A Joint Parameter Estimation Method Based on 3D Matrix Pencil for Integration of Sensing and Communication
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摘要: 作为一种基于软硬件资源共享和信息共享的新型信息通信技术,通感一体化(ISAC)可将无线感知集成到Wi-Fi平台,为低成本的室内定位提供一种高效的方法。针对室内定位参数估计实时性与准确性问题,该文提出一种基于3维矩阵束(MP)联合参数估计算法。首先,对信道状态信息(CSI)数据进行分析,构建包含到达角(AoA)、飞行时间(ToF)和多普勒频移(DFS)的3维矩阵。其次,对3维矩阵进行平滑处理并利用3维MP算法进行参数估计,通过聚类找到直达径。最后,利用双角定位法进行定位,验证该文所提算法的有效性。实验结果表明,与多重信号分类(MUSIC)参数估计算法相比,无需复杂的峰值搜索步骤,降低了90%计算复杂度。与2维MP算法相比,加入多普勒参数,使AoA估计误差均值在会议室和教室两种场景下分别降低了1.45°和2°。该文通过实际测试验证了所提算法在室内可以达到在置信度67%处平均0.56 m的定位精度。因此,该文所提算法有效地改善了现有室内定位参数估计的实时性和准确性。Abstract:
Objective Integration of Sensing and Communication (ISAC) is an emerging technology that leverages the sharing of software and hardware resources, as well as information exchange, to integrate wireless sensing into Wi-Fi platforms, providing a cost-effective solution for indoor positioning. Existing Wi-Fi-based Channel State Information (CSI) positioning technologies are advantageous in resolving multipath signals in indoor environments, offering finer sensing granularity and higher detection accuracy. These features make them suitable for high-precision target detection and positioning in complex indoor environments, enabling the estimation of parameters such as Angle of Arrival (AoA), Time of Flight (ToF), and Doppler Frequency Shift (DFS). However, CSI-based indoor positioning faces significant challenges. On one hand, the complexity of indoor environments, including reflections from walls and pedestrian movement, reduces the Signal-to-Noise Ratio (SNR), leading to difficulties in effectively estimating signal parameters using traditional algorithms. On the other hand, indoor positioning requires high real-time performance, but most algorithms suffer from high computational complexity, resulting in low efficiency and poor real-time performance. To address these issues, this paper proposes a positioning method based on the three-dimensional (3D) Matrix Pencil (MP) algorithm, which improves the real-time performance and accuracy of existing indoor positioning parameter estimation techniques. Methods To address the real-time and accuracy issues in indoor positioning parameter estimation, a joint parameter estimation algorithm based on the 