Low sidelobe constant modulus phase coding sequence set design method

Abstract

The invention discloses a low sidelobe constant modulus phase coding sequence set design method. The method comprises the following steps: S1, carrying out random initialization on a constant modulus phase coding sequence set according to an expected signal parameter; s2, constructing a minimum sidelobe horizontal objective function based on a p norm and performing simplification; s3, deriving the gradient of the target function relative to the constant modulus phase coding sequence set; and S4, solving the minimization problem of the target function by using a conjugate gradient descent algorithm to obtain a low-sidelobe constant-modulus phase coding sequence set. According to the constant modulus sequence set design method disclosed by the invention, the design of a low-sidelobe single waveform and a multi-orthogonal waveform can be realized at the same time, and by changing the value of p, the method can meet the requirements of integral sidelobe level minimization and peak sidelobe level minimization sequence set design at the same time. The method provided by the invention not only can obtain a very low related sidelobe level, but also has very high calculation efficiency, and has a wide application prospect in pulse system radars.

Description

Technical Field

[0001] The invention relates to a low sidelobe constant modulus phase coding sequence set design method, and belongs to the field of radar signal processing. Background Art

[0002] The waveform design of pulse compression radar has been studied for decades and plays a very important role in controlling radar performance. Generally speaking, radar systems use matched filters to improve higher range resolution and better signal-to-noise ratio. Therefore, for the waveform after pulse compression, the sidelobe level of its correlation function is a key criterion in waveform design. A lower correlation sidelobe level can not only avoid the situation where the main lobe of a weak target is submerged by the side lobe of a strong target as much as possible, but also help to reduce the impact of fuzzy energy caused by non-complete orthogonality between multiple transmitted waveforms.

[0003] In the early years, designing constant modulus sequences with low autocorrelation sidelobes was the main research direction of researchers, and with the continuous deepening of research, a large number of advanced methods have emerged. Among them, some researchers have constructed two-phase and multi-phase sequences with good autocorrelation characteristics based on heuristic search methods. In addition, the cyclic algorithm has also been used to reduce the autocorrelation sidelobes of constant modulus sequences and achieved excellent performance. Compared with directly solving the problem of minimizing autocorrelation sidelobes, some researchers use algorithms such as the alternating direction multiplier method (ADMM), coordinate descent method (CD) and MM to transform a complex problem into a series of sub-problems that are easy to solve, thereby obtaining a constant modulus phase coded sequence with very low autocorrelation sidelobe levels.

[0004] On the other hand, in addition to the above-mentioned constant modulus sequence design method for minimizing autocorrelation sidelobes, some practical applications also require the use of orthogonal sequences to improve the performance of radar systems. For example, a set of sequences with high orthogonality can utilize the characteristics of multiple-input multiple-output (MIMO) to obtain more information as much as possible without crosstalk between sequences. In addition, orthogonal waveforms can be used in waveform agile radar systems to achieve waveform agility, which is crucial to improving anti-interference and detection capabilities. Generally, the acquisition of orthogonality can be divided into three main methods: time domain multiplexing, frequency domain multiplexing, and code domain multiplexing. Among them, the orthogonal waveform obtained by the last scheme not only has the flexibility of the entire pulse width, an advantage that the time domain multiplexing method does not have, but also works in the same frequency band, which is exactly what is needed in a crowded spectrum. However, this code domain method also faces a thorny problem, that is, how to establish an optimization model and design a corresponding optimization algorithm to obtain a sequence set with desired correlation characteristics. In addition, since the power amplifier in the radar system usually works in a saturated state, it is necessary to ensure that the sequence set has a constant amplitude.

[0005] In order to solve the above problems, researchers have made a lot of contributions over the years and proposed many excellent constant modulus sequence set design methods to minimize the correlation sidelobe level. Among them, algorithms such as CAN, ISL-New and MM-Corr use loop iteration and MM framework to obtain constant modulus sequence sets with an integrated sidelobe ratio (ISLR) close to the theoretical limit, but their peak sidelobe ratio (PSLR) is high. Therefore, some constant modulus sequence set design methods based on PSL minimization have been proposed to obtain lower PSLR. However, since the PSL minimization problem is more difficult to solve than the ISL minimization problem, most of these methods have high computational complexity, resulting in low computational efficiency, which limits their real-time application in radar systems. Summary of the invention

[0006] Purpose of the invention: Based on the p-norm, the present invention proposes a method for designing a low sidelobe constant modulus phase coding sequence set, which can not only obtain a very low correlation sidelobe level, but also greatly improve the calculation efficiency.

