the theory of quantum measurement has been mired in controversy for the majority of its history . in a large partthis is due to the philosophical difficulties , usually referred to as the `` quantum measurement problem , '' with which it is associated  .however , in most areas of physics it has been generally possible to ignore quantum measurement theory , because it was largely irrelevant to real measurements that were being made by experimentalists  . in particular ,if needed at all , an ensemble picture in which quantum measurement theory merely predicted average statistics provided a sufficient description .however , experimental technology has now advanced to the point where repeated measurements may be made on a _ single _ quantum system as it evolves  . as a result ,not only does quantum measurement theory become directly relevant to experimental physics , but we can consider testing the predictions of this theory regarding the effects of measurement on the single system itself .while the theory of quantum measurement is now widely accepted , some still remain unconvinced that the conditioned state describes what is really happening to an _ individual _ system when it is measured .how might we verify these predictions , and bring at least this part of the controversy to an end ?one way to do this is to attempt to perform feedback control on an individual quantum system .such an experiment involves taking into account the changes in the system due to the measurement results , and using this knowledge to alter inputs to the system in order to control its dynamics .this would therefore provide a direct verification of quantum measurement theory : if the feedback algorithm works , then the theory is correctly predicting the behavior of the system ; conversely , if the theory is not giving accurate predictions , the feedback algorithm will be ineffective .cavity quantum electrodynamics ( cqed ) is one area in which the motion of a single quantum system , an individual atom in an optical cavity , can be monitored experimentally in real time  .not only does the light emitted from the optical cavity provide a means to track the atomic motion , but the laser driving the cavity provides a means to apply forces to the atom .this system is therefore an excellent candidate for implementing real - time quantum feedback control .it is our purpose here to examine the implementation of such a process in this system , and to show that by using approximate estimation techniques , which will be essential in such experiments for the foreseeable future , a measurable degree of control over the atomic motion can be realized .in particular , we will simulate a control algorithm designed to cool the atomic motion .this might be referred to as `` active '' cooling ( control ) , as opposed to passive cooling schemes which have also been the focus of recent theoretical and experimental work .  in general , feedback control of the atomic motional staterequires the real - time estimation of the quantum state from the continuous measurement , thus providing the information needed to decide how the system should be perturbed to bring it to the target state .the theoretical tools required for obtaining this continuous estimate are those of continuous quantum measurement  .an estimate of the state of the quantum system ( in our case the motional state of the atom ) , conditioned continuously upon the results of measurement , is given by a stochastic master equation ( sme ) .the concept of an sme was first developed by belavkin in  ( as the quantum equivalent of the classical kushner - stratonovich equation ) , and a derivation of an sme in the language of quantum optics may be found in  . in principlethe observer may calculate this state estimate by integrating the relevant sme using the measurement results , and from this determine the values of the inputs to the system to effect control .however , in practice the complexity of the sme will prevent the observer from calculating the state estimate in real time for systems where the dynamics are sufficiently fast , such as the atomic dynamics we consider here . as a result ,a simplified state - estimation procedure is essential .the theory of feedback control for classically observed linear systems is well developed  .this theory considers the optimality and robustness of feedback algorithms for given resources , and in some special cases may be applied directly to the control of quantum systems  .discussions of the application of classical feedback control techniques to quantum systems may be found in refs .however , for nonlinear control problems , such as that of the atomic motion we consider , no general results regarding the optimality of control algorithms exist as of yet . herewe will be concerned with presenting simple control algorithms for cooling the atoms , and demonstrating that along with an approximate estimation algorithm , a substantial intervention into the dynamics of the atom using real - time feedback control should be achievable experimentally in the near future . in doing so, we build upon previous results by providing a detailed analysis of the cooling performance in this system , including its robustness to parameter variations and real - world limitations on computational speed and detection efficiency .furthermore , we are able to anticipate several difficulties in an experimental implementation and suggest solutions to overcome them.  owing to the computationally expensive nature of simulating an atom in an optical cavity , including the atomic motion in one dimension , supercomputers are invaluable for the task of evaluating the performance of the control algorithm over many realizations .this is the first time that full quantum simulations of such a cqed system , including the quantized , one - dimensional motion of the atom , have been performed .we are thus able to confirm the results of various approximations which have been used previously to provide a simplified analysis of this system  .the system we consider here is that of a two - level atom in a driven , single - mode optical cavity , where the output from the cavity is monitored using homodyne detection  . due to the continuous observation ,the evolution of the system is described by a stochastic master equation ( sme ) for the density operator , which can be written in it form as          + \gamma dt \int_{-\hbar k}^{\hbar k } n(u )           \mathcal{d}[\sigma e^{-iux/\hbar}]\rho \ , du \\     & & \displaystyle + \kappa \mathcal{d}[a]\rho\ , dt       + \sqrt{\eta\kappa } \mathcal{h}[a]\rho\ , dw ,  \end{array }  \label{sme}\ ] ] where the hamiltonian is  and we have defined the superoperators  ] by \rho & { } : = { } & c\rho c^\dagger - \frac{1}{2}(c^\dagger c \rho    + \rho c^\dagger c)\\   \mathcal{h}[c]\rho & { } : = { } & c\rho + \rho c^\dagger - \expct{c+c^\dagger}\rho   \end{array}\ ] ] for an arbitrary operator .in addition , the measured homodyne signal , given by continuously provides information about the optical phase shift due to the atom - cavity system and hence , as we will see below , information about the atomic position . in the above equations, is an operator representing the atomic position along the length of the cavity , is the conjugate momentum operator , is the annihilation operator for the cavity mode , and is the lowering operator for the atomic internal states .the energy loss rate of the cavity ( or equivalently , the full - width - at - half - maximum cavity transmission ) is , and gives the strength of the driving laser , and is related to the laser power by , where is the angular frequency of the laser light .we take the laser frequency to be resonant with the cavity mode .the laser angular wave number is given by , the decay rate of the atomic excited state is denoted by , is the cqed coupling constant between the atomic internal states and the cavity mode , and is the detuning between the atom and the cavity mode , given by , where is the atomic transition frequency .we will assume a positive ( red ) detuning , corresponding to a trapping optical potential . also , is the probability that the atomic momentum recoil along the -direction is due to a spontaneous emission event , and is the differential of a stochastic wiener process . note that we have written the stochastic master equation in the interaction picture where the free oscillation of the light and the dipole rotation of the atom are implicit .the density operator obtained as a solution to the sme condenses our knowledge of the initial condition and the measurement record , allowing a prediction of the statistics of any future measurement on the system . as we show numerically in section [ sec : estimation - cooling - dynamics ] , the solution to the equation is , generically , insensitive to the initial conditions except at very early times .we therefore refer to the solution of the sme as the observer s _ best estimate _ or _ state of knowledge _ of the quantum system .  in order to perform numerical calculations , it is sensible to choose units for position , momentum and time that are relevant for the corresponding scales of the system . when the detuning of the atom from the cavity is large compared to the cavity decay rate and the cqed coupling constant , both the internal atomic states and the cavity mode may be adiabatically eliminated , and in this case the atom sees an effective potential , given by where .since this regime is the one in which we will be primarily interested , convenient units for and are given respectively by the width in position and momentum of the ground - state wave function in the harmonic approximation to one of the cosine wells . a convenient unit for timeis set by the frequency of oscillation in the same harmonic potential .thus , given the three scales of the problem , we will use the scaled variables and , and we will measure time in units of .the sme for the density matrix may now be written as + \tilde{\gamma } \,dt          \int_{-\tilde{k}}^{\tilde{k } } n(\tilde{u } )      \mathcal{d}(\sigma e^{-i\tilde{u}x})\rho \ , d\tilde{u } \\     & & + \tilde{\kappa } \mathcal{d}[a]\rho\ , dt +        \sqrt{\eta\tilde{\kappa } } \mathcal{h}[a]\ , dw ,  \label{scsme }  \end{array}\ ] ] where and with .the scaled rate constants , and are obtained from their absolute counterparts by dividing by .the scaled wave number is naturally .note that in these scaled units we have scaled out of the problem , so that =i ] .we can now identify a regime in which is large compared to . in this casethe cavity mode damps quickly to a steady state , and we can approximate the solution by ^{-1 } ,  \\     & { } \approx { } & \displaystyle i\alpha  - \alpha \left (         \frac{2\tilde{g}^2}{\tilde{\delta}\tilde{\kappa } }         \right ) \cos^2(\tilde{k}x ) , \\a^\dagger a & { } \approx { } & \displaystyle\alpha \frac{(a - a^\dagger)}{2i } .