3D MP algorithm is proposed. First, the CSI data is analyzed, and Doppler parameters are integrated into the two-dimensional (2D) MP algorithm to construct a 3D matrix that includes AoA, ToF, and DFS. The 3D matrix is then smoothened, and the 3D MP algorithm is applied for parameter estimation. Clustering methods are used to obtain the AoA of the direct path, and a weighted least squares method is applied for final target position estimation, while also achieving AoA, ToF, and DFS estimation. This approach effectively improves the resolution and accuracy of parameter estimation. A two-angle positioning method is used for localization to validate the proposed algorithm. By using multiple CSI packets to construct the 3D Hankel Matrix (HM), parameter estimation accuracy is improved compared to using a single CSI packet. Compared to the 3D Multiple Signal Classification (MUSIC) algorithm, the proposed method reduces computational complexity. Incorporating the DFS parameter enhances path resolution, leading to improved AoA parameter estimation accuracy compared to the 2D MP algorithm. Results and Discussions Experiments are conducted in two different scenarios ( Fig. 1 ), with the detailed experimental parameters provided in the table. The two scenarios tested 21 and 13 target positions, respectively. The receiver and transmitter were positioned at the same height, and their geometric relationship was confirmed using a laser rangefinder to determine positioning and direction on the ground. The results indicate that in the conference room scenario, the AoA accuracy and positioning accuracy of the 3D MP algorithm are comparable to those of the MUSIC algorithm, with the 3D MP algorithm showing a significant improvement over the 2D MP algorithm. This is because the 3D MP algorithm introduces an additional dimension to parameter estimation, improving signal resolution and making it easier to identify the direct path of the target (Fig.3). In the classroom scenario, cumulative distribution functions are used to represent overall AoA and positioning errors. For an estimation error of 0.667, the positioning accuracy of the 2D MP, MUSIC, and 3D MP algorithms are 0.73 m, 0.44 m, and 0.48 m, respectively. To observe the real-time performance, each algorithm