CN107677995B - OFDM waveform design method based on PMEPR-PSLR joint optimization - Google Patents

Abstract

The invention discloses an OFDM waveform design method based on PMEPR-PSLR joint optimization. Existing waveform designs for OFDM signals all consider the optimization of a single parameter, that is, only the peak-to-average envelope ratio (PMEPR) or the peak-to-side lobe level ratio (PSLR) is optimized. Considering that the modern electronic warfare system tends to be comprehensive and integrated, that is, the research on the integrated signal design of radar communication is becoming more and more mature, not only the radar detection performance of the signal, but also its application in communication should be considered. Therefore, the present invention considers using the convex optimization theory to construct an optimization problem with PMEPR as the objective function and PSLR and energy as constraints. First, using the relaxation method of SDP, the problem is simplified and related processing is carried out using the CVX toolbox. Next, the optimal solution is approximated by a randomization method. The feasibility of the design method can be further verified by simulation experiments, and the corresponding PMEPR and PSLR values have also been improved accordingly.

Description

An OFDM waveform design method based on PMEPR-PSLR joint optimization

technical field

The invention belongs to the field of radar communication integrated waveform design, and particularly relates to the field of OFDM (Orthogonal Frequency Division Multiplexing) signal modeling, optimization theory and radar detection.

Background technique

With the continuous development of modern science and technology, modern military warfare is changing in the direction of systematic confrontation. This requires the modern combat platform to be equipped with numerous electronic combat equipment to improve the survivability and comprehensive combat capability of the combat platform. That is, how to realize the horizontal integration of various systems and equipment has become a common concern of many research experts and scholars. Exploring and researching related technologies of comprehensive integrated electronic warfare system has become one of the key issues to be solved in modern warfare. OFDM signal is a typical communication signal with good anti-interference ability, high-speed transmission ability and easy implementation, so it is widely used in integrated signal design. But its biggest defect is that the signal envelope fluctuates greatly, which will cause the power amplifier to enter a saturation state in the waveform transmission to cause signal distortion, and may lead to interference problems between sub-carriers, so how to smooth the envelope of the OFDM signal is a key issue of the research.

At present, the concept of integrated design of radar communication has received extensive attention and attention at home and abroad, and the research on integrated waveform based on OFDM has been relatively complete. IEEE Radar Conference, Cincinnati, USA, 2014: 1424-1429.", the constraints of the OFDM signal spectrum and the optimization of the corresponding autocorrelation function are considered, and the optimization of the fluctuation of the signal envelope is not considered. In the document "Low PMEPROFDM radar waveform design using the iterative least squares algorithm. IEEE Signal Processing Letters, 22.11(2015): 1975-1979", a method to reduce the PMEPR (peak average envelope power) of OFDM signals based on subcarrier reservation is proposed. ratio) algorithm, which transforms the optimization problem into a least squares problem and uses an iterative algorithm to solve it, but only considers a single parameter optimization, and does not consider the suppression of peak side lobes.

SUMMARY OF THE INVENTION

The purpose of the present invention is to design an OFDM signal integrated with radar and communication to realize the peak-average envelope and peak value of the signal considering that the modern electronic warfare system tends to be comprehensive and integrated, that is, the research on the integrated signal design of radar communication is becoming more and more mature. The joint optimization of the side lobes improves the radar target detection performance, and the envelope of the smooth signal is easier to implement in hardware.

Solution of the present invention: First, the waveform optimization problem is designed. Consider using convex optimization theory to construct an optimization problem with PMEPR as the objective function and PSLR (peak sidelobe level ratio) and energy as constraints; use the relaxation method of SDP (positive semi-definite programming) to simplify the problem and use convex The optimization toolbox performs related processing; uses the randomization method to approximate the optimal solution. Next, use the designed waveform to measure the target distance and speed. The method effectively solves the joint optimization problem of PSLR and PMEPR, and the effectiveness of the algorithm is verified by relevant simulation experiments.

