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目前自适应阵列信号处理技术在雷达,声纳和移动通信系统中有着越来越多的重要应用[1]。宽带信号波束形成算法可以采用时域和频域分析方法。频域方法是对阵列接收数据按照频率进行分块,并通过DFT变换将时域数据转换成频域数据,由于在子块中信号频宽较窄,可以通过独立的窄带波束形成器实现波束成形,最后在通过合成实现宽带信号波束成形。频域方法复杂度较小,但不能得到连续的时域波束输出[2]。文献[3]和文献[4]分别采用频率矩和滑动窗函数的方法,在有效地进行宽带波束形成的同时也抑制了宽带干扰。文献[5]将分数阶傅里叶变换应用于宽带信号频域多波束形成中,从而提高了运算的速度。文献[6]采用基于均匀一致性旁瓣级的设计方法,将宽带信号用一组FIR(finite impulse response)滤波器拟合,从而实现恒定束宽的波束形成,但该方法存在着迭代收敛的步长难以确定的问题,因而算法的精度不高,收敛速度慢。文献[7-9]基于高斯模型的超宽带信号,提出数字延迟线和分数时延滤波器相结合的方法,但由于在时域处理方法中往往需要较多的延时单元,并且涉及到高阶协方差矩阵的求逆问题,因此对于宽带或超宽带信号处理,该方法采样频率高,数据量大,难于在工程中实现。在宽带信号处理中,有限脉冲响应(FIR)抽头系数较多,为了减轻矩阵运算的高复杂性可以采用无限脉冲响应(IIR),然而IIR算法的不稳定性也使得收敛速度变慢[10]。
对于阵列信号处理技术中波束形成目前最广泛采用的是最小方差无失真响应(minimum variance distortionless response, MVDR)波束合成、对角加载方法、旁瓣对消器(generalized sidelobe canceller, GSC)[11-14]。MVDR是通过调节阵列的输出功率达到最佳的阵列权重向量。对角加载算法是通过对投影变换技术对阵列接收数据协方差矩阵进行预处理,在约束区域形成稳定的响应幅度。GSC是一个自适应波束形成器,通过求解线性约束得最小方差合成波束实现。最小方差的求解可以利用众所周知的自适应滤波算法,如最小均方(least mean square, LMS)、递归最小二乘(recursive least square, RLS)进行无约束的自适应优化[15-16]。然而,这些优化方法都涉及对矩阵求逆的计算,对于宽带接受信号,矩阵计算量非常大。因此,本文提出了一种基于小波滤波器组的改进GSC自适应波束形成方法,该方法在降低计算低复杂度和提高收敛速度方面具有一定优势。
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图 1为GSC的结构示意图。GSC自适应的原理是:将接收的已知期望信号方向信息的阵列信号$\mathit{\boldsymbol{x}}(n)$变换为上下两个支路信号。上支路信号通过归一化期望信号的导向矢量${\mathit{\boldsymbol{h}}_0}$变换为参考信号${\mathit{\boldsymbol{d}}_0}(n)$,${\mathit{\boldsymbol{d}}_0}(n)$同时含期望信号和干扰信号;下支路信号通过正交阻塞矩阵${\mathit{\boldsymbol{B}}_0}$阻塞掉期望信号,这样${\mathit{\boldsymbol{x}}_0}(n)$中就只含干扰,再将变换后的上下支路信号进行维纳滤波${\mathit{\boldsymbol{w}}_{x0}}$,从而自适应的抵消干扰。
GSC波束形成结构中阵列相关输出量为:
$${\mathit{\boldsymbol{d}}_0}(n) = \mathit{\boldsymbol{h}}_0^{\rm{H}}\mathit{\boldsymbol{x}}(n)$$ (1) $${\mathit{\boldsymbol{x}}_0}(n) = {\mathit{\boldsymbol{B}}_0}\mathit{\boldsymbol{x}}(n)$$ (2) $$y(n) = {\varepsilon _0}(n)$$ (3) 式中,${\mathit{\boldsymbol{h}}_0}$为归一的期望信号矢量,这里取$\mathit{\boldsymbol{h}}_0^{\rm{H}}{\mathit{\boldsymbol{h}}_0} = 1$,归一化为:
$${\mathit{\boldsymbol{h}}_0} = \frac{{\mathit{\boldsymbol{a}}({\theta _0})}}{{\left\| {\mathit{\boldsymbol{a}}({\theta _0})} \right\|}} = \frac{{\mathit{\boldsymbol{a}}({\theta _0})}}{{\sqrt {N - 1} }}$$ (4) 式中,$\left\| {} \right\|$表示向量求模;${\mathit{\boldsymbol{B}}_0}$为$N \times (N + 1)$行滿秩阻塞矩阵;${\mathit{\boldsymbol{x}}_0}(n)$为$N \times 1$维矢量。${\mathit{\boldsymbol{B}}_0}$满足:${\mathit{\boldsymbol{B}}_0}{\mathit{\boldsymbol{h}}_0} = 0$, 由${\mathit{\boldsymbol{B}}_0}$和${\mathit{\boldsymbol{h}}_0}$构造$(N + 1) \times (N + 1)$变换矩阵T:
$$\mathit{\boldsymbol{T}} = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{h}}_0^{\rm{H}}}\\ {{\mathit{\boldsymbol{B}}_0}} \end{array}} \right]$$ (5) 则T为非奇异阵,得:
