There are two basic types of problems of interest in this context. Signal detection problems are concerned primarily with situations in which the information to be extracted from a signal is discrete in nature. That is, signal detection procedures are techniques for deciding among a discrete(usually finite) number of possible alternatives. An example of such a problem is the demodulation of a digital communication signal, in which the task of interest is to decide which of several possible transmitted symbols has elicited a given received signal. Estimation problems, on the other hand, deal with the determination of some numerical quantity taking values in a continuum. An example of an estimation problem is that of determining the phase or frequency of the carrier underlying a communication signal. Although signal detection and estimation is an area of considerable current research activity, the fundamental principles are quite well developed. These principles, which are based on the theory of statistical inference. plain and motivate most of the basic signal detection and estimation procedures used in practice. In this ction,we will give a brief overview of the basic principles underlying the field of signal detection. Estimation is treated elsewhere this volume, notably in Section 16. 2. A more complete introduction to these subjects found in Poor[1994] General Considerations The basic principles of signal detection can be conveniently discussed in the context of decision-making between two possible statistical models for a set of real-valued measurements, Y,, Y2,..., Ym. In particular, on observing Y Y,..., Y,, we wish to decide whether these measurements are most consistent with the m Yk=Nk,k=1,2,,,n (73.1) Yk=N+ Sk, k=l, 2,...,n (73.2) where N, N2,..., N is a random sequence representi d where S, S,,...,S, is a sequence In deciding between Eqs. (73. 1)and(73. 2), there are two types of errors possible: a false alarm, in which (73. 2)is falsely chosen, and a miss, in which(73. 1) is falsely chosen. The probabilities of these two types of errors can be used as performance indices in the optimization of rules for deciding between(73. 1)and(73. 2) Obviously, it is desirable to minimize both of these probabilities to the extent possible. However, the minimi- zation of the false-alarm probability and the minimization of the miss probability are opposing criteria. So, it is necessary to effect a trade-off between them in order to design a signal detection procedure. There are several ways of trading off the probabilities of miss and false alarm: the Bayesian detector minimizes an average of the two probabilities taken with respect to prior probabilities of the two conditions(73. 1)and(73. 2), the minimax detector minimizes the maximum of the two error probabilities, and the Neyman-Pearson detector mizes the miss probability under an upper-bound constraint on the false-alarm probabili If the statistics of noise and signal are known, the Bayesian, minimax, and Neyman-Pearson detectors are all of the same form. Namely, they reduce the measurements to a single number by computing the likelihood ratio I(Y,12…,Y)P、Yn (73.3) Px(Y1,Y2,……,Yn) where PsN and px denote the probability density functions of the measurements under signal-plus-noise(73. 2) and noise-only(73. 1)conditions, respectively. The likelihood ratio is then compared to a decision threshold, with the signal-present model(73. 2) being chosen if the threshold is exceeded, and the signal-absent model (73. 1)being chosen otherwise. Choice of the decision threshold determines a trade-off of the two error probabilities, and the optimum procedures for the three criteria mentioned above differ only in this choice e 2000 by CRC Press LLC© 2000 by CRC Press LLC There are two basic types of problems of interest in this context. Signal detection problems are concerned primarily with situations in which the information to be extracted from a signal is discrete in nature. That is, signal detection procedures are techniques for deciding among a discrete (usually finite) number of possible alternatives. An example of such a problem is the demodulation of a digital communication signal, in which the task of interest is to decide which of several possible transmitted symbols has elicited a given received signal. Estimation problems, on the other hand, deal with the determination of some numerical quantity taking values in a continuum. An example of an estimation problem is that of determining the phase or frequency of the carrier underlying a communication signal. Although signal detection and estimation is an area of considerable current research activity, the fundamental principles are quite well developed. These principles, which are based on the theory of statistical inference, explain and motivate most of the basic signal detection and estimation procedures used in practice. In this section, we will give a brief overview of the basic principles underlying the field of signal detection. Estimation is treated elsewhere this volume, notably in Section 16.2. A more complete introduction to these subjects is found in Poor [1994]. General Considerations The basic principles of signal detection can be conveniently discussed in the context of decision-making between two possible statistical models for a set of real-valued measurements, Y1, Y2, . . ., Yn. In particular, on observing Y1, Y2, . . ., Yn, we wish to decide whether these measurements are most consistent with the model Yk = Nk , k = 1, 2, . . . , n (73.1) or with the model Yk = Nk + Sk , k = 1, 2, . . . , n (73.2) where N1, N2 , . . ., Nn is a random sequence representing noise, and where S1, S2, . . ., Sn is a sequence representing a (possibly random) signal. In deciding between Eqs. (73.1) and (73.2), there are two types of errors possible: a false alarm, in which (73.2) is falsely chosen, and a miss, in which (73.1) is falsely chosen. The probabilities of these two types of errors can be used as performance indices in the optimization of rules for deciding between (73.1) and (73.2). Obviously, it is desirable to minimize both of these probabilities to the extent possible. However, the minimi￾zation of the false-alarm probability and the minimization of the miss probability are opposing criteria. So, it is necessary to effect a trade-off between them in order to design a signal detection procedure. There are several ways of trading off the probabilities of miss and false alarm: the Bayesian detector minimizes an average of the two probabilities taken with respect to prior probabilities of the two conditions (73.1) and (73.2), the minimax detector minimizes the maximum of the two error probabilities, and the Neyman-Pearson detector minimizes the miss probability under an upper-bound constraint on the false-alarm probability. If the statistics of noise and signal are known, the Bayesian, minimax, and Neyman-Pearson detectors are all of the same form. Namely, they reduce the measurements to a single number by computing the likelihood ratio (73.3) where pS+N and pN denote the probability density functions of the measurements under signal-plus-noise (73.2) and noise-only (73.1) conditions, respectively. The likelihood ratio is then compared to a decision threshold, with the signal-present model (73.2) being chosen if the threshold is exceeded, and the signal-absent model (73.1) being chosen otherwise. Choice of the decision threshold determines a trade-off of the two error probabilities, and the optimum procedures for the three criteria mentioned above differ only in this choice. L Y Y Y p Y Y Y p Y Y Y n S N n N n ( , , , ) ( , , , ) ( , , , ) 1 2 1 2 1 2 ... ... ... D +