[0007] Technical solution: A method for designing a low sidelobe constant modulus phase coding sequence set, characterized by comprising the following steps:

[0008] S1: Obtain a constant modulus phase coding sequence set according to the expected signal parameters;

[0009] S2: Construct the objective function of minimizing the sidelobe level based on the p-norm and simplify the objective function;

[0010] S3: derive the gradient of the objective function with respect to the set of constant modulus phase coded sequences;

[0011] S4: Use the conjugate gradient descent algorithm to solve the minimization problem of the objective function and obtain a set of low sidelobe constant modulus phase coding sequences.

[0012] Preferably, the implementation process of step S1 is: generating a constant modulus phase coding sequence set according to the expected sequence length V and the number of sequence groups l, wherein the phase coding is presented in the form of a polyphase code with a value between 0 and 2π.

[0013] Preferably, in step S2, firstly, an objective function for minimizing the sidelobe level is established based on the p-norm as follows:

[0014]

[0015] In the formula, V and l represent the expected sequence length and the number of sequence groups, respectively, i and ζ represent variables, and r ii (ζ) and r ij (ζ) are the autocorrelation and cross-correlation functions of the constant modulus phase coding sequence set, respectively, and the parameter p ranges from 2 to positive infinity;

[0016] When p = 2, the objective function of minimizing the sidelobe level becomes the ISL criterion of the integrated sidelobe level. When p approaches positive infinity, the minimization of the sidelobe level converges to the PSL criterion of the peak sidelobe level. Therefore, the balance between ISL and PSL is achieved by changing the value of p.

[0017] Based on the matrix change, the objective function of minimizing the sidelobe level is changed to the following form:

[0018]

[0019] Where U is the constant modulus phase coding sequence set, T ζ is the transformation matrix, which is expressed as:

[0020]

[0021] Where, T -ζ Represents the transfer matrix when the delay is negative, and the objective function of minimizing the sidelobe level is further simplified as follows:

[0022]

[0023] In the formula, F and F H Respectively represent Fourier transform and inverse Fourier transform, R ζ =U H T ζ U is the correlation function of the constant modulus phase coding sequence set, is the kth phase coding sequence after zero padding. In addition,

[0024] R ζ =(U H T ζ U) T ,r ij =[r ij (0),...,r ij (V-1)] T ,

[0025] P=[I l×l ,...,I l×l (k),...,I l×l (l-1)],Q=[I l×l ,I l×l ,...,I l×l ]

[0026] Among them I l×l (k) indicates I l×l Circularly shift k units along the row, P and Q represent matrices.

[0027] Preferably, in step S3, the gradient solution factor is first established as follows:

[0028]

[0029] Where ψ is the polyphase coding matrix. The gradient solution factor is used to further simplify the minimization of the sidelobe level objective function, and the minimization of the sidelobe level objective function relative to the polyphase coding matrix gradient is obtained as follows:

[0030]

[0031] In the formula, Indicates the virtual operation of the matrix, M1, M2, Z, C, and B are all matrices:

[0032]

[0033] Preferably, the implementation process of step S4 is as follows: after determining the value of p and the preset parameters in the conjugate gradient descent algorithm, firstly solve the minimized sidelobe level objective function and its gradient size of the constant modulus phase coding sequence set in S1 according to steps S2 and S3 respectively, then update the descent direction of the minimized sidelobe level objective function according to the calculated gradient and the preset parameters in the conjugate gradient descent algorithm, and select a suitable descent step size based on the minimized sidelobe level objective function and the updated gradient; finally, update the constant modulus phase coding sequence set, if the algorithm meets the preset conditions, stop the iteration and output the low sidelobe constant modulus phase coding sequence set, otherwise, repeat step S4 until the preset conditions are met and output the low sidelobe constant modulus phase coding sequence set.

