CN107991655B - An LFM-PC Signal and Its Fuzzy Function Optimization Method - Google Patents

An LFM-PC Signal and Its Fuzzy Function Optimization Method Download PDF

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CN107991655B
CN107991655B CN201711314502.1A CN201711314502A CN107991655B CN 107991655 B CN107991655 B CN 107991655B CN 201711314502 A CN201711314502 A CN 201711314502A CN 107991655 B CN107991655 B CN 107991655B
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CN107991655A (en
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张劲东
陈婉迎
张超
徐乃清
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Nanjing University of Aeronautics and Astronautics
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
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    • G01S13/89—Radar or analogous systems specially adapted for specific applications for mapping or imaging
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Abstract

本发明公开了一种LFM‑PC信号及其模糊函数优化方法，基于极小化峰值旁瓣电平准则建立优化模型，采用序列二次规划法优化模糊函数，最后通过仿真，验证了该方法的可行性。本发明采用一组正交的相位编码信号作为雷达发射信号，提高SAR抗干扰性能；与LFM相比，新信号具有更好的抗干扰性能，与相位编码信号相比，具有更大的多普勒容限；采用极小化峰值旁瓣电平准则，应用序列二次规划法优化模糊函数，从而增大信号的多普勒容限。

Figure 201711314502

The invention discloses an LFM-PC signal and a fuzzy function optimization method thereof. An optimization model is established based on the minimization peak side lobe level criterion, and a sequence quadratic programming method is used to optimize the fuzzy function. Finally, simulation is carried out to verify the effectiveness of the method. feasibility. The present invention adopts a set of orthogonal phase-encoded signals as radar transmit signals to improve SAR anti-jamming performance; compared with LFM, the new signal has better anti-jamming performance, and has greater Doppler performance compared with phase-encoded signals Tolerance; using the minimization of the peak side lobe level criterion, applying the sequential quadratic programming method to optimize the fuzzy function, thereby increasing the Doppler tolerance of the signal.