\end{array}\ ] ] there will be fluctuations about these values due to the noise with a spectrum with a width of the order of .when ( or ) is sufficiently large , then , and the height of the potential will therefore be determined by the strength of the driving . as a consequence, one can manipulate the potential that the atom sees by changing the strength of the driving .the measured homodyne signal is given in the adiabatic approximation by where ; the position - information content of this signal is more clear in this form .as mentioned in the introduction , in implementing feedback control , two considerations are paramount .the first is the ability to effectively track the evolution of the system in real time ( state estimation ) .the second is to provide an algorithm which determines how this information about the system should be used to alter the inputs to the system to effect control .we now consider these two questions in turn .the sme given in eq .( [ sme ] ) continuously provides the observer s best estimate of the state of the system from the information provided by the measurement record [ eq . ( [ mr ] ) ] .therefore , it is possible , in principle , to use the measurement record to integrate the full sme to obtain a continuous state estimate .however , the resources required to do this in real time are prohibitive in practice .this is because the sme , being a partial stochastic differential equation , contains a large effective number of variables , and is almost impossible to integrate in real time , given that the time scale of the atomic motion is typically a few . as a result, we must obtain an estimation equation which contains only a small number of variables , but is nevertheless a good approximation to the full sme .  to obtain a compact system of estimation equations, we first note that initially gaussian states remain close to gaussian for a fairly large number of oscillations in a sufficiently strong sinusoidal potential .we thus make a gaussian approximation for the density matrix , which amounts to ignoring all cumulants higher than second order . in the present system , which effectively measures , we expect this approximation to remain valid as long as the wave packet is much narrower than a single well ( ) , and the wave - packet momentum width is much larger than two photon recoils ( ) . to further simplify the estimation equations , we consider the sme after adiabatic elimination of the cavity and internal atomic degrees of freedom , given by \ , dt    + 2 \gamma\mathcal{d}[\cos^2(\kt x)]\rho\ , dt \\    & & \displaystyle -\sqrt{2\eta\gamma}\mathcal{h}[\cos^2(\kt x)]\rho\ , dw ,  \end{array }  \label{adiabaticsme}\ ] ] which gives a three - parameter model ( , , and ) . here, we have defined the effective hamiltonian where the maximum light shift is not an independent parameter in this static problem , but we will treat it as such when considering feedback via a time - dependent potential .we have also defined the effective measurement strength we further simplify the estimation model by making a gaussian ansatz for the density operator .the resulting estimation equations contain merely 5 variables : the estimated mean position and momentum ,  and , the estimated variances , and , and the estimated symmetrized covariance .( due to the information content of the measurement process , the two variances and the covariance are algebraically independent , in contrast to a closed system . ) using the equation of motion for the expectation value of an arbitrary operator , \\ & { } = { } & \displaystyle    -i\expct{[a,\heff]}dt    + 2 \gamma \expct{d[\cos^2(\kt x)]a}dt\\ & & \displaystyle    { } - \sqrt{2\eta\gamma}\expct{\mathcal{h}[\cos^2(\kt x)]a}dw ,  \end{array}\ ] ] we can calculate the system of five gaussian estimation equations , which are dt\\   & & - 8\eta\gamma \kt^2 \ce^2 \exp(-4\kt^2\vxe)\sin^2(2\kt\xe ) dt\\   & & - \sqrt{2\eta\gamma}\kt^2[1 - 4       ( \ce^2+\kt^2\vxe)\exp(-2\kt^2\vxe)]\\     & & \hspace{6mm}\times\cos(2\kt\xe)\dwe   \\\dce & { } = { } & \displaystyle    2\pi\vpe dt\\   & & \displaystyle -2 \vmax       \kt^2\vxe\exp(-2\kt^2\vxe)\cos(2\kt\xe ) dt\\   & & - 8\eta\gamma \kt^2 \vxe \ce \exp(-4\kt^2\vxe)\sin^2(2\kt\xe ) dt\\   & & + 2\sqrt{8\eta\gamma}\kt^2\vxe\ce\exp(-2\kt^2\vxe)\cos(2\kt\xe)\dwe .\end{array }  \label{gest}\ ] ] although we can eliminate from these estimator equations ( since ) , we keep the optical potential explicit here because both and will effectively become time - dependent quantities when we consider feedback control by modulating the amplitude of the optical potential .the reconstructed `` wiener increment '' arises by making the gaussian approximation to the photocurrent equation ( [ mradiab ] ) : .\ ] ] these estimation equations resemble stochastic differential equations , but it is important to recognize that does not have the same statistics as the usual wiener increment .in particular , the reconstructed increment is the usual wiener increment with an extra deterministic component , dt ,  \end{array }  \label{dwefromdw}\ ] ] which reflects the difference between the actual and estimated quantum states .this is the route by which the measurement information is incorporated into the estimator evolution .thus , these equations can not be treated by the usual high - order numerical methods for stochastic differential equations , which assume that the deterministic and stochastic parts of the differential equation can be explicitly separated .another important issue to note is that because the estimator only has information about , it can not tell us in which well the atom is located , and therefore , while feedback will allow us to cool an atom within a well , we can not know within _ which _ well the atom is confined .moreover , the estimator can not tell us on what side of a well the atom is located .note that this degeneracy is not a consequence of the gaussian approximation , but also applies to an observer using the full sme for estimation . as a consequence ,the estimator will show us the motion of the atom up to a phase factor of : the estimated state will either be in phase with the true motion , or completely out of phase , since these two motions are mirror images of one another .thus , for the feedback algorithm to be effective for every run , it is important that it works for both cases without change .the algorithms that we describe in the next section have this property .it is worth noting that the insensitivity of the feedback algorithm to the side of the well the atom is on has a potential advantage . in simulating the dynamics of the atom, we find that if the initial energy of the atom is large enough for it to move from one well to another , then as it does so the wave function tends to split across the interwell barrier , with part going back down into the original well , and part going down into an adjacent well . in this casethe wave function is no longer gaussian , but consists of two or more humps ." however , even though the two humps are on opposite sides of their respective wells , they have approximately the same downward motion .thus , even though the estimator is gaussian , the fact that the humps are in different wells is immaterial to the measurement of , and the estimator effectively tracks the evolution of an `` equivalent '' quantum state localized within a single well in the sense of generating the same measurement record as the fragmented state . as a result , we can expect the gaussian estimator to continue to work even though the atom is distributed across more than one well .in order to perform the continuous estimation ( whether using the full sme or the above gaussian approximate estimator ) one chooses an initial state for the estimator that reflects the initial ignorance of the true quantum state and integrates the estimator equations using the measurement record as this record is obtained . in our case , when the atom is dropped into the cavity we have very little idea of where it is . it is therefore sensible to choose as the initial state for the estimator a gaussian broad enough to cover one side of a well , and centered somewhere on that side ( it is not important where precisely ) .we only need to include one side of the well in our initial uncertainty , since as mentioned above , both sides are equivalent from the point of view of the estimator . in practice , though , it is sufficient to pick a state with a much smaller uncertainty product , centered on one side of a potential well . even if this estimator state does not substantially overlap the actual quantum state , as we will see below, it will converge to the `` true '' state in a time of order as the measurement information is incorporated .   as discussed in section [ thesys ] , it is possible to raise and lower the height of the potential by changing the input laser power .thinking of the atom as a classical particle , one would expect to be able to cool the atom using the following simple algorithm : when the atom is moving toward the center of the well , it is being accelerated by the potential , and so we can lower the laser power to reduce this acceleration . on the other hand, when the atom is moving away from the center of the well , then the potential decelerates the atom , and in this case we can increase the laser power so as to increase the deceleration . writing the optical field amplitude in the absence of feedback as , we will denote the value of the field during times when it is `` switched high '' as and the value when it is `` switched low '' as .thus the potential height will have the respective values and , where is the unmodulated potential depth . in this way, we repeatedly switch the potential between a high value and a low value , which both slows the atom and moves it in towards center of the well , reducing the total energy and hence cooling the atom .this algorithm also has the nice property that all the information it requires is whether the atom is climbing or descending the well .therefore , the question of whether the estimated state is in phase or completely out of phase with the true motion is immaterial , since in both cases the relevant information is known .however , it turns out that this algorithm is not very effective at cooling the atomic motion , because while it does cool the centroid , it simultaneously feeds energy into the motion by squeezing the atomic wave function .a detailed discussion of this effect is given in appendix a. the result is that this simple centroid - only cooling algorithm is not adequate to cool the atom to the ground state .to do so we must develop a more sophisticated algorithm which can reduce the motional energy associated with the variances of the wave function .  