Computer Science Mathematics
Computer Science-2021.gz:143320
Computer Science
2f04342c48292e192f0bdf52af441b4f07615071
The Theoretical Limit of Radar Target Detection

In the field of radar target detection, the false alarm and detection probabilities are used as the universal indicator for detection performance evaluation so far, such as Neyman Person detector. In this paper, inspired by the thoughts of Shannon's information theory, the new system model introducing the target existent state variable v into a general radar system model is established for target detection in the presence of complex white Gaussian noise. The equivalent detection channel and the posterior probability distribution are derived based on the priori statistical characteristic of the noise, target scattering and existent state. The detection performance is measured by the false alarm and detection probabilities and the detection information that is defined as the mutual information between received signal and existent state. The false alarm theorem is proved that false alarm probability is equal to the prior probability of the target existence if the observation interval is large enough and the theorem is the basis for the performance comparison proposed detector with Neyman-Person detector. The sampling a posterior probability detector is proposed, and its performance is measured by the empirical detection information. The target detection theorem is proved that the detection information is the limit of the detection performance, that is, the detection information is achievable and the empirical detection information of any detector is no greater than the detection information. Simulation results verify the correctness of the false alarm and the target detection theorems, and show that the performance of the sampling a posterior probability detector is asymptotically optimal and outperforms other detectors. In addition, the detector is more favorable to detect the dim targets under the detection information than other detectors.

other detectors. In addition, the detector is more favorable to detect the dim targets under the detection information than other detectors.

Index Terms
Target detection, detention information, theoretical limit, sampling a posteriori probability detector, Neyman-Person detector.

I. Introduction
T HE target detection problem of the known signal in the presence of noise has evolved significantly during the last decades. The distance, the amplitude, and the direction, of the target information can be obtained from the received signal of radar systems. As the primary link of radar systems, the radar target detection, whose detection performance is of great importance and the detection results will have a significant impact on the subsequent signal processing. Therefore, it has been the main concern of radar signal processing.
The purpose of target detection is to determine the existence or absence of a target based on the receiver signal [1], which is usually judged by comparing the amplitude of the receiver signal with a predetermined threshold and the target exists if the amplitude is greater [2]. Setting the appropriate threshold to obtain the maximum detection probability given false alarm probability is a traditional indicator for evaluating a target detector. It is generally known that the Neyman-Pearson(NP) detector is optimal under the false alarm and detectoin probabilities [3]. Therefore, NP detection method is widely used in the design of target detection systems [4], target detectors [5]- [8] and radar waveform [20], and it is also used in joint detection work with tracking [9], classification [10] and estimation [11]. However, the operation of threshold comparison may produce wrong decisions. Moreover, this method only gives the probability of whether the target exists or not, and does not provide more information about the target for the subsequent work, such as, localization, classification, etc.
The application of information theory to radar can be traced to the early 1950s when Shannon published the paper [12]. Shortly after the pioneering work, Woodward and Davies investigated the distance information of a single target using the principle of inverse probability, and used information-theoretic principles to obtain a posteriori radar receiver [13]- [15]. In 1988, Bell [16], [17] first applied mutual information metrics to the waveform design of radar systems, and used the mutual information between the target impulse response and the received signal as a measure to improve the performance of the information extraction [18]- [20]. Unlike Bell's defination for the mutual information, J. R. Roman et al. [21] select the optimal waveform to be transmitted by minimizing the mutual information between the last-received signal and the to-be-received signal. S. Sen and A. Nehorai [22] used the mutual information between the state and the measurement vectors as the criterion to select the optimal waveform for improved tracking performance. Therefore, it is no longer immature to use the information-theoretic approach to solve radar problems.
In recent years, we made many achievements about spatial information theory of radar [23]- [28]. The concept of the spatial information was proposed firstly [23], which is a unified framework consist of distance information and scattering information, and the closed expressions for them are given [24]- [27].
Moreover, the entropy error is proposed to evaluate the estimation performance [24]. In addition, the theoretical limit of parameter estimation that is one of the two basic tasks of radar detection is proved [28], and another is target detection.
In this paper, the contributions can be summarized as follows: (1) Modeling: We establish the system model for target detection by introducing the target existent state variable into a general radar system model and derive the posterior probability distribution of the existent state conditioned on the received signal in the presence of complex additive white Gaussian noise.
(2) Performance Indicator: As a new performance evaluation indicator proposed, the detection information(DI) is defined as the mutual information between the received signal and existent state. The calculation of the DI is based on the priori probability of the existent state.
(3) False Alarm Theorem: We prove false alarm theorem that the false alarm probability is equal to the prior probability of the target existence if the observation interval is large enough. Due to NP detector maximizes the detection probability given the false alarm probability, thus the proof of the false alarm theorem is the premise for the performance comparison between detectors.
(4) Target Detector: Sampling a posterior(SAP) probability detector that a probability is drawn randomly from a set consist of posterior probability of existent states is proposed, and the state corresponding to the probability is the decision outputed by the detector. SAP detector is more favorable for the detection of dim targets than NP and maximum a posterior probability detectors.
(5) Target Detection Theorem: The fact that DI is the theoretical limit of detection performance is proved in target detection theorem based on the central limit theorem and the property of the jointly typical sequence, which also states that the DI is achievable. None of the target information obtained by the various detectors in the simulation results exceeds the DI.
The rest of the paper is organized as follows. In Section II, the system model is given. The DI, the false alarm and detection probabilities are presented in Section III. Section IV proves the false alarm theorem. Section V gives the detectors and their performance evaluation. The detection theorem is proved in Section VI. Section VII presents the simulation results and Section VIII concludes this paper.
Notations: We use lower case letters to signify variable, upper case letters to denote random variable, bold italic lower case letters to signify column vectors and bold italic upper case letters to denote random column vectors. E(·) denotes the expectation operator. The operators (·) T and (·) H denote transpose and conjugate transpose of a vector respectively. A Gauess variable with expectation a and variance σ 2 is denoted by N(a, σ 2 ). I 0 (·) stands for the zero-order modified Bessel function of the first kind. π(·) denotes the priori probability. d(·) denotes the detection function.