[0034] Compared with the prior art, the beneficial effect of a low sidelobe constant modulus phase coding sequence set design method disclosed in the present invention is: the proposed scheme utilizes the Fourier relationship between the correlation function and the spectral density, and proposes a sidelobe optimization model based on the p-norm metric. The model converges to PSL minimization when p approaches infinity, and converges to ISL minimization when p=2, and can simultaneously meet the design requirements of PSL minimization and ISLR minimization constant modulus phase coding sequence sets. In addition, the model adopts fast Fourier transform (FFT) and inverse FFT (IFFT) operations to reduce the computational complexity. Compared with the current method for minimizing PSL constant modulus phase coding sequence sets, the proposed method can not only obtain a lower PSLR, but also has a higher computational efficiency. In addition, when the number of desired sequence sets is large, the proposed method can provide an ISLR that is almost the same as the ISL minimization method, while having a PSLR value much lower than it. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 is the convergence curve of the objective function under different p values;

[0036] Figure 2(a) is the convergence curve of PSLR under different p values;

[0037] Figure 2 (b) is the convergence curve of ISLR under different p values;

[0038] Figure 3 (a) is a comparison diagram of the autocorrelation function of waveform 1 between the proposed method and the comparative method when V = 128 and l = 2;

[0039] Figure 3 (b) is a comparison diagram of the cross-correlation functions of waveforms 1 and 2 between the proposed method and the comparative method when V = 128 and l = 2;

[0040] Figure 3 (c) is a comparison diagram of the waveform 2, 1 cross-correlation function of the proposed method and the comparative method when V = 128, l = 2;

[0041] Figure 3 (d) is a comparison diagram of the autocorrelation function of waveform 2 between the proposed method and the comparative method when V = 128 and l = 2;

[0042] Figure 4 It is an overall flow chart of the method proposed in the present invention. DETAILED DESCRIPTION

[0043] A method for designing a low sidelobe constant modulus phase coding sequence set proposed by the present invention is described in detail below in conjunction with the accompanying drawings.

[0044] Consider a radar system that needs to transmit l sets of orthogonal phase-coded sequence sets, where the length of each sequence is V and the expression of the sequence set is as follows:

[0045]

[0046] In the formula, g(t) is expressed as a matrix window function, T s and T p are the duration of the sub-pulse and the complete pulse, respectively, and both satisfy T p =NT s Therefore, the sequence u i =[u i (1),...,u i (N)] T and u j =[u j (1),...,u j (N)] T The expression of the non-periodic correlation function when the delay is ζ is as follows:

[0047]

[0048] Where, Tζ is the transformation matrix, and its expression is

[0049]

[0050] The PSL and ISL guidelines can be obtained as follows:

[0051]

[0052] Based on this, the objective function is defined using the p-norm as follows:

[0053]

[0054] In the formula, V and l represent the sequence length and the number of sequence sets respectively, r ii (ζ) and r ij (ζ) are the autocorrelation and cross-correlation functions of the sequence set, respectively, and the value of p ranges from 2 to positive infinity. In particular, when p = 2, the above formula becomes the integrated sidelobe level (ISL) criterion. When p approaches positive infinity, the above formula converges to the peak sidelobe level (PSL) criterion. Therefore, the balance between ISL and PSL can be achieved by changing the value of p. Based on the matrix change, U = [u1,…,u] is defined, and the target can be changed to the following form:

[0055]

[0056] Obviously, once the value of p is given, the constant term lV p The exponential term 2 / p has no effect on the minimization of the objective function. This is because when x+a p >0 (x+a p ) 2 / p is monotonic with respect to x, and here Υ p p / 2 There is no doubt that this condition is met. Therefore, the objective function can be further simplified as follows:

[0057]

[0058] In the formula, F and F H denote Fourier transform and inverse Fourier transform respectively, is the i-th phase coding sequence after zero padding. In addition,

[0059] R ζ =(U H T ζ U) T ,r ij =[r ij (0),...,r ij (V-1)] T ,

[0060] P=[I l×l ,I l×l (1),…,I l×l (l-1)],Q=[I l×l ,I l×l ,…,I l×l ]

[0061] Among them I l×l (i) by I l×l It is obtained by cyclically shifting i units along the row. So far, the design of low sidelobe constant modulus phase coding sequence set is equivalent to solving the following problem:

[0062]

[0063]

[0064] To use the gradient descent algorithm to solve the above problem, we need to find the gradient first. The specific steps are as follows. First, rewrite the objective function to get

[0065]

[0066] Afterwards, p The differential can be expressed as

[0067] d p =tr(C T dC+CdC T )=2tr(C T dC)

[0068] Based on the chain rule, we can get

[0069]

[0070] In the formula,

[0071]

[0072] Therefore, the gradient of the objective function with respect to the sequence set can be obtained as