to proceed , we consider the change in the motional energy due to time dependence of the optical - potential amplitude : it is clear from this expression that the `` bang - bang '' feedback strategy from the simple cooling algorithm , where the potential amplitude is switched cyclically and suddenly between two values , maximizes the energy extraction rate if we consider only algorithms with a cyclic modulation of the potential within that range .in fact , we should switch the potential low when the value of is maximized ( i.e. , the modulus is minimized ) , thus maximizing the energy extracted from the atom .then we should switch the potential high when is minimized , thus minimizing the energy transferred to the atom when completing the modulation cycle .when we must rely on the gaussian estimator to determine when to switch the potential , we switch the potential low or high when the estimated quantity is maximized or minimized , respectively .doing so involves an additional technical complication , however , since at any given instant , we must predict whether or not is an extremal value . to do thiswe dynamically fit a quadratic curve to a history of these values ( typically the last several hundred ) , and then trigger the feedback transitions on the slope of the fitted curve .this procedure is a convenient method for predicting the times when is maximized or minimized , and helps to reject the residual noise in the time evolution of this quantity .more formally , given a set of estimates corresponding to times , we implement the curve fit of the function to the last values of .we will defer the discussion of the computational efficiency of this algorithm until section [ section : speed ] .it may seem strange that this algorithm makes use of past state information , when all the necessary information should in principle be contained in the current estimate of the state .indeed , it is clear from the quantum bellman equation that the optimal control algorithm is given by only looking forward in time given the current state . in practice , such an algorithm is difficult to realize , and the present algorithm is likely suboptimal but robust and effective .we also note that since we are triggering in this algorithm on the estimated value of , it seems that it could be much simpler to simply trigger on the detector photocurrent , which from eq .( [ mradiab ] ) we know is a direct measure of this quantity plus noise .in fact , we can use the same curve - fitting procedure on to reject the noise on the signal .however , as we will see , this procedure does not work nearly as well as with the gaussian estimator , because the signal is noise - dominated , and the gaussian estimator acts as a nearly optimal filter for the useful information from this signal , in the same sense as a kalman filter .at this point , we can work out a simple theory of cooling for the improved cooling algorithm developed in the previous section .when we switch the optical potential low , the change in energy from eq .( [ coolingenergy ] ) is given by where we have taken , so that the potential amplitude is switched from to .when we switch high , the expression for the energy change is the same except for an overall minus sign .now we consider these expectation values for a particle very close to the ground state , and we will first consider the case where all the excess motional energy is associated with the motion of the wave - packet centroid .for the ground state itself , an equal part of the ground - state energy is associated with each of the kinetic and potential parts of the effective hamiltonian .if we define as the small , additional motional energy above the ground state , then as the displaced wave packet ( coherent state ) evolves in the nearly harmonic potential , the value \rangle ] ) , and large position variance ( , indicating the atom is not localized within the cavity ) . if any of these conditions are detected , the variances are reset according to the initial values , , while the means and are not modified ; the measurement information will quickly restore the correct estimator values .for a typical cooling simulation with the canonical parameters , the estimator must be reset in about 95% of the trajectories , with trajectories on average requiring 3 resets due to the area condition ( the other conditions producing negligible numbers of resets in comparison ) .also , to help alleviate the number of resets caused by this condition , we instead implement the condition that resets only occur when .this situation may seem curious , as the gaussian area evolves according to dt\\   & & -8\eta\gamma\kt^2\vxe\ae\exp(-4\kt^2\vxe)\sin^2(2\kt\xe ) dt\\   & & -8\eta\gamma\kt^4\vxe^2[1 - 4\kt^2\vxe\exp(-2\kt^2\vxe)]\\     & & \hspace{2mm}\times \exp(-2\kt^2\vxe )     \cos^2(2\kt\xe ) dt\\   & &  -\sqrt{2\eta\gamma}\kt^2\vxe[1 - 4(\kt^2\vxe+\ae)\exp(-2\kt^2\vxe)]\\     & & \hspace{2 mm }  \times   \cos^2(2\kt\xe ) \dwe  ,  \end{array}\ ] ] whence it follows after some examination that for all time if this is true initially .thus , these resets are numerical artifacts of evolving the estimator using a finite time step and a low - order numerical method ; in the simulations , reducing the time step by a factor of 10 reduces the average number of resets due to the area condition to 1.1 ( with 0.8 resets on average due to the condition ). however , we continue to use the larger step size , which is more reasonable to implement in practice given technological constraints on the speed of calculations for the forseeable future . for somewhat smaller measurement efficiencies ,the estimator requires fewer resets ( e.g. , about 12% of the trajectories for , and only 1.7 resets on average per trajectory ) , since the steady - state area is larger ( corresponding to a state with lower purity , as = 1/(2\ae) ] below , whereas energy in only the centroid can not .this subtle unraveling of the measurement information into the estimated state is automatic in the measurement formulation we have used here .  , comparing the performance of the improved cooling algorithm with different initial ( gaussian ) state estimates .heavy solid line ( a ) : estimate matches the canonical parameter set . solid line ( b ) : initial estimate is much more impure and substantially overlaps . dashed line ( c ) : same as solid line , but centered on .[ fig : initcondcheck ] ]  fig .[ fig : basiccooling ] compares the ensemble - averaged evolution of the energy in the case where the atoms are not cooled to the cases where the atoms are cooled by the centroid - only and improved cooling algorithms .when the atoms are not actively cooled , the energy simply increases linearly in time .the initial heating rate of predicted by eq .( [ uniformheating ] ) for measurement - backaction is substantially smaller than the initial simulated heating rate of 2.1(1 ) .we can understand this discrepancy , since the initial conditions in the simulation are uniformly distributed in phase space along a constant - energy surface , weighting more heavily the gradients of the potential where heating is maximized ; eq .( [ uniformheating ] ) underestimates the heating rate by assuming a uniform distribution in configuration space .the simple , centroid - based cooling algorithm cools the atoms rapidly for short times , but around , the cooling stops and the atoms begin to heat back up , apparently without bound . clearly, we expect some heating from the squeezing effects discussed above .furthermore , we expect that when the centroid cools and the variance becomes much larger than the mean - square amplitude of centroid motion , the centroid evolution will be dominated by measurement ( projection ) noise , and thus the feedback signal will also be determined mostly by random noise from the measurement . in this state ,the cooling algorithm is ineffective and leads to heating .this process runs away , since there is no mechanism to cool the energy associated with the variances ( squeezing ) .  , comparing the performance of the improved cooling algorithm ( heavy solid line , a ) with that of cooling based directly on the homodyne signal ( solid line , b ) .heavy solid line : improved cooling algorithm .each line is an average of 128 trajectories ; parameters are for the canonical set .[ fig : directcooling ] ]  on the other hand , the improved cooling algorithm also rapidly cools the atoms and the atomic energies settle to a well - defined steady state , which already demonstrates the potential utility of this cooling algorithm for long - term storage of the atoms . for the moment, we will defer the question of how closely the atoms are cooling to the lowest energy band of the optical potential until we take a closer look at the atomic dynamics in the next section .the cooling behavior that we see is also insensitive to the exact choice of the initial ( gaussian ) state estimate , so long as we have made a reasonable choice . to illustrate this statement , fig .[ fig : initcondcheck ] shows the ensemble - averaged cooling dynamics for two initial state estimates in addition to the canonical estimate that we use for the rest of the simulations . in the first modified initial estimate , we have selected , , , , and for a much larger uncertainty product ( lower purity  ] , which explicitly maintains the identity without requiring the evaluation of a square root ._ * execution times . * _ to evaluate how quickly the control algorithm can be iterated , we implemented the control algorithm on a modern , general - purpose , serial microprocessor ( 1.25 ghz alpha ev68 ) .a single iteration of the estimator according to eqs .( [ gest ] ) and a curve fit , including all steps necessary to compute the control output , can be completed in about 260 ns .switching to the faster method of eqs .( [ fastgest ] ) with the more stable updates ( [ fastgestmod ] ) can be completed in 140 ns with a truncation at fourth order in , and 120 ns with a second - order truncation ( including a normalization step ) .the cooling performance is essentially the same with all these algorithms , in that they all reach the final temperature ; the exception is an algorithm that uses eqs .( [ fastgest ] ) without the modifications of eqs .( [ fastgestmod ] ) , which has substantially worse cooling performance .the tradeoff in the truncated - expansion algorithms is that lower - order truncations are faster but also more unstable and thus require more resets of the estimator .based on this performance , and assuming moore s law , general - purpose hardware will not be able to iterate this algorithm in 3 ns for about a decade .however , even presently available , specialized technologies such as multicore processors and field - programmable gate arrays ( fpgas ) can likely be used to significantly decrease the execution time .for example , because the curve fit can be moved onto a separate logical processor , the estimator can be updated more quickly ( 84 ns for the fourth - order version , 70 ns for the second - order version ) ; the extra 60 ns required to generate the feedback value via the curve fit then represents an additional delay , but does not limit the rate at which the estimator can be iterated .further parallelization should bring these times down substantially .   to , as a function of the feedback delay .( time is measured in units of the 5.8  harmonic - oscillator period . )cooling is according to the improved algorithm based on the gaussian estimator , with extrapolation implemented in the curve fit ; error bars reflect standard errors from averages over 128 trajectories .other parameters are as in the canonical set .