In this paper, in order to reduce the probability of the radar waveform intercepted by the passive detection system, the time-bandwidth product of the radar waveform is increased, and the detection probability of the radar waveform to the target is improved. This paper tackles the holographic RF stealth radar and proposes a joint coding waveform based on the linear frequency modulation (LFM) waveform. Joint coding uses complementary codes to perform phase-coding, and combines the codewords optimized by genetic algorithm in order to perform discrete frequency-coding waveform. The joint coding waveform model is theoretically analyzed, and the ambiguity function, pulse compression and target detection probability of the joint coding waveform are obtained by numerical simulation. In addition, the complexity of the algorithm and the low probability of intercept (LPI) characteristic of the joint coding waveform are analyzed. The results show that the joint coding waveform has an approximate "pushpin" ambiguity function, ultra-low sidelobe characteristics, better RF stealth and target detection performance. Finally, it has good application prospects in the current battlefield environment.
Keywords: RF stealth; joint coded waveform; ambiguity function; pulse compression; ultra-low sidelobe
As a kind of radio detection equipment, radar has the advantages of all-weather, all-time and robust penetration ability. It plays a crucial role in war, disaster relief and civil use [[
In order to reduce the risk of radar signals being detected by interceptors and gain the initiative on the battlefield, the RF stealth technology has been developed in recent years [[
The radar waveform theory is an essential branch of radar theory. The radar waveform directly or indirectly determines the radar's primary performance and signal processing method [[
At present, the radar mainly transmits the LFM waveform, and it is one of the earliest and most widely used LPI pulse compression signals [[
In this paper, a phase-frequency joint coding radar LPI waveform is designed using the characteristic of complete software of holographic radar [[
- (
1 ) Using the characteristics of complementary inter-code cancellation and zero autocorrelation sidelobe, the phase-coding waveform is designed to increase the peak sidelobe ratio of the transmitted waveform. GA optimizes the DFC to form a new GADFC codeword, which increases the orthogonality between the codewords and reduces the autocorrelation sidelobe level. - (
2 ) A CPC-GADFC joint coding waveform based on LFM is designed. The joint coding waveform can make up for the drawbacks of a single modulation waveform, and enhance the radar waveforms' LPI and anti-interference abilities to ensure the radar range and speed measuring resolution. - (
3 ) Through the combination of CPC and GADFC, the echo pulse compression of the radar transmit waveform has the characteristics of ultra-low sidelobes, and the main lobe width is narrower so that the designed waveform has more advantages in target detection.
The remainder of this paper is organized as follows. Section 2 introduces the algorithm model, which focuses on the phase encoding of the complementary code sequence and the frequency encoding using the discrete frequency sequence optimized by the GA. Section 3 presents the CPC-GADFC joint coding waveform design, performance analysis and processing method analysis. The CPC-GADFC joint coding waveform expression is also deduced, and the performance analysis and processing method design are performed. Section 4 details the simulation experiment which verifies the performance of the designed CPC-GADFC joint coding waveform based on LFM. Finally, the conclusions are drawn in Section 5.
Based on the LFM waveform, this paper designs a CPC-GADFC joint coding waveform. In phase, the complementary code sequence is used for phase-coding, while in frequency, the discrete frequency sequence optimized by the GA is used for frequency-coding waveform. This section mainly introduces the complementary code phase-coding model and GA for DFC optimization.
Golay and Erickson [[
The definition of complementary two-phase code is given by: there are two-phase sequences of length N, positive code
(
then
(
The complementary code's autocorrelation function is determined (cf. Figure 1) according to Equation (
It can be seen from Figure 1 that the autocorrelation function of the
The characteristics of complementary code have two sequences. In order to reduce the influence of the echo performance fluctuation caused by angular scintillation of the target effective reflection area, in a pulse group transmitted by radar, the pulse is transmitted by alternating phase modulation of positive code
It can be observed from Figure 2 that in a pulse group, the radar pulse alternately transmits the positive code
The GA is based on Darwin's theory of evolution and Mendel's genetics theory. The theory of evolution proposes that species constitute a complex process that is not random through step-by-step evolution. This paper uses the GA to find the optimal codeword combination of DFC, increase the orthogonality between codewords, and reduce the sidelobe level of the GADFC autocorrelation function.