The present invention provides an OFDM waveform design method based on PMEPR-PSLR joint optimization, which comprises the following steps:

Step 1. Initialize system parameters

The initialized system parameters include: bandwidth B of OFDM signal, time width T, number of subcarriers N, subcarrier frequency interval Δf, sampling rate f _s , peak sidelobe constraint parameter γ, and energy constraint parameter P;

Step 2. Build a signal model

The discrete processing of the corresponding OFDM signal can be expressed as the following form:

故信号的矩阵表达形式为Among them: s[l] represents the lth sampling point of the discretized OFDM signal sequence, and a _n represents the weight corresponding to the corresponding nth subcarrier; rewrite the discrete sequence into a vector form, that is, let

Therefore, the matrix representation of the signal is

s=Fa (2)

表示的是傅里叶变化矩阵，

表示的是OFDM信号对应的权值向量/码字序列；信号的设计直接转换为对码字a的计算；in,

represents the Fourier transform matrix,

It represents the weight vector/codeword sequence corresponding to the OFDM signal; the design of the signal is directly converted into the calculation of the codeword a;

Step 3, PSLR matrix

Since the expression of the discretized PSL (peak sidelobe level) is,

Among them, the autocorrelation function R(m) after discretization is expressed as,

m represents the delay unit;

In order to derive the discretized autocorrelation function, a delay operator is introduced, which is

The above formula represents the expression of the vector x after m time delays; according to formula (5), the Fourier matrix is divided into columns, and F=[f ₀ , f ₁ ,...f _N-1 ] The expression after the delay is:

where m=0,1,2,...,N-1; the discrete form of the autocorrelation function of the signal is further obtained as,

则最终的自相关表达式以及PSLR为，make

Then the final autocorrelation expression and PSLR are,

Step 4. PMEPR matrixing

Denote s[ _l ] as sl, then the discretized PMEPR expression is,

E _l [|s _l | ² ] represents the mean of |s _l | ² ; it is obtained by dividing the rows of matrix F into blocks,

l＝1,2,…N；所以，可以得到，in,

l=1,2,...N; therefore, it can be obtained,

and,

The expression for PMEPR is obtained as,

Step 5. Create an optimization problem

所以可以建立以PMEPR为目标函数，以PSLR为约束的优化问题为，make

Therefore, the optimization problem with PMEPR as the objective function and PSLR as the constraint can be established as,

where ao _pt represents the optimal solution corresponding to the optimization problem, γ, P are given constants, m=1,2,...,N-1;

Step 6. SDP optimization algorithm based on randomization

Using the relaxation method of SDP to relax the above problem as,

where tr( ) represents the trace operation of the corresponding matrix,

Let A=aa ^H be a positive definite Hermitian matrix, simplify the above problem as,

The solution is solved by the convex optimization toolbox to obtain the code word a.

Further, the specific method of the step 6 is:

其中

表示随机变量的协方差矩阵；Step 6.1: Generate a random vector of length W

in

represents the covariance matrix of random variables;

判断该向量是否满足步骤6问题中的约束；如果不等式是满足的，那么将a^(w)记录下来，并存入空矩阵B中；如果不满足，则跳出本次循环；Step 6.2: Make

Determine whether the vector satisfies the constraints in the problem in step 6; if the inequality is satisfied, record a ^(w) and store it in the empty matrix B; if not, jump out of this loop;

Step 6.3: Let the column vector of matrix B be represented as b ^(q) , where q=1,2,...Q, Q represents the number of column vectors contained in matrix B, bring b ^(q) into the objective function to get

Step 6.4: Calculate the optimal q ^* ,

Step 6.5: The optimal vector can be obtained

The method of the invention solves the joint optimization problem of the signal peak-average envelope and the peak side lobes, improves the radar target detection performance, and at the same time smoothes the signal envelope. The randomization-based SDP algorithm is used to solve the optimization problem of PSLR and PMEPR, so as to obtain the optimal weight of the corresponding OFDM waveform. The advantage of the present invention is that the optimization of two performance indicators is jointly considered, the realization is relatively easy, and the feasibility of the algorithm for radar target detection is verified through simulation.

Description of drawings

FIG. 1 is a schematic diagram of the time domain-frequency domain of a basic OFDM pulse signal.

The abscissa represents the time dimension, and the ordinate represents the frequency dimension.