[ fig : delaysweep ] ]   to , as a function of the gaussian estimator time step .( time is measured in units of the 5.8  harmonic - oscillator period . )shown are results obtained by reducing the number of samples to keep the time interval for the quadratic curve constant ( circles ) as well as results obtained via the multiple - estimator staggering method to maintain a 300-point fit ( triangles ) .error bars reflect standard errors from averages over 128 trajectories .other parameters are as in the canonical set .[ fig : updatesweep ] ]  , feedback delay , and estimator time step .( time is measured in units of the 5.8  harmonic - oscillator period . )these parameter values for the curves ( a)-(d ) are given in the text ; others are as in the canonical set .extrapolation is implemented in the curve fit , and multiple - estimator staggering is used to maintain a 300-point fit .each curve represents an average over 128 trajectories .[ fig : realistic ] ]  _ * impact on cooling performance . *_ given these limitations in computing power it is necessary to evaluate how quickly the algorithm needs to be iterated in order to achieve good cooling performance . the cooling performance in the presence of feedback delay is shown in fig .[ fig : delaysweep ] .the cooling is very robust to the delay , since we can compensate by extrapolating the curve fit .the cooling behavior is essentially unaffected out to a delay of about 460 ns , which is well above the 60 ns figure that we mentioned for generating the feedback value , and thus additional signal propagation and processing delays should not be too problematic .limitations on the time required to iterate the gaussian estimator can also seriously impact the cooling performance .not only does a large step reduce the quality of the estimated solution , but in fitting the quadratic curve over a finite interval , there are fewer points involved in the fit , reducing the effectiveness of the fit s noise reduction .these effects on the cooling behavior are illustrated in fig .[ fig : updatesweep ] , which shows that the late - time ensemble energies grow approximately linearly with . here, the 70 ns figure mentioned above for yields a final energy of around 25 , compared to around 9 for the smallest value of ( 3 ns ) shown , a substantial decrease in the cooling effectiveness .one possible way to address this problem is to use multiple gaussian estimators that are staggered in time .for example , defining , and given that the computational hardware is limited to for some integer , we can arrange to have independent implementations of the gaussian estimator executing in parallel .they all receive the same measurement record as input , but they are staggered in time such that one produces output at each small time step . in this way , each of the individual solutions is of lower quality than a fine - time - step solution , but the curve fit can be fit to all of them ( restoring a full 300 point curve fit ) , thus helping to stabilize the cooling algorithm . the implementation of this method is also displayed in fig . [fig : updatesweep ] , which shows that this method restores much of the quality of the cooling that was lost in using the simpler algorithm .the rightmost point in this figure corresponds to operating 55 parallel estimators .so far , we have studied the performance impact on cooling of each of the three equipment - limitation effects ( detection efficiency , feedback delay , and estimator time step ) separately .some representative examples of their combined effect are shown in fig .[ fig : realistic ] . the four curves , in order of increasing departure from the ideal case ,correspond to the following parameters : ( a ) , , ( i.e. , 3 ns in physical time ) ; ( b ) , ( i.e. , 29 ns ) ; ( c ) , ( i.e. , 72 ns ) ; ( d ) , ( i.e. , 116 ns ) ; ( e ) , ( i.e. , 145 ns ) .the sums of the populations in the lowest two bands for these cases , computed between and , are , , , , and , respectively .the cooling performance , even for the case that should be within reach of current technology ( d ) , is not unreasonable in comparison to the ideal case ( a ) , although cooling quickly becomes worse as these parameter values are relaxed ( e ) .even with these effects , this method is certainly suitable for long - time storage of atoms in the optical potential and can still reach the ground state with serviceable efficiency .

Abstract: we consider the problem of controlling the motion of an atom trapped in an optical cavity using continuous feedback . in order to realize such a scheme experimentally , one must be able to perform state estimation of the atomic motion in real time . while in theory this estimate may be provided by a stochastic master equation describing the full dynamics of the observed system , integrating this equation in real time is impractical . here we derive an approximate estimation equation for this purpose , and use it as a drive in a feedback algorithm designed to cool the motion of the atom . we examine the effectiveness of such a procedure using full simulations of the cavity qed system , including the quantized motion of the atom in one dimension .


with the advent of complex cyber - physical systems ( cps ) and industrial internet - of - thing , the safety of integrated cyber - physical systems has become an important design feature that must be incorporated in all software levels . in particular, this feature must also be present in the low - level control systems where both aspects of safety and stability are integrated in the control design for safety - critical systems , such as , biomedical devices , smart infrastructure systems , and smart energy systems .  for the past few years, a number of control design methods have been proposed in literature on the design of feedback controller that can guarantee both the safety and stability , simultaneously . to name a few ,we refer interested readers to , , and .in and , the authors proposed an optimization problem , in the form of a quadratic programming , where both control lyapunov and control barrier inequalities are formulated in the constraints .the proposed method generalizes the well - known pointwise min - norm control method for designing a control law using control lyapunov functions via an optimization problem .it has been successfully implemented in the cruise control of autonomous vehicle as reported in .another direct approach is pursued in which is based on the direct merging of control lyapunov function and control barrier function .the merging process results in a control lyapunov - barrier function which can be used to stabilize the system with guaranteed safety by using sontag s universal control law .  despite the appealing idea in the aforementioned works for guaranteeing stability and safety ,it remains unclear on how to analyze the robustness of the closed - loop system in the presence of external ( disturbance ) input signals .there are many tools available for analyzing the robustness of systems stability , including , and -stability theories , absolute stability theory , input - to - state stability ( iss ) theory and many others .however , analogous tools for systems safety are still lacking which makes it difficult to carry out robustness analysis to the aforementioned results that deal with the problem of stabilization with guaranteed safety .  the seminal work in on the characterization of input - to - state stability has been one of the most important tools in the stability analysis of nonlinear systems .it has allowed us to study stability of interconnected systems , to quantify systems robustness with respect to external disturbances and to provide means for constructing a robustly stabilizing control law .the use of iss lyapunov function is crucial in all of these applications . in the following decade, the concept of iss has been used and/or generalized in various directions with a commonality on the robustness analysis of systems stability .however , safety and constraint aspects have not been considered in this framework . by considering the complement of the set of unsafe state, one might consider to apply recent generalization of iss to the stability of invariant sets as in .but it may not give us an insightful detail on the influence of external disturbance signals to the state of safety of the system .in this case , the resulting iss inequality will only provide us information on the effect of external input to the systems trajectory with respect to the complement set of unsafe state , but not on how far it is from being unsafe .  in this paper, we propose a new notion of input - to - state safety which is an adaptation of iss inequality to the systems safety case .in particular , instead of the usual iss inequality where the state trajectory of the system can be bounded from above by a term that depends on initial condition and decays to zero and another term that depends on the -norm of the external input signal , we look at the following inequality where is the set of unsafe state , denotes the distance of to , the function is strictly increasing function , is strictly increasing function in both arguments , and as the gain function that is dependent on input , akin to the iss case . as will be discussed later in section 3 , the inequality : will be called input - to - state safety ( issf ) inequality .roughly speaking , this inequality can be interpreted as follows .when there is no external input signal , then the state trajectory will never get closer to .on the other hand , if there is an external input signal then it may jeopardize the systems safety when the input signal is taken sufficiently large .the above interpretation serves very well with what we can expect in real systems where external disturbance input can potentially bring the system into the unsafe state ._ in has presented also a preliminary study on the robustness aspect for systems safety where they provide an indirect relationship between the external input norm to the admissible initial conditions such that the system remains safe .this relationship is also captured in : where if the bound on the input signal is known then the inequality : will make sense only if the initial conditions are bounded away from by a constant that depends on the input norm .complementary to the work of xu _ etal ._ in , we adapt the iss framework ala sontag to the systems safety case through the use of issf barrier function which implies : .preliminary work on this concept has been presented in which is restricted to the case of exponential input - to - state safety . in this paper, we extend it to general nonlinear case , as well as to the analysis of feedback interconnection .this paper is organized as follows . in section 2 ,we briefly recall the notion of stabilization with guaranteed safety , of iss and of barrier certificate . in section 3 , we introduce formally the notion of input - to - state safety and its characterization using issf barrier function . in section 4 , we provide a numerical example of the aforementioned results for a simple mobile robot navigation system .