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相比窄带雷达,宽带雷达具有距离分辨率高、抗干扰能力强等优点[1-2],因此在各领域中获得广泛应用。然而,由于宽带雷达接收机需对宽带信号进行采样,对模数转换(analog to digital converter, ADC)单元的性能提出更高要求,使系统成本显著增加[3-4]。对于一些超宽带雷达系统,无法提供采样率足够高的ADC进行采样处理,严重限制了宽带雷达的发展应用。
考虑到宽带相控阵雷达通常采用线性调频(linear frequency modulated, LFM)信号作为发射信号,根据LFM信号的特点[3],文献[5-7]提出了一系列去斜处理算法以降低信号带宽,从而降低对ADC采样率的要求。文献[8]提出了一种基于时域变换技术的去斜算法,利用3个色散网络和一个混频器实现降低信号带宽的目的。文献[9]研究了一种改进的宽带时延波束形成算法,通过在每个阵元上进行模拟去斜处理,省去了模拟时延单元,是一种低成本去斜解决方案。基于此架构,文献[10]研究其在自适应干扰抑制方面的应用。文献[11]则采用数字去斜处理结合数字下变频代替模拟去斜处理,不仅提升了响应速度,而且具有更高的距离分辨率。然而,上述这些去斜算法在完成混频之后均需使用数字时延滤波器,在实际应用中实现难度较大、成本高,不适合大规模阵列雷达的设计。
为了解决宽带大规模阵列雷达的应用需求,本文提出了一种无数字时延滤波器的宽带雷达去斜处理算法,利用划分子阵的方式,减少了通道数量,并通过对各子阵输出通道信号进行频率和相位的补偿,确保各个通道的信号一致性,从而去掉了数字时延滤波器的使用,使整个系统的实现成本更低。通过仿真分析表明,在相同的时延估计误差下,相比传统去斜算法,本文算法在抗干扰应用中具有更高的输出信噪比。
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基本去斜处理算法的核心是通过二次混频和数字时延滤波器对各通道信号进行调整,使输出信号与参考信号一致,以实现在后期信号处理中获得更高的相干增益。
假设雷达阵列为具有两个阵元的均匀线阵,阵元的间距为d,采用单个数字式频率合成器(direct digital synthesis, DDS)时的基本去斜处理流程如图1所示[12]。接收到的回波LFM信号的方位角为
$\theta $ ,载波频率为${f_0}$ ,信号带宽为B,LFM信号调频斜率为$\mu = {B \mathord{\left/ {\vphantom {B T}} \right. } T}$ 。阵元1的回波可以表示为:
$$ {x_1}\left( t \right) = \exp \left[ {{\rm{j}}2{\text{π}} \left( {{f_0}\left( {t - {t_0}} \right) + \frac{{\mu {{\left( {t - {t_0}} \right)}^2}}}{2}} \right)} \right] $$ (1) 式中,
$R$ 表示目标到阵元1的距离;回波时延为$ {t_0} = {{2R} \mathord{\left/ {\vphantom {{2R} c}} \right. } c} $ ,c为光速。阵元2的回波信号可以表示为:
$$ {x_2}\left( t \right) = {x_1}\left( {t - \tau } \right) $$ (2) 式中,
$\tau = {{d\sin \theta } \mathord{\left/ {\vphantom {{d\sin \theta } c}} \right. } c}$ 为阵元1与阵元2之间的时延。设
${t_0}$ 表示回波时延,${t_r}$ 为回波时延估计值,则DDS根据回波时延所产生的本振信号为:$$ {S_{{\text{ref}}}}\left( t \right) = \exp \left[ {{\rm{j}}2{\text{π}} \left( {{f_0}\left( {t - {t_r}} \right) + \frac{{\mu {{\left( {t - {t_r}} \right)}^2}}}{2}} \right)} \right] $$ (3) 经过混频和低通滤波器以后,以采样间隔为
${T_s} = {1 \mathord{\left/ {\vphantom {1 {{f_s}}}} \right. } {{f_s}}}$ 的ADC进行采样后的信号可以写为:$$ \begin{split} & {z_1}\left( n \right) = \exp \left[ {{\rm{j}}\left( {2{\text{π}} {f_1}{T_s}n + {\psi _1}} \right)} \right] \hfill \\& {z_2}\left( n \right) = \exp \left[ {{\rm{j}}\left( {2{\text{π}} {f_2}{T_s}n + {\psi _2}} \right)} \right] \end{split}$$ (4) 其中,
