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characteristics are varied in time or are functions of time. In real practice, noise jumping can reach several tens of dB. For example, jumping on 2 ÷ 3 dB only under the fixed detection threshold reduces to changes in the probability of false alarm PF on four orders by magnitude [78]. Thus, a synthesis of robust signal processing and detection algorithms possessing a sufficiently stable quality of service (QoS) under changes in environment conditions is a special interest. As a rule, there is a need to ensure the stability of important QoS only or the probability of false alarm PF only. In this case, we must solve the problem of constant false alarm rate or, in other words, the CFAR problem.

Depending on a priori information about the target return signals and noise, we distinguish the parametric and nonparametric uncertainty. For the parametric uncertainty, the pdf pXN1 {xi} under the hypothesis 1 (a “yes” signal) and the pdf pN 0 {xi} under the hypothesis 0 (a “no” signal) are considered as known pdf, and only some parameters of these pdf are considered as unknown. Changes in environment and external operation conditions are changes in the noise statistical characteristics such as the mean, variance, covariance function, and so on. The robust signal detection algorithms ensuring CFAR are considered, in this case, as adaptive algorithms allowing us to obtain estimations of unknown statistical characteristics of the noise and to employ these estimations to normalize the input signal or to control the threshold. In the second case, the target return signal and noise pdf shape is, as a rule, unknown both at the hypothesis 1 (a “yes” signal) and at the hypothesis 0 (a “no” signal). In this case, a synthesis of robust signal detection algorithms is carried out based on methods of inspection of nonparametric statistical hypotheses. The obtained nonparametric signal detection algorithms bring about the probability of false alarm PF that is independent (or invariant) of the noise envelope pdf pN 0 {xi}. However, a statistical independence of the target return signal sample values is the indispensable condition of invariance of the nonparametric signal detection algorithms. If there is a correlation between sample values of the target return signal, the mixed signal detection algorithms using parametric and nonparametric statistic are implemented.

The robust signal detection algorithms are able to ensure the best detection performance under the presence of some information about the noise pdf in comparison with the invariant signal detection algorithms and the best stability in comparison with the optimal signal detection algorithms if, in reality, the noise pdf differs from the adopted noise pdf model that is used under synthesis of signal detection algorithms. Ambiguity of the noise statistical characteristics can be given, for example, in the following form for the case of one-dimensional sample:

When pˆ N 0 {xi} is the normal Gaussian pdf and the noise samples are uniform, the DGD constructed based on the robust signal detection algorithm must accumulate statistic at the output that is the nonlinear function of Ref. [44], that is,

There are specific procedures and methods to reduce an impact of the nonstationarity of continuous noise and interference and chaotic pulse noise and interference in the adaptive DGD. Discuss some of them.

•To determine, for example, the total variance σ2Σ = σ2n + σ2in of the noise and interference, assuming that the noise and interference are independent of each other, the noise and interference sampling is carried out under each scanning and in immediate proximity with a resolution element by radar range investigated to give an answer a “yes” or a “no” target. In other words, we employ m time sampling intervals neighboring with the studied resolution element, which are considered as interference. At the same time, we assume that noise samples are uncorrelated and possess a definite stationary by power interval (the quasistationary interval). To reduce an effect of jumping in noise amplitude on the shift of

estimation of the variance σ2Σ, for example, to test a “yes” or a “no” target, we select the resolution element (elementary cell) that is in the center of m + 1 adjacent cells or resolu-

tion elements. The procedure to estimate the total variance σ2Σ and to normalize the voltage of a “yes” signal elementary cell is explained by Figure 3.13.

•To reduce a sensitivity of the total variance σ2Σ to stimulus of powerful interference, for example, chaotic pulse interference, we use a preliminary comparison of the target return signal samples obtained in two adjacent elements of sampling in radar range with followup restriction to the threshold used to detect chaotic pulse interference (the method of

contrasts) [89]. In accordance with this method, the sampling value xi comes in at the input of total noise variance estimation network if there is no exceeding of the threshold;

that is, the condition xi < Cxi−1 0 < C < 1, is satisfied. If this condition is not satisfied, then a previous result is checked and at xi−1 < Cxi−2 the value xi is excluded and the sample size of training sample is decreased on unit. When xi−1 > Cxi−2 the sampled value is replaced by the threshold value Cxi−1. The constant factor C is selected in such a way that it will be possible to conform the tolerable losses in the case of interference absence to the required accuracy of estimation of the noise variance in the given range of the off-duty ratio and energy of chaotic pulse interference. Choice of the value C is made, as a rule, based on simulation.

xi+ m2

xi– m2

FIGURE 3.13  Procedure of estimation definition of the total variance σ2Σ.