Figure 201711314502

Description

LFM-PC signal and fuzzy function optimization method thereof
Technical Field
The invention relates to the technical field of waveform optimization design, in particular to a linear frequency modulation and phase coding composite modulation LFM-PC signal and a fuzzy function optimization method thereof.
Background
The LFM signal is easy to generate and process and is not sensitive to doppler, which makes it the most commonly used radar transmission signal. After decades of research and development, the technology of SAR imaging based on LFM signals has become mature and sophisticated. However, the LFM signal has poor anti-interference performance, and cannot achieve good imaging effect today with increasingly strong electronic countermeasure.
A group of orthogonal phase coding signals are adopted as radar emission signals, and the anti-interference performance of the SAR can be improved. The phase encoded signal has a pin-shaped blur function and therefore has good autocorrelation properties, but is also sensitive to doppler, and the dominant-to-sidelobe ratio of the filter output signal decreases rapidly as soon as the echo signal does not match the filter.
At present, many researches are carried out on optimizing fuzzy functions, but the obtained research results are not many. An optimization method based on minimization of the integrated sidelobe energy can be adopted, however, the optimized waveform may have higher peak sidelobe because the objective function is the integrated sidelobe energy. Or optimizing a cross-blur function, and obtaining a set of optimized waveforms and filters by minimizing the difference between the cross-blur function and the desired blur function.
Disclosure of Invention
The invention aims to solve the technical problem of providing an LFM-PC signal and a fuzzy function optimization method thereof, which can improve the anti-interference performance of SAR and increase the Doppler tolerance of the signal.
In order to solve the above technical problem, the present invention provides an LFM-PC signal and a method for optimizing its fuzzy function, comprising the steps of:
(1) the LFM-PC signal is obtained by modulating an LFM signal by a phase coding signal, and the two signals are directly multiplied in a time domain; the ambiguity function χ of the LFM-PC signal is obtained according to the definition of the ambiguity functionu(τ, ξ) is
Figure BDA0001503561620000011
Where τ is time, ξ is Doppler frequency, u isPC(t) is a phase encoded signal, K ═ BL/TpIs the chirp rate, BL、TpBandwidth and time width of the LFM signal, respectively; p is the code length and is the code length,
Figure BDA0001503561620000012
is the value of the n-th sub-pulse,
Figure BDA0001503561620000021
is a rectangular pulse pk(t) and pl(t) may be expressed as:
Figure BDA0001503561620000022
in the formula (I), the compound is shown in the specification,
Figure BDA0001503561620000023
tbis the sub-pulse width, and Tp＝Ptb，
Figure BDA0001503561620000024
(2) Introducing vectors
Figure BDA0001503561620000025
And a subpulse cross-ambiguity function matrix H, where ψ is a vector of 1 XP dimension and the phase set of the phase encoded signal
Figure BDA0001503561620000026
One-to-one correspondence, each element in ψ has a value range of [0,2 π ].
Figure BDA0001503561620000027
The blur function can be expressed in the form of vector multiplication according to equations (1) and (3):
χu(τ,ξ)＝|SHH(τ,ξ)S|2 (4)
(3) and (3) performing waveform optimization by using a sequence quadratic programming method, and increasing the Doppler tolerance of the LFM-PC signal.
Preferably, in the step (3), the waveform optimization by using the sequential quadratic programming method specifically includes the following steps: (31) defining a normalized fuzzy function;
Figure BDA0001503561620000028
in the formula, τξxi/K is the position of the main peak of the autocorrelation function at the Doppler frequency xi, | SHH(τξξ) S | is the dominant peak of the autocorrelation function at the Doppler frequency ξ, which for a given Doppler frequency ξ, | SHH(τξXi) S | is oneA constant;
(32) the above problem can be regarded as a constrained nonlinear programming problem, and the following objective function is established:
Figure BDA0001503561620000031
wherein, IΩFor the paravalvular region of pulse pressure, let's assume0Is the width of the main lobe, then IΩIn the range of
Figure BDA0001503561620000032