[0073]

[0074] After obtaining the objective function size and its gradient relative to the constant modulus sequence set from the above formula, the conjugate gradient descent algorithm can be used to complete the sequence set search. The specific process is given in Table 1. First, the constant modulus phase coding sequence set needs to be randomly initialized according to the required sequence length (generally the time-bandwidth product) and the number of sequence sets to obtain ψ0, sequence

[0075] Table 1: Low sidelobe phase coding sequence set design algorithm

[0076]

[0077]

[0078] Each phase encoding value in the set is presented in the form of a polyphase code, with a size between 0 and 2π. Then, the objective function and gradient size of the initialized constant modulus phase coding sequence set are solved, and the decision update is made based on the calculated gradient and the preset parameters to obtain a faster descent direction of the objective function. In addition, a suitable descent step size is selected based on the objective function and the updated gradient to improve the convergence speed and operation efficiency of the algorithm. Finally, the constant modulus phase coding sequence set can be updated. If the algorithm converges, the iteration is stopped and the constant modulus sequence set is output. Otherwise, the above process needs to be repeated until the preset conditions are met. In this experiment, we choose the gradient descent algorithm parameters as follows: ρ = 10 -4 , c = 10 -2 , μ=0.9, β down =0.9,β up =1.01.

[0079] The proposed algorithm uses the p-norm to approximate the PSL criterion and combines the gradient descent algorithm to minimize the objective function, so it is named PGD-PSL here. In order to evaluate the sidelobe performance and computational efficiency of the proposed PGD-PSL algorithm, we conducted multiple sets of simulation experiments and compared the proposed scheme with the CAN, ISL-New and p-MM algorithms. In order to more accurately evaluate the performance of the sequence set, the numerical evaluation indicators and their corresponding theoretical lower limits are given as follows:

[0080] Table 2: Comparison of PSLR and ISLR results for different waveform design methods

[0081]

[0082]

[0083]

[0084] ISLR Bound =10log10(l(l-1))

[0085] Since the optimization model to be solved is based on the p-norm, it is instructive to explore the relationship between the objective function, PSLR and ISLR and the value of variable p. Here, the values ​​of p are selected as 2, 5, 10, 20, 50 and 100, and the preset stop condition is |Y p (Ψ k )-Y p (Ψ k-1 )| / |Yp (Ψ k-1 )|≤10 -9 , and fix the sequence length and the number of waveform sets to 256 and 2 respectively. Figure 1 First, the objective function changes with the number of iterations under the above p value. It can be seen that as the p value increases, the algorithm needs more iterations to converge. This is because the larger the p value, the closer the objective function is to the PSL criterion, and the PSL criterion is more difficult to solve than the ISL criterion. Even so, the optimization can still be completed within a few hundred iterations, which verifies the efficiency of the proposed algorithm. Figure 2 (a) and (b) show the variation curves of PSLR and ISLR under different p values. Interestingly, the value of PSLR does not decrease monotonically with the increase of the number of iterations.

[0086] Table 3: Comparison of operation time of different sequence set design methods (seconds)

[0087]

[0088] The optimization model with a larger p value is only an approximate version of PSL minimization, while when p = 2, the objective function is exactly the same as ISL, making the value of ISLR in a monotonically decreasing state during the iteration process. Therefore, the balance between PSLR and ISLR can be achieved by presetting an appropriate p value, which can be seen from Figure 2 This is verified in (a), that is, as the p value increases, the performance of PSLR is improved, while the ISLR of the related function becomes higher.

[0089] In order to evaluate the PSLR and ISLR performance of the proposed method under different sequence lengths and numbers of sequence sets, we set its p value to 100 and compare it with existing advanced methods, including CAN, ISL-New and p-MM algorithms, as shown in Table 2. Compared with the first two ISL minimization methods, a large p value can ensure a lower PSLR, but it will lose some ISLR performance. However, it can be found that as the number of sequence sets increases, the gap between PGD-PSL and other methods in ISLR gradually narrows, while still maintaining a very low PSLR. For example, when the number of sequence sets is 16, PGD-PSL can not only reduce the PSLR by about 6 dB, but also has almost the same ISLR as other methods. Therefore, when the number of expected sequence sets is large, the proposed PGD-PSL method is undoubtedly superior to the ISL minimization method, because it can not only bring lower PSLR, but also almost does not need to sacrifice ISLR performance. In addition, compared with the PSL minimization method (i.e., p-MM), the proposed method can obtain a lower PSLR and maintain a similar ISLR.

