From Formula (13), it can be seen that the biggest advantage of instantaneous autocorrelation is that there is no time integration. In other words, the instantaneous information of correlation processing is well preserved.
3.1. Simulation Analysis
In order to analyze the difference of coherent characteristics between the coherent and noncoherent pulse trains, three simulation experiments were performed based on time-frequency correlation. We took two sets of pulse trains: the first one was a coherent pulse train and the second a noncoherent pulse train. The signal type and the parameters of both pulse trains were the same. It is noteworthy that the number of pulses used for emitter identification must be greater than or equal to 4.
Simulation Experiment 1: Signal modulation type is continuous wave (CW), CF is 30 MHz, sampling frequency (Fs) is 100 MHz, PRI is 60 μs, PW is 10 μs, pulse number is 10. Autocorrelation results of the two groups, without noise, are shown in
Figure 2.
Simulation Experiment 2: Signal modulation type is linear frequency modulation (LFM), CF is 30 MHz, bandwidth (BW) is 10 MHz, Fs is 100 MHz, PRI is 150 μs, PW is 15 μs, pulse number is 8. Autocorrelation results of the two groups, without noise, are shown in
Figure 3.
Simulation Experiment 3: Signal modulation type is nonlinear frequency modulation (NLFM), CF is 100 MHz, BW is 20 MHz, Fs is 200 MHz, PRI is 50 μs, PW is 5 μs, pulse number is 15. Autocorrelation results of the two groups, without noise, are shown in
Figure 4.
As shown in
Figure 2,
Figure 3 and
Figure 4, it can be seen that the three experiments of different modulation types have similar rules: the correlation degree between the coherent and noncoherent pulse trains is almost the same near the zero point, while outside the zero point, the correlation of the coherent signals is much better than that of the noncoherent signals. With increasing time delay, the correlation of the coherent signals decreases periodically and steadily, while it changes irregularly in the noncoherent signals. This is due to the strong correlation and continuity of the phase information of the coherent pulse train, while that of the noncoherent pulse train is random. Therefore, we can use this difference to distinguish the coherent and noncoherent signals. The next task is to define a feature parameter to describe this difference.
3.2. The Model of Coherent Feature Extraction
According to the simulation analysis above, define the maximum peak at the zero point as the main peak. Except for the maximum peak, the maximum of the remaining peaks is defined as the secondary peak. The discrimination model of coherent characteristics can be named as the ratio of the secondary peak value to the main peak value (SMR). The model parameter is given as below.
where
F(1) is the main peak value at the zero point.
In terms of the conditions of Experiments 1–3 to calculate the SMR of different signal types, take the average of 100 times of calculation results. SMRs are computed without noise in
Table 1.
As shown in
Table 1, it can be seen that the SMRs of the coherent pulse train are close to 1, while the SMRs of the noncoherent pulse train are much smaller than 1.
In order to verify the rationality of the discrimination threshold of SMR in different SNRs, according to the conditions of Experiments 1 to 3, SMRs of different signal types are computed based on the Monte-Carlo for 100 times in different SNRs. The results are shown in
Figure 5.
As can be seen in
Figure 5, when SNR = −5 dB, SMRs of different signal types are overlapped as SMR cannot distinguish the difference very well. When SNR ≥ 0 dB, the difference in SMRs between the coherent and noncoherent pulse trains is obvious. This is due to the instantaneous autocorrelation processing, which can weaken the noise and strengthen the signal. To a certain extent, SNR of the signal has been improved. From the simulation analysis above, we can conclude that SMR can be used as a feature parameter for REID, and it is insensitive to SNR and signal type changes.
Furthermore, we can see when SNR ≥ −5 dB, SMR of the coherent pulse train is always above 0.6, while SMR of the noncoherent pulse train is always below 0.6. Additionally, each signal type has the same rule. Therefore, we can set the discrimination threshold at 0.6. When SMR > 0.6, the pulse train is coherent, otherwise the pulse train is noncoherent.
