随机投影的观测方法及其在超宽带信号采样中的应用
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摘要
在现代信号处理中，人们对信息需求量的不断增长使得携带信息的信号带宽变得越来越宽，对信号数字化带来了极大的挑战。因此对超宽带信号的数据获取方法研究已成为信号处理领域中一个非常重要的研究内容。如果根据Nyquist采样定理对超宽带信号采样，鉴于现有ADC（Analog-to-Digital Converter）芯片的工艺水平，单个ADC芯片所获得的样本无法同时具有高采样率和高分辨率的性质。在本文中，我们针对超宽带信号的数据获取方法展开研究探讨，一方面，在具有信号先验的合作性环境下，研究了基于压缩感知理论，利用主动式雷达的先验知识，低速率获取超宽带回波信号的方法。为了有效地应用压缩感知理论对超宽带雷达信号进行低速获取，分别对雷达回波信号的稀疏表示方法和模拟信号的随机观测方法进行了研究。另一方面，在无法获得信号先验的非合作环境下，研究超宽带信号的高速率高分辨率获取方法。
1．我们从模拟信息转换器（Analog-to-Information Converter, AIC）入手，研究其构成组件和等效的观测矩阵。然后将其应用于雷达回波信号的低速获取，提出了基于AIC结构的雷达回波信号获取系统。在构造了回波信号的稀疏字典的前提条件下，所提出的系统可以实现对回波信号的随机观测与优化重构。该系统的应用场景是使用主动式雷达对静态点目标进行探测，因此可以根据感兴趣目标的特点确定回波信号的数学模型。在通常使用的主动式探测模型中，由多个目标回波构成的回波信号可以表示为发射信号移位形式的线性叠加。因此，把发射信号离散化后进行能量归一化处理，利用处理后信号的多个时移形式构成字典，可达到稀疏表示回波信号的目的。这样，通过构造回波信号的稀疏字典，可以满足应用压缩感知理论对雷达回波信号进行低速观测的前提条件，保障回波信号低速观测后能够被高概率地成功重构。
2．考虑到AIC结构对信号的稀疏变换矩阵缺乏普适性，为了在理论上得到对信号变换具有普适性的观测矩阵，即不论稀疏表示处理采用的基函数具有何种时长特性，对系数向量的观测矩阵（即信号的观测矩阵与信号变换矩阵的乘积）都具有良好的性质，提出了基于随机投影的并行采样结构（Parallel Sampling StructureBased on Random Projection, PSRP）。在该多通路结构中，每个通道对信号进行随机调制、积分与采样，得到信号的一个观测数据。由于PSRP结构的等效观测矩阵具有良好性能，因此，把该结构用作观测结构对雷达回波信号进行观测时，与基于AIC结构的雷达回波获取系统相比，为了达到同等的重构质量，采用所提出的PSRP结构的系统需要的观测数量会大大减少。而且后者的噪声鲁棒性能更优。
3．为了在实践中降低观测系统的复杂度，使其与常规的A/D转换电路具有兼容性，提出了直接欠采样率随机感知（Direct Sub-Nyquist Random Sampling, DSRS）方法，通过调整常规基于Nyquist定理的A/D电路的采样时序即可实现对信号的随机观测。除了需要重新设计采样时钟，传统采样电路的其他部分并不需要重新设计就可以用于压缩感知框架下的信号观测处理。利用DSRS结构对信号进行观测处理，大大降低了压缩感知框架下信号采集系统的尺寸、重量、耗能以及成本。把该结构用于超宽带雷达接收系统对回波信号进行观测，仿真实验验证了在一定的条件下，该结构可以在压缩感知的框架下实现对信号随机观测的目的。
4．在非合作性环境下，研究了超宽带信号的高速率高分辨率采样方法。对一般的超宽带信号，我们仍然沿用传统的信号模型，即待采样信号是带限的。为了同时达到高速率和高分辨率，本文中超宽带信号的采样使用了所提出的基于随机投影的并行采样结构，即PSRP结构。使用M通道的PSRP结构进行信号采样时，由于每个通道中ADC的速率是整个结构采样速率的1/M，因此每个通道可使用低速率、高分辨率的ADC芯片，该结构的输出是高速率高分辨率的信号样本。不同于基于压缩感知的采样系统，该系统既没有利用信号的先验信息，也不把降低采样数量作为目的，所以获取的数据量与未知的信号样本量相同。因此信号重构可通过快速求解一个线性方程组实现。
为了在并行系统中控制通道数，进而降低系统实现复杂度，对信号需要进行分段处理，分段处理会带来截断误差。为了抑制截断误差，本文给出了两种解决策略。一种是基于软件计算的方法，采用反对称的信号扩展技术，该方法虽然不能彻底消除截断误差，但不会带来硬件设备上的负担。另一种方法是在信号投影采样处理前，采用一个高速的采样保持器（Sampling-and-Hold，S/H）对信号进行预处理，该处理可以完全消除截断误差，但S/H电路会在一定程度上提高系统成本。可以根据实际情况与需要，选择使用反对称扩展技术或高速S/H电路的并行采样系统以满足不同精度的高速率信号获取需求。
In modern signal processing, since the requirement of information amountincreases continually, the band of signal for carrying information becomes wider andwider, which brings a huge challenge for signal discretization. Therefore the research ondata acquisition for ultra wideband (UWB) signal became very important in the area ofsignal processing. If the sampling of UWB signal is implemented based on the Nyquisttheorem, the properties of high speed and high resolution for the samples cannot bepossessed at the same time, considering the technology of current ADC chips. Thispaper studies the data acquisition of UWB signal for two different situations. First,considering a cooperative environment in which some prior knowledge can be obtained,we study the measuring method for the echo signal in UWB active radar based on thetheory of compressed sensing, in which the prior knowledge of active radar is usedappropriately. In order to effectively measure the echo signal of UWB radar at low rate,we focus on the study of the method for representing the echo signal and the measuringmethod for analog signal. Second, for a non-cooperative environment in which no priorknowledge can be obtained besides the signal’s bandwidth, we study the approach tosampling and reconstructing UWB signal with properties of high speed and highresolution.