* notation*. throughout this paper , we consider an affine non - linear system described by where denotes a state vector , denotes an ( external ) input or disturbance to the system .the functions and are where the space consists of all continuously differentiable functions . without loss of generality and for simplicity of presentation, we will assume throughout that the solution to : is complete ( i.e. , it exists for all ) for any bounded signal .this assumption holds when the system has the input - to - state stability property which we will recall shortly .  for a given signal ,its norm is given by for and its norm is defined by . for a given bounded set , we define the distance of a point with respect to by where is a metric norm .we define an open ball centered at a point with radius by and its closure is denoted by .we define the class of continuous strictly increasing functions by and denote by all functions which satisfy .moreover , denotes all functions which satisfy as .by we denote all functions such that for a fixed and and is decreasing and converging to zero for a fixed .correspondingly , we also denote by all functions such that and is srictly increasing in both arguments .let be the set of initial conditions and let an open and bounded set be the set of unsafe states , where we assume that .for a given set , we denote the boundary of by and the closure of by .  following safety definition in , the ( autonomous ) system : with is called _ safe _ if for all and for all , . additionally , : with is called ( asymptotically ) stable with guaranteed safety if it is both ( asymptotically ) stable and safe . based on these notions ,the problem of stabilization with guaranteed safety has been investigated in where the control problem is to design a feedback law such that the closed loop system is safe and asymptotically stable , i.e. for all , we have that for all and . moreover , when the problem is called _ the global stabilization with guaranteed safety_.  as discussed briefly in the introduction , analyzing the robustness of systems stability in the presence of an ( external ) input signal can be done using the input - to - state stability ( iss ) framework .let us briefly recall the iss concept from .the system : is called _ input - to - state stable _ if there exist a and such that for any and , the following inequality holds for all : in this notion , the functions and in : describe the decaying effect from a non - zero initial condition and the influence of a bounded input signal to the state trajectory , respectively . the lyapunov characterization of iss systems is provided in the following well - known theorem from .[ thm1 ] the system : is iss if and only if there exists a smooth , functions and a function such that and hold for all and for all .  the notion of iss andits lyapunov characterization as above have been seminal in the study of nonlinear systems robustness with respect to the uncertainties in the initial conditions and to the external disturbance signals .for instance , a well - known nonlinear small - gain theorem in is based on the use of and .the study of convergence input convergence state property as in is based on the use of iss lyapunov function . however , as mentioned in the introduction , existing results on robustness have focused on the systems stability and there is not many attention on the robustness analysis on systems safety .let us recall few main results in literature on safety analysis . in order to verify the safety of system : with respect to a given unsafe set , a lyapunov - like function which is called barrier certificate has been introduced in where the safety of the system can be verified through the satisfaction of a lyapunov - like inequality without having to explicitly evaluate all possible systems trajectories .such barrier certificate is a reminiscent of chetaev function for analyzing instability of nonlinear systems .while the chetaev instability theorem is used to show that the trajectory of an autonomous system always escapes any compact set , the barrier certificate is mainly applied to show that a trajectory does not enter a given compact set .the barrier certificate theorem is summarized in following theorem .[ bcthm ] consider the ( autonomous ) system : with , i.e. , where , with a given unsafe set and set of initial conditions .assume that there exists a barrier certificate  then the system is safe .the proof of this theorem is based on the fact that the evolution of starting from a non - positive value ( c.f .:) will never cross the zero level set due to : , i.e. , the state trajectory will always be safe according to : .although the safety result as in theorem [ bcthm ] is formulated only for autonomous systems , an extension to the non - autonomous case has also been presented in . for the case where an external input is considered , e.g. , the complete system as in : , the safety condition : becomes where denotes the admissible set of input . however , the condition : is a very restrictive assumption since it must hold for all including the case when the initial condition is very close to .it means that when we start very close to the unsafe state , the system must always remain safe for whatever type of input signals as long as it has values in . in this case , we can say that such system is very robust with respect to bounded external input signals . in practice, we should expect a certain degree of fragility in the system , in the sense that , if we start very close to the unsafe state , a small external input signal can already jeopardize the systems safety ; a feature that is not captured in : .  instead of considering the inequality :, we will consider a more restrictive condition on for our main results later , where the non - increasing assumption of as in : is replaced by a strict inequality as follows where is a function .  in , the use of such barrier function for control design that guarantees safety has been presented .it is shown in these works that the standard lyapunov - based control design can directly be extended to solving the safety problem by replacing the lyapunov function with the barrier one .interested readers are referred to for control design methods that solve the stabilization with guaranteed safety by merging the control lyapunov function with the control barrier function .in this section , we will explore a new notion of input - to - state safety as a tool to analyze the robustness of systems safety .in particular , we focus our study on extending existing results on barrier certificate to the input - to - state safety framework ; akin to the role of lyapunov stability theory in the input - to - state stability results .[ isfdef ] the system : is called _ input - to - state safe _ ( issf ) locally in and with respect to the set of unsafe state if for all , there exist , and such that holds for almost all and for all admissible , we mean that the tuple is such that the rhs of : is strictly positive for almost all . ] , where the constant can be dependent on boundary of .if a system is issf , we can infer from : that the system : may be brought to the unsafe state if the -norm of is sufficiently large such that the rhs of : is negative .hence one can quantify the robustness of the system s safety with respect to an external input signal using this notion .for instance , if the initial condition is in the neighborhood of the boundary of unsafe state then : shows that a small external input signal may steer the state trajectory to enter ; even when the autonomous case is safe .since the first element on the rhs of : is a function , it implies that the distance between and is lower - bounded by a strictly increasing function until leaves .as this lower - bound of the distance is non - decreasing with time , : means that the system can eventually withstand larger input signal .we can also take a different view to the issf inequality above . if is considered to be a disturbance signal with known magnitude , e.g. , with , then : provides us with information on the admissible such that the rhs of : remains positive so that the system under such external disturbance will remain safe .let us now investigate the iss - lyapunov like condition for input - to - state safety of system : .[ issf_nl_pro ] consider system : with a given unsafe set .suppose that there exists an issf barrier function satisfying where , i=1, .. 4 . assume further that the system is iss .then the system is input - to - state safe locally in and w.r.t .in particular , for any and for all , the issf inequality : holds for all and for all admissible where , , and with be the solution of the following initial value problem so that for all .prior to proving this proposition , a few remarks can be made on the relation between the issf barrier function satisfying : - : and the barrier certificate satisfying : - : .first , it is easy to see that the condition : implies : where in : is .second , when we consider the autonomous case ( i.e. , ) , then : implies the strict version of : ( c.f . , :) .let us first evaluate the solution of : with . from : it follows that , thus evaluating the time derivative of gives us with which holds whenever .thus for almost all such that , inequality : implies that holds whenever . by letting ,the last inequality becomes note that function belongs to function and the function is positive definite .hence , the rhs of : is always positive .now by the comparison lemma , where is the solution of i.e. , for any positive initial condition .  by subtituting : into the lower bound and upper bound of in: it follows that which holds for almost all s.t . and whenever .now , let us consider the other case where . in this case, it follows immediately that   we will now combine these two cases as follows .firstly , from : , it follows that where .this inequality is obtained by adding both sides of : by and substracting the right - hand side of : by which is non - positive for all . on the other hand , by multiplying both sides of : by and then by adding both sides by , we get thus , : ( which holds for ) and : ( which is true for ) imply that holds for all s.t . .since the state trajectory starts from the safe region , then for a given initial condition and bounded input , there exists sufficiently small  and such that the right hand side of : and each term on the left - hand side are positive for all .thus , since for , : implies that holds for almost all .we will prove now that we can extend the time interval , where : is valid , to with finite if leaves the set at time , or when stays in at all time .in particular , we show that we can choose and such that both terms on the lhs of : are positive for almost all , so that : holds accordingly .firstly , let us show that for any , there exists such that latexmath:[\[\label{condition_1 }  since the system is iss , there exists and such that by triangular inequality and by denoting , it follows that where the last inequality is due to the fact that for all .thus , by taking ,\ ] ] the inequality : implies that : holds for all .hence , the second term on the lhs of : is always positive for all .it remains now to check whether for all .we will show this by contradiction .suppose that there is a finite that defines the time when . in this case ,: still holds and we have that since for all ( by hypothesis of the proposition on the admissibility of with and ) , it follows from the above inequality that which is a contradiction .thus , we have that : holds for almost all .  