .a common approach is to test for an excursion that is a few noise standard deviations above the pedestal level to define a signal .this method , however , can be compromised by additional random fluctuations in the data , such as accidentals , or the pedestal level .also , excursions a few time samples from the correct time can occur .  if the signals have an approximately fixed shape apart from magnitude ( or can be transformed to such a shape ) , a correlation comparison against a reference peak is a far better way to identify a real signal . to dig a signal pattern out of noise fluctuations with some level of confidence ,there is little choice but to use such comparisons . at the lower values of ,the cc method is effectively a ` bump ' detector for low signals .detailed studies with simulated data have established that the correlation - coefficient ( or similarity ) method is very successful in identifying energy deposits ( hits ) in detector elements well into the noise region .it provides great benefits especially for offline analysis .it easily provides very good estimates of the signal energy and time that are needed for subsequent detailed fitting , and optionally for reducing the amount of stored data .  for online use in fadc modules ,the application of the method should be sufficiently conservative to preserve possible hits of small signals for later analysis , but also to provide a high level of data compression . as with other possible methods of working in the noise region ,there is some risk of losing a real signal .the use of the method must be carefully tailored to the design requirements of the experiment , especially the balance between the required signal range , photo - statistics , and other sources of noise .we thank our colleagues on the koto experiment for many thoughtful questions and comments during the development .this work was supported in part by the doe award de - sc0002644 through a subcontract from the university of michigan and doe award de - sc0006497 to arizona state university .9 e. frle , d. po , and s. ritt , nucl .instrum .phys . res .a * 463 * , 341 ( 2001 ) .

Abstract: the large growth of flash adc techniques for processing signals , especially in applications of streaming data , raises issues such as data flow through an acquisition system , long - term storage , and greater complexity in data analysis . in addition , experiments that push the limits of sensitivity need to distinguish legitimate signals from noise . the use of correlation coefficients is examined to address these issues . they are found to be quite successful well into the noise region . the methods can also be extended to field programmable gate array modules for compressing the data flow and greatly enhancing the event rate capabilities .   flash adc , daq , koto , correlation coefficient