is run ten times under identical conditions on the same computer, and the average runtime (Fig.5) is recorded. The 2D MP algorithm has the shortest runtime, while the MUSIC algorithm has the longest. The runtime of the 3D MP algorithm is approximately 90% shorter than that of the MUSIC algorithm.Conclusions This paper presents a localization method based on a 3D MP parameter estimation algorithm. A data model for the receiver is first established, and the 3D MP algorithm is introduced. Using a clustering method, the AoA of the direct path is estimated, and multiple Access Points (APs) are combined for target localization. Experimental results show that the proposed algorithm achieves an average localization accuracy of 0.56 m with an estimation error ratio of 0.667, while reducing computational complexity by 90% compared to the MUSIC algorithm. This makes the algorithm highly practical for real-time localization. The results demonstrate that the proposed method significantly reduces computational complexity while maintaining minimal positioning error when compared to the MUSIC algorithm. Although the 3D MP algorithm introduces some computational overhead compared to the 2D MP algorithm, it improves localization accuracy. Parameter estimation and localization experiments in two typical environments confirm that the proposed algorithm outperforms current systems, extending the application of Wi-Fi sensing technology within ISAC. -
1. 引言
强海杂波背景下的弱目标检测是雷达目标检测中较为重要的难题,其检测性能主要受到杂波、噪声和其他干扰的限制,而海杂波的影响最为显著。在现有的检测方法中,基于快速傅里叶变换(Fast Fourier Transform, FFT)滤波器组的单元平均法(Cell Averaging, CA),即FFT-CA法被广泛应用[1]。然而,在杂波谱展宽、多普勒滤波器组能量泄露以及多普勒分辨率较低的短脉冲序列情况下,FFT-CA算法的性能受到严重的影响[1]。为了克服这些缺点,Conte等人[2]采用增加相干积累时间的相干检测算法,提出了自适应归一化匹配滤波器(Adaptive Normalized Matched Filter, ANMF)检测方法,该方法对杂波的结构分量和协方差矩阵都具有恒虚警性,且相对于非相参雷达,显著提高了检测性能。随后一系列基于相干积累检测算法被相继提出并得以应用[3-5]。
对于目标多普勒矢量未知的情况下,相干积累检测算法会存在失配损失。为了克服这一缺点,一类捕获数据之间相关性的矩阵CFAR检测方法被提出[6-17]。类比于常规FFT+CFAR检测方法,Arnaudon等人[6]提出的矩阵CFAR检测器利用检测单元回波相关矩阵与参考单元相关矩阵之间的黎曼均值的测地线距离作为检测统计量,并将其应用于其他雷达目标检测场景中[7]。为了克服测地线距离的能量积累性能有限的缺陷,赵兴刚等人[8-10]利用具有更优积累性能的KLD (Kullback-Leibler Divergence)代替测地线距离,提出了一种改进的矩阵CFAR检测器和基于AR模型的矩阵CFAR检测器,取得了较好的检测和恒虚警性能。Ye等人[11]利用信息几何将角度和多普勒域的多维信息映射到厄米特正定矩阵空间,提出局部矩阵CFAR算法用于低信杂比的高频天波雷达中,获得了不错的检测效果。