GA is used to represent the problem as a "chromosome" -like representation of strings coding in binary or floating-point numbers. A group of "chromosomes", namely the initial population (hypothetical solution set), is then given. These hypothetical solutions are placed in the "environment" of the problem. According to the principle of survival, the fittest and its survival, as well as the "chromosomes" that are more suitable for the environment, are considered for the process of replication, crossover and variation in order to produce a new generation of "chromosome" groups that are more suitable for the environment. It finally converges to a "chromosome" that is most suitable for the environment through continuous evolution. After decoding, an approximately optimal solution to the problem is obtained. The basic steps of the GA are summarized as follows:
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(1) Chromosome coding. The optimal arrangement between DFC codewords is converted into a search space that the GA can process with Gray encoding. Since the codewords of DFC are decimal, it is necessary to first convert them into binary representation. The variation range of the DFC codewords is _HT_ _ht_ , the encoding length is _I_l_i_ and the encoding precision of the binary encoding is _HT_ _ht_ . The conversion relationship between the DFC binary codewords and its decimal is given by: (3) _HT_ _ht_ where _I_a_i_ is an argument between _HT_ _ht_ and _Ii_ is a m-bit binary code represented as
4 ) - (
2 ) Define the fitness function and generate the initialization population. Before using the GA, the fitness function should be used to determine the final goal. In this study, the maximum Euclidean distance of DFC should be found. That is, the objective function is the optimal codeword combination mode to form a new GADFC codeword and achieve a lower sidelobe autocorrelation level. The objective function is given by: (5 )6 ) - (
3 ) Select replication, crossover and mutation operations on the obtained population to generate the next generation population. This selection is the key to GA. It is based on the evaluation of individual fitness, while the purpose is to avoid gene deletion and improve the global convergence. The crossover consists in selecting a more significant value of the random number than the crossover probability for the next step. The mutation first selects some individuals from the population with a certain high probability, and performs the inverse operation on each chosen individual. In this paper, the probability of mutation is 0.001. The global search ability of the GA is mainly provided by selection, crossover and mutation. The mutation ensures that the algorithm can search every point from the problem space to the solution space, making the algorithm have global optimality and enhancing the GA's robustness. - (
4 ) Determine whether the algorithm satisfies the stopping criterion. If it is not satisfied, then repeat step (3 ). - (
5 ) The algorithm ends, and the optimal GADFC codewords are obtained.
The DFC coding and GADFC coding autocorrelation functions are then compared (cf. Figure 3).
Figure 3 shows that the GADFC coding autocorrelation sidelobe level generated by the GA is significantly lower than that of the DFC coding. The waveform generated by frequency-coding LFM waveform with GADFC has satisfactory low sidelobe characteristics. Considering that the optimization of the DFC codeword by GA can improve the orthogonality between radar transmitted pulses, the waveform correlation coefficients of two adjacent transmitted pulses shown in Figure 2 are used for analysis (cf. Table 1).
It can be seen from Table 1 that GADFC efficiently reduces the correlation coefficient between adjacent pulses, increases the orthogonality, and more efficiently improves the radar waveform target detection performance.
The expression of the CPC-GADFC joint coding waveform is deduced, and the performance analysis and processing method design are performed, according to section II of the complementary code phase-coding model and the results of DFC optimization by GA.
Based on the LFM waveform, GADFC is performed on the waveform. The time domain expression of the waveform is given by:
(
where
It can be seen that the GADFC's envelope is similar to the LFM waveform (
Based on
(
(
where
At this point, the complex envelope of the joint coding waveform can be expressed as:
(
(
Equations (
(
where ⊗ represents the convolution operation.
Therefore, the time-domain expression of the CPC-GADFC joint coding waveform based on LFM, is given by:
(
Figure 3 and Equation (
Figure 4 shows that each pulse in a pulse group is alternately transmitted with the positive code and complement code of the CPC (cf. Figure 4b,c). Transmitting alternately aims at reducing the influence of the fluctuation of echo ability caused by the angular scintillation of the effective reflection area of the target. It can be seen from Figure 5 that, since the GADFC coding sequence performed by each pulse is the same within a pulse group, the time-frequency distribution between the pulses is also the same. The design flow of the LFM-based CPC-GADFC joint modulation radar waveform is shown in Figure 6.
According to the Fourier transform properties, the spectrum of the combined signal is the product of the LFM-GADFC waveform and the CPC waveform spectrum:
(
where
Due to the fact that in the frequency-phase-coding process based on the LFM waveform, the basic shape of the signal will not be changed by the phase-coding process, the factor affecting the frequency spectrum is the frequency-coding one. The frequency spectrum of the single pulse of the joint coding waveform is shown in Figure 7.