Figure 2 compares the envelope fluctuations of the random phase encoding sequence and the sequence optimized by the randomized SDP algorithm.

Figure 3 compares the improvement of PSL of signal autocorrelation under different optimization algorithms.

FIG. 4 is a ranging result based on time-domain correlation processing of OFDM signals.

Figure 5 shows the speed measurement results based on moving target detection.

Detailed ways

The present invention mainly adopts the method of computer simulation for verification, and all steps and conclusions are verified correctly on MATLAB-R2014a. The specific implementation steps are as follows:

Step 1. Initialize system parameters

为20dB。Assume that the bandwidth of the OFDM signal is B=100MHZ, the time width is T=2.075μs, the number of subcarriers is N=128, the subcarrier frequency interval Δf=B/N, the number of sampling points is K=N=128, and the sampling rate f _s =N/ T, the pulse accumulation number is 10 ⁴ , and the corresponding randomization vector number is 10 ⁵ . Assuming that the distance and velocity information of the two targets are (100m, 3m/s), (200m, 4m/s), the signal-to-noise ratio is

is 20dB.

Step 2, signal model establishment

表示的是傅里叶变化矩阵。The corresponding OFDM signal is discretely processed, and its matrix expression s=Fa is obtained, where a represents the weight vector to be optimized,

represents the Fourier transform matrix.

Step 3. Optimization problem construction

其中R_m，R₀以及α_l均为已知的。以PMEPR为目标函数，PSLR为约束建立优化问题，利用随机化SDP算法进行求解。According to the above, the discrete expressions of PSLR and PMEPR can be further obtained, which are respectively

where R _m , R ₀ and α _l are all known. The optimization problem is established with PMEPR as the objective function and PSLR as the constraint, and the randomized SDP algorithm is used to solve it.

Step 4. Solve the optimization problem

进一步可以求解出最优的OFDM序列。According to the PMEPR-PSLR joint optimization problem designed in step 3, the SDP algorithm is first used to relax it into a convex optimization problem, and the CVX optimization toolbox is used to solve the optimal matrix. Then, use the randomization method to decompose the matrix into rank one, construct and solve the optimal weight vector that satisfies the constraints

Further, the optimal OFDM sequence can be solved.

Step 5: Analyze and study the peak side lobes and the peak-to-average envelope ratio of the signal autocorrelation function after optimization.

Step 6: The optimized signal is transmitted through a Gaussian white noise channel for moving target detection.

的秩为1，则可以通过等式

直接解算出最优的权向量

其中λ为对应的特征值，是一个常数。如果解得的

的秩是大于1的，那么此时秩一的分解是不适用的。所以要利用随机化的方法进行解的逼近，因为要将问题中的约束考虑到解逼近的过程，所以具体步骤如下所述：According to the above algorithm, the optimal matrix can be solved. At this time, it is analyzed whether the matrix satisfies the rank one constraint. If solved

is of rank 1, then the equation

Directly solve the optimal weight vector

where λ is the corresponding eigenvalue and is a constant. If solved

The rank of is greater than 1, then the decomposition of rank one is not applicable at this time. Therefore, the randomization method should be used to approximate the solution, because the constraints in the problem should be considered in the process of solution approximation, so the specific steps are as follows:

Step 7. Radar performance analysis

In the radar receiving part, the principle of matched filtering is used to study the autocorrelation characteristics of the signal, and the peak sidelobe level of the signal and the envelope fluctuation of the signal are further verified; the time domain correlation processing and MTD (moving target detection) are used to detect the moving target. The initial position and movement speed are detected.

The three curves in Figure 2 represent the PMEPR-based optimization, the PMEPR-PSLR-based joint optimization, and the random phase encoding sequence. The corresponding PMEPR values are: 3.7672, 4.9584, 6.6496. It can be seen that although the smooth level of the envelope decreases after adding the autocorrelation constraint, the effectiveness of the optimization algorithm in the suppression of envelope fluctuations can be obtained by comparing with the random phase encoding sequence.

In Figure 3, comparing the two different optimization situations, it can be clearly seen that the peak sidelobe of the signal autocorrelation under the joint optimization is improved compared with the former, about 4dB lower, which further verifies the effectiveness of the algorithm.