(33) Introducing variables, t further converting equation (6) into an inequality constrained nonlinear programming problem:
Figure BDA0001503561620000033
in the formula, t is both an objective function and a variable, and the physical meaning of t is the upper bound of the peak value side lobe ratio of the normalized fuzzy function; (34) the nonlinear programming problem of the inequality constraint can be solved by adopting a sequential quadratic programming method, and the optimization problem can be solved by directly adopting an fmincon function in an MATLAB optimization tool.
Preferably, the feasibility of the optimization problem in step (34) that can be solved by the sequential quadratic programming method is demonstrated as follows: since the denominator of F (τ, ξ) is a constant for a given doppler frequency ξ, it is only necessary to consider the quadratic differentiability of the numerator to make γ (τ, ξ) ═ SHH (t, ξ) S, then
Figure BDA0001503561620000034
Wherein
Figure BDA0001503561620000035
As an hadamard product;
Figure BDA0001503561620000036
wherein
Figure BDA0001503561620000037
Equation (9) can be simplified as:
Figure BDA0001503561620000041
therefore, the objective function and the constraint in the formula (7) both satisfy quadratic continuous differentiability, and can be solved by adopting a sequential quadratic programming method.
The invention has the beneficial effects that: according to the invention, a group of orthogonal phase coding signals are used as radar emission signals, so that the anti-interference performance of the SAR is improved; compared with the LFM, the new signal has better anti-interference performance and has larger Doppler tolerance compared with the phase coding signal; and optimizing a fuzzy function by adopting a minimization peak sidelobe level criterion and applying a sequence quadratic programming method, thereby increasing the Doppler tolerance of the signal.
Drawings
FIG. 1 is a schematic view of the process of the present invention.
FIG. 2(a) is a schematic diagram of the ambiguity of the LFM-PC signal before optimization according to the present invention.
FIG. 2(b) is a schematic diagram of the ambiguity of the LFM-PC signal after optimization according to the present invention.
Fig. 2(c) shows the pattern of different doppler slices of the LFM-PC signal before optimization according to the present invention as a function of the doppler value.
Fig. 2(d) shows the change of the pattern of the LFM-PC signal with the doppler value after the optimization.
Detailed Description
As shown in fig. 1, an LFM-PC signal and its fuzzy function optimization method includes the following steps:
(1) the LFM-PC signal is obtained by modulating an LFM signal by a phase coding signal, and the two signals are directly multiplied in a time domain; the ambiguity function χ of the LFM-PC signal is obtained according to the definition of the ambiguity functionu(τ, ξ) is
Figure BDA0001503561620000042
Where τ is time, ξ is Doppler frequency, u isPC(t) is a phase encoded signal, K ═ BL/TpIs the chirp rate, BL、TpBandwidth and time width of the LFM signal, respectively; p is the code length and is the code length,
Figure BDA0001503561620000043
is the value of the n-th sub-pulse,
Figure BDA0001503561620000044
is a rectangular pulse pk(t) and pl(t) may be expressed as:
Figure BDA0001503561620000051
in the formula (I), the compound is shown in the specification,
Figure BDA0001503561620000052
tbis the sub-pulse width, and Tp＝Ptb，
Figure BDA0001503561620000053
(2) Introducing vectors
Figure BDA0001503561620000054
And a subpulse cross-ambiguity function matrix H, where ψ is a vector of 1 XP dimension and the phase set of the phase encoded signal
Figure BDA0001503561620000055
One-to-one correspondence, each element in ψ has a value range of [0,2 π ].
Figure BDA0001503561620000056
The blur function can be expressed in the form of vector multiplication according to equations (1) and (3):
χu(τ,ξ)＝|SHH(τ,ξ)S|2 (4)
(3) and (3) performing waveform optimization by using a sequence quadratic programming method, and increasing the Doppler tolerance of the LFM-PC signal.
The simulation data of this embodiment is set as follows: the time width of the signal is 40 mus, the bandwidth of the LFM signal is 20MHz, and the modulation frequency is 5 multiplied by 1011Hz/s, a sampling frequency of 40MHz, a code length of the phase encoded signal of 160, and a symbol width t of the phase encoded signalb＝Tp/P, assuming the Doppler range to be optimized is (-B)L/P,BL/P) according to a speed resolution of 0.5/TpAnd (6) sampling.
