not possible to specify accurate statistical models for signals or noise, and so it is of interest to design detection procedures that do not rely heavily on such models. Of course, the parametrized models described in the foregoing paragraphs allow for uncertainty in the statistics of the observations. Such models are known parametric models, because the set of possible distributions can be parametrized by a finite set of real parameters e While parametric models can be used to describe many types of modeling uncertainty, composite models in which the set of possible distributions is much broader than a parametric model would allow are sometimed more realistic in practice. Such models are termed nonparametric models. For example, one might be able to ssume only some very coarse model for the noise, such as that it is symmetrically distributed. a wide variety of useful and powerful detectors have been developed for signal-detection problems that cannot be parame trized. These are basically of two types: robust and nonparametric. Robust detectors are those designed to perform well despite small, but potentially damaging, nonparametric deviations from a nominal parametric model, whereas nonparametric detectors are designed to achieve constant false-alarm probability over very wide classes of noise statistics Robustness problems are usually treated analytically via minimax formulations that seek best worst-case performance as the design objective. This formulation has proven to be very useful in the design and charac erization of robust detectors for a wide variety of detection problems Solutions typically call for the intro- duction of light limiting to prevent extremes of gain dictated by an(unrealistic)nominal model. For example, the correlation detector of Fig. 73. 1 can be made robust against deviations from the Gaussian noise model by introducing a soft-limiter between the multiplier and the accumulator. Nonparametric detection is usually based on relatively coarse information about the observations, such a he algebraic signs or the ranks of the observations. One such test is the sign test, which bases its decisions on the number of positive observations obtained. This test is nonparametric for the model in which the noise samples are i.i. d. with zero median and is reasonably powerful against alternatives such as the presence of a positive constant signal in such noise. More powerful tests for such problems can be achieved at the expense of complexity by incorporating rank information into the test statistic. Distributed and Sequential Detection The detection procedures discussed in the preceding paragraphs are based on the assumption that all measure- ments can and should be used in the detection of the signal, and moreover that no constraints exist on how measurements can be combined. There are a number of applications, however, in which constraints apply to the information pattern of the measurements One type of constrained information pattern that is of interest in a number of applications is a network consisting of a number of distributed or local decision makers, each of which processes a subset of the measurements, and a fusion center, which combines the outputs of the distributed decision makers to produce a global detection decision the communication between the distributed decision makers and the fusion center is constrained so that each local decision maker must reduce its subset of measurements to ng local decision to be transmitted to the fusion center. Such structures arise in applications such as the testing of large-scale integrate circuits, in which data collection is decentralized, or in detection problems involving very large data sets, in which is desirable to distribute the computational work of the detection algorithm. Such problems lie in the field of distributed detection. Except in some trivial special cases, the constraints imposed by distributing the detection algorithm introduce a further level of difficulty into the design of optimum detection systems. Nevertheless, considerable progress has been made on this problem, a survey of which can be found in Tsitsiklis [ 1993] Another type of nonstandard information pattern that arises is that in which the number of measurements potentially infinite, but in which there is a cost associated with taking each measurement. This type of model arises in applications such as the synchronization of wideband communication signals. In such situations, the error probabilities alone do not completely characterize the performance of a detection system, consideration must also be given to the cost of sampling. The field of sequential detection deals with the optimization of detection systems within such constraints. In sequential detectors, the number of measurements taken becomes a random variable depending on the measurements themselves. a typical performance criterion for optimizing such a system is to seek a detector that minimizes the expected number of measurements for given levels of miss and false-alarm probabilities e 2000 by CRC Press LLC© 2000 by CRC Press LLC not possible to specify accurate statistical models for signals or noise, and so it is of interest to design detection procedures that do not rely heavily on such models. Of course, the parametrized models described in the foregoing paragraphs allow for uncertainty in the statistics of the observations. Such models are known as parametric models, because the set of possible distributions can be parametrized by a finite set of real parameters. While parametric models can be used to describe many types of modeling uncertainty, composite models in which the set of possible distributions is much broader than a parametric model would allow are sometimed more realistic in practice. Such models are termed nonparametric models. For example, one might be able to assume only some very coarse model for the noise, such as that it is symmetrically distributed. A wide variety of useful and powerful detectors have been developed for signal-detection problems that cannot be parame￾trized. These are basically of two types: robust and nonparametric. Robust detectors are those designed to perform well despite small, but potentially damaging, nonparametric deviations from a nominal parametric model, whereas nonparametric detectors are designed to achieve constant false-alarm probability over very wide classes of noise statistics. Robustness problems are usually treated analytically via minimax formulations that seek best worst-case performance as the design objective. This formulation has proven to be very useful in the design and charac￾terization of robust detectors for a wide variety of detection problems. Solutions typically call for the intro￾duction of light limiting to prevent extremes of gain dictated by an (unrealistic) nominal model. For example, the correlation detector of Fig. 73.1 can be made robust against deviations from the Gaussian noise model by introducing a soft-limiter between the multiplier and the accumulator. Nonparametric detection is usually based on relatively coarse information about the observations, such as the algebraic signs or the ranks of the observations. One such test is the sign test, which bases its decisions on the number of positive observations obtained. This test is nonparametric for the model in which the noise samples are i.i.d. with zero median and is reasonably powerful against alternatives such as the presence of a positive constant signal in such noise. More powerful tests for such problems can be achieved at the expense of complexity by incorporating rank information into the test statistic. Distributed and Sequential Detection The detection procedures discussed in the preceding paragraphs are based on the assumption that all measure￾ments can and should be used in the detection of the signal, and moreover that no constraints exist on how measurements can be combined. There are a number of applications, however, in which constraints apply to the information pattern of the measurements. One type of constrained information pattern that is of interest in a number of applications is a network consisting of a number of distributed or local decision makers, each of which processes a subset of the measurements, and a fusion center, which combines the outputs of the distributed decision makers to produce a global detection decision. The communication between the distributed decision makers and the fusion center is constrained, so that each local decision maker must reduce its subset of measurements to a summarizing local decision to be transmitted to the fusion center. Such structures arise in applications such as the testing of large-scale integrated circuits, in which data collection is decentralized, or in detection problems involving very large data sets, in which it is desirable to distribute the computational work of the detection algorithm. Such problems lie in the field of distributed detection. Except in some trivial special cases, the constraints imposed by distributing the detection algorithm introduce a further level of difficulty into the design of optimum detection systems. Nevertheless, considerable progress has been made on this problem, a survey of which can be found in Tsitsiklis [1993]. Another type of nonstandard information pattern that arises is that in which the number of measurements is potentially infinite, but in which there is a cost associated with taking each measurement. This type of model arises in applications such as the synchronization of wideband communication signals. In such situations, the error probabilities alone do not completely characterize the performance of a detection system, since consideration must also be given to the cost of sampling. The field of sequential detection deals with the optimization of detection systems within such constraints. In sequential detectors, the number of measurements taken becomes a random variable depending on the measurements themselves. A typical performance criterion for optimizing such a system is to seek a detector that minimizes the expected number of measurements for given levels of miss and false-alarm probabilities