II. System Model
Consider a radar system with the ideal low-pass transmitted signal ψ(t) = sinc(Bt) is limited bandwidth B/2. Suppose that possible K targets are completely contained in the observed interval, from which we obtain the received signal is given by (1) Where v k ∈ {0, 1} is the discrete variable of existent state, v k = 0 and v k = 1 corresponding to nullhypothesis H 0 and alternative hypothesis H 1 in a binary hypothesis testing, respectively. s k is the complex reflection coefficient, τ k is the time-delay. w(t) is the complex additive white Gaussian noise process of limited bandwidth B/2 with zero mean and N 0 /2 variance in its real and imaginary parts, respectively.
Assume that sample the received signal at the rate of B samples, and correspondingly, T is the sampling time. Define the observed sample number N as the time-bandwidth product N = T B. The discrete-time equivalent of the received signal is given by Where x k = Bτ k is the regularized delay. Due to the sample values are independent complex gaussian random variables, so are the real part and the imaginary part each other.
For briefness, (2) can be expressed as Where v = (v 1 , v 2 , · · · , v K ) is the state vector, U (x) = [u (x 1 ) , u (x 2 ) , · · · u (x K )] is position matrix of targets, u T (x k ) = (· · · , sinc(n − x k ), · · ·) is the sampling waveform of the k-th target. Equation (3) is a system model containing the state variable and target parameter, also known as the system equation, which central feature is directly introduce the state vector into the general radar system model.
The probability density function of the N-dimensional complex Gaussian noise vector w is given as Given the state vector v, the regularized delay vector x and the scattering vector s, the multidimensional conditional probability density function of y is obtained as The constant module is a typical non-fluctuation, also known as the swerling 0, in which the complex scattering coefficient is constant and the phases are distributed uniformly in [0, 2π]. Since the performance measure of multi-target for the constant module is very complicated, we only consider the case of a single target in the observed interval. The complex scattering coefficient is expressed as s = αe jϕ , where α denotes the constant modulus and ϕ denotes the phase. Suppose that the target position is also distributed uniformly within the observed interval [−N/2, N/2]. Averaging over the random phase and target position, the conditional probability density function of a single target is given by Equation (6) depicts the detection channel. Note that u H (x)y is the module of the matched filter output.