It can be seen from Figure 7 that the LFM-CPC waveform can still maintain the basic form of the LFM waveform spectrum after phase-coding. However, the new joint coding waveform spectrum after CPC-GADFC joint coding, loses the basic form of the LFM waveform. When the interceptor receives the radar transmission waveform, it is not easy to accumulate for a long time. It is easier to consider noise, which improves the radar LPI performance.
Section 3.1 introduced the joint coding waveform's design process and time-frequency domain expression. In Section 3.2, the ambiguity function analysis, LPI characteristic analysis, algorithm complexity analysis and signal processing flow of the joint coding waveform, are introduced.
Because the LFM-based CPC-GADFC joint coding waveform uses complementary phase-coding sequences, it has two sequences of positive code and complement code. Therefore, according to the definition of ambiguity function, the ambiguity function of the joint coding waveform also consists of two parts:
(
(
By substituting Equations (
(
(
Equations (
(
According to Equation (
(
According to Equation (
(
It can be deduced from this equation derivation that the ambiguity function of the CPC-GADFC joint coding waveform based on LFM, is composed of two parts. Each part is composed of the convolution superposition of two ambiguity functions. According to the autocorrelation property and the ambiguity function expression of complementary code, the autocorrelation characteristics can be obtained using complementary code for the phase-coding. Moreover, the influence of the range-Doppler coupling on the signal is reduced. The ambiguity function graph of the joint coding waveform is shown in Figure 8.
It can be seen from Figure 8 that the ambiguity function of the LFM-based CPC-GADFC joint coding waveform presents a good "pushpin" shape. Based on the high Doppler tolerance of the LFM signal, the joint coding waveform reduces the Doppler range coupling. It has a high range and velocity resolution, as well as low sidelobe characteristics, which improves the radar's ability to detect targets. Complex waveforms have high modulation complexity, and the phase-frequency joint coding increases the difficulty of demodulation and accumulation of radar waveforms by interceptors. For the wideband interception receivers, the intra-pulse phase-frequency agility can make the radar waveform spectral form similar to the noise frequency. This makes the waveform submerged in the noise and challenging to capture. Due to the complex phase-frequency hopping rules and moderate hopping performance for narrowband receivers, the joint coding waveform significantly improves the LPI performance of the radar system.
In order to quantitatively analyze the LPI of the joint coding radar, the perspective of the Schleher interception factor can first be used:
(
where
When the radar parameter settings are the same and the parameter settings of the interception receiver are certain, the interception factor can be simplified and expressed in the form of time-bandwidth product:
(
When the radar uses LFM as the transmitting waveform, since its time-bandwidth product is
(
If the radar uses the LFM-based CPC-GADFC joint coding waveform as the transmit waveform, since the phase-coding and the frequency-coding simultaneously encode the intra-pulse LFM waveform, the impact on the waveform time-bandwidth product is consistent, which is
(
By comparing Equations (
The complexity of the radar waveform can reflect the used modulation mode, and the complexity characteristics of the waveform are less affected by the signal-to-noise ratio. Therefore, the sparsity of the waveform frequency domain can be used to express the complexity [[
(
where N is the frequency domain waveform length, and
(
where M is the total number of 1 element in the frequency domain waveform.
It can be seen that compared with the traditional radar LFM waveform, the CPC-GADFC waveform carries out phase and frequency joint coding in the pulse. As shown in the spectrum of the single pulse of the joint coding waveform in Figure 7, the sparse complexity of the CPC-GADFC frequency domain is higher than that of the LFM waveform.
The time complexity of the CPC-GADFC waveform is higher than those of the LFM waveform and DFC waveform. This is due to the fact that the CPC-GADFC waveform is simultaneously coded with phase and frequency, and GA is used for DFC codeword optimization. In addition, compared with the CPC waveform, the CPC-GADFC waveform has optimized frequency-coding on the basis of the CPC waveform. Therefore, the time complexity of the CPC-GADFC is higher. The complex radar waveform can increase the accumulation and deciphering difficulty of the interceptor, and significantly improve the LPI performance of the radar.
Because the joint coding waveform uses a complementary code sequence in phase modulation, the waveform in the pulse group is alternately transmitted with phase-coding positive code and complementary code in the transmission process. The two coding waveforms should be processed together in echo processing. When an echo is received, the positive code and the complement code are first used as a priori to analyze the correlation coefficient of the echo. This step aims at determining the type of echo phase-coding, which is convenient for subsequent analysis. Assuming that the echo received at the current moment is a joint coding waveform encoding with positive code when performing pulse compression, the two-phase codes should be subjected to matched filtering. The results are simultaneously superimposed. This process is referred to as joint coding echo pulse compression processing.