上述矩阵CFAR检测器利用迭代算法估计均值矩阵,其较高的计算复杂度限制了其在实际武器装备中的应用。为了降低这类算法的计算复杂度,赵文静等人[12-15]基于特征值分解,提出了采用最大特征值作为检测统计量的矩阵CFAR检测方法(Matrix CFAR Detection method based on the Maximum Eigenvalue, MEMD),在保证恒虚警性能的前提下,利用目标导向矢量的先验信息对数据进行预处理,提出了将频域相干积分与最大特征值方法相结合的预处理的最大特征值矩阵CFAR检测(Maximum Eigenvalue Matrix CFAR Detection Using Pre-Processing, P-MEMD)方法[14],适用于目标多普勒频率偏离杂波中心频谱的场景。随后,赵文静等人[16,17]利用矩阵谱范数来测量矩阵的非相似性提出了两个矩阵CFAR检测器,均获得了较好的检测性能。
当目标的多普勒频率在杂波中心频谱附近时,MEMD的检测性能较好。然而在实际环境中,目标多普勒频率可能在杂波频谱的任何位置出现,故当目标的多普勒频率偏离杂波中心频谱时,其检测问题也亟待解决。于是本文从这一实际环境出发,提出滤波器组与矩阵CFAR相结合的思想,采用滤波器组作为预处理过程[18],并利用最大特征值简化滤波器组的求门限过程,实现对通带外杂波的抑制,对目标多普勒频率进行了准确的定位,结合矩阵CFAR调整门限提高检测性能,对目标能量进行最大限度的积累,实现目标多普勒频率在杂波频谱内和远离杂波频谱时检测性能的提升。
本文的主要结构安排如下:第2节主要介绍海杂波背景下的目标检测模型,第3节提出基于矩阵CFAR的最大特征值检测与滤波器组子带分鲜相结合的FD-MEMD检测算法,第4节通过仿真实验验证本文算法的有效性,第5节给出结论。
2. 检测模型
假设雷达接收回波
y ,在不失一般性的情况下,依据文献[18],雷达信号检测问题可以通过二元假设检验模型表述H0:{y=c˜yk=ck,k=1,2,···,KH1:{y=s+c˜yk=ck,k=1,2,···,K} (1) 其中,在零假设
H0 下,接收到的N 维脉冲数据y∈CN×1 仅由杂波c 组成;在备择假设H1 下,接收到的N 维脉冲数据y 不仅包含杂波c ,还包含目标信号s ,目标信号与杂波是统计独立的。˜yk 为辅助数据,仅由杂波样本组成,K 为参考单元的个数。主要数据y 和辅助数据˜yk 是独立同分布的。借助文献[18],通过式(2)模拟目标回波
s(n)=ˉA√Pca(n)exp⋅{j[4πλ(v0(1−nN)+v1nN)nΔt+φ0]},n=1,2,···,N (2) 其中,
Pc 是海杂波的功率,ˉA 是调整信杂比(Signal to Clutter Ratio, SCR)的正因子,a(n) 是高度相关的幅度序列,λ 是雷达波长,v0 和v1 分别是初始和最终的径向速度,Δt 是雷达的脉冲重复周期,φ0 是随机初始相位。对于目标回波模型式(2)的参数选择,幅度序列
a(n) 是单位平均正随机序列。将模拟目标回波与功率为Pc 的海杂波相加,雷达回波的SCR为20lg(ˉA) 。参数ˉA 由间隔[10−12,101] 的均匀分布产生,即ˉA∼U(10−12,101) ,对应于−10 ~20dB 的SCR范围。实际场景中,海面小目标一般具有较小的速度和加速度[19,20]。对于海面小目标,当观测时间在几秒以内时,简单的恒加速度模型就足够了。假设目标速度服从
[−α,α] 的均匀分布,目标加速度在长度为NΔt 的观测时间间隔内限制在[−β,β] 内,运动方向与雷达视线的夹角服从[−π,π] 的均匀分布。在这些假设下,初始和最终的径向速度为u0=αx,u1=αy,|u0−u1|≤βNΔt,v0=u0cosθ,v1=u1cosθ,θ=πz,x,y,z∼U(−1,1) (3) 其中,随机数
x ,y 和z 相互独立。注意,如果不满足对加速度的约束,则再次生成随机数x 和y 。a(n) 被建模为非负、高度相关的幂次随机序列,依据文献[19],a(n) 的表达形式为a(n)=√3(1+ρ)(1−ρ)√2(2+ρ)(11−ρ+g(n+W)),n=1,2,··· (4) 其中,
g(n) 为通过1阶自回归系统生成的高度相关的序列,g(n+1)=ρg(n)+f(n+1),n=1,2,···;ρ∈(0,1) (5) 独立样本
f(n) 满足f(n)∼U(−1,1) ,其均值为0 ,方差为1/3 ,g(1)=f(1) ,g(n)∈[−11−ρ,11−ρ] 。在自回归系统中,采用足够大的整数W 来避免初始值的过渡效应。3. 基于子带分解最大特征值的矩阵CFAR检测器
基于最大特征值矩阵CFAR检测方法(MEMD)的设计是为了在目标频谱与杂波频谱重叠时获得更好的检测性能和相对其他CFAR检测方法具有更低的计算复杂度。然而,依据文献[14],当目标多普勒频率严重偏离杂波频谱时,MEMD算法的检测性能不佳。在此基础上,一些学者考虑增加预处理过程来减弱杂波对检测性能的影响,并提出了将频域相干积分与最大特征值方法结合在一起的基于预处理的最大特征值检测(P-MEMD)方法。其使用目标导向矢量的先验信息来减弱杂波的影响,进一步提高了MEMD的检测性能。由于海表面的无规则波动,海杂波的能量比较分散,然而海面慢速小目标的回波信号的能量在频域上往往集中在特定频段内。此外,动目标检测依据目标与杂波的能量差异进行目标检测,一般使用各种滤波器,滤去海浪等背景产生的杂波而取出运动目标的回波[21]。基于这一现象,本文利用动目标检测原理,提出了利用滤波器组对接收数据进行滤波处理,然后再与最大特征值相结合的矩阵CFAR检测方法,将其命名为FD-MEMD,以达到保留潜在的有用目标回波信号,同时抑制带外杂波的目的,由此即可提升雷达回波信号的信杂比。后续的仿真实验表明,该方法适用于海面慢速小目标,目标多普勒频率在杂波频谱内和远离杂波频谱均获得了较好的检测性能。