III. Performance Evaluation of Target Detection System
In this section, the posterior probability distribution of the existence state is derived firstly. Then the DI, the false alarm and detection probabilities are derived based on the posterior probability distribution.
The DI is a novel evaluation indicator. Specially, we consider the general case and the matched case that the target position is known by detector. It is strictly that the assumption of the known target position is less than reasonable, since the task of target detection is to detect the target existence or absence in the observation interval, and the position is not known at all. However, the correlation peak is usually the largest when perfectly matched at high SNR region, and it is reasonable to use the largest peak for detection.

A. Posteriori Probability Distribution of Target Existent State
For a specific snapshot, assuming the target position is x 0 and the scattering signal is αe jϕ 0 , the received signal is given by Substituting (7) into (6), one has According to the autocorrelation of sin Where ρ 2 = α 2 N 0 denotes the signal to noise ratio(SNR).
It can be seen that the conditional probability density function mainly depends on SNR.
Based on the Bayes formula and (9), the posteriori probability distribution function is given by where denotes the regularized factor. Equation (10) can also be expressed as where denotes the detection statistics and is the average over the Bessel function with respect to time in the observed interval.

B. The False Alarm and The Detection Probabilities
The theoretical false alarm and the theoretical detection probabilities are differ from the false alarm and the detection probabilitie of NP detector in that the former are the corresponding posterior probability obtained from the priori probability of existent state. Let y 0 and y 1 denote the received signal with only noise and signal-plus-noise, respectively. According to (9), the outputs of the matched filter are According to (12), the probability of the target existence are where the detection statistics are The false alarm and the detection probabilities are , According to (9), the likelihood function for a specific position x 0 is given by Based on the Bayes formula and (18), the posteriori probability distribution function is given by where the detection statistics are The false alarm and the detection probabilities are

C. The Detection Information
Based on the posteriori probability distribution function of (10), the definition of the DI is as following.
Definition 1. Suppose V is the existent state vector and Y is the received signal vector, the mutual information I (Y; V) between the existent state and the received signal is defined as the DI, Where is the priori entropy of the existent state, is the posteriori entropy of the existent state.
The physical meaning of I (Y; V) is the difference between the priori entropy and the posteriori entropy and means the reduction in the uncertainty of V due to the knowledge of received signal Y.

IV. False Alarm Theorem
In this section, we point out the ralation between the theoretical false alarm probability and priori probability π (1) by the theorem as follows.
Proof: See appendix A.
We obtain the following corollary according to the condition of Theorem 1.
It is clear that the theoretical false alarm probability depends on the prior probability of the target existence from theorem 1 and corollary 1.
The prior probability is used as known conditions in the analysis of the III section. According to the Bayesian principle, the prior probability is obtained based on historical experience and statistical data. However, it is difficult to determine π (1) precisely to the prior probability of target existence is particularly small in a complicated practical environment, such as the radar target detection. Then, we can apply the false alarm theorem in reverse, assuming the false alarm probability according to the application needs and using it to replace the prior probability of the target existence. Moreover, the performance comparison of the detector proposed in the next section with NP detector is also based on the theorem.

V. Radar Target Detector
In this section, we introduce several detectors, mainly including NP detector [29], maximum a posteriori(MAP) probability detector, and SAP detector. The last one is new.

A. NP Detector
The Neyman-Pearson detector maximizes the detection probability for a given false alarm constraint η (η ∈ (0, 1)) by the likelihood ratio test. According to (9), the likelihood ratio is Where T h denotes the detection threshold given by the false alarm probability.

The false alarm and the detection probabilities
According to (16), the detection threshold given false alarm probability is obtained as and the detection probability based on the detection threshold is It is note that Υ (1, y 0 ) = 1 is given in the proof of false alarm theorem, thus NP detector does not work in the general case. To compare NP detector with MAP and SAP detectors below, we derive the false alarm and detection probabilitie for specific position under the mutual information method in the section III.
The detection threshold for a specific position x 0 is given by [29] T h = − N 0 ln P FA , the detection probability is where Q M (•) is Marcum function.