The echo pulse compression processing of modern radar mainly uses digital signal processing. Two approaches exist for performing it. The time-domain correlation processing method is often used when the demand is a small pulse pressure ratio. When a large pulse pressure ratio is required, it is usually performed in the frequency domain by Fourier transform. Since the matched filter is a linear time-invariant system, it can be determined from the nature of Fourier change:
(
(
where
(
When both signals are correctly sampled, the matched filter output signal of the positive and complement phase joint coding signal can be expressed as:
(
Finally, the matched filter requires to superimpose the
It can be seen from Figure 9 that the pulse compression of the LFM-based CPC-GADFC joint coding waveform consists of a matched filter by the positive phase and the complement phase, respectively. The results are then superimposed to obtain the joint coding waveform pulse compression output. The advantage of the alternately transmitted process is that the low sidelobe level of the complementary code autocorrelation can be used to obtain a pulse compression result with low sidelobe characteristics, in order to achieve a better target detection performance.
In order to verify the robustness of the LFM-based CPC-GADFC joint coding waveform, the experiment performs a time-frequency analysis of the waveform from the perspective of a pulse waveform. Pulse compression is performed on the echo, and the obtained result is compared with the LFM waveform and CPC-DFC joint coding waveform. Finally, the target detection performance of the joint coding waveform is analyzed. In this paper, the complementary code sequence, DFC sequence and GADFC sequence are all of 32 bits. Their codewords are shown in Table 2.
In one transmission pulse, the time domain waveform and time-frequency analysis of the CPC-GADFC joint coding waveform based on LFM, are shown in Figure 10 and Figure 11.
It can be seen from Figure 10 that the complementary code sequence efficiently encodes the LFM, and the purpose of increasing the time-bandwidth product is achieved by changing the phase structure of the signal. Moreover, it can be observed from Figure 11 that GADFC changes the frequency distribution of the LFM waveform, and simultaneously performs frequency-coding based on phase-coding intra-pulse, which increases the complexity of the joint coding waveform.
In order to verify the performance of the designed waveform, the echo pulse compression results of the LFM-DFC waveform, CPC-DFC waveform and CPC-GADFC waveform are compared (cf. Figure 12, Figure 13 and Figure 14).
It can be seen from Figure 12, Figure 13 and Figure 14 that the CPC-GADFC joint coding waveform uses the complementary code for phase-coding, which can significantly reduce the sidelobe through the characteristics of the autocorrelation and sidelobe cancellation of the complementary code. Compared with DFC, the joint coding GADFC can find the optimal distance between each codeword in DFC encoding by GA, which increases the orthogonality between the codewords. After coding, the maximum main side ratio can be reduced by 3.5 dB, compared with the unoptimized joint coding waveform. The joint coding waveform has the characteristics of ultra-low sidelobes, and the distribution of side lobes is more uniform. The jointly coding waveform improves the LPI performance of the signal while improving the radar's ability to detect targets, as shown in Figure 15.
It can be seen from Figure 15 that the pulse compression sidelobe level of CPC-GADFC waveform is significantly lower than that of other waveforms and lower than that of the DFC waveform with Hamming window. It can also be seen that the pulse compression main lobe of the CPC-GADFC waveform is significantly narrower than other waveforms, which indicates that the waveform designed in this paper can concentrate the radar energy and have more accurate target detection performance. Moreover, it can be observed that the detection probability of the CPC-GADFC waveform is much higher than that of the DFC and PC-DFC waveform, and the detection probability of the CPC-GADFC waveform is higher than that of the CPC-DFC waveform.
This paper proposes a CPC-GADFC joint coding holographic RF stealth radar waveform, based on the LFM waveform. The time-frequency domain and ambiguity function expressions of the joint coding waveform are derived. The LPI characteristic of the joint coding waveform is derived by the interception factor. In addition, the complexity of the algorithm is analyzed. Finally, a matched filter is designed for the joint coding waveform in order to achieve echo pulse compression. The experiments show that the CPC-GADFC joint coding waveform has the advantages of phase-coding waveform and frequency-coding waveform. This large time-bandwidth product waveform has better LPI and low sidelobe characteristics. Similarly, the CPC-GADFC joint coding waveform has better target detection performance and higher practical value.
Graph: Figure 1 Complementary Code Autocorrelation Function.
