The most commonly used sequential detection procedure is the sequential probability ratio test, which operates by recursive comparison of the likelihood ratio (73.3)to two thresholds In this detector, if the likelihood ratio for a given number of samples exceeds the larger of the two thresholds, then the signals presence is announced and the test terminates. Alternatively, if the likelihood ratio falls below the smaller of the two thresholds, the e. &nal's absence is announced and the test terminates. However, if neither of the two thresholds is crossed, then another measurement is taken and the test is repeated Detection with Continuous Time measurements Note that all of the preceding formulations have involved the assumption of discrete-time (i.e, sampled-data) measurements. From a practical point of view, this is the most natural framework within which to consider these problems, since implementations most often involve digital hardware. However, the procedures discussed this section all have continuous-time counterparts, which are of both theoretical and practical interest. Mathematically, continuous-time detection problems are more difficult than discrete-time ones, because they involve probabilistic analysis on function spaces. The theory of such problems is quite elegant, and the interested reader is referred to Poor[ 1994] or Grenander [1981] for more detailed exposition. Continuous-time models are of primary interest in the front-end stages of radio frequency or optical communication receivers. At radio frequencies, continuous-time versions of the models described in the preceding paragraphs can be used. For example, one may consider the detection of signals in continuous-time Gaussian white noise. At optical wavelengths, one may consider either continuous models(such as Gaussian processes)or point-process models(such as Poisson counting processes), depending on the type of detection used(see, e.g., Snyder and Miller[1991 ) In the most fundamental analyses of optical detection problems, it is sometimes desirable to consider the quantum mechanical nature of the measurements [ Helstrom, 1976] Defining Terms Bayesian detector: A detector that minimizes the average of the false-alarm and miss probabilities, weighted with respect to prior probabilities of signal-absent and signal-present conditions Correlation detector: The optimum structure for detecting coherent signals in the presence of additive white Discrete-time white Gaussian noise: Noise samples modeled as independent and identically distributed Envelope detector: The optimum structure for detecting a modulated sinusoid with random phase in the presence of additive white Gaussian noise. False-alarm probability: The probability of falsely announcing the presence of a signal Likelihood ratio: The optimum processor for reducing a set of signal-detection measurements to a single Miss probability: The probability of falsely announcing the absence of a signal. Neyman-Pearson detector: A detector that minimizes the miss probability within an upper-bound constraint Quadratic detector: A detector that makes use of the second-order statistical structure(e.g, the spectral characteristics)of the measurements. The optimum structure for detecting a zero-mean Gaussian signal in the presence of additive Gaussian noise is of this form. Related Topics 16.2 Parameter Estimation. 70.3 Spread Spectrum Communications U Grenander, Abstract Inference, New York: Wiley, 1981 C W. Helstrom, Quantum Detection and Estimation Theory, New York: Academic Press,1976 S.A. Kassam, Signal Detection in Non-Gaussian Noise, New York: Springer-Verlag, 1988 e 2000 by CRC Press LLC© 2000 by CRC Press LLC The most commonly used sequential detection procedure is the sequential probability ratio test, which operates by recursive comparison of the likelihood ratio (73.3) to two thresholds. In this detector, if the likelihood ratio for a given number of samples exceeds the larger of the two thresholds, then the signal’s presence is announced and the test terminates. Alternatively, if the likelihood ratio falls below the smaller of the two thresholds, the signal’s absence is announced and the test terminates. However, if neither of the two thresholds is crossed, then another measurement is taken and the test is repeated. Detection with Continuous-Time Measurements Note that all of the preceding formulations have involved the assumption of discrete-time (i.e., sampled-data) measurements. From a practical point of view, this is the most natural framework within which to consider these problems, since implementations most often involve digital hardware. However, the procedures discussed in this section all have continuous-time counterparts, which are of both theoretical and practical interest. Mathematically, continuous-time detection problems are more difficult than discrete-time ones, because they involve probabilistic analysis on function spaces. The theory of such problems is quite elegant, and the interested reader is referred to Poor [1994] or Grenander [1981] for more detailed exposition. Continuous-time models are of primary interest in the front-end stages of radio frequency or optical communication receivers. At radio frequencies, continuous-time versions of the models described in the preceding paragraphs can be used. For example, one may consider the detection of signals in continuous-time Gaussian white noise. At optical wavelengths, one may consider either continuous models (such as Gaussian processes) or point-process models (such as Poisson counting processes), depending on the type of detection used (see, e.g., Snyder and Miller [1991]). In the most fundamental analyses of optical detection problems, it is sometimes desirable to consider the quantum mechanical nature of the measurements [Helstrom, 1976]. Defining Terms Bayesian detector: A detector that minimizes the average of the false-alarm and miss probabilities, weighted with respect to prior probabilities of signal-absent and signal-present conditions. Correlation detector: The optimum structure for detecting coherent signals in the presence of additive white Gaussian noise. Discrete-time white Gaussian noise: Noise samples modeled as independent and identically distributed Gaussian random variables. Envelope detector: The optimum structure for detecting a modulated sinusoid with random phase in the presence of additive white Gaussian noise. False-alarm probability: The probability of falsely announcing the presence of a signal. Likelihood ratio: The optimum processor for reducing a set of signal-detection measurements to a single number for subsequent threshold comparison. Miss probability: The probability of falsely announcing the absence of a signal. Neyman-Pearson detector: A detector that minimizes the miss probability within an upper-bound constraint on the false-alarm probability. Quadratic detector: A detector that makes use of the second-order statistical structure (e.g., the spectral characteristics) of the measurements. The optimum structure for detecting a zero-mean Gaussian signal in the presence of additive Gaussian noise is of this form. Related Topics 16.2 Parameter Estimation • 70.3 Spread Spectrum Communications References U. Grenander, Abstract Inference, New York: Wiley, 1981. C.W. Helstrom, Quantum Detection and Estimation Theory, New York: Academic Press, 1976. S.A. Kassam, Signal Detection in Non-Gaussian Noise, New York: Springer-Verlag, 1988