Referring to fig. 1, an LFM-PC signal and its fuzzy function optimization method includes the following steps:
step 1: the LFM-PC signal is obtained by modulating the LFM signal by the phase coding signal, and the two signals are directly multiplied in a time domain. The ambiguity function χ of the LFM-PC signal is obtained according to the definition of the ambiguity functionu(τ, ξ) is
Figure BDA0001503561620000061
Where τ is time, ξ is Doppler frequency, u isPC(t) is a phase encoded signal with chirp rate K ═ BL/Tp＝5×1011The bandwidth and the time width of the Hz/s and LFM signals are respectively BL＝20MHz、Tp40 μ s; the code length P is 160 which is,
Figure BDA0001503561620000062
is the value of the n-th sub-pulse,
Figure BDA0001503561620000063
is a rectangular pulse pk(t) and pl(t) may be expressed as:
Figure BDA0001503561620000064
in the formula (I), the compound is shown in the specification,
Figure BDA0001503561620000065
sub-pulse width tb0.25. mu.s, and Tp＝Ptb＝40μs，
Figure BDA0001503561620000066
Step 2: introducing vectors
Figure BDA0001503561620000067
And a subpulse cross-ambiguity function matrix H, where ψ is a vector of 1 XP dimension and the phase set of the phase encoded signal
Figure BDA0001503561620000068
One-to-one correspondence, each element in ψ has a value range of [0,2 π ].
Figure BDA0001503561620000069
The blur function can be expressed in the form of vector multiplication according to equations (1) and (3):
χu(τ,ξ)＝|SHH(τ,ξ)S|2 (4)
and step 3: the method for optimizing the waveform by using the sequence quadratic programming method to increase the Doppler tolerance of the LFM-PC signal comprises the following specific steps:
step 3-1: defining a normalized blur function
Figure BDA0001503561620000071
In the formula, τξxi/K is the position of the main peak of the autocorrelation function at the Doppler frequency xi, | SHH(τξξ) S | is the dominant peak of the autocorrelation function at the doppler frequency ξ. For a given Doppler frequencyThe ratio xi, | SHH(τξξ) S | is a constant.
Step 3-2: the above problem can be regarded as a constrained nonlinear programming problem, and the following objective function is established
Figure BDA0001503561620000072
Wherein, IΩThe paravalvular region of pulse pressure. Let τ be0Is the width of the main lobe, then IΩIn the range of
Figure BDA0001503561620000073
Figure BDA0001503561620000074
Step 3-3: introducing variables, t further converting equation (6) into an inequality constrained nonlinear programming problem:
Figure BDA0001503561620000075
where t is both the objective function and the variable, its physical meaning is the upper bound of the peak-to-side lobe ratio of the normalized blur function.
Step 3-4: the nonlinear programming problem of the inequality constraint can be solved by adopting a sequential quadratic programming method, and the optimization problem can be solved by directly adopting an fmincon function in an MATLAB optimization tool.
The feasibility that the optimization problem in the step 3-4 can be solved by adopting a sequence quadratic programming method is proved as follows: since the denominator of F (τ, ξ) is a constant for a given doppler frequency ξ, it is only necessary to consider the quadratic differentiability of the numerator to make γ (τ, ξ) ═ SHH (t, ξ) S, then
Figure BDA0001503561620000076
Wherein
Figure BDA0001503561620000077
As an hadamard product;
Figure BDA0001503561620000081
wherein
Figure BDA0001503561620000082
Equation (9) can be simplified as:
Figure BDA0001503561620000083
therefore, the objective function and the constraint in the formula (7) both satisfy quadratic continuous differentiability, and can be solved by adopting a sequential quadratic programming method.
FIG. 2(a) is an ambiguity plot of the LFM-PC signal before optimization; FIG. 2(b) is an ambiguity diagram of the LFM-PC signal after optimization; FIG. 2(c) shows the shape of the different Doppler frequency cuts | χ (τ, ξ) | before optimization; fig. 2(d) shows the shape of the different doppler frequency cuts χ (τ, ξ) after optimization.
While the invention has been shown and described with respect to the preferred embodiments, it will be understood by those skilled in the art that various changes and modifications may be made without departing from the scope of the invention as defined in the following claims.