$$\left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{d}}_0}(n)}\\ {{\mathit{\boldsymbol{x}}_0}(n)} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{h}}_0^{\rm{H}}}\\ {{\mathit{\boldsymbol{B}}_0}} \end{array}} \right]\mathit{\boldsymbol{x}}(n)$$ (6) 由GSC波束形成中的自适应对消干扰原理,得参考信号表达式为:
$$\begin{array}{c} {\mathit{\boldsymbol{d}}_0}(n) = \mathit{\boldsymbol{h}}_{\rm{0}}^{\rm{H}}\mathit{\boldsymbol{x}}(n) = \\ \mathit{\boldsymbol{h}}_0^{\rm{H}}[\mathit{\boldsymbol{a}}({\theta _0}){\mathit{\boldsymbol{s}}_0}(n) + \sum\limits_{j = 1}^M {\mathit{\boldsymbol{a}}({\theta _j}){\mathit{\boldsymbol{s}}_j}(n) + \mathit{\boldsymbol{N}}(n)} ] = \\ \sqrt {N + 1} {\mathit{\boldsymbol{s}}_0}(n) + \sum\limits_{j = 1}^M {{\phi _j}{\mathit{\boldsymbol{s}}_j}(n) + \mathit{\boldsymbol{h}}_0^{\rm{H}}\mathit{\boldsymbol{N}}(n)} \end{array}$$ (7) 式中,${\phi _j} = \mathit{\boldsymbol{h}}_0^{\rm{H}}\mathit{\boldsymbol{a}}({\theta _j})$为期望信号与干扰的空间相关系数。由式(2) 得:
$$\begin{array}{c} {\mathit{\boldsymbol{x}}_0}(n) = {\mathit{\boldsymbol{B}}_0}[\mathit{\boldsymbol{a}}({\theta _0}){\mathit{\boldsymbol{s}}_0}(n) + \sum\limits_{j = 1}^M {\mathit{\boldsymbol{a}}({\theta _j}){\mathit{\boldsymbol{s}}_j}(n)} + \mathit{\boldsymbol{N}}(n)] = \\ \sum\limits_{j = 1}^M {{\mathit{\boldsymbol{B}}_0}\mathit{\boldsymbol{a}}({\theta _j}){\mathit{\boldsymbol{s}}_j}(n)} + {\mathit{\boldsymbol{B}}_0}\mathit{\boldsymbol{N}}(n) \end{array}$$ (8) 由式(8) 可知:GSC波束形成结构中上下支路的变换是把阵列数据向量向两个正交的子空间投影,${\mathit{\boldsymbol{h}}_0}$将$\mathit{\boldsymbol{x}}(n)$投影到期望信号对应的一维子空间中,而$\mathit{\boldsymbol{B}}_0^{\rm{H}}$将$\mathit{\boldsymbol{x}}(n)$投影到与期望信号子空间正交的N维子空间,输出信号$\mathit{\boldsymbol{x}}(n)$经正交投影变换后进行维纳滤波器实现干扰对消。
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GSC波束结构中存在构造满秩阻塞矩阵${\mathit{\boldsymbol{B}}_0}$的问题,在实际系统中由于各种误差的存在,如阵元位置误差、阵元幅相误差、阵元之间的互耦和通道频率特性失配等,使得构造的${\mathit{\boldsymbol{B}}_0}$矩阵难以满足条件,从而存在导向矢量失配的问题,导致阻塞矩阵${\mathit{\boldsymbol{B}}_0}$不能完全阻塞掉期望信号,造成部分有用信号互相抵消,输出信干噪比(signal interference noise ration, SINR)下降,波束图中的旁瓣电平被升高的现象。
改进的GSC波束形成算法采用小波滤波器组在频域中阻塞接收信号的有用分量,然后对输出信号进行重构并求出协方差矩阵,再用重构的协方差矩阵形成自适应波束的权向量,其结构图如图 2所示。
改进的GSC波束形成算法结构中,${\mathit{\boldsymbol{B}}_0}$为行滿秩正交阻塞矩阵,其作用是阻塞接收信号中的期望信号,但在实际构造${\mathit{\boldsymbol{B}}_0}$阵列中如果存在误差,则将有一部分期望信号泄露到辅助支路中,从而引起期望信号相消的现象。对于阵列接收信号$x(t)$的协方差矩阵为:
$$\begin{array}{c} {\mathit{\boldsymbol{R}}_{xx}} = {\rm{E}}[\mathit{\boldsymbol{x}}(t){\mathit{\boldsymbol{x}}^{\rm{H}}}(t)] = \\ {\mathit{\boldsymbol{R}}_{i + n}} + \sigma _s^2\mathit{\boldsymbol{a}}({\theta _0}){\mathit{\boldsymbol{a}}^{\rm{H}}}({\theta _0}) \end{array}$$ (9) 对式(9) 进行矩阵求逆,得:
$$\mathit{\boldsymbol{R}}_{xx}^{ - 1} = \mathit{\boldsymbol{R}}_{i + n}^{ - 1} - \sigma _s^2\frac{{\mathit{\boldsymbol{R}}_{i + n}^{ - 1}\mathit{\boldsymbol{a}}({\theta _0}){\mathit{\boldsymbol{a}}^{\rm{H}}}({\theta _0})\mathit{\boldsymbol{R}}_{i + n}^{ - 1}}}{{1 + \sigma _s^2{\mathit{\boldsymbol{a}}^H}({\theta _0})\mathit{\boldsymbol{R}}_{i + n}^{ - 1}\mathit{\boldsymbol{a}}({\theta _0})}}$$ (10) GSC波束形成器的权矢量为:
$$\begin{array}{c} {\mathit{\boldsymbol{w}}_{GSC}} = \frac{{\mathit{\boldsymbol{R}}_{xx}^{ - 1}\mathit{\boldsymbol{a}}({\theta _0})}}{{{\mathit{\boldsymbol{a}}^{\rm{H}}}({\theta _0})\mathit{\boldsymbol{R}}_{xx}^{ - 1}\mathit{\boldsymbol{a}}({\theta _0})}} = \\ \frac{{\mathit{\boldsymbol{R}}_{i + n}^{ - 1}\mathit{\boldsymbol{a}}({\theta _0})}}{{{\mathit{\boldsymbol{a}}^{\rm{H}}}({\theta _0})\mathit{\boldsymbol{R}}_{xx}^{ - 1}\mathit{\boldsymbol{a}}({\theta _0})[1 + \sigma _s^2{\mathit{\boldsymbol{a}}^{\rm{H}}}({\theta _0})\mathit{\boldsymbol{R}}_{i + n}^{ - 1}\mathit{\boldsymbol{a}}({\theta _0})]}} \end{array}$$ (11) 通过式(11) 可知:
$$\begin{array}{c} {\mathit{\boldsymbol{a}}^{\rm{H}}}({\theta _0})\mathit{\boldsymbol{R}}_{xx}^{ - 1}\mathit{\boldsymbol{a}}({\theta _0})[1 + \sigma _s^2{\mathit{\boldsymbol{a}}^{\rm{H}}}({\theta _0})\mathit{\boldsymbol{R}}_{i + n}^{ - 1}\mathit{\boldsymbol{a}}({\theta _0})] = \\ {\mathit{\boldsymbol{a}}^{\rm{H}}}({\theta _0})\mathit{\boldsymbol{R}}_{i + n}^{ - 1}\mathit{\boldsymbol{a}}({\theta _0}) \end{array}$$ (12) 故式(12) 可以改写为:
$${\mathit{\boldsymbol{w}}_{{\rm{GSC}}}} = \frac{{\mathit{\boldsymbol{R}}_{i + n}^{ - 1}\mathit{\boldsymbol{a}}({\theta _0})}}{{{\mathit{\boldsymbol{a}}^{\rm{H}}}({\theta _0})\mathit{\boldsymbol{R}}_{i + n}^{ - 1}\mathit{\boldsymbol{a}}({\theta _0})}}$$ (13) 小波滤波器组能够有效地对期望信号进行检测,因此小波滤波器对阵列信号接收的数据进行频域滤波后,可以有效阻塞期望信号。相应的输出干扰和噪声的协方差矩阵${\mathit{\boldsymbol{R}}_{i + n}}$特征值分解为:
$$\begin{array}{c} {\mathit{\boldsymbol{R}}_{i + n}} = \sum\limits_{m = 1}^M {{\lambda _{i + n}}{\mathit{\boldsymbol{u}}_{i + n}}\mathit{\boldsymbol{u}}_{i + n}^{\rm{H}}} = \\ {\mathit{\boldsymbol{E}}_i}{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}\mathit{\boldsymbol{E}}_i^{\rm{H}} + {\mathit{\boldsymbol{E}}_n}{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_n}\mathit{\boldsymbol{E}}_n^{\rm{H}} \end{array}$$ (14) 式中,${u_{i + n}}$为干扰和噪声的值;${\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i} = {\rm{diag}}\{ {\lambda _1},{\lambda _2}, \cdots ,$ ${\lambda _{P - 1}}\} $包含P-1个干扰信号的特征值;${\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_n} = {\rm{diag}}\{ {\lambda _P},$ ${\lambda _{P + 1}}, \cdots ,{\lambda _M}\} $为噪声的特征值;${\mathit{\boldsymbol{E}}_i}$为干扰子空间;${\mathit{\boldsymbol{E}}_n}$为噪声子空间。对${\mathit{\boldsymbol{E}}_i}$干扰子空间重构得到干扰空间的协方差矩阵为:
$${\mathit{\boldsymbol{R}}_{ii}} = \sum\limits_{p = 1}^{P - 1} {{\lambda _p}{\mathit{\boldsymbol{u}}_p}\mathit{\boldsymbol{u}}_p^{\rm{H}}} $$ (15) 当对阵列天线噪声估计不足时会造成协方差矩阵特征值分散,使得自适应波束的旁瓣升高。可以采用用噪声的平均功率$\sigma _n^2 = ({\lambda _p} + {\lambda _{p + 1}} + \cdots + {\lambda _M})/$ $(M - P + 1)$进行对角加载,加载量的运用对协方差矩阵中的大特征值不产生影响,但使噪声对应的小特征值加大并近似等于加载量,从而抑制或降低了小特征值的扰动,得到较好的旁瓣压低性能。有:
$${\mathit{\boldsymbol{R}}_{ss}} = \sum\limits_{p = 1}^{P - 1} {{\lambda _p}} {\mathit{\boldsymbol{u}}_p}\mathit{\boldsymbol{u}}_p^{\rm{H}} + \sigma _n^2\mathit{\boldsymbol{I}}$$ (16) 转换到阵元空间为:
$${\mathit{\boldsymbol{\tilde R}}_{ii}} = \mathit{\boldsymbol{G}}{\mathit{\boldsymbol{R}}_{ii}}{\mathit{\boldsymbol{G}}^{\rm{H}}}$$ (17) 式中,G为小波综合滤波器的矩阵形式。根据式(13) 在期望信号方向矢量和协方差矩阵准确已知的条件下,干扰噪声的协方差矩阵可以代替阵列的协方差矩阵,权矢量保持不变,对干扰子空间重构后的自适应波束的权向量表示为:
$$\mathit{\boldsymbol{w}} = \frac{{\mathit{\boldsymbol{\tilde R}}_{ii}^{ - 1}\mathit{\boldsymbol{a}}({\theta _0})}}{{{\mathit{\boldsymbol{a}}^{\rm{H}}}({\theta _0})\mathit{\boldsymbol{\tilde R}}_{ii}^{ - 1}\mathit{\boldsymbol{a}}({\theta _0})}}$$ (18) 由于阵列信号经小波滤波器组后协方差矩阵中不包含期望信号分量,从而克服了存在系统误差时自适应波束形成中期望信号相消的问题,解决了在实际应用中GSC波束形成算法需要构造与期望信号的导向矢量完全正交的阻塞矩阵的难点问题。
Improved Sidelobe Cancellation Algorithm for Wideband Beamforming Based on Wavelet
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摘要: 研究了改进的旁瓣相消算法,该算法利用小波滤波器组阻塞接收信号的期望信号分量进行自适应的波束形成。经过改进算法处理后,阵列信号的协方差矩阵中不包含期望信号分量,从而克服了当存在系统误差时自适应波束形成中期望信号相消的问题,并解决了在实际应用中旁瓣相消波束形成算法GSC波束形成算法需要构造与期望信号的导向矢量完全正交的阻塞矩阵的难点问题,从而达到提高系统稳健性能的目的,仿真结果验证了该算法的性能优于传统的MVDR方法和基于MVDR的对角加载方法。Abstract: The orthogonal block matrix is needed in the generalized sidelobe canceller algorithm to offset the received signal and the desired signal. But there is a limitation on constructing the orthogonal block matrix in the actual pattern synthesis processing. In this paper, an improved sidelobe cancellation algorithm is proposed. The wavelet filter bank is used to block the desired signal component in received signals of antenna array for adaptive beamforming. The covariance matrix of array signal does not contain the desired signal component in the improved algorithm. So the desired signal cancellation problem in adaptive beamforming with the system error can be overcome. Also the proposed algorithm solves the problem that the generalized sidelobe canceller (GSC) beamforming algorithm needs to construct the blocking matrix which is orthogonal with the desired signal, thus the robust performance of the system is achieved. The effectiveness of the improved algorithm is verified by the simulation results. The simulation results show that the performance of the improved algorithm is better than the traditional minimum variance distortionless response (MVDR) method and the diagonal loading method based on MVDR algorithm.
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