$$ \begin{split} &\qquad {f_1} = \mu \left( {{t_r} - {t_0}} \right) \hfill \\ &\;\;\; {f_2} = \mu \left( {{t_r} - {t_0}} \right) - \mu \tau = {f_1} - \mu \tau \hfill \\ &\;\; {\psi _1} = 2{\text{π}} {f_0}\left( {{t_r} - {t_0}} \right) + {\text{π}} \mu ( {t_0^2 - t_r^2} ) \hfill \\ & {\psi _2} = {\psi _1} + \left[ {{\text{π}} \mu ( {2{t_0}\tau + {\tau ^2}} ) - 2{\text{π}} {f_0}\tau } \right] \end{split} $$ (5) 由式(5)能够得出,两个阵元所接收信号的频率、相位均不相同。为了使得阵元2与阵元1的信号完全一致,可以在数字域对阵元2的回波信号进行二次混频处理,并通过数字时延滤波器进行补偿,以补偿阵元1和阵元2之间的频率和相位偏差。
根据式(5)可得,阵元2的通道在数字域进行二次混频的本振信号可表示为:
$$ {s'_{{\text{ref}}}}\left( n \right) = {{\text{e}}^{{\text{j}}2{\text{π}} ( { - \mu \tau {T_s}n + \mu {t_r}\tau + \tfrac{1}{2}{\tau ^2} - {f_0}\tau } )}} $$ (6) 二次混频后,采用的数字时延滤波器的冲击响应为
$\delta \left( {n + {\tau \mathord{\left/ {\vphantom {\tau {{T_s}}}} \right. } {{T_s}}}} \right)$ ,则处理后的输出信号为:$$ {z_2}^{\prime \prime }\left( n \right) = \left\{ {{z_2}\left( n \right){{\left[ {{{s'}_{{\text{ref}}}}\left( n \right)} \right]}^ * }} \right\} * \delta \left( {n + {\tau \mathord{\left/ {\vphantom {\tau {{T_s}}}} \right. } {{T_s}}}} \right) = {z_1}\left( n \right) $$ (7) 式中,
$ {(·)}^{*} $ 表示取共轭复数;“$ * $ ”为卷积。至此便实现了阵元2与阵元1的输出信号完全一致。然而,上述去斜算法为了使其他通道的相位与参考通道保持一致,必须使用数字时延滤波器对参考通道(阵元2)的信号进行调整。而数字时延滤波器的实现成本高、设计复杂,不适合大规模阵列雷达的应用。为了解决该问题,本文设计了一种无数字时延滤波器的分子阵宽带雷达去斜处理算法,所提方法通过频率和相位补偿,代替二次混频和数字时延滤波器,从而将其他通道的频率和相位调整为与参考通道一致,简化了系统设计复杂度,降低了实现成本。
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针对宽带大规模阵列雷达,本节首先通过子阵划分,以减少通道数量,然后利用频率和相位补偿的方式,实现其他通道的频率和相位调制,进而使参考通道与其他通道的频率和相位一致。另外,还具体分析了回波时延误差对输出信噪比的影响。
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假设大规模阵列雷达为均匀线阵,将整个阵列划分为L个子阵,每个子阵由M个阵元组成,若相邻阵元间距为d,则相邻子阵的间距D=Md。通过子阵划分,可降低系统设计复杂度,继而降低设计成本。同时,为了去掉数字时延滤波器,采用频率补偿和相位补偿,以保证其他通道与参考通道的输出信号频率和相位一致。本文提出的去斜处理流程如图2所示。
为便于推导,将子阵1中第一个阵元接收脉冲的前沿时刻作为时间参考点,则其接收信号为:
$$ {x_{1,1}}\left( t \right) = \exp \left[ {{\text{j}}2{\text{π}} \left( {{f_0}t + \frac{{\mu {t^2}}}{2}} \right)} \right] $$ (8) 则第
$l$ 个子阵中第$m$ 个阵元的接收信号可以表示为:$$ {x_{l,m}}\left( t \right) = \exp \left[ {{\rm{j}}2{\text{π}} \left( {{f_0}\left( {t - {\tau _{l,m}}} \right) + \frac{{\mu {{\left( {t - {\tau _{l,m}}} \right)}^2}}}{2}} \right)} \right] $$ (9) 式中,
