A minimum main lobe frequency invariant beamforming method based on SRV relaxation convex optimization

Abstract

The present application belongs to the technical field of signal processing, and relates to a minimum main lobe frequency invariant beamforming method based on SRV relaxation convex optimization. The method of the present application firstly divides the space domain of a wideband beam pattern into a relaxation region and a strict region, so as to avoid the problem that the feasible solution of the weight value cannot be obtained due to improper main lobe selection of the traditional convex optimization technology. Then, a relaxation convex optimization problem is constructed to minimize the main lobe width, and the SRV constraint based on the infinite norm is considered to obtain a smaller frequency invariance of the pattern. Compared with the traditional SRV convex optimization method, the FFT-based method and the least square-based method, the present application can design a wideband pattern without relying on a reference narrowband pattern, and can obtain the narrowest main lobe width. Moreover, the present application can make a trade-off between the main lobe width and the frequency invariance performance. Finally, the space-frequency null control performance of the present method is illustrated.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of signal processing, and relates to a minimum main lobe frequency invariant beamforming technology based on SRV relaxation convex optimization. BACKGROUND

[0002] Now, array antennas are widely used in radar, communication and electronic countermeasure systems. Beamforming technology is to enhance the desired signal and suppress the interference signal by adjusting the weight of different array elements. Generally, beamforming needs to assume that the signal meets the narrowband condition. If the signal is a wideband signal, the main lobe width and frequency are inversely proportional, and the main lobe width problem will appear at low frequencies, which will affect the effect of beamforming. Therefore, it is particularly important to study the wideband constant beamwidth beamforming technology.

[0003] In order to realize wideband beamforming, the wideband signal can be divided into multiple sub-band frequency points, and the spacing of each sub-array can be changed according to different frequency points. The frequency invariance of the wideband pattern is realized by nesting sub-arrays, and each sub-band is processed by different sub-arrays, so that the pattern of each sub-array is approximately constant. However, this method requires a large number of array elements, which greatly increases the system complexity, and the frequency invariance between different frequency points cannot be guaranteed. Then, Ward deduced a system design method according to the frequency invariance of the array response. And proposed to realize wideband beamforming in the digital domain using FIR filter, which overcomes the shortcomings of the above method. Then, there are design methods of frequency invariant beamformers based on spatial resampling, based on FFT and based on least square method, but these methods are limited to uniform arrays or cannot accurately control the sidelobe.

[0004] In order to solve the problem of frequency invariant beamforming of arbitrary structure array, Yang Yixin et al. proposed a design method based on Bessel function decomposition. This method uses Bessel series to represent the array steering vector, decouples the spatial frequency of the array steering vector, and then eliminates the frequency components in the steering vector according to the least square method to realize the frequency invariance of the pattern within the working bandwidth. However, when the number of array elements is small, the truncation error of Bessel series is large, which leads to the decline of algorithm performance.

[0005] Boyd first modeled the beamforming problem as a mathematical convex optimization problem. Since then, beamforming algorithms based on convex optimization have been extensively studied. However, this method must artificially set the appropriate main lobe width. If the main lobe width is too narrow or too wide, it will lead to the decline of beam performance. SUMMARY

[0006] The application provides a wideband minimum constant beam width beamforming technology based on SRV relaxation convex optimization, and compared with the above-mentioned subarray division method, the method based on FFT and the traditional convex optimization algorithm, the application can realize arbitrary beamforming design of the narrowest main lobe width not depending on a reference direction pattern, and can make a choice between the main lobe width and frequency invariability.

[0007] Suppose that a wideband signal passes through an M-element linear array, the array element positions are x1=0, x2=d1,..., x M M M-1 , each array element is connected with an N-order FIR filter, the sampling frequency of the signal is f s , then the direction pattern of the array at the frequency f and the angle θ is defined as

[0008] P(f,θ)=|w H a(f,θ)|

[0009] Wherein,

[0010]

[0011] w m =[w m,0 ,w m,1 ,...,w m,N-1 ] T

[0012]

[0013]

[0014]

[0015] The technical scheme of the application is as follows:

[0016] The wideband minimum constant beam width beamforming technology based on SRV relaxation convex optimization carries out distortionless beamforming on a wideband signal through the SRV-based relaxation convex optimization technology, and makes the main lobe width minimum to achieve the maximum enhancement of the signal at the expected direction θ0.

[0017] S1, the space domain of the wideband array direction pattern is divided into a relaxation region φ slack And a strict region φ strict , to avoid the problem that a feasible solution cannot be obtained due to improper selection of the main lobe of the traditional convex optimization technology:

[0018] φ slack =[θ0-δ,θ0)∪(θ0,θ0+δ]

[0019]

[0020] where θ0is the desired beam pointing, δ is a normal number, which can be chosen to be larger than the desired main lobe width (the first zero width), and subsequent iterations will shrink it to the minimum.