Figure 4 and Figure 5 mainly reflect the radar detection performance of the OFDM waveform. Only considering the influence of white noise on radar detection, through the analysis of the principle of OFDM signal ranging and speed measurement, the distance and speed can be measured more accurately, in which the target distance measurement error is 4.5%, 2.75%, and the target speed measurement error Both are 3.1%. The validity of the OFDM signal based on PMEPR-PSLR joint optimization in radar detection is verified through the above, which lays a certain foundation for the further research of the radar communication integrated system.

It can be seen from the specific implementation of the present invention that the present invention realizes the joint optimization of the peak-average envelope and the peak side lobes of the signal by designing an OFDM signal integrated with radar communication, improves the radar target detection performance, smoothes the signal envelope, and suppresses the signal. Distortion and interference between sub-carriers ensure a strict orthogonal relationship between sub-carriers. The implementation of the present invention also provides a possibility for the design of radar communication integrated signals.

Claims (2)

1. An OFDM waveform design method based on PMEPR-PSLR joint optimization, which comprises the following steps: Step 1. Initialize system parameters The initialized system parameters include: bandwidth B of OFDM signal, time width T, number of subcarriers N, subcarrier frequency interval Δf, sampling rate f _s , peak sidelobe constraint parameter γ, and energy constraint parameter P; Step 2. Build a signal model The discrete processing of the corresponding OFDM signal can be expressed as the following form:

Among them: s[l] represents the lth sampling point of the discretized OFDM signal sequence, and a _n represents the weight corresponding to the corresponding nth subcarrier; rewrite the discrete sequence into a vector form, that is, let

Therefore, the matrix representation of the signal is s=Fa (2) in,

represents the Fourier transform matrix,

It represents the codeword sequence corresponding to the OFDM signal; the design of the signal is directly converted into the calculation of the codeword sequence a; Step 3, PSLR matrix Since the expression of the discretized PSL is,

Among them, the autocorrelation function R(m) after discretization is expressed as,

m represents the delay unit; In order to derive the discretized autocorrelation function, a delay operator is introduced, which is

The above formula represents the expression of the vector x after m time delays; according to formula (5), the Fourier matrix is divided into columns, and F=[f ₀ , f ₁ ,...f _N-1 ] The expression after the delay is:

where m=0,1,2,...,N-1; the discrete form of the autocorrelation function of the signal is further obtained as,

make

Then the final autocorrelation expression and PSLR are,

Step 4. PMEPR matrixing Denote s[ _l ] as sl, then the discretized PMEPR expression is,

E _l [|s _l | ² ] represents the mean of |s _l | ² ; it is obtained by dividing the rows of matrix F into blocks,

in,

So, you can get,

and,

The expression for PMEPR is obtained as,

Step 5. Create an optimization problem make

Therefore, the optimization problem with PMEPR as the objective function and PSLR as the constraint can be established as,

where a _opt represents the optimal solution corresponding to the optimization problem, γ, P are given constants, m=1,2,...,N-1; Step 6. SDP optimization algorithm based on randomization Using the relaxation method of SDP to relax the above problem as,

where tr( ) represents the trace operation of the corresponding matrix, Let A=aa ^H be a positive definite Hermitian matrix, simplify the above problem as,

The solution is solved by the convex optimization toolbox, and the codeword sequence a is obtained. 2. a kind of OFDM waveform design method based on PMEPR-PSLR joint optimization as claimed in claim 1 is characterized in that the concrete method of described step 6 is: Step 6.1: Generate a random vector of length W

in

represents the covariance matrix of random variables; Step 6.2: Make

Determine whether the vector satisfies the constraints in the problem in step 6; if the inequality is satisfied, record a ^(w) and store it in the empty matrix B; if not, jump out of this loop; Step 6.3: Let the column vector of matrix B be represented as b ^(q) , where q=1,2,...Q, Q represents the number of column vectors contained in matrix B, bring b ^(q) into the objective function to get

Step 6.4: Calculate the optimal q ^* ,

Step 6.5: The optimal vector can be obtained

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