Graph: Figure 2 Pulse Alternate Transmission Mode.
Graph: Figure 3 DFC Coding and GADFC Coding Autocorrelation Function Comparison.
Graph: Figure 4 CPC-GADFC Joint Coding Radar Transmit Waveform, (a) a pulse group is alternately transmitted by positive code phase-coding and complementary code phase-coding, (b) the positive code phase-coding time domain waveform, (c) the complementary code phase-coding time domain waveform.
Graph: Figure 5 Time-Frequency Analysis of Joint Coding Waveform.
Graph: Figure 6 Design Process of CPC-GADFC Joint Coding Radar Waveform Based on LFM.
Graph: Figure 7 Spectrum comparison of waveform single pulse, (a) the LFM waveform spectrum, (b) the LFM-CPC waveform spectrum, (c) the CPC-GADFC waveform spectrum.
Graph: Figure 8 Ambiguity Function of the Joint coding Waveform.
Graph: Figure 9 Pulse Compression Model of Joint coding Waveform Based on LFM.
Graph: Figure 10 CPC -GADFC Time Domain Waveform.
Graph: Figure 11 Time -Frequency Analysis of CPC-GADFC Waveform.
Graph: Figure 12 LFM -DFC Waveform Pulse Compression.
Graph: Figure 13 CPC -DFC Waveform Pulse Compression.
Graph: Figure 14 CPC -GADFC Waveform Pulse Compression.
DIAGRAM: Figure 15 Comparison Diagram of Several Waveform Pulse Compressions and Target Detection Probabilities, (a) comparison of the waveform pulse compression, (b) enlarged view of the main lobe position of the waveform pulse compression, and (c) comparison of the waveform target detection performance. Note that "win" denotes the Hamming window.
Table 1 Correlation Coefficient Between Adjacent Transmitted Pulses.
Radar Waveform Correl. Coeff. DFC Waveform 0.558 GADFC Waveform 0.164
Table 2 Complementary Code, DFC Sequence and GADFC Sequence Codewords.
Coding Mode Codewords Positive Code 1, 1, 1, 1, 1, −1, 1, −1, −1, −1, 1, 1, 1, −1, −1, 1, 1, 1, −1, −1, 1, −1, −1, 1, −1, −1, −1, −1, 1, −1, 1, −1 Complement Code −1, −1, −1, −1, −1, 1, −1, 1, 1, 1, −1, −1, −1, 1, 1, −1, 1, 1, −1, −1, 1, −1, −1, 1, −1, −1, −1, −1, 1, −1, 1, −1 DFC 12, 13, 25, 15, 6, 10, 19, 9, 18, 16, 5, 22, 7, 30, 32, 1, 11, 24, 27, 17, 14, 31, 2, 4, 3, 20, 29, 23, 8, 21, 28, 26 GADFC 31.5455, 31.5152, 10.9091, 20.5455, 28, 9.2424, 31.9091, 5.9394, 20.4848, 10.6364, 18.7273, 17.7273, 9, 17.6667, 17.3939, 6, 12.3939, 11.2121, 5, 4.8788, 4.5152, 3.1212, 9.7576, 7.0606, 18.0303, 5.2424, 7.6364, 4.9394, 12.0606, 28.8181, 16.9394, 31.3636
Conceptualization, Y.S., B.T. and S.X.; methodology, Y.S. and B.T.; software, Y.S. and Y.W.; validation, Y.S., J.X., Y.Y. and B.T.; formal analysis, Y.S.; investigation, Y.S.; resources, Y.S. and B.T.; data curation, Y.S.; writing—original draft preparation, Y.S., J.X. and B.T.; writing—review and editing, Y.S.; visualization, Y.S.; supervision, B.T. and S.X.; project administration, B.T. and S.X.; funding acquisition, B.T. and S.X. All authors have read and agreed to the published version of the manuscript.
Not applicable.
The authors declare no conflict of interest.
The following abbreviations are used in this paper:
LPI Low Probability of Intercept UWB Ultra-Wide Band LFM Linear Frequency Modulation FSK Frequency Shift Keying PSK Phase Shift Keying CPC Complementary Phase-Coding DFC Discrete Frequency-Coding GA Genetic Algorithm GADFC Discrete Frequency-Coding Optimized by Genetic Algorithm
The authors acknowledge the funding, equipment and technical support provided by the School of Electronics and Communication Engineering of Sun Yat-sen University.
By Yuxiao Song; Yu Wang; Jingyang Xie; Yiming Yang; Biao Tian and Shiyou Xu
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