The Detection Information
The calculating method for the posteriori entropy of NP detector for a specific position x 0 is proposed by M. Kondo [30], which is based on the false alarm and detection probabilities, i.e., where PV(0) is the probability that the target is detected as non-existent, i.e., Where A denotes the probability of the target real existence under the decision signal "target existence", and D denotes the get non-existence probability under the decision signal "target non-existence", i.e.,

B. MAP Detector
The detector that the obtained decision maximizes the posterior probability distribution function P(v|y) is called MAP probability detector, i.e.,v where P(v|y) = p(y|v)π(v)/p(y) . If the priori probabilities of the target existence states are equal, we can detect the target by the maximum likelihood detector, i.e.,v ML = arg max v p (y|v). But, the probability of target existence or absence is likely to be unequal in a real target detection environment. Hence, the method of the maximum likelihood is not applicable to the design of actual detector.
The false alarm probability of MAP detector is and the detection probability is The detection statistics of MAP detector is accurate in the considerable sample space. Hence, the DI can be gained according to the calculating method of NP detector, that is using the false alarm and detection probabilities.

C. SAP Detector
The decision of SAP detector is corresponding to the "lots" drawn from a set consist of the posterior probability of existence states. Therefore, SAP detection is stochastic. Conversely, MAP and NP detectors are deterministic since they judge the target must be existent or not. In other words, the probabilisitic decision of the target existence of SAP detector may be 0.2 or 0.7, while the that of MAP and NP detectors can only be 0 or 1. Thus the deterministic method is relatively crude since the target detection is essentially a probabilistic event. SAP detector is defined as following.
where P(v|y) is the posteriori probability distribution of existence state conditioned receiver signal, and arg smp v {·} denotes the sampling operation.
The stochastic feature of SAP detector makes its performance infinitely close to the theoretical performance with the sufficient samples. The average performance of SAP detector is dependent on the P(v|y) and the posteriori probability distribution function of SAP detector is obtained by Therefore, the detection statistics, the DI, the false alarm probability and the detection probability of SAP detector is same as the corresponding theoretical value.

VI. Target Detection Theorem
We introduced the concepts of DI in section III, can it be used to evaluate an optimal detector? what kind of detector is optimal, in what sense optimal, and what is optimal performance? In this section, we introduce most of the basic definations required for the subsequent developement of the target detection theorem firstly, and then prove the theorem.
We denote a target detection system as (A, π(v), p(y|v),v = d(y), B), consists of an input set A = { v| v = 0, 1}, the priori probability π (v) of existence state, the probability density function p(y|v) that determine the characteristic of the detection channel, a detectorv = d(y), and a output set B. Where (A, π(v), p(y|v), B) denotes the joint target and channel and depicts the detection environment. It is assumed that the statistical characteristics of the joint target and channel is known absolutely, although the detector usually knows only a part of that in practice.
One detection process consists of several parts: the target, the channel, the detector, and one snapshot.
The extension of target and channel are generated by multiple snapshots whose detection process is shown in Fig.1   It should be noted that the counterpart to the target detection system of M snapshots is empirical DI.
The definition of empirical DI is given below.
Definition 5. The empirical DI obtained for M snapshots of the detector is defined as End of the proof.
Based on Lemma 2 and the definition of the jointly typical sequences, we can obtain the following theorem.
Conversely, the empirical DI of any detector is no greater than the DI.
Proof: See appendix B.
Remark 2. The application of target detection theorem is extensive and the proof is given only for the detection of a single target, but its framework of proof can be extended to multi-target, while it is not considered for the sake of brevity.
Remark 3. The proof of the target detection theorem is constructive, that is, SAP detection is an achievable method of target detection, which performance is asymptotically optimal. In addition, it is more common to detect dim targets in the radar actual work. SAP detection is more conducive to the detection of dim targets, so that radar detection can be applied to more complex scenarios.

VII. Simulation Results
In this section, numerical simulations are performed to verify the false alarm and target detection theorems, and to evaluate the performances of proposed detectors.