${\tau _{l,m}} = \left( {l - 1} \right){\tau _D} + {\tau _m}$ 为第l个子阵中第$m$ 个阵元的相对于参考阵元的传播时延,且${\tau _D} = {{Md\sin \theta } \mathord{\left/ {\vphantom {{Md\sin \theta } c}} \right. } c}$ 为相邻子阵之间的传播时延,$ {\tau _m} = {{\left( {m - 1} \right)d\sin \theta } \mathord{\left/ {\vphantom {{\left( {m - 1} \right)d\sin \theta } c}} \right. } c} $ 为子阵内部第m个阵元相对于第1个阵元的传播时延。那么对于第l个子阵中第m个阵元的移相因子为:
$$ {w_{l,m}} = {{\text{e}}^{ - {\text{j}}2{\text{π}} {f_0}{\tau _{l,m}} + {\rm{j}}2{\text{π}} \mu \left( {l - 1} \right){\tau _D}{\tau _m} + {\text{j}}{\text{π}} \mu \tau _m^2}} $$ (10) 同一子阵内部,各阵元信号经过移相器后合成为一个通道,则第l个子阵的输出信号为:
$$ {u_l}\left( t \right) = \sum\limits_{m = 1}^M {\left[ {{x_{l,m}}\left( t \right)w_{l,m}^*} \right]} $$ (11) 在实际运用中,式(3)中时延的估计值与真实值总会存在偏差,设估计误差为
$\Delta t$ ,则DDS产生的本振信号为:$$ {S_{{\text{ref}}}}\left( t \right) = {x_{1,1}}\left( {t + \Delta t} \right) $$ (12) 于是,第
$l$ 个子阵混频后的输出信号为:$$ \begin{split}& \;\;\;\;\;\;\;\; {u_l}^\prime \left( t \right) = {u_l}\left( t \right) \times S_{{\text{ref}}}^*\left( t \right) = \\ & {{\text{e}}^{{\text{j}}{\psi _l}}}{{\text{e}}^{{\text{j}}{\varOmega _l}t}}\sum\limits_{m = 1}^M {\left\{ {\exp \left[ {{\text{j}}2{\text{π}} {f_{1,m}}t + {\psi _{1,m}}} \right]} \right\}} \\[-8pt] \end{split} $$ (13) 式中,
${\psi _l} = {\text{π}} \mu {\left( {l - 1} \right)^2}\tau _D^2,{\varOmega _l} = - 2{\text{π}} \mu \left( {l - 1} \right){\tau _D}$ 与划分子阵方法有关系;$ {f_{1,m}} = - \mu \left( {{\tau _m} + \Delta t} \right) $ 为第1个子阵中第m个阵元的信号频率;$ {\psi _{1,m}} = - {\text{π}} \mu {\left( {\Delta t} \right)^2} - 2{\text{π}} {f_0}\Delta t $ 为该阵元的相位。根据傅里叶变换可得:$$ {U_l}^\prime \left( {{\rm{j}}\varOmega } \right) = {{\rm{e}}^{{\text{j}}{\psi _l}}}{U_1}^\prime \left[ {{\rm{j}}\left( {\varOmega - {\varOmega _l}} \right)} \right] $$ (14) 可发现,第1个子阵与第l个子阵经混频之后的输出信号仅存在频率和相位的差别。因此,可通过频移和相移,使这两个通道的信号完全一致。
在应用中仅需保留混频之后的低频成分,而高频成分需要利用低通滤波器滤除。低通滤波器也会对信号的频率和相位产生影响,为了保证频率和相位补偿的准确性,下面将详细分析低通滤波器对混频后信号的影响。
设低通滤波器的时域单位冲击响应为h(t),低通滤波器的时域和频域输出信号分别可表示为:
$$ {u_l}^{\prime \prime }\left( t \right) = {u_l}^\prime \left( t \right) * h\left( t \right) $$ (15) $$ {U_l}^{\prime \prime }\left( {{\text{j}}\varOmega } \right) = {{\text{e}}^{{\text{j}}{\psi _l}}}{U_1}^\prime \left[ {{\text{j}}\left( {\varOmega - {\varOmega _l}} \right)} \right]H\left( {{\text{j}}\varOmega } \right) $$ (16) 则经ADC采样后的数字信号在时域和频域分别表示为:
$$ \begin{split}& {r_l}\left( n \right) = {u_l}^{\prime \prime }\left( {n{T_s}} \right) = {u_l}^\prime \left( {n{T_s}} \right) * h\left( {n{T_s}} \right) = \\ & \left\{ {{{\text{e}}^{{\text{j}}{\omega _l}n + {\text{j}}{\psi _l}}}\sum\limits_{m = 1}^M {\exp \left( {{\text{j}}\frac{{2{\text{π}} {f_{1,m}}}}{{{f_s}}}n} \right)} } \right\} * h\left( n \right) \\[-8pt] \end{split} $$ (17) $$ R\left( {{\text{j}}\omega } \right) = \frac{1}{{{T_s}}}\left\{ {{U_1}^\prime \left[ {{\text{j}}\left( {\omega - {\omega _l}} \right)} \right]{e^{{\text{j}}{\psi _l}}}} \right\}H\left( {{\text{j}}\omega } \right) $$ (18) 式中,
${\varOmega _l} = {{{\varOmega _l}} \mathord{\left/ {\vphantom {{{\varOmega _l}} {{f_s}}}} \right. } {{f_s}}}$ 为对应于${\varOmega _l}$ 的数字频率。为了实现去斜,即其他子阵的输出信号与第1个子阵相同,可在式(17)或式(18)上引入频率和相位补偿项。
为了补偿第l个子阵与第1个子阵间的频率偏差,其频率补偿项为:
$$ {r_l}^\prime \left( n \right) = {r_l}\left( n \right){{\text{e}}^{ - {\text{j}}{\omega _l}n}} $$ (19) 则频率补偿后输出信号的频域表示为:
$$ \begin{split}& \;\;\;\;\;\; {R_l}^\prime \left( {{\text{j}}\omega } \right) = {R_l}\left[ {{\text{j}}\left( {\omega + {\omega _l}} \right)} \right] = \\ & \frac{1}{{{T_s}}}\left\{ {{U_1}^\prime \left[ {{\text{j}}\omega } \right]{{\text{e}}^{{\text{j}}{\psi _l}}}} \right\}H\left[ {{\text{j}}\left( {\omega + {\omega _l}} \right)} \right] \\[-8pt] \end{split} $$ (20) 为了补偿第l个子阵与第1个子阵间的相位偏差,其相位补偿项为:
$$ {r_l}^{\prime \prime }\left( n \right) = {r_l}^\prime \left( n \right){( {{{\text{e}}^{{\text{j}}{\psi _l}}}} )^*} $$ (21) 则相位补偿后输出信号的频域表示为:
$$ {R_l}^{\prime \prime }\left( {{\text{j}}\omega } \right) = {R_l}^\prime \left( {{\text{j}}\omega } \right){( {{{\text{e}}^{{\text{j}}{\psi _l}}}} )^*} = \frac{1}{{{T_s}}}{U_1}^\prime \left[ {{\text{j}}\left( \omega \right)} \right]H\left[ {{\text{j}}\left( {\omega + {\omega _l}} \right)} \right] \\ $$ (22) 而第1个子阵的数字频率为
${\omega _l}\left| {_{l = 1}} \right. = - 2{\text{π}} \left( {1 - 1} \right) $ $ {{\tau _D}} \mathord{\left/ {\vphantom {{ - 2{\text{π}} \left( {1 - 1} \right){\tau _D}} {{f_s}}}} \right. } {{f_s}} = 0$ ,即第1个子阵的输出信号为:$$ {R_1}^{\prime \prime }\left( {{\text{j}}\omega } \right) = {R_l}^\prime \left( {{\text{j}}\omega } \right){{\text{e}}^{ - {\text{j}}{\psi _l}}} = \frac{1}{{{T_s}}}{U_1}^\prime \left[ {{\text{j}}\left( \omega \right)} \right]H\left[ {{\text{j}}\left( \omega \right)} \right] \\ $$ (23) 对比式(22)和式(23)可以得出,经过上述的频率和相位补偿以后,子阵之间输出信号的差距仅由低通滤波器所致,于是余下的工作就是考虑如何补偿低通滤波器引入的差距。