令
{y,˜y1,···,˜yK} 是接收到的数据矩阵,它分为主要数据y 和辅助数据˜y1,˜y2,···,˜yK 。这里,y 表示待检测单元的N 维接收数据(主要数据),˜y1,˜y2,···, ˜yK 表示参考单元中由杂波组成的接收数据(辅助数据)。图1显示了所提出的FD-MEMD方法的检测框图。本文将分DFT调制滤波器组、数据滤波处理、确定目标所在的子带和FD-MEMD检测统计量这4个小节进行介绍。3.1 DFT调制滤波器组
本文使用线性相位DFT调制滤波器组实现子带分解,采用线性相位是为了保持目标回波的相位结构不发生改变,同时,DFT调制滤波器组具有高的阻带抑制能力。
DFT调制滤波器组由一个低通原型滤波器
h(l) 调制而成。不失一般性,假定低通原型滤波器是因果2L 阶线性相位滤波器[18],且满足h(2L−l)= h(l) ,l=0,1,···,2L ,它的频率响应为H(ω)=e−jLω{h(L)+L−1∑l=0h(l)cos(L−l)ω}≡e−jLωdTL(ω)q (6) 其中,
q=[h(0),h(1),···,h(L)]T (7) dL(ω)=[2cos(Lω),···,2cos(ω),1]T (8) 其中,上标
T 表示转置。针对本文的应用,期望原型滤波器h(l) 具有尽可能平坦的通带和高的阻带抑制性,平坦的通带保证了目标回波的幅度和相位具有较小的失真,高的阻带抑制性保证了通带外的杂波能够被抑制。依据文献[18],
2P+1 个通道的DFT调制滤波器组的结构为Hp(ω)=H(ω−2pπ2P+1) (9) 其中,
p=−P,−P+1,···,0,1,···,P ,H(ω)= ∑h(l)e−jωl ,h(l) 是通带位于[−π2P+1,π2P+1] 的低通原型滤波器。只要h(l) 是一个线性相位、因果的、有限冲激响应滤波器,所有子带滤波器hp(l) 都是线性相位、因果的、有限冲激响应滤波器。3.2 数据滤波处理
如图1所示,虚线框为预处理部分,包括一个前置的
2P+1 个通道的线性相位DFT调制滤波器组,其中Xp(n) ,p=−P,−P+1,···,P−1,P 是滤波后的子带时间序列,其表达形式为Xp(n)=hp(l)∗y(n),p=−P,−P+1,···,P−1,P;l=0,1,···,2L (10) 其中,
∗ 表示卷积运算,y(n) 为雷达接收的待检测单元的回波数据。滤波器组主要达到提升雷达回波信杂比的目的,无论目标回波的能量在频域上较为集中或是较为分散,在
2P+1 个滤波器中总会有一个或多个滤波器的通带内保留了目标信号。3.3 确定包含目标信号的子带
ˆp 求出待检测单元的子带信号所对应的相关矩阵
Cp 的最大特征值ζp ,找出2P+1 个最大特征值的最大值η 所对应的子带ˆp ,将第ˆp 个子带指定为包含目标信号的子带,从而实现了抑制杂波。待检测单元的相关矩阵
Cp 表述为Cp=E[XpXHp]=[rp,0···ˉrp,N−1⋮⋱⋮rp,N−1···rp,0] (11) 其中,
Cp 是一个特普利茨厄米特正定矩阵,CHp= Cp ,(.)H 表示共轭转置。根据广义平稳的遍历性,数据Xp 的相关系数可以通过随时间平均而非其统计平均来计算,即rp,i=1NN−1−|i|∑n=0xp(n)ˉxp(n+i),|i|≤N−1 (12) 其中,
xp(n) 是向量Xp 的元素,ˉx 表示x 的复共轭。对
Cp 进行特征值分解,可得Cp=UpΛpUHp (13) 其中,
Up 是由特征向量组成的特征矩阵,Λp 是由特征值[λp,1,λp,2,···,λp,M] 组成的对角矩阵。第p 个子带相关矩阵Cp 对应的最大特征值ζp 为ζp=max(λp,1,λp,2,···,λp,M) (14) 于是,所有
2P+1 个子带的特征值的最大值η 为η=max(ζ−P,ζ−P+1,···,ζP) (15) 因此,确定最大特征值
η 对应的子带为ˆp ˆp:η≡ζˆp (16) 其中,
ˆp∈{−P,−P+1,···,0,1,···,P} ,即为目标信号所在的子带。于是,利用
η≡ζˆp 可以确定出第ˆp 个子带的接收回波的滤波数据Xˆp 。3.4 FD-MEMD的检测统计量
利用第
ˆp 个子带的滤波器对所接收到的参考数据˜yk(n) 进行滤波处理,即˜Xˆp,k=hˆp(l)∗˜yk(n),k=1,2,···,K (17) 参考数据第
ˆp 个子带的滤波数据˜Xˆp,k 的相关矩阵˜Cˆp,k 为˜Cˆp,k=E[Xˆp,kXHˆp,k],k=1,2,···,K (18) 借助式(11)、式(12)和式(13),对
˜Cˆp,k 进行特征值分解,并求得第ˆp 个子带第k 个距离单元的最大特征值˜ξk ,则所有K 个参考单元中最大特征值˜ξk 的算术平均值为δ=1KK∑k=1˜ξk (19) 同理,待检测单元第