A. Verification of False Alarm Theorem
For modern radar detection systems, the number N of sampling points in one fast snapshot can reach tens to hundreds of thousands, so the condition of Theorem 1 and Corollary 1 for sufficiently long observation are usually satisfied. Fig. 2 depicts the relationship between the false alarm probability and the priori probability π (1) about S NR = 0dB and S NR = 5dB respectively. It is seen that the conclusions match well at low SNR region even with a narrower observation interval (N = 64 ). At medium SNR region, it also fits well when the number of sampling points N > 1000. Therefore, the false alarm theorem that the probability of false alarm is equal to the priori probability of the target existence is true.

B. Verification of Detection Theorem
The analytic and numerical curves of the DI and the empirical DI of the SAP detector for different snapshot numbers are compared in Fig. 3, in which, the empirical DI gained by 10,000 snapshots, 1000   snapshots and 100 snapshots respectively. As seen, the empirical DI gradually approaches the DI as the number of snapshots increases, and it almost overlaps with the DI for the 10,000 snapshots, which indicates that the DI is achievable.

C. Receiver Operating Characteristic for Detectors
It is universally acknowledged that NP detector is optimal when false alarm and detection probabilities pair is selected as the evaluation indicator of the radar target detectors. Fig. 4 illustrates the view and depicts the comparison of the receiver operating characteristic received by the method of information theory and NP detector for different SNR. It can be seen that the detection probability of NP detector is always greater for the same false alarm probability whether in the medium or low SNR region. In other words, the part below the green line is the achievable area of the NP detector. It can be seen that the achievable area of SAP detector is greater than that of NP. This is to be expected since the DI is the theoretical value under the information-theoretic method. Moreover, the figure also shows that the amount of information obtained by SAP detector is greater than that of MAP and NP detectors. Due to the stochastic feature of SAP detector making its performance infinitely close to the theoretical performance, therefore, SAP detector outperforms other detectors. In addition, as shown in Fig. 5(b), the detection performance of SAP detector significantly outperforms NP and MAP detectors in the low SNR region, which indicates that the detector is more favorable for detecting dim targets in complex scenario.

D. The Detection Information for Detectors
We can also see that the performance curves of MAP and NP detectors have a crossover point. Since the threshold values are generated in MAP and NP detection processes, however, the generating methods of which are different resulting in different slope of performance curves for the same received signal.
While SAP detector is independent of the detection threshold and gains a detection probability no matter how small the SNR is.
At present, there is no conclusion on which of the two indicators is optimal, but the information theory

VIII. Conclusion and Further Work
In this paper, we make contributions to radar target detection from the perspective of information theory. Firstly, we develop a system model for target detection, propose an equivalent detection channel, and derive the posterior probability distribution of the existence states. Then, DI is proposed as a novel indicator to measure the detection performance. Moreover, the false alarm theorem is proved, stating that the false alarm probability is equal to the prior probability π (1). In addition, SAP detector is proposed and the detection performance outperforms the MAP and NP detectors from the perspective of DI. Finally, the target detection theorem is proved that the DI is reachable and that the empirical DI of any detector does not exceed the DI. Our contributions are worthy of further investigation, such as the performance measure of detectors for multi-targets detection.

Appendix
Proof of the theorems A. The proof of false alarm theorem Proof: The received signal is y = vsu(x) + w for arbitrary scattered signal s , according to (6), the conditional probability density function of y is Then, based on the Bayes formula, the posteriori probability distribution function is derived as or, When the received signal is y 0 , the output of the matched filter is u H (x)y 0 = w(x), then The time average of the stationary process is equal to the ensemble average if the observation interval is large enough, then Since w (t) is the complex additive white Gaussian noise with zero mean and N 0 variance, one has so Υ (1, y 0 ) = π(s) exp − |s| 2 N 0 exp |s| 2 N 0 ds = π(s)ds = 1. (57) Substituting equation (57) into equation (15) yields P (1 |y 0 ) = π(1)Υ (1, y 0 ) π(0) + π(1)Υ (1, y 0 ) = π(1).
According to the definition of false alarm probability, one has P FA = E y 0 P(1|y 0 ) .
End of the proof.

B. The proof of target detection theorem
Proof: The proof of theorem is constructive and divided into two parts, the positive to and the converse to theorem, and the former is proved first.
Proof of the positive to theorem: We namely, the empirical DI of arbitrary detector is no greater than the DI.
End of the proof.