•If in the course of training sample analysis there is a possibility not only to detect the chaotic pulse noise but also to measure and define their amplitude, then we can define the value of normalized factor in the adaptive DGD in the following manner. For instance, let l samples of noise from the sample size m are affected by the chaotic pulse noise. Then the estimation of the continuous noise variance (under the normal Gaussian pdf with zero mean) takes the following form:

and the estimation of the chaotic pulse interference variance is represented in the following form:

l

σˆ 2chp = 1l ∑(Achpj )2 , (3.91) j =1

where Achpj is the amplitude of chaotic pulse interference. In the considered case, the target return signal samples must be normalized by the weight

This method is effective when the chaotic pulse noise sample is no more than 25%–30% of the training sample and the sample size of the chaotic pulse noise sample is defined by m ≥ 15 … 20.

The considered methods and procedures of adaptation to the noise and interference have a general disadvantage—the number of false signals at the detector output is not registered by any way. Thus, variations in this number of false signals are not detected and the detector is not controlled. In other words, there is no feedback between the detector and the number of false signals. By this reason, the hardware designed and constructed using the principle of closed loop system or open-loop one to control the threshold and decision function and ensuring the CFAR along with normalization is employed in radar systems with the automatic signal processing and signal detection subsystems where CFAR stabilization is very important.

3.3.3  Nonparametric DGD

Under the nonparametric ambiguity a shape of the pdf p(x) is unknown both in the case of a “yes” signal and in the case of a “no” signal. At this condition, the nonparametric methods of the theory of statistical decisions are employed. This approach allows us to synthesize and design the signal detection algorithms with the predetermined probability of false alarm PF independent of the pdf p(x) shape, that is, with the CFAR in a wide class of unknown pdf p(x) of input signals. Since CFAR is essential for CRS digital signal processing and detection subsystems, there is a great interest to investigate all possible ways to realize the nonparametric signal detection algorithms.

Note that direct values of sample readings of the target return signal are not used by the nonparametric digital detectors. Reciprocal order of the direct values of sample readings of the target

interval of the input signal) at one of them. Delayed and not delayed signals are compared using a comparator. The signals at the comparator output

are transformed into signs by the following law:

For each kth sampling interval (the range ring) by a set of sign sampling

the signal detection algorithm is realized by the N adjacent scanning periods based on the linear sign statistic

i=1

where K is the detection threshold defined by the predetermined and acceptable probability of false

alarm PF.

The discussed signal detection algorithm implements the method of one-sided contrast [89]. The high value of the contrast function between samples of the target return signal mixed with noise and the noise only is the main precondition under employment of the sign nonparametric generalized signal detection algorithm. There are many modified sign nonparametric signal detection algorithms [70]. QoS of the sign nonparametric signal detection algorithms is characterized by coefficients of asymptotic and relative effectiveness. It is well known that in the case of nonfluctuating target return signal, the coefficient of asymptotic and relative effectiveness of the sign nonparametric generalized signal detection algorithm with respect to the optimal generalized signal detection algorithm is equal to 2/π = 0.65 under the normal Gaussian pdf of the noise; that is, we see that the use of the sign nonparametric generalized signal detection algorithm has a loss rate of 35% in comparison with the optimal generalized signal detection algorithm. Moreover, when the noise is subjected to non-Gaussian pdf, the effectiveness of the sign nonparametric generalized signal detection algorithm can be higher in comparison with the optimal generalized signal detection algorithm.

3.3.3.2  Rank-Nonparametric DGD

To ensure the CFAR under the arbitrary noise pdf, the rank nonparametric detectors are employed. The rank-nonparametric detectors use rank information contained in the sampled sequence of input signals to make a decision. At the same time, the condition of independence of elements of the ranked sample is the indispensable condition also for the sign nonparametric detectors. In practice, under detection of radar signals, when the number of resolution elements (the number of channels) in radar range, in which a “no” signal exists, is much more than the number of resolution signal elements in radar range, in which a “yes” signal exists, we employ the contrast method [89], an essence of which is the following. Each ranked reading si, (i = 1, 2,…, N), which is considered as the sample reading of the target return signal, is compared with a set of reference (noise) sample readings

wi1, wi2,…, wim taken from the adjacent resolution elements in the radar range. As a result, we determine the reading rank si in the following form:

The ranked and reference samples as well as the results of rank determination can be presented in the following form:

i=1

where

ZDGDout (ri ) is the known rank function

Kg is the threshold defined based on the acceptable value of the probability of false alarm PF

The simplest rank statistic is the Wilcoxon statistic defined by summation of ranks. By Wilcoxon criterion, the decision a “yes” signal is made in accordance with the following signal detection algorithm:

i=1

Relative effectiveness of the rank DGD is higher in comparison with the sign DGD. In the case of the nonfluctuating target return signal and normal Gaussian noise pdf, the relative effectiveness is estimated by 3/π ≈ 0.995. Thus, the rank-generalized signal detection algorithms are practically effective like the optimal generalized signal detection algorithm. A higher level of efficacy of the rank-generalized signal detection algorithms in comparison with the sign generalized signal detection algorithms is obtained owing to the complexity of signal detection algorithm, since ranking of single sample element requires m + 1 summations (subtractions) instead of a single summation (subtraction) per one sample element in the sign signal detection algorithm. Under sequential (moving) rank computation for all resolution elements in radar range, m + 1 summations (subtractions) must be made within the limits of the sampling interval Ts.