Claims (2)

1.一种LFM-PC信号及其模糊函数优化方法，其特征在于，包括如下步骤：1. a LFM-PC signal and fuzzy function optimization method thereof, is characterized in that, comprises the steps: (1)LFM-PC信号由相位编码信号调制LFM信号得到，在时域表现为两种信号直接相乘；根据模糊函数的定义得，LFM-PC信号的模糊函数χu(τ,ξ)为(1) The LFM-PC signal is obtained by modulating the LFM signal with the phase-encoded signal, and in the time domain, the two signals are directly multiplied; according to the definition of the ambiguity function, the ambiguity function χ u (τ,ξ) of the LFM-PC signal is
Figure FDA0003089613460000011
Figure FDA0003089613460000011
 其中，τ为时间，ξ为多普勒频率，uPC(t)为相位编码信号，K＝BL/Tp为调频斜率，BL、Tp分别为LFM信号的带宽和时宽；P为码长，
Figure FDA0003089613460000012
为第n个子脉冲的值，n＝k或1，
Figure FDA0003089613460000013
为矩形脉冲pk(t)和pl(t)的互模糊函数，表示为：Among them, τ is the time, ξ is the Doppler frequency, u PC (t) is the phase-encoded signal, K =BL /T p is the frequency modulation slope, BL and T p are the bandwidth and time width of the LFM signal, respectively; P is the code length,
Figure FDA0003089613460000012
is the value of the nth sub-pulse, n=k or 1,
Figure FDA0003089613460000013
is the mutual ambiguity function of the rectangular pulses p k (t) and p l (t), expressed as:
Figure FDA0003089613460000014
Figure FDA0003089613460000014
 式中，
Figure FDA0003089613460000015
tb是子脉冲宽度，且Tp＝Ptb，
Figure FDA0003089613460000016
In the formula,
Figure FDA0003089613460000015
t b is the sub-pulse width, and T p =Pt b ,
Figure FDA0003089613460000016
 (2)引入向量
Figure FDA0003089613460000017
和子脉冲互模糊函数矩阵H(τ,ξ)，其中，ψ是一个1×P维的向量，与相位编码信号的相位集
Figure FDA0003089613460000018
一一对应，ψ中每个元素的取值范围为[0,2π)；(2) Import vector
Figure FDA0003089613460000017
and the subpulse mutual ambiguity function matrix H(τ,ξ), where ψ is a 1×P-dimensional vector, and the phase set of the phase-encoded signal
Figure FDA0003089613460000018
One-to-one correspondence, the value range of each element in ψ is [0, 2π);
Figure FDA0003089613460000019
Figure FDA0003089613460000019
 根据式(1)和式(3)将模糊函数表示为向量相乘的形式：According to formula (1) and formula (3), the fuzzy function is expressed as the form of vector multiplication: χu(τ,ξ)＝|SHH(τ,ξ)S|2 (4)χ u (τ,ξ)＝|S H H(τ,ξ)S| 2 (4) (3)利用序列二次规划法进行波形优化，增大LFM-PC信号的多普勒容限；利用序列二次规划法进行波形优化具体包括如下步骤：(3) Using the sequential quadratic programming method to optimize the waveform to increase the Doppler tolerance of the LFM-PC signal; using the sequential quadratic programming method to optimize the waveform includes the following steps: (31)定义归一化的模糊函数；(31) define a normalized fuzzy function;
Figure FDA0003089613460000021
Figure FDA0003089613460000021
 式中，τξ＝ξ/K是多普勒频率ξ处自相关函数的主峰位置，|SHH(τξ,ξ)S|是多普勒频率ξ处自相关函数的主峰值，对于给定的多普勒频率ξ，|SHH(τξ,ξ)S|为一常数；where τ ξ =ξ/K is the main peak position of the autocorrelation function at the Doppler frequency ξ, |S H H(τ ξ ,ξ)S| is the main peak value of the autocorrelation function at the Doppler frequency ξ, for Given Doppler frequency ξ, |S H H(τ ξ ,ξ)S| is a constant; (32)建立如下目标函数：(32) Establish the following objective function:
Figure FDA0003089613460000022
Figure FDA0003089613460000022
 其中，IΩ为脉压旁瓣区，假设τ0是主瓣宽度，则IΩ的范围为
Figure FDA0003089613460000023
Among them, I Ω is the pulse pressure side lobe region, and assuming that τ 0 is the main lobe width, the range of I Ω is
Figure FDA0003089613460000023
 (33)引入变量h，进一步将式(6)转化为不等式约束的非线性规划问题：(33) The variable h is introduced, and the equation (6) is further transformed into an inequality-constrained nonlinear programming problem:
Figure FDA0003089613460000024
Figure FDA0003089613460000024
 式中，h既是目标函数也是变量，其物理意义是归一化的模糊函数的峰值旁瓣比的上界；In the formula, h is both the objective function and the variable, and its physical meaning is the upper bound of the peak sidelobe ratio of the normalized fuzzy function; (34)上述不等式约束的非线性规划问题采用序列二次规划法求解，直接采用MATLAB优化工具中的fmincon函数求解该优化问题。(34) The above inequality-constrained nonlinear programming problem is solved by sequential quadratic programming, and the optimization problem is solved directly by the fmincon function in the MATLAB optimization tool. 2.如权利要求1所述的LFM-PC信号及其模糊函数优化方法，其特征在于，步骤(34)中优化问题采用序列二次规划法求解的可行性证明如下：由于给定多普勒频率ξ，F(τ,ξ)的分母为一常数，因此讨论式(7)中不等式约束是否二次连续可微，只需要考虑分子的二次可微性，令γ(τ,ξ)＝SHH(t,ξ)S，则2. LFM-PC signal as claimed in claim 1 and fuzzy function optimization method thereof, it is characterized in that, in step (34), the feasibility proof that optimization problem adopts sequential quadratic programming method to solve is as follows: because given Doppler The denominator of frequency ξ, F(τ,ξ) is a constant, so to discuss whether the inequality constraint in equation (7) is quadratic continuous differentiability, only the quadratic differentiability of the numerator needs to be considered, let γ(τ,ξ)= S H H(t,ξ)S, then
Figure FDA0003089613460000025
Figure FDA0003089613460000025
 其中
Figure FDA0003089613460000031
⊙为哈达玛积；in
Figure FDA0003089613460000031
⊙ is the Hadamard product;
Figure FDA0003089613460000032
Figure FDA0003089613460000032
 其中
Figure FDA0003089613460000033
in
Figure FDA0003089613460000033
 式(9)简化为：Equation (9) is simplified to:
Figure FDA0003089613460000034
Figure FDA0003089613460000034
 因此，式(7)中的目标函数和约束均满足二次连续可微，采用序列二次规划法求解。Therefore, the objective function and constraints in equation (7) satisfy the quadratic continuous differentiability, and the sequential quadratic programming method is used to solve it.
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