finally , we will derive the conservative lower bound of : such that it will no longer depend on ( which is currently dependent on and as in :) . by the definition of in : , it is trivial to check that , thus , : implies that for almost all .  on the other hand , by defining , we have that when ( including for the second case when ) , latexmath:[\[\label{outsidex_eq }  once leaves and enters again at a later time interval , then we can use again the argument as before where the initial condition is taken in the neighborhood of the boundary of . indeed , suppose that enters again at time . then by following the same argument as before , we get for almost all where is the maximum time where remains in .  since in all of these cases , satisfies either : , : or : in different time intervals , we can combine them by taking the minimum of their lower bounds . thus by defining with as defined before : , holds for almost all .hence , we have issf with and where and are as in : and : , respectively , and as defined above .note that the choice of is , in this case , independent of admissible tuple .the iss assumption in this proposition can be relaxed by weaker conditions that can guarantee the boundedness of so that the inequality : in the proof of proposition [ issf_nl_pro ] holds .for instance , we can assume that the system is integral input - to - state stable or it is practically input - to - state stable .one can see from proposition [ issf_nl_pro ] that the inequalities in : and : are reminiscent to those used in the study of iss lyapunov function . in this context, the inequality : resembles the dissipation inequality in the iss lyapunov function and the growth of as in : can be likened to the growth of as in : , albeit they grow with different sign as well as with different metric norm .we can now combine the notion of input - to - state stability and that of input - to - state safety which allows us to study the robustness of a stable and safe system with respect to an external input signal .system : is called iss with guaranteed safety ( iss - gs ) with respect to if there exists such that the system : is both input - to - state stable and input - to - state safe locally in and w.r.t . .it is trivial to show that if there exist both an iss lyapunov function satisfying : : and an issf barrier function satisfying : : locally on with then the system is input - to - state stable with guaranteed safety . instead of considering two separate functions and as suggested before , we can also consider combining the iss lyapunov inequality : and issf barrier inequality : as shown in the following proposition .[ wprop ] suppose that there exists and such that where is an indicator function for , , the functions for .then it is iss with guaranteed safety with respect to .it is trivial to check that qualifies as an iss lyapunov function satisfying : : and as an issf barrier function satisfying : : locally in .the iss property follows trivially from : and : and theorem [ thm1 ] .let for all .subsequently , let the function be extended smoothly to so that : holds for all .it follows from : that holds for all and for all . by proposition [ issf_nl_pro ], it implies that it is issf .in this section , we consider an example of a simple mobile robot navigation described by the following equations where ^t ] is its velocity which is used as a feedback control input , and ^t\in l^\infty ] with . on the other hand, one can evaluate that the control law }=-\nabla_x b(x)$ ] with ensures that the closed - loop system is issf .we will now try to combine both control laws following the same construction as in where a control lyapunov function and a control barrier function can be combined .firstly , we will modify such that it will be a compactly - supported function in the neighborhood of , which will later be related to the set in : and : in corollary [ wprop ] . for numerical simulation, we will consider .using the above , we can define a compactly - support function as follows . where is any point in the boundary of , is any path from point to any point , and . for , is defined as .following the same procedure discussed in , we can merge both and into where we can choose and such that : - : hold .using , the control law which achieves iss with guaranteed safety of the closed - loop system is given by the gradient control law .its explicit formula is given by figure [ x1-x2_iss - gs ] shows the evolution of state and starting from four different initial conditions .it can be seen from the figure that all state trajectories of the closed - loop system converge to origin while avoiding the unsafe state despite being perturbed by an external disturbance signal .when we evaluate the evolution of and that is started from , figure [ time_iss - gs ] shows clearly the iss with guaranteed safety property of the closed - loop system .   starting from four different initial conditions .the set of unsafe state is shown in red area , and the boundary of is shown by dashed line.,width=336,height=240 ]   and started from , and disturbance signal .,width=336,height=288 ]in this paper , we have presented a novel notion of input - to - state safety which is complementary to the well - known input - to - state stability notion .the new notion has allowed us to characterize the evolution of the state distance to the set of unsafe state whose lower bound depends on the initial condition and the external input signal .it can be used for the robustness analysis of systems safety against external disturbances .we would like to thank anonymous reviewers for their suggestions and technical comments which have improved the paper .a. banerjee , k.k .venkatasubramanian , t. mukherjee , s.k.s .gupta , `` ensuring safety , security , and sustainability of mission - critical cyber - physical systems , '' proc .ieee , vol .283 - 299 , 2011 .a. mehra , w - l .ma , f. berg , p. tabuada , j. w. grizzle , & a.d .ames , `` adaptive cruise control : experimental validation of advanced controllers on scale - model cars '' , proc .1411 - 1418 , chicago , 2015 .s. prajna , a. jadbabaie , `` safety verification of hybrid systems using barrier certificates , '' hybrid systems : computation and control , lecture notes in computer science , pp .477 - 492 , springer - verlag , 2004 .

Abstract: in this paper , we propose a new robustness notion that is applicable for certifying systems safety with respect to external disturbance signals . the proposed input - to - state safety ( issf ) notion allows us to certify systems safety in the presence of the disturbances which is analogous to the notion of input - to - state stability ( iss ) for analyzing systems stability .   input - to - state safety , input - to - state stability , robust control , safety control .


an information - bearing signal encountered in communications and signal processing often exhibits periodicity in its mean and auto - covariance functions and thus it is well modeled by a wide - sense cyclostationary ( wscs ) random process .this structure in the first - order and the second - order statistics has long been exploited in the design of many communications and signal processing systems .one of the classical problems related to the processing of wscs random processes is a joint optimization of the transmitter ( tx ) and receiver ( rx ) in a communication system . in , real - baseband pulse amplitude modulation ( pam ) of a wide - sense stationary ( wss ) real - valued data symbol sequenceis considered with a linear rx for use over an additive wss colored noise channel . under the minimum mean - squared error ( mmse )optimality criterion and the average transmit power constraint , the jointly optimal transmit and receive waveforms are derived .it is shown that , interestingly , the waveforms have nonzero spectral values only on a generalized nyquist interval with length equal to the minimum bandwidth required to satisfy the nyquist condition for zero intersymbol interference ( isi ) .this joint optimization problem is extended in to complex - baseband quadrature amplitude modulation ( qam ) of a wss complex - valued data symbol sequence . under the linear mmse ( lmmse )optimality criterion and the average transmit power constraint , the jointly optimal transmit and receive waveforms are derived for use over an additive wscs noise channel .it is well known that a wscs noise model is better than a wss model for the case in which data - like qam interferences are present as well as an ambient gaussian noise .in contrast to the previous results only with an additive wss noise , the optimal waveforms are shown in general to have nonzero spectral values on a frequency interval whose length is greater than that of the generalized nyquist interval .this is because , unlike a wss random process , a wscs random process possesses non - zero correlation in the frequency domain among the components that are spaced integer multiples of the symbol rate apart . to exploit such spectral correlation of the wscs random process , a vectorized fourier transform ( vft ) techniqueis employed in .this technique is motivated by the harmonic series representation of a wscs random process , and the use of that representation for joint tx and rx optimizations in cyclostationary interference and noise has been examined in and .the results in , however , have considered only the real passband or , equivalently , the complex baseband transmission of a proper - complex data sequence .hence , these results are not directly applicable to , e.g. , the real passband transmission of a bpsk data sequence , which is an improper - complex data sequence in complex baseband .recall that complex - valued random variables , vectors , and processes are called proper if their complementary covariance , complementary covariance matrix , and complementary auto - covariance function ( a.k.a .pseudo - covariance , pseudo - covariance matrix , and pseudo - covariance function ) vanish , respectively . otherwise , they are called improper . although the complex envelopes of the majority of digitally modulated signals are proper , there still remain other digitally modulated signals whose complex envelopes have non - vanishing complementary auto - covariance functions . for example , the complex envelopes of pam , vestigial sideband pam , unbalanced qam , offset quaternary phase - shift keying ( oqpsk ) , and gaussian minimum shift keying are improper .  