[0090] After verifying the excellent performance of the proposed PGD-PSL method in terms of correlation sidelobes, the operation time comparison of different methods is given in Table 3. It can be seen that when the number of expected sequence sets is small, the proposed method can even achieve higher computational efficiency than the ISL minimization method, and is much faster than the p-MM algorithm under any parameters, which verifies the high efficiency of the proposed algorithm and can meet the real-time application requirements of the radar system.

[0091] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.

Claims

1. A method for designing a low sidelobe constant modulus phase coding sequence set, characterized by: The following steps are involved: S1: Obtain a constant modulus phase coding sequence set according to the expected signal parameters; S2: Construct the objective function of minimizing the sidelobe level based on the p-norm and simplify the objective function; S3: derive the gradient of the objective function with respect to the set of constant modulus phase coded sequences; S4: Use the conjugate gradient descent algorithm to solve the minimization problem of the objective function and obtain a set of low sidelobe constant modulus phase coding sequences.

2. A method for designing a low sidelobe constant modulus phase coding sequence set as claimed in claim 1, characterized in that: The implementation process of step S1 is: generating a constant modulus phase coding sequence set according to the expected sequence length V and the number of sequence groups l, wherein the phase coding is presented in the form of a polyphase code with a value between 0 and 2π.

3. A method for designing a low sidelobe constant modulus phase coding sequence set as claimed in claim 1, characterized in that: In step S2, firstly, the objective function of minimizing the sidelobe level is established based on the p-norm as follows: In the formula, V and l represent the expected sequence length and the number of sequence groups, respectively, i and ζ represent variables, and r ii (ζ) and r ij (ζ) are the autocorrelation and cross-correlation functions of the constant modulus phase coding sequence set, respectively, and the parameter p ranges from 2 to positive infinity; When p = 2, the objective function of minimizing the sidelobe level becomes the ISL criterion of the integrated sidelobe level. When p approaches positive infinity, the minimization of the sidelobe level converges to the PSL criterion of the peak sidelobe level. Therefore, the balance between ISL and PSL is achieved by changing the value of p. Based on the matrix change, the objective function of minimizing the sidelobe level is changed to the following form: Where U is the constant modulus phase coding sequence set, T ζ is the transformation matrix, which is expressed as: Where, T -ζ Represents the transfer matrix when the delay is negative, and the objective function of minimizing the sidelobe level is further simplified as follows: In the formula, F and F H Respectively represent Fourier transform and inverse Fourier transform, R ζ =U H T ζ U is the correlation function of the constant modulus phase coding sequence set, is the kth phase coding sequence after zero padding. In addition, R ζ =(U H T ζ U) T ,r ij =[r ij (0),...,r ij (V-1)m T , P=[I l×l ,...,I l×l (k),...,I l×l (l-1)],Q=[I l×l ,I l×l ,...,I l×l ] Among them I l×l (k) indicates I l×l Circularly shift k units along the row, P and Q represent matrices.

4. A method for designing a low sidelobe constant modulus phase coding sequence set as claimed in claim 3, characterized in that: In step S3, the gradient solution factor is first established as follows: Where ψ is the polyphase coding matrix. The gradient solution factor is used to further simplify the minimization of the sidelobe level objective function, and the minimization of the sidelobe level objective function relative to the polyphase coding matrix gradient is obtained as follows: In the formula, Indicates the virtual operation of the matrix, M1, M2, Z, C, and B are all matrices:

5. A method for designing a low sidelobe constant modulus phase coding sequence set as claimed in claim 4, characterized in that: The implementation process of step S4 is as follows: after determining the value of p and the preset parameters in the conjugate gradient descent algorithm, firstly solve the minimum sidelobe level objective function and its gradient size of the constant modulus phase coding sequence set in S1 according to steps S2 and S3 respectively, then update the descent direction of the minimum sidelobe level objective function according to the calculated gradient and the preset parameters in the conjugate gradient descent algorithm, and select a suitable descent step size based on the minimum sidelobe level objective function and the updated gradient; finally, update the constant modulus phase coding sequence set, if the algorithm meets the preset conditions, stop the iteration and output the low sidelobe constant modulus phase coding sequence set, otherwise, repeat step S4 until the preset conditions are met and output the low sidelobe constant modulus phase coding sequence set.

Images