ˆp 个子带的滤波数据Xˆp 的相关矩阵Cˆp 为Cˆp=E[XˆpXHˆp] (20) 依据相同的处理,对
Cˆp 进行特征值分解,并求得该第ˆp 个子带的最大特征值为ξ 。最后,依据矩阵CFAR的原理,FD-MEMD检测统计量可表示为
ξδ≷ (21) 其中,
\hat \eta 为FD-MEMD的检测阈值。当检测统计量大于阈值\hat \eta 时,判断为目标出现在检测单元中,否则判为目标在检测单元中未出现。4. 仿真实验及结果分析
该节主要基于真实海杂波数据进行了仿真实验,验证本文提出的FD-MEMD算法的有效性。在仿真实验中,使用由加拿大McMaster大学提供的IPIX海杂波数据(19980205_185111_ANTSTEP.CDF)。该数据由27个距离单元组成,每个距离单元包含60000个脉冲数据。对于这个数据集,本文获取实测数据的多普勒频谱图,如图2所示,经过FFT变换,估计海杂波的平均多普勒频率约为–222 Hz。考虑到目标多普勒在杂波频谱内外的情况,利用式(2),添加仿真目标之后的海杂波和目标的混合频谱图如图3所示。
由于有限的实测数据,在本文的仿真实验中,虚警概率被设置为
{10^{{\rm{ - }}3}} ,脉冲数N = 8 ,脉冲重复频率为1000 Hz,雷达射频频率{{F}} 为9.39 GHz。参考单元个数K = 16 ,极化方式为HH, HV, VH和VV。对于滤波器组仿真模型,P = 8 ,L = 25 。对于目标仿真模型,\rho 取0.96,整数W 取500。考虑了目标多普勒频率在杂波频谱内外的两种情况,利用公式{{f_{\rm{d}}}} = \dfrac{{vF}}{\rm{c}} ,目标多普勒频率分别被设置为–222 Hz和215 Hz。其中{{f_{\rm{d}}}} 是目标多普勒频率,v 是目标相对雷达的径向速度,F 是雷达射频,{\rm{c}} 是电磁波的传播速度。图4(a)、图4(c)、图4(e)和图4(g)对应于杂波频谱与目标频谱重叠的检测场景。可以看出FD-MEMD方法的检测性能明显优于ANMF, sub-ANMF以及P-MEMD。值得注意的是,上述图中的FD-MEMD方法相对于P-MEMD方法的性能有明显的改善,这主要是因为P-MEMD利用对检测单元和参考单元接收到的数据进行FFT变换,将其在频域上分解成N个多普勒频段,利用峰值搜索找到目标所在的多普勒频段,从而抑制杂波。而FD-MEMD方法利用滤波器组将接收到的数据分解成2P+1个子带信号,通过查找最大特征值的最大值所在的子带信号从而确定目标所在的子带,目标子带之外的杂波全被滤除,相对于P-MEMD方法,杂波抑制更明显。图4(b)、图4(d)、图4(f)和图4(h)描述了目标多普勒频率在杂波频谱之外的情况下的检测性能的比较。可以看出,FD-MEMD方法的检测性能同样优于ANMF, sub-ANMF和P-MEMD。总之,FD-MEMD方法在目标多普勒频率位于杂波频谱之内和之外的两种场景下都具有稳健的性能,更适用于在实际环境中应用。
5. 结束语
本文主要解决海杂波背景下海面目标检测问题,旨在改善目标多普勒频率严重偏离杂波频谱时矩阵CFAR检测性能较低的状况。本文依据目标回波的能量在频域上相对集中而海杂波能量在频域上相对分散这一特点,提出一种基于子带分解的矩阵CFAR最大特征值检测方法。该方法有效地抑制了杂波,提高了信杂比及检测性能。仿真结果表明,该方法不仅在强杂波的环境下具有良好的检测性能,在目标多普勒频率偏离杂波频谱的情况下也具有较好的检测性能,能够在实际海洋环境中应用。
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表 1 实验参数
参数名称 符号 数值 接收天线数量 N 4 子载波数量 M 49 包的数量 B 10 矩阵束参数1 {M_{\rm p}} 25 矩阵束参数2 {N_{\rm p}} 2 矩阵束参数3 {B_{\rm p}} 5 表 2 实验参数
算法 主要步骤 算法复杂度 参考数值 MUSIC算法 特征值分解 \begin{gathered} \left\{ {{{(BMN)}^2}\left( {BMN - q} \right) + {{(BMN - q)}^2}BMN + {{(BMN)}^2}} \right\} \\ \times {\mathrm{sr}}\_{\mathrm{AoA}} \times {\mathrm{sr}}\_{\mathrm{ToF}} \times {\mathrm{sr}}\_{\mathrm{DFS}} \\ \end{gathered} 1.25×1016 峰值搜索 2维MP算法 离散傅里叶变换 \dfrac{{11}}{4}{({M_{\rm p}}{N_{\rm p}})^3} + 4{({M_{\rm p}}{N_{\rm p}})^2}{K_M}{K_N} 1.28×106 奇异值分解 3维MP算法 奇异值分解 11{({B_{\rm p}}{M_{\rm p}}{N_{\rm p}})^3} + 4{({B_{\rm p}}{M_{\rm p}}{N_{\rm p}})^2}2{K_B}{K_M}{K_N} 2.84×108 -
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