among these improper - complex signals , we focus in this paper on a linear modulation of an improper - complex data sequence using only one transmit waveform . in particular, we consider an improper - complex data sequence that is well modeled by a zero - mean improper - complex second - order stationary ( sos ) random process for which the auto - covariance and the complementary auto - covariance functions depend only on the time difference .this results in an improper - complex second - order cyclostationary ( socs ) transmitted signal . for example , pam , vestigial sideband pam , and unbalanced qam fall into this category .it is assumed that such an improper - complex socs signal is transmitted over a strictly band - limited frequency - selective linear time - invariant ( lti ) channel whose output is corrupted by an additive proper - complex socs random process .as already mentioned , proper - complex socs random processes well model the complex envelopes of the majority of digitally modulated signals as well as the complex envelope of an additive gaussian noise .  our objective is to extend the aforementioned joint optimizations of the tx and rx for proper - complex wscs signaling to a joint tx and rx optimization problem for improper - complex socs signaling under the mmse optimality criterion and the average transmit power constraint .it is well known that the second - order properties of an improper - complex signal are not well captured by a linear rx , but instead by a class of nonlinear rx s called widely linear rx s .there are two types of widely linear rx s .the first one linearly processes the signal augmented by its complex conjugate , whereas the second one linearly processes the real part of the signal augmented by the imaginary part . in this paper ,the first type of widely linear processing also referred to as the linear - conjugate linear ( lcl ) filtering is employed .it is noteworthy that , unlike the joint optimizations in , we now need to find two receive waveforms under the widely linear mmse ( wlmmse ) optimality criterion , where one is employed to filter the complex envelope of the received signal and the other to filter its complex conjugate .the vft technique again enables us to convert the objective function and the average transmit power constraint described initially in the time domain into those in the frequency domain .unlike the previous joint optimizations , the objective function is now expressed in terms of the vft of the transmit waveform augmented by the vft of its complex conjugate and the vft of a receive waveform augmented by the vft of the other receive waveform .using these augmented vector - valued functions , we derive the optimal waveforms of the wlmmse rx in a straightforward way as a function of the transmit waveform .it is shown that the two receive waveforms of the wlmmse rx exploit not only the periodic spectral correlation due to the cyclostationarity , but also the symmetric spectral correlation about the origin due to the impropriety .  to derive the optimal transmit waveform, we devise the notion of the impropriety frequency function of the transmitted improper - complex sos data sequence by using the relation between the power spectral density ( psd ) and the complementary psd of the random process .this real - valued non - negative function converts the transmit waveform optimization problem into an equivalent convex optimization problem to find the optimal energy density of the transmit waveform .then , a line search combined with an iterative algorithm is proposed to solve the problem . after finding the optimal energy density , the optimal transmit and receivewaveforms are obtained .numerical results provide an example of joint waveform design and also show the effect of the impropriety frequency function on the mean - squared error ( mse ) performance .the rest of this paper is organized as follows . in sectionii , the system model is described and the problem is formulated in the time domain . in section iii ,the problem is reformulated in the frequency domain . in section iv, the impropriety frequency function is introduced and the jointly optimal transmit and receive waveforms are derived .numerical results are provided in section v , and concluding remarks are offered in section vi .in this section , we describe the system model and formulate the optimization problem in the time domain .the system model is an extension of that in , which only considers the transmission and reception of a proper - complex sos data sequence , to now allow improper - complex sos sequences .the optimality criterion of the joint optimization problem is also extended from the lmmse criterion to the wlmmse criterion .a tx and an rx operate over a real passband to transmit a data sequence \}_{l\in\mathbb{z}} ] is assumed well modeled by a zero - mean improper - complex sos random process with auto - covariance and complementary auto - covariance functions given , respectively , by \triangleq  { \mathbb{e}}\{b[k+l]b[l]^*\} ] , where the superscript denotes complex conjugation . by applying the discrete - time fourier transform ( dtft ) operations to  ] ,the psd and the complementary psd of the data sequence \}_{l\in\mathbb{z}} ] and e^{-{\rm j } 2 \pi fk} ]is passed through a strictly band - limited channel that is modeled by an lti system with impulse response having the one - sided bandwidth [ hz ] in complex baseband .  the received signal denoted by consists of the signal from the tx and an additive interference - plus - noise signal , where the latter is modeled by a zero - mean proper - complex socs random process with fundamental cycle period .it is assumed that the multiplicative inverse of the symbol transmission rate of the desired signal is chosen as an integer multiple of .thus , can be written as p(t - kt ) + n(t),\vspace{-4mm}\ ] ] where denotes the overall response with the operator denoting the convolution integral .there should be no confusion from the superscript that denotes the complex conjugation .in ( [ eq : signal_model ] ) , it can be easily shown that the desired signal component p(t - kt) ] .in other words , the mean , the auto - covariance , and the complementary auto - covariance functions of satisfy , respectively , , , and , . in what follows ,we also call and the auto - correlation and the complementary auto - correlation functions , respectively , because has mean zero .in ( [ eq : signal_model ] ) , it can be straightforwardly shown that the interference - plus - noise signal is socs with mean zero and cycle period , because is assumed to be an integer multiple of , i.e. , , , and , . now that is a summation of two uncorrelated zero - mean socs random processes with cycle period , it is also a zero - mean socs random processes with cycle period .it is well known that , for a vector - valued signal model , a widely linear rx employing two linear filters outperforms a linear rx employing only one linear filter when either the desired signal or the interference - plus - noise signal is improper .thus , in this paper , we employ two lti filters with impulse responses and to process the improper - complex socs process and its complex conjugate , respectively .the two lti filters are followed by uniform samplers with rate [ samples / sec ] , and then the sequence of decision statistics \}_{l\in\mathbb{z}} ] and  ] for , where .given a pair and an socs random process with cycle period having the auto - correlation function , the matrix - valued psd of is defined as a matrix - valued function of , whose entry is given by {k , l}\triangleq r_n^{(k - l ) } ( f+({l - l-1})/{t}) ] for .note that the matrix - valued complementary psd of the interference - plus - noise signal becomes an all - zero matrix because is modeled by a zero - mean proper - complex socs random process .note also that the effective matrix - valued complementary psd can be defined similarly to the effective matrix - valued psd . in what follows , each of the vft ,the matrix - valued psd , and the matrix - valued complementary psd is an effective one .  by using the above definitions , the matrix - valued psd and the matrix - valued complementary psd of the desired signal component in ( [ eq : signal_model ] ) are derived as follows .[ lemma : mv - psd_2 ] the -by- matrix - valued psd and the -by- matrix - valued complementary psd of the desired signal p(t - lt) ] , ^ * z_1[l]\ } = \int_{{\mathcal{f } } }  \bm{w}_1(f)^{\mathcal{h } } m(ft ) \bm{p}(f ) df ] , respectively , where is the vft of and .see ( * ? ? ?* proposition 1 - 4 ) .[ proposition : proof2 ] .the last three terms of the right side of ( [ eq : mse wl ] ) can be rewritten as ^ * z_2[l]\ } = \int_{{\mathcal{f } } }  \bm{w}_2(f)^{\mathcal{h}} ] , and |^2\ } = \int_{{\mathcal{f } } }  \bm{w}_2(f)^{\mathcal{h } }  \bm{j}(-f ) ] that the pre - multiplication of the backward identity matrix reverses the order of the rows whereas the post - multiplication reverses that of the columns .note also that , where is the vft of the transmit waveform and is defined as with representing the vft of .  to simplify the expression of the objective function , we define ,  c[eq : def_augmented_vft ] |(f ) ^ |(f ) ^ ,  where denotes transposition . here , the length- vector - valued functions and are the vft of the transmit waveform augmented by the vft of its complex conjugate and the vft of a receive waveform augmented by the vft of the other receive waveform , respectively .also , let the -by- matrices , , and be defined , respectively , as  ,  , and \vspace{-4mm}\ ] ] with denoting the -by- identity matrix and denoting the block diagonal matrix whose diagonal blocks are the matrices and .these notions enable us to derive the optimal receive waveforms in a straightforward way .  by substituting the results of propositions [ proposition : proof1 ] and [ proposition : proof2 ] into ( [ eq : mse wl ] ) , we can rewrite the objective function as which is a function of and .also , by using ( * ? ? ?* eq . ( 32 ) ) and the definition of , we can rewrite the average transmit power defined in ( [ eq : power_constraint_td ] ) as   this leads to the equivalent joint optimization problem to find and as  [ problem : fd ]  ll & ( |(f ) , |(f ) ) [ eq : problem_mse_fd ] + & |p = p_t .[ eq : problem_constraint_fd ]  in the next section , we solve this optimization problem to obtain the vfts of the optimal receive and transmit waveforms .in this section , we first derive the optimal that minimizes the objective function in ( [ eq : problem_mse_fd ] ) for a given .then , by substituting this and introducing the notion of the impropriety frequency function , we obtain the optimization problem over . by solving this problem, we finally obtain the optimal transmit and receive waveforms .   as in (* theorem 2 ) , to find the optimal for given , an unconstrained quadratic optimization problem is solved .thus , by using the first - order necessary condition , we have the solution by substituting the above solution into ( [ eq : mse_freq ] ) , we can rewrite the mse as  l[eq : rx_opt_mse ] ( |(f ) ) = _ ( tm(ft ) - |(f ) ^ |(ft)^ |(f)^ |(f)^-1 |(f )   which is a function only of .   