1. We start with the analog-to-information converter (AIC) by analyzing itscomponents and equivalent matrices, and then we apply it to measure the echo signal ofUWB radar at low rate. If the echo signal can be sparsely represented in a dictionary, theproposed system can achieve the random measuring of signal and the recovering ofsignal by optimization. The scene of this problem is set to be utilizing an active radar todetect static scattered targets. So the mathematical model of the echo signal can beproperly designed based on the feature of the interested targets. The conventional modelfor active detection is employed in this paper, in which the echo signal containingmultiple target echoes can be represented as the linear summation of the shifted versionof the transmitted signal. A dictionary can be constructed with the shifted versions of aprototype atom which is generated by discretizing and normalizing the transmittedsignal. Therefore, the constructed dictionary can represent the echo signal sparsely. As aresult, the precondition of applying compressed sensing theory on the echo signal inUWB active radar is satisfied, and the successful reconstruction of signal with highprobability can be guaranteed.
2. Considering the disadvantages of the AIC structure, i.e., the lack of universality forthe transform space of sparse signal representation, in order to get a measurementmatrix with universality for signal representation, that is to say, the equivalentmeasurement matrix for the sparse coefficients vector will have good properties, nomatter how long the duration of the basis functions is, we propose a parallel samplingstructure based on random projection (PSRP). Here the equivalent measurement matrixfor the coefficients vector is the product of the measurement matrix and the transformmatrix for the signal. Each channel of the proposed PSRP structure is composed with arandom modulator, an integrator and an ADC. The output of each channel is ameasurement of input signal. Since the equivalent measurement matrix of the PSRPstructure is with good properties, when the PSPR structure is used as a measuringstructure in a data acquisition system for the echo signal in radar, compared with theacquisition system using AIC structure, the acquisition system using the PSRP structurerequires less number of measurements to achieve the same performance of signalreconstruction. In addition, the latter is more robust to noise.
3. In order to reduce the complexity of the measuring structure and make itcompatible with the conventional sampling circuit, the direct sub-Nyquist randomsampling (DSRS) method is presented. In the proposed DSRS method, the randommeasurement of signal can be realized by adjusting the trigger time of the conventionalNyquist-based sampling circuit. For a conventional sampling circuit, besides theredesign of sampling clock, there is no need to change the other components to measurea signal based on compressed sensing theory. When the proposed DSRS structure isutilized in the measuring process in the framework of compressed sensing, the size,weight, energy consumption and cost of signal measuring will decrease greatly. Whenthe DSRS structure is employed in the data acquisition system for the echo signal basedon compressed sensing, under certain conditions, it can accomplish the low-ratemeasurement of signal, which is demonstrated by the simulation results.
