a detector based on(73. 15)simply measures the energy in the measurements and then announces the presence of the signal if this energy is large enough. This type of detector is known as a radiometer. Thus, radiometry is optimum in the case in which both signal and noise are i.i. d Gaussian sequences with zero means. Since in this case the presence of the signal is manifested only by an increase in energy level, it is intuitively obvious that radiometry is the only way of detecting the signals presence. More generally, when the ignal is correlated, the quadratic function(73.13)exploits both the increased energy level and the correlation structure introduced by the presence of the signal. For example, if the signal is a narrowband Gaussian then the quadratic function(73. 13)acts as a narrowband radiometer with bandpass characteristic that imately matches that of the signal. In general, the quadratic detector will make use of whatever properties the signal exhibits If the signal is random but not Gaussian, then its optimum detection [described by(73.6)] typica more complicated nonlinear processing than the quadratic processing of (73. 13)in order to exploit butional differences between signal and noise. This type of processing is often not practical for implementation, and thus approximations to the optimum detector are typically used. An interesting family of such detectors ses cubic or quartic functions of the measurements, which exploit the higher-order spectral properties of the ignal [Mendel, 1991. As with deterministic signals, random signals can be parametrized. In this case, however, is the distribution of the signal that is parametrized. For example, the power spectrum of the signal of interest may be known only up to a set of unknown parameters. Generalized likelihood ratio detectors(73.9)are often used to detect such signals Deciding Among Multiple Signals The preceding results have been developed under the model (73. 1)-(73. 2)that there is a single signal that is either present or absent. In digital communications applications, it is more common to have the situation in which we wish to decide between the presence of two(or more) possible signals in a given set of measurement The foregoing results can be adapted straightforwardly to such problems. This can be seen most easily in the case of deciding among known signals. In particular, consider the problem of deciding between two alternatives (73.16) Y=N+s,k=1,2,,n (73.17) where,s,……, s(o)and s出,s2,…, s(are two known signals. Such problems arise in data transmission problems, in which the two signals sfo) and s()correspond to the waveforms received after transmission of a logical"zero"and"one, "respectively. In such problems, we are generally interested in minimizing the average robability of error, which is the average of the two error probabilities weighted by the prior probabilities of straightforward extension of the correlation detector based on (73.5). In particular, under the assumptIons t, ccurrence of the two signals. This is a Bayesian performance criterion, and the optimum decision rule is a the two signals are equally likely to occur prior to measurement, and that the noise is white and Gaussian, the optimum decision between(73. 16)and (73. 17) is to choose the model(73.16)if 2 a s(l, is larger than 2,=l s Y, and to choose the model(73.17)otherwise probability is to choose the signal si,, ,...,s, where j is a solution of the maximization problem ∑Y= max (73.18) 0≤m≤M=1 e 2000 by CRC Press LLC© 2000 by CRC Press LLC A detector based on (73.15) simply measures the energy in the measurements and then announces the presence of the signal if this energy is large enough. This type of detector is known as a radiometer. Thus, radiometry is optimum in the case in which both signal and noise are i.i.d. Gaussian sequences with zero means. Since in this case the presence of the signal is manifested only by an increase in energy level, it is intuitively obvious that radiometry is the only way of detecting the signal’s presence. More generally, when the signal is correlated, the quadratic function (73.13) exploits both the increased energy level and the correlation structure introduced by the presence of the signal. For example, if the signal is a narrowband Gaussian process, then the quadratic function (73.13) acts as a narrowband radiometer with bandpass characteristic that approx￾imately matches that of the signal. In general, the quadratic detector will make use of whatever spectral properties the signal exhibits. If the signal is random but not Gaussian, then its optimum detection [described by (73.6)] typically requires more complicated nonlinear processing than the quadratic processing of (73.13) in order to exploit the distri￾butional differences between signal and noise. This type of processing is often not practical for implementation, and thus approximations to the optimum detector are typically used. An interesting family of such detectors uses cubic or quartic functions of the measurements, which exploit the higher-order spectral properties of the signal [Mendel, 1991]. As with deterministic signals, random signals can be parametrized. In this case, however, it is the distribution of the signal that is parametrized. For example, the power spectrum of the signal of interest may be known only up to a set of unknown parameters. Generalized likelihood ratio detectors (73.9) are often used to detect such signals. Deciding Among Multiple Signals The preceding results have been developed under the model (73.1)–(73.2) that there is a single signal that is either present or absent. In digital communications applications, it is more common to have the situation in which we wish to decide between the presence of two (or more) possible signals in a given set of measurements. The foregoing results can be adapted straightforwardly to such problems. This can be seen most easily in the case of deciding among known signals. In particular, consider the problem of deciding between two alternatives: (73.16) and (73.17) where s 1 (0), s 2 (0), . . ., s n (0) and s 1 (1), s 2 (1), . . ., s n (1) are two known signals. Such problems arise in data transmission problems, in which the two signals s(0) and s(1) correspond to the waveforms received after transmission of a logical “zero’’ and “one,’’ respectively. In such problems, we are generally interested in minimizing the average probability of error, which is the average of the two error probabilities weighted by the prior probabilities of occurrence of the two signals. This is a Bayesian performance criterion, and the optimum decision rule is a straightforward extension of the correlation detector based on (73.5). In particular, under the assumptions that the two signals are equally likely to occur prior to measurement, and that the noise is white and Gaussian, the optimum decision between (73.16) and (73.17) is to choose the model (73.16) if (k n =1 s k (0)Yk is larger than (k n =1 s k (1)Yk , and to choose the model (73.17) otherwise. More generally, many problems in digital communications involve deciding among M equally likely signals with M > 2. In this case, again assuming white Gaussian noise, the decision rule that minimizes the error probability is to choose the signal s 1 (j) , s 2 (j) , . . ., s n (j) , where j is a solution of the maximization problem (73.18) Y Ns k n k kk =+ = ( ), , ,..., 0 1 2 Y Ns k n k kk =+ = ( ), , ,..., 1 1 2 sY s Y k j k k n m M k m k k n ( ) ( ) max = ££ - = Â Â = 1 0 1 1