3.3. Coherent Feature Evaluation
To evaluate the comprehensive identification performance of SMR and the typical five feature parameters, AHP [
25] is introduced in this paper. AHP is a systematic method that takes a complex multiobjective decision-making problem as a system and decomposes the objective into several levels, namely, multiobjective, multicriteria, and multiattribute. Through quantitative and qualitative analyses, the single ranking and total ranking of the levels are calculated as a systematic method of objective and optimized decision-making.
Evaluation steps based on AHP are given as below.
Step 1: Establish a model of the evaluation system
To establish a suitable feature evaluation system as the basis of efficient evaluation, the feature evaluation system needs to follow certain evaluation criteria, such as purposefulness, a scientific nature, and systematization. In this paper, the feature evaluation system is designed as a three-layer structure, namely, a target layer, an index layer, and a scheme layer. The target layer denotes the radar emitter identification. In the index layer, three metrics—reliability, accuracy, and robustness—are adopted to evaluate the performance of the parameters. The scheme layer is the parameters to be evaluated, which are CF, PRI, PW, AOA, PA, and SMR. The model of the evaluation system is shown in
Figure 6.
Step 2: Construct the judgment matrix.
According to the Saaty relative importance hierarchy table [
26], reliability, accuracy, and robustness in the index layer are taken to make a comprehensive evaluation of the evaluated feature. The judgment matrix is constructed in terms of these three indexes, which are compared with each other by the expert’s scoring. Judgment matrix
A is computed below.
where
denotes that the influence of reliability on the evaluation system is much higher than that of robustness for radar emitter identification. The definition of the other elements in matrix
A can be deduced in the same way.
In the same way, judgment matrices
B1,
B2,
B3 can be obtained below.
In judgment matrix B1, denotes the reliability of CF is weaker than that of PW, denotes the reliability of CF is better than that of AOA. The definition of other elements in matrices B1, B2, B3 can be deduced in this way.
Step 3: Calculate the weight vector of judgment matrices A, B1, B2, B3 and check their consistency.
In this step, we need to introduce three definitions [
27]: compatibility index (
CI), random consistency index (
RI), and compatibility ratio (
CR). The mathematical expression of
CI and
CR are given below.
In the formulas mentioned above,
n denotes the order of matrix
A,
is the maximum eigenvalue of the judgment matrix
A. When
n = 3,
RI = 0.58;
n = 4,
RI = 0.96;
n = 5,
RI = 1.12;
n = 6,
RI = 1.24 [
28].
The rules for consistency testing are as follows: when CR < 0.1, it is considered that the inconsistency scale of judgment matrix A is in the allowable range and can be accepted. If CR ≥ 0.1, go back to Step 2 to adjust judgment matrix A again, then calculate and check the consistency until it meets the consistency condition (CR < 0.1).
According to the Formulas (19) and (20) and judgment matrix
A, the maximum eigenvalue and the normalized eigenvector of judgment matrix
A can be computed as below.
where
< 0.1, it denotes that matrix
A meets the requirement of consistency testing.
The maximum eigenvalues
,
,
and the normalized eigenvectors
,
,
of matrices
B1,
B2,
B3 can be computed in the same way. The computation results are shown in
Table 2.
As shown in
Table 2, it can be seen that judgment matrices
B1,
B2,
B3 all meet the consistency testing requirement (
).
Step 4: Calculate the total weight of each parameter in the scheme layer and check the consistency of the evaluation system.
The calculation results of each parameter weight can be shown in
Table 3.
The results of the consistency testing of hierarchical total sorting are as below.
Where, when
CR < 0.1, it denotes that the weights of each parameter in the scheme layer are reasonable and effective. As shown in
Table 3, it can be seen that the order of weight from large to small is SMR > PRI > CF > PW > PA > AOA, and the SMR feature has the largest weight among these six characteristic parameters.
According to the comprehensive evaluation results above, a sketch map of using SMR in REID is shown in
Figure 7.