to convert into a function only of , the notion of the impropriety frequency functionis introduced as follows .given a discrete - time improper - complex sos random process with psd and complementary psd , its impropriety frequency function is defined as   the above definition is motivated by the impropriety coefficient of an improper - complex random variable ( * ? ? ?* definition 3.1 ) and by a relation between and shown in ( * ? ? ?* eq . ( 5 ) ) . by using the phase of , we can rewrite the complementary psd as , where . in the next lemma ,the properties of the impropriety frequency and the phase functions are provided .[ lemma : psd_com ] the impropriety frequency function and the phase function satisfy  c 0 k(f ) 1 , k(-f)=k(f ) , ( -f)=(f ) , f.  since =\tilde{m}[k] ] , where .for each  ] .the candidate function that satisfies the power constraint ( [ eq : power constraint_tx_power2 ] ) is the optimal density function .see appendix c.  note that any line search algorithm can be used to find in proposition [ proposition : optimize_power ] .note also that the algorithm in table [ table : 1 ] allows the construction of an approximate solution with arbitrary accuracy if the interval is partitioned finely enough .now , by using , we can find the vfts of the optimal transmit and receive waveforms as follows .[ thm : 1 ] the vft of the jointly optimal transmit waveform as the solution to problem [ problem : 1 ] are given by  rcl _opt(f ) & = & \ {  ll + ( f)e^j ( f ) , & f_m , + , & f ( _ m ) ^c , +  .[ eq : opt_tx_final ]  where can be chosen arbitrarily .then , the vfts and of the jointly optimal receive waveforms and can be found by using ( [ eq : rx_opt ] ) .the conclusion immediately follows from the relation ( [ eq : def_augmented_vft ] ) among , , , , and , and propositions [ proposition : inner_optimization ] and [ proposition : optimize_power ] .as already mentioned , cyclostationarity and impropriety , respectively , imply the periodic spectral correlation and the symmetric spectral correlation about the origin ( * ? ? ?10 ) , .theorem 1 vividly shows these structures in the optimal transmitted signal .specifically , the use of the vft technique and the augmentation of and to form take care of the periodic spectral correlation and the symmetric spectral correlation , respectively .in this section , numerical results are provided that show the magnitude square of the optimal transmit and receive waveforms and that show the mse performance achieved by the optimal waveforms as a function of the amount of impropriety . for illustrative purposes ,it is assumed throughout this section that an interferer linearly modulates a data sequence consisting of uncorrelated zero - mean proper - complex qpsk symbols with [ db ] and a square - root raised cosine transmit waveform having excess bandwidth .it is assumed that the tx linearly modulates a data sequence consisting of uncorrelated zero - mean improper - complex qam symbols with uncorrelated in - phase and quadrature components .it is also assumed that all the channels are frequency flat and corrupted by awgn .the first results are to compare the psd of data - like interference with the squared magnitudes of the optimal transmit and receive waveforms .there is a single interferer in figs .[ fig : spectrum]-(a ) and ( b ) , whereas there are two uncorrelated interferers in fig .[ fig : spectrum]-(c ) .the qam symbols of the tx have [ db ] . for fig .[ fig : spectrum]-(a ) , the qam symbols have the in - phase variance the same as the quadrature variance , which implies . for figs .[ fig : spectrum]-(b ) and ( c ) , the qam symbols have the in - phase variance -times the quadrature variance , which implies .it can be seen that , processing the complex conjugate of the received signal , is zero for the data sequence having , but it is non - zero for the data sequence having .the next results are to compare the mses achieved by the optimal transmit and receive waveforms for different levels of impropriety .we consider the same number of interferers and interference parameters as fig .[ fig : spectrum]-(c ) . in fig .[ fig : mse]-(a ) , the qam symbols of the tx have from to [ db ] and have , or . in fig .[ fig : mse]-(b ) , the qam symbols of the tx have or [ db ] and have from to . in both cases , as the amount of impropriety increases , the optimal pair of the tx and rx more exploits impropriety and cyclostationarity of the desired signal in suppressing the data - like interference and , consequently , the mse performance monotonically improves .in this paper , we have considered a joint optimization of the tx and rx for the transmission of an improper - complex sos data sequence over an additive proper - complex cyclostationary noise channel .an mse minimization problem is formulated under the average transmit power constraint to find the jointly optimal transmit waveform of a linear modulator and the receive waveforms of a widely linear rx .this problem is converted into an equivalent problem described in the frequency domain with the help of the vft technique and solved by introducing the notion of the impropriety frequency function .it is shown that the optimal transmit and receive waveforms well exploit the frequency - domain second - order structure of the improper - complex sos data sequence and the additive proper - complex socs noise .then , we can rewrite as . also , define the -by- matrix and the -by- matrix as and , respectively . due to the ambient noise component in , and positive definite for all . by using , define the -by- matrix as .then , it can be shown that  .thus , the second term of the integrand in ( [ eq : rx_opt_mse ] ) , which contains , can be rewritten as   where conveniently denotes the appropriately sized identity matrix throughout this proof .let .then , we can rewrite and defined in ( [ eq : c_f ] ) as  and , respectively .if , then it can be shown that ( [ eq : prp_1 ] ) leads to ( [ eq : mse_simple ] ) by using the matrix inversion lemma showing for any vector .if , then , since is invertible , it can be shown that , where is defined as .since and are not zero , we can rewrite as  .thus , we now can rewrite the right side of ( [ eq : prp_1 ] ) as . by using the matrix inversion lemma, the conclusion follows .   for convenience ,the integration interval is partitioned into equal - length subintervals .then , the solution can be straightforwardly extended to the original problem by letting tend to infinity .let , , , , , , , and .then , the original optimization problem can be approximated by   where is given by for non - negative real numbers , and with , .it can be easily shown that , if , can be chosen arbitrarily because does not affect both the objective function and the constraint .it can be also easily shown that results in to keep from wasting the transmit power .similarly , if , then can be chosen arbitrarily , and if , then .the case of is discussed after solving the optimization problem for .thus , in what follows , we assume that , , and , .define and as ^{\mathcal{t}} ] , respectively .then , it can be easily shown that the hessian of the objective function is a positive definite matrix for each and the equality constraint is an affine function of .thus , the problem in ( [ eq : dt opt problem ] ) is a strictly convex optimization problem . since the karush - kuhn - tucker ( kkt ) condition is necessary and sufficient for a point to be the unique solution of a strictly convex optimization problem ( * ? ? ?* theorem 22.9 ) , we first need to find the kkt condition .the lagrangian function of ( [ eq : dt opt problem ] ) can be written as by introducing the multipliers and ^{\mathcal{t}} ] , where two steps are needed to construct a candidate solution and the multiplier at each .first , a candidate solution associated with is constructed as follows .given , we need to find the pair of satisfying the kkt condition , i.e. , and with , , , , , , which can be rewritten as    if , and if .it can be easily shown that is a decreasing function of and is a decreasing function of .thus , becomes an increasing function of .it is noteworthy that the non - negative numbers and are upper - bounded by and .thus , when we alternately update and from by using ( [ eq : kkt_1_update ] ) and ( [ eq : kkt_2_update ] ) , respectively , both and converge to the solution satisfying the kkt conditions .this iteration algorithm can be also used to find the candidate solution and for the case of .note that , if and , any pair of and satisfying ^+$ ] can be the candidate solution associated with . after finding and , and can be computed by substituting , , and into the kkt condition .second , after constructing the candidate solution associated with , we check whether the candidate solution satisfies the power constraint .if so , then the candidate solution associated with is the optimal solution . if not , then the line search continuestherefore , the conclusion follows .t. berger and d. w. tufts , `` optimum pulse amplitude modulation part i : transmitter - receiver design and bounds from information theory , '' _ ieee trans .inf . theory _196 - 208 , apr .1967 .j. yeo and j. h. cho , `` asymptotic fresh properizer for block processing of improper - complex second - order cyclostationary random processes , '' submitted to _ieee trans .inf . theory _ , apr .2013 , also available online at _http://arxiv.org / abs/1304.7375_.            * choose .* construct and as follows . ** * set . * * * * repeat * * * * update as by using defined in ( [ eq : optimal_a_f_update ] ) . * * * update as . * * * * until * and converge . *repeat lines for all .

Abstract: in this paper , the transmission of an improper - complex second - order stationary data sequence is considered over a strictly band - limited frequency - selective channel . it is assumed that the transmitter employs linear modulation and that the channel output is corrupted by additive proper - complex cyclostationary noise . under the average transmit power constraint , the problem of minimizing the mean - squared error at the output of a widely linear receiver is formulated in the time domain to find the optimal transmit and receive waveforms . the optimization problem is converted into a frequency - domain problem by using the vectorized fourier transform technique and put into the form of a double minimization . first , the widely linear receiver is optimized that requires , unlike the linear receiver design with only one waveform , the design of two receive waveforms . then , the optimal transmit waveform for the linear modulator is derived by introducing the notion of the impropriety frequency function of a discrete - time random process and by performing a line search combined with an iterative algorithm . the optimal solution shows that both the periodic spectral correlation due to the cyclostationarity and the symmetric spectral correlation about the origin due to the impropriety are well exploited .   cyclostationarity , improper - complex , joint transmitter and receiver optimization , mean - squared error ( mse ) , vectorized fourier transform ( vft ) .