4. In non-cooperative (or passive) environment, we research the method for acquiringgeneral UWB signal with high speed and high resolution. For general UWB signal, wefollow the conventional signal model in which the signal to be sampled is assumed to beband-limited. In order to get the high sampling rate and high resolution simultaneously,the sampling of the UWB signal in this paper is achieved by a sampling system with theparallel sampling structure based on random projection (PSRP). When M-channel PSRPstructure is used to sample signal, since the speed of ADC in channel is1/M of thespeed of the whole structure, the low-rate high-resolution ADC chip can be adopted, but the output of the PSRP structure is with properties of high rate and high resolution.Different from the data acquisition system based on compressed sensing, for theproposed system, no prior knowledge on signal besides the signal’s bandwidth is usedand the reduction of sample amount is not the purpose. So the number of themeasurement equals to the number of unknown Nyquist samples. Therefore the signalreconstruction can be accomplished by solving a linear equation, which can beimplemented very fast.
In order to make the channel number in the PSRP structure as small as possible andmake the complexity of the sampling system as low as possible, the signal is segmentedinto pieces with equal length, which will cause the segmentation error. In this paper, weprovide two strategies to suppress the segmentation error. One is software-basedanti-symmetric signal extension technique which will not bring any burden on hardwareimplement. However, the segmentation error cannot be eliminated completely. Theother is to preprocess the input signal with a high-rate sampling-and-hold circuit, whichcan eliminate the segmentation error completely. But the system cost will increase atsome extent due to the exact device. According to the practical situation and needs, thechoice of parallel sampling system using the ASE technique or the high rate S/H canmeet the different precision of high-rate data acquisition.
1．我们从模拟信息转换器（Analog-to-Information Converter, AIC）入手，研究其构成组件和等效的观测矩阵。然后将其应用于雷达回波信号的低速获取，提出了基于AIC结构的雷达回波信号获取系统。在构造了回波信号的稀疏字典的前提条件下，所提出的系统可以实现对回波信号的随机观测与优化重构。该系统的应用场景是使用主动式雷达对静态点目标进行探测，因此可以根据感兴趣目标的特点确定回波信号的数学模型。在通常使用的主动式探测模型中，由多个目标回波构成的回波信号可以表示为发射信号移位形式的线性叠加。因此，把发射信号离散化后进行能量归一化处理，利用处理后信号的多个时移形式构成字典，可达到稀疏表示回波信号的目的。这样，通过构造回波信号的稀疏字典，可以满足应用压缩感知理论对雷达回波信号进行低速观测的前提条件，保障回波信号低速观测后能够被高概率地成功重构。
2．考虑到AIC结构对信号的稀疏变换矩阵缺乏普适性，为了在理论上得到对信号变换具有普适性的观测矩阵，即不论稀疏表示处理采用的基函数具有何种时长特性，对系数向量的观测矩阵（即信号的观测矩阵与信号变换矩阵的乘积）都具有良好的性质，提出了基于随机投影的并行采样结构（Parallel Sampling StructureBased on Random Projection, PSRP）。在该多通路结构中，每个通道对信号进行随机调制、积分与采样，得到信号的一个观测数据。由于PSRP结构的等效观测矩阵具有良好性能，因此，把该结构用作观测结构对雷达回波信号进行观测时，与基于AIC结构的雷达回波获取系统相比，为了达到同等的重构质量，采用所提出的PSRP结构的系统需要的观测数量会大大减少。而且后者的噪声鲁棒性能更优。
3．为了在实践中降低观测系统的复杂度，使其与常规的A/D转换电路具有兼容性，提出了直接欠采样率随机感知（Direct Sub-Nyquist Random Sampling, DSRS）方法，通过调整常规基于Nyquist定理的A/D电路的采样时序即可实现对信号的随机观测。除了需要重新设计采样时钟，传统采样电路的其他部分并不需要重新设计就可以用于压缩感知框架下的信号观测处理。利用DSRS结构对信号进行观测处理，大大降低了压缩感知框架下信号采集系统的尺寸、重量、耗能以及成本。把该结构用于超宽带雷达接收系统对回波信号进行观测，仿真实验验证了在一定的条件下，该结构可以在压缩感知的框架下实现对信号随机观测的目的。
4．在非合作性环境下，研究了超宽带信号的高速率高分辨率采样方法。对一般的超宽带信号，我们仍然沿用传统的信号模型，即待采样信号是带限的。为了同时达到高速率和高分辨率，本文中超宽带信号的采样使用了所提出的基于随机投影的并行采样结构，即PSRP结构。使用M通道的PSRP结构进行信号采样时，由于每个通道中ADC的速率是整个结构采样速率的1/M，因此每个通道可使用低速率、高分辨率的ADC芯片，该结构的输出是高速率高分辨率的信号样本。不同于基于压缩感知的采样系统，该系统既没有利用信号的先验信息，也不把降低采样数量作为目的，所以获取的数据量与未知的信号样本量相同。因此信号重构可通过快速求解一个线性方程组实现。
为了在并行系统中控制通道数，进而降低系统实现复杂度，对信号需要进行分段处理，分段处理会带来截断误差。为了抑制截断误差，本文给出了两种解决策略。一种是基于软件计算的方法，采用反对称的信号扩展技术，该方法虽然不能彻底消除截断误差，但不会带来硬件设备上的负担。另一种方法是在信号投影采样处理前，采用一个高速的采样保持器（Sampling-and-Hold，S/H）对信号进行预处理，该处理可以完全消除截断误差，但S/H电路会在一定程度上提高系统成本。可以根据实际情况与需要，选择使用反对称扩展技术或高速S/H电路的并行采样系统以满足不同精度的高速率信号获取需求。
