Optimal Signal Sets for Non-Gaussian Detectors

OPTIMAL SIGNAL SETS FOR NONGAUSSIAN DETECTORS y z MARK S GOCKENBACH AND ANTHONY J KEARSLEY Abstract Identifying a maximallyseparated set of signals is imp ortant in the design of mo dems The notion of optimality is dep endent on the mo del chosen to describ e noise in the measurements while some analytic results can b e derived under the assumption of Gaussian noise no suchtechniques are known for choosing signal sets in the nonGaussian case To obtain numerical solutions for non Gaussian detectors minimax problems are transformed into nonlinear programs resulting in a novel formulation yielding problems with relatively few variables and many inequality constraints Using sequential quadratic programming optimal signal sets are obtained for a variety of noise distributions Key words Optimal Design Inequality Constraints Sequential Quadratic Programming of the National Institute of Standards and Technology and not sub ject to copyright Contribution in the United States y Department of Mathematics Universityof Michigan Ann Arb or MI z Mathematical and Computational Sciences Division National Institute of Standards and Tech nology Gaithersburg MD Introduction The transmission of digital information requires signals nite time series that can b e distinguished from one another in the presence of noise These signals may b e constrained by b ounds on their energy or amplitude the degree to which they can b e distinguished dep ends on the distribution of noise We study the design of optimal signal sets under amplitude constraints and in the presence of nonGaussian noise We will call a signal set optimal when roughly sp eaking the largest probability of mistaking any one signal for any other is minimal For this reason as weshowbelow this problem is naturally formulated as a smo oth and twice continuously dierentiable minimax problem We shall assume that M signals s s s M are to b e constructed where each signal is to b e a linear combination of K given signals K Moreover it is assumed that each is a time series of length N andthat f g is an orthonormal set k k under the Euclidean inner pro duct We denote the comp onents of by n n N k k and the comp onents of s similarly The unknowns to b e determined are the weights f g dening m mk signals the K X m M s m mk k k A signal set is often referred to as a constel lation In the sp ecial case of only two basis functions the signals can b e represented in the plane by the co ecients In the case of Gaussian noise it turns out M M that the problem reduces to maximizing the minimum Euclidean distance b etween anytwo signals sub ject to constraints on the energy or amplitudes of the signals For this reason heuristic metho ds have b een used to design go o d signals sets on a lattice and Typically these heuristics have taken the form of cho osing the points lying w densely packing them within a xed region of the plane Identifying these latticebased someho constellations asso ciated with lowaverage energies has b een an active area of research examples and pictures of these constellations can b e found in and Although wearenotaware of previous attempts to nd optimal constellations according to the criterion we describ e b elow related problems have b een investigated The most famous is the sphere packing problem of communication theory see this requires a constellation which maximizes the probability of detection under Gaussian noise Mo dern researchinto this question has fo cused on the case in which the signals are chosen from a large dimensional space in particular an imp ortant research topic has b een the Strong Simplex Conjecture which deals with the case M K see We are concerned with the case in which the signals are chosen from a small dimensional space K or In the remainder of this pap er we show that this problem can be formulated as a smo oth nonlinear programming problem with relatively few variables but many inequality constraints This problem is solved using an SQP algorithm In the following second section we explain the formulation of the optimization problem that describ es optimal signal sets In Section we describ e the SQP algorithm employed to solve these problems The noise distributions used in our computations are describ ed in Section and in Section numerical tests and results are presented We conclude the pap er with observations and comments ork in Section on future w Problem formulation As mentioned ab ove we wish to nd a signal set which minimizes the largest probability of mistaking any one for any other This notion of optimality is therefore based on hypothesis testing see for example We consider for the moment that one of two signals s and s is to b e transmitted and that the received signal is denoted by y We assume further that the transmitted signal is corrupted by indep endent identically distributed iid additive noise drawn from some xed distribution with probability density function p df p In other words N y s m or m We assume that the a priori probabilities of s and s b eing transmitted are P and P respec tively and that there is a cost C asso ciated with detecting signal s when s is actually presenta m miss and a cost C asso ciated with detecting s when s is presentafalse alarm It is then easy f to show see that the exp ected cost or risk is minimized by detecting s whenever py js P C m py js P C f and otherwise detecting s By taking the logarithm of b oth sides and using the fact that the noise is iid we obtain the following optimal detector Detect s whenever N X p y n s n N log N p y n s n N n where is the threshold determined by the a priori probabilities and the costs of errors and otherwise detect s Now assume that s was actually transmitted so that y s and let s s s the optimal detector then computes N X p n N log s n N p n N n The exp ected value of the n th term is Z h i p N K sn log p d N N p sn N this quan tity is the KullbackLeibler distance b etween the noise density and the noise density shifted bysn see Thus if s is actually transmitted the exp ected value of the sum in is N X K sn N N n one assumes that p is symmetric as we will it is easy to show that if s is transmitted the If N ected value of the sum in is the negative of Therefore the probability of detecting exp the correct signal increases with From this discussion we see that wewishtocho ose the signals s s s so that m N X n n s min K s m m N m m n is maximized For physical reasons either the average p ower energyL norm or the p eak p ower amplitudeL norm of the signals must b e constrained In this pap er we are concerned with amplitude constraints We are thus faced with a constrained minimax problem Due to the diculty of solving such problems we rewrite it as a smo oth NLP b ariable t yintro ducing an auxiliary v min t t N X n t m m n s st K s m m N n s n C n N m M m t where C is the b ound on the amplitudes of the signals This problem has MK variables and M M MN inequality constraints A typical problem would have K M N giving variables and constraints Diculties arise when one tries to solve the ab ove NLP with standard algorithms The fact that there are far few er variables than constraints results in three sp ecic diculties There are many almost binding constraints at the solution The linearized constraints are often inconsistent The b oundary of the feasible region is noticeably nonlinear The proximity of nearbinding constraints to the solution suggests that correct identication of the active set b ecomes more dicult near the solution In turn this suggests that the region of rapid lo cal convergence of the iteration sequence fx g will b e small The lack of consistent linear k k inequalities complicates the calculation of the SQP step the solution of the quadratic subproblem As observed in previous works and others nonlinear feasible regions can result in small ac ceptable steps if p enalty parameters b ecome small Iterates will follow to o closely a feasible region b oundary that leads away from optimality All three of these issues were relevanttothesolutionof the optimal signal set problem presented here Name Density K s exp s p Gaussian p jsj expj j jsj p p p xp Laplacian e sech Hyp erb olic Secant lnsech s s s Generalized Gaussian exp A A Cauchy s r ln Table Noise densities and associatedKullbackLeibler distances Nonlinearly Constrained Optimization Many algorithms have b een develop ed for the solution of smo oth inequalityconstrained optimization problems Among the most p opular metho ds is the Sequential Quadratic Programming SQP family of algorithms see for a review of these metho ds Given an estimate of the solution an SQP algorithm progresses by solving a quadratic program QP which is dened by a lo cal quadratic mo del of the ob jective function and linearized constraints The solution to the quadratic program is then used to construct an improved estimate of the solution Many dierent SQP algorithms can b e constructed byvarying the algorithm for solving the QP the Lagrange multiplier estimates and the globalization strategy One version of the SQP algorithm prop osed by Boggs Kearsley and Tolle see for example app ears to b e well suited for solving this class of NLPs The algorithm employs a combined trustregion and merit function see line search pro cedure for globalization Amoderninterior point metho d called OD

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