The detection of signals in the presence of noise is an important problem in the field of communications. The detection problem is concerned with the design of systems which determine only the presence or absence of a signal which occurs with background noise. An example of considerable importance, which is encountered quite frequently in practice, is the radar detection problem. In the radar problem it is desired to determine the presence of a target by detecting the presence of a radar return signal in noise. Another example is the seismic exploration problem which utilizes the detection of reflected signals from different depths of rock formations.
In the past, most of the work in signal detection (1)-(5) has been limited to situations in which the signals were assumed to be of known deterministic form and the noise was assumed to be of known statistical form. This type of detection, which deals with signals and noise of known form, will be denoted as parametric detection (2). In some detection problems the information required by the detectors may be available. However, in other situations there may be much less information; for example, the statistical form of the noise may be unknown. In such cases, the parametric detectors become inappropriate.
Thus, there is a need for a theory of detection systems which require much less a priori information than the parametric detectors. This detector, when the signal and noise are not completely known is called a nonparametric detector (2). The nonparametric detector originates from nonparametric statistical methods which are well covered in the literature (15)-(48). These nonparametric statistical methods (nonparametric detectors) have previously been applied to the problem of signal detection (6)-(14), but not nearly as extensively as the parametric detectors. These nonparametric detectors however, have been limited to the detection of signals which only change the dc level of the noise distribution when the signals are added to the noise.
In this paper, nonparametric detectors will be considered, which not only will detect one specific class of signals, for example, signals which change the dc level of the noise distribution, but in general will detect signals which change the noise distribution in any respect. The signals will be assumed to be of the familiar additive variety (60). Using the goodness criterion of Asymptotic Relative Efficiency (A.R.E.) (9, 10, 14, 20), these nonparametric detectors are compared to the corresponding optimum parametric detectors, which are optimum in gaussian noise and gaussian signal plus noise. The A.R.E. of one detector with respect to another is an indication of how many more sample points on the observation interval one detector requires than the other to detect a weak signal with the same error levels. For a given noise distribution and a given signal plus noise distribution, the optimum detector is defined as that detector which requires the least number of sample points to achieve the desired accuracy.
After the theory of these nonparametric detectors is developed, a nonparametric detector is used for the detection of a Frequency Modulated (FM) signal (49)-(59) in the presence of background noise under conditions of low(\u3c1) and extremely low (\u3c\u3c1) signal to noise ratios. Since many FM detection problems restrict attention to messages which are assumed to be expressed in a binary coded form (Binary Frequency Shift Keyed) or in r-ary coded form (Multiple Frequency Shift Keyed) (49)-(59), attention will be limited to FM signals of this form --Introduction, pages 1-3