在低通滤波器的通带内,幅度响应可设定恒为1,相位响应为线性,即:
$$ H\left( {{\text{j}}\omega } \right) = \left| {H\left( {{\text{j}}\omega } \right)} \right|{{\text{e}}^{{\text{j}}K\omega }} $$ (24) 式中,
$K = {{ - {N_1}} \mathord{\left/ {\vphantom {{ - {N_1}} 2}} \right. } 2}$ ,${N_1}$ 为低通滤波器的抽头数。将其代入式(22)可得:$$ {R_l}^{\prime \prime }\left( {{\text{j}}\omega } \right) = \frac{1}{{{T_s}}}{U_1}^\prime \left[ {{\text{j}}\left( \omega \right)} \right] \left| {H\left[ {{\text{j}}\left( {\omega + {\omega _l}} \right)} \right]} \right|{{\text{e}}^{{\text{j}}K\left( {\omega + {\omega _l}} \right)}} $$ (25) 通常频率偏移项
${\omega _l}$ 较小,在通带范围内,不会严重影响低通滤波器的幅度相应,可将其忽略,即:$$ \left| {H\left( {{\text{j}}\omega } \right)} \right| = \left| {H\left[ {{\text{j}}\left( {\omega + {\omega _l}} \right)} \right]} \right| $$ (26) 那么,式(25)可表示为:
$$ {R_l}^{\prime \prime }\left( {{\text{j}}\omega } \right) = \frac{1}{{{T_s}}}{U_1}^\prime \left[ {{\text{j}}\left( \omega \right)} \right] \left| {H\left[ {{\text{j}}\omega } \right]} \right|{{\text{e}}^{{\text{j}}K\left( {\omega + {\omega _l}} \right)}} $$ (27) 此时,第l个子阵与第1个子阵间的关系可表示为:
$$ {R_l}^{\prime \prime }\left( {{\text{j}}\omega } \right) \approx {R_1}^{\prime \prime }\left( {{\text{j}}\omega } \right){{\text{e}}^{{\text{j}}K{\omega _l}}} $$ (28) 即第l个子阵与第1个子阵之间仅存在相位差异。
为补偿低通滤波器引入的相位变化,将式(21)的相位补偿因子
${\psi _l}$ 修正为:$$ {\psi _l}^\prime = {\psi _l} + \left( {{{{\text{π}} {N_1}\left( {l - 1} \right){\tau _D}} \mathord{\left/ {\vphantom {{{\text{π}} {N_1}\left( {l - 1} \right){\tau _D}} f}} \right. } f}} \right) $$ (29) 此时,相位补偿项也可将低通滤波器引入的相位偏差一起消除,最终使得第l个子阵与第1个子阵的输出一致,实现去斜目的。
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通常情况下,式(12)表征的目标回波时延估计误差
$\Delta t \ne 0$ 。根据式(13)可知,$\Delta t$ 会导致频率和相位的偏移,从而影响最终去斜效果。下面分析时延估计误差的边界条件。设回波时延的相对估计误差为:
$$ \Delta e = {{\Delta t} \mathord{\left/ {\vphantom {{\Delta t} {{t_0}}}} \right. } {{t_0}}} $$ (30) 那么第1个子阵内阵元m所接收信号的频率为:
$$ {f_{1,m}} = - \mu \left( {{\tau _m} + \Delta e{t_0}} \right) = \hfill {\text{ }} - \frac{B}{T}\left( {\frac{{\left( {m - 1} \right)d\sin \theta }}{c} + \Delta e\frac{{2R}}{c}} \right) \hfill \\ $$ (31) 设低通滤波器的截止频率为