In modern signal processing, since the requirement of information amountincreases continually, the band of signal for carrying information becomes wider andwider, which brings a huge challenge for signal discretization. Therefore the research ondata acquisition for ultra wideband (UWB) signal became very important in the area ofsignal processing. If the sampling of UWB signal is implemented based on the Nyquisttheorem, the properties of high speed and high resolution for the samples cannot bepossessed at the same time, considering the technology of current ADC chips. Thispaper studies the data acquisition of UWB signal for two different situations. First,considering a cooperative environment in which some prior knowledge can be obtained,we study the measuring method for the echo signal in UWB active radar based on thetheory of compressed sensing, in which the prior knowledge of active radar is usedappropriately. In order to effectively measure the echo signal of UWB radar at low rate,we focus on the study of the method for representing the echo signal and the measuringmethod for analog signal. Second, for a non-cooperative environment in which no priorknowledge can be obtained besides the signal’s bandwidth, we study the approach tosampling and reconstructing UWB signal with properties of high speed and highresolution.
1. We start with the analog-to-information converter (AIC) by analyzing itscomponents and equivalent matrices, and then we apply it to measure the echo signal ofUWB radar at low rate. If the echo signal can be sparsely represented in a dictionary, theproposed system can achieve the random measuring of signal and the recovering ofsignal by optimization. The scene of this problem is set to be utilizing an active radar todetect static scattered targets. So the mathematical model of the echo signal can beproperly designed based on the feature of the interested targets. The conventional modelfor active detection is employed in this paper, in which the echo signal containingmultiple target echoes can be represented as the linear summation of the shifted versionof the transmitted signal. A dictionary can be constructed with the shifted versions of aprototype atom which is generated by discretizing and normalizing the transmittedsignal. Therefore, the constructed dictionary can represent the echo signal sparsely. As aresult, the precondition of applying compressed sensing theory on the echo signal inUWB active radar is satisfied, and the successful reconstruction of signal with highprobability can be guaranteed.
2. Considering the disadvantages of the AIC structure, i.e., the lack of universality forthe transform space of sparse signal representation, in order to get a measurementmatrix with universality for signal representation, that is to say, the equivalentmeasurement matrix for the sparse coefficients vector will have good properties, nomatter how long the duration of the basis functions is, we propose a parallel samplingstructure based on random projection (PSRP). Here the equivalent measurement matrixfor the coefficients vector is the product of the measurement matrix and the transformmatrix for the signal. Each channel of the proposed PSRP structure is composed with arandom modulator, an integrator and an ADC. The output of each channel is ameasurement of input signal. Since the equivalent measurement matrix of the PSRPstructure is with good properties, when the PSPR structure is used as a measuringstructure in a data acquisition system for the echo signal in radar, compared with theacquisition system using AIC structure, the acquisition