[0021] S2, according to the characteristics of the left main lobe increase and the right main lobe decrease, a series of relaxation variables r l ≥ 0, l = 1, 2,..., L, and

[0022] r1≤ r2≤...≤ r k k+1 ≥ r k+2 ≥... ≥ r L

[0023] To characterize this characteristic, a block diagonal matrix R is constructed

[0024]

[0025] where L is the number of discrete sampling points of the relaxation region. Therefore, this characteristic can be expressed as

[0026]

[0027] where r = (r1, r2,..., r L ), b = (b1, b2,..., b L ).

[0028] S3, given the reference level, the constraints of the relaxation region and the strict region can be written as

[0029] |w H a(f j , θ l )|≤ ρ(f j , θ l ) + r l , θ l ∈ φ slack , f j ∈ [f min , f max ]

[0030] |w H a(f j , θ s )|≤ ρ(f j , θ s ), θ i ∈ φ strict , f j ∈ [f min , f max ]

[0031] where [f min , f​max ] is the frequency invariant range, p(f j , θ s ) is the desired level at (f j , θ s ).

[0032] S4, considering the frequency invariance of the wideband pattern, introduce the spatial response variation (SRV) constraint

[0033]

[0034] where B is the signal bandwidth, f0is an arbitrarily chosen reference frequency within [f min , f max ].

[0035]

[0036] φ is the set of angles to control the frequency invariance, the selection of φ can be traded off according to the main lobe width and the frequency invariance, ψ is the size of the angle range to control the frequency invariance.

[0037] In order to get smaller frequency invariance, define the infinite norm of d(w) as a function of w

[0038]

[0039] S5, given the upper bound ε of , in order to reduce the amount of calculation, constrain Optimization problem can be solved only at the reference frequency f0, through the sparsification of the relaxation variable to obtain the minimum main lobe width.

[0040]

[0041] s.t.w H a(f0, θ0) = 1,

[0042] r l ≥ 0, l = 1, 2,..., L,

[0043]

[0044] |w H a(f0, θ l )| ≤ p(f0, θ l ) + r l ,

[0045] |w H a(f0, θ s )| ≤ p(f0, θ s ), ​

[0046]

[0047] The above optimization problem is converted into a set of convex optimization problems to be solved iteratively, q = 1, 2,... Q:

[0048]

[0049]

[0050] r l q ≥ 0, l = 1, 2,..., L,

[0051]

[0052]

[0053]

[0054]

[0055] where ζ is a small positive number to avoid denominator being zero. This problem can be solved effectively using existing convex optimization algorithms or toolboxes (e.g. interior point method, subgradient method).

[0056] Or only the main lobe frequency invariance is reserved, in the strong interference environment to generate a null frequency scenario, the optimization problem is as follows:

[0057]

[0058]

[0059] r l q ≥ 0, l = 1, 2,..., L,

[0060]

[0061]

[0062]

[0063]

[0064] S6, the beamforming vector w is obtained by solving the above sequence of convex optimization problems Q , so as to complete the beamforming by weighting the wideband signal according to the obtained weight vector.

[0065] The present application has the beneficial effects that the present application can realize wideband beamforming to enhance signals in the desired direction and suppress signals in the interference direction, uses SRV constraints to realize the frequency invariance of the wideband directional pattern independent of the reference narrowband directional pattern, avoids distortion of the received signal, and achieves the narrowest main lobe width to improve target detection performance. The main lobe width and frequency invariance can also be traded off according to the selection of the SRV angle range. BRIEF DESCRIPTION OF DRAWINGS

[0066] Figure 1 A flowchart for implementing the present application is shown.

[0067] Figure 2 A traditional convex optimization beam main view and three-dimensional view are shown.

[0068] Figure 3 A convex optimization beam main view and three-dimensional view of the present application are shown.

[0069] Figure 4 A comparison chart for setting the relaxation region [-20°, 20°] to trade off the main lobe width and frequency invariance is shown.

[0070] Figure 5 A comparison chart for setting the relaxation region [-90°, 90°] to trade off the main lobe width and frequency invariance is shown.

[0071] Figure 6 A directional pattern for three different frequencies is shown. DETAILED DESCRIPTION

[0072] The technical solutions of the present application will be further described below in conjunction with the drawings and examples.

[0073] Example 1

[0074] The purpose of this example is to compare the traditional SRV convex optimization method with the proposed method to verify that the method of the present application can realize a narrower main lobe width. In this example, 16 array elements are set, each followed by a 32-order FIR filter, the sidelobe is set to -20 dB, the normalized frequency invariance range is set to [0.2π, 0.5π], the beam pointing direction is set to -10°, and the reference frequency is set to 0.3π. The traditional SRV convex optimization sets the sidelobe region to [-90°, -31°]∪[11°, 90°], and the present method sets the strict region to [-90°, -31°]∪[11°, 90°], and ε = 10 -4 The beam main view and three-dimensional view of the two methods are shown in Figure 2 , Figure 3 The results show that the traditional convex optimization main lobe width is 46°, and the present method main lobe width is 31°, which has a narrower main lobe width.