${B_f}$ ,而在式(31)中由${\tau _m}$ 决定的频率项可以忽略,因此,为确保回波信号在低通滤波器的通带范围内,应满足:$$ \Delta f = \Delta e\frac{B}{T}\frac{{2R}}{c} < {B_f} \Rightarrow \Delta e < {B_f}\frac{T}{B}\frac{c}{{2R}} $$ (32) 即去斜算法所选择的参数保证式(32)成立,回波时延估计误差将不会影响目标回波顺利通过低通滤波器。
在改进的去斜处理中,采用将整个阵列分割为多个子阵的思想,解决了大规模阵列中数据量大的问题,同时通过适当的频率和相位补偿,确保各子阵的输出数据一致,从而实现大规模阵列各输出通道的相干叠加。
A Stretch Processing Method without Digital Delay Filter for Large-Scale Broadband Array Radar
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摘要: 在宽带雷达接收机中,为了降低宽带信号采样对模数转换单元性能的要求,需对接收的宽带信号进行去斜处理。传统去斜处理算法需使用数字时延滤波器,具有设计复杂、成本高的缺点,因此不适合大规模阵列应用。为了解决此问题,提出一种无数字时延滤波器的去斜处理算法,通过频率和相位补偿,使各通道数据达到一致,从而省去数字时延滤波器,简化了系统的设计复杂度。结合子阵划分技术,减少处理通道数量,降低系统硬件成本与数据处理难度。同时,还分析了所提算法中回波时延估计误差的边界条件。仿真结果表明,相比传统算法,该算法输出信噪比更高,且对回波时延估计误差的敏感度更低。Abstract: In a wideband radar receiver, in order to reduce the performance requirements of the analog-to-digital conversion unit for wideband signal sampling, a stretch processing method is required to debase the sampling signal bandwidth for the received broadband signal. Usually, conventional stretch processing algorithms contain a digital delay filter module. Whereas, it is not suitable for large-scale array radar due to its shortcomings of high complexity and expensive cost. To deal with this problem, this paper proposes a stretch processing algorithm without digital delay filter. Specifically, the proposed algorithm exploits frequency and phase compensation to replace the digital delay filter to ensure the consistency of the output of each channel, thereby significantly reducing the complexity of the system. Meanwhile, subarrays are leveraged to reduce the number of channels to be processed, which further reduces the complexity of the system. In addition, the proposed algorithm also equips with the ability to analyze the boundary conditions of the echo delay estimation error. Finally, simulation results show that the proposed algorithm can obtain a higher output signal-to-noise ratio compared with the conventional stretch processing methods, and is less sensitive to echo delay estimation errors.
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