[0075] Example 2

[0076] The purpose of this example is to show the impact of the selection of different angle ranges on the main lobe width and frequency invariance. 20 array elements are set, each followed by a 32-tap FIR filter, with a side lobe of -30dB, a normalized frequency invariance range of [0.2π, 0.5π], and ε = 10 -3 , with the beam pointing at 0°, and the reference frequency Figure 4 The relaxed region is set to [-20°, 20°], and the resulting main lobe width is 31.9°, Figure 5 The relaxed region is set to [-90°, 90°], and the resulting main lobe width is 34.7°, but the frequency invariance is ensured over the whole spatial domain, which shows that the method can trade off between frequency invariance and main lobe width.

[0077] Example 3

[0078] The purpose of this example is to show the spatial-frequency null control performance of the method. 16 array elements are set, each followed by a 32-tap FIR filter, with a side lobe of -20dB, a frequency invariance range of [0.125, 0.25]GHz, and -40dB spatial-frequency nulls set in the range of [0.15, 0.2]GHz, [30°, 50°], as shown in Figure 6 , which shows that the method has good spatial-frequency null control performance.

Claims

1. A minimum main lobe frequency-invariant beamforming method based on SRV relaxation convex optimization, defining a broadband signal passing through an M-element linear array, with element positions x1=0, x2=d1,...,x M =d M-1 d represents the element spacing, each element is connected to an Nth-order FIR filter, and the signal sampling frequency is f. s Then the array pattern at frequency f and angle θ is: P(f,θ)=|w H a(f,θ)| in, In m =[in m,0 ,In m,1 ,...,In m,N-1 ] T Its key feature is that it performs distortion-free beamforming on broadband signals using SRV-based relaxation convexity optimization techniques, minimizing the main lobe width to maximize signal enhancement in the desired direction θ0; specifically, it includes the following steps: S1. Divide the spatial domain of the broadband array pattern into relaxation regions φ slack and strict region φ strict : f slack =[θ0-δ,θ0)∪(θ0,θ0+δ] Where θ0 is the desired beam direction and δ is a positive constant; S2. Based on the characteristic of the main lobe increasing on the left and decreasing on the right, construct a series of slack variables r. l ≥0, l=1,2,...,L, where L is the number of discrete sampling points in the relaxation region, we get: r1≤r2≤...≤r k ,r k+1 ≥r k+2 ≥...≥r L To characterize the left-increase-right-decrease characteristic of the main lobe, a block-diagonal matrix is ​​constructed. in ν = (1, -1), thus representing the characteristic of the main lobe increasing on the left and decreasing on the right as follows: Where r = (r1, r2, ..., r L b = (b1, b2, ..., b) L ); S3. Given a reference level, the constraints between the relaxed and strict regions are: |w H a(f j ,i l )|≤ρ(f j ,i l )+r l ,i l ∈φ slack ,f j ∈[f min ,f max ] |w H a(f j ,i s )|≤ρ(f j ,i s ),θ i ∈φ strict ,f j ∈[f min ,f max ] Where [f] min ,f max ] represents the frequency-invariant range, ρ(f) j ,θ s ) is (f j ,θ s The desired level at () S4. Considering the frequency invariance of the broadband radiation pattern, introduce the spatial response variation (SRV) constraint. Where B is the signal bandwidth, and f0 is [f min ,f max The reference frequency can be arbitrarily selected within the range; φ represents the set of angles that control the frequency to remain constant, and ψ represents the range of angles within which the frequency invariance is desired to be controlled. Will Defined as the infinite norm of the function d(w) with respect to w: S5, Given The upper bound ε, constraint Solving the optimization problem only at the reference frequency f0, by sparsifying the slack variables. To obtain the minimum main lobe width, we establish the optimization problem: stw H a(f0,θ0)=1, r l ≥0,l＝1,2,...,L, |w H a(f0,θ l )|≤ρ(f0,θ l )+r l , |w H a(f0,θ s )|≤ρ(f0,θ s ), There are two main ways to solve optimization problems: 1) Transform the optimization problem into a set of convex optimization problems and solve them iteratively, q = 1, 2, ... Q: r l q ≥0,l＝1,2,...,L, in ζ is a small positive number to avoid the denominator being 0; 2) Retaining only the invariant main lobe frequency, in a strong interference environment with a null space frequency, the optimization problem is transformed into the following form before solving: r l q ≥0,l＝1,2,...,L, S6. Based on the beamforming vector w obtained from solving the optimization problem in step S5. Q Then, the broadband signal is weighted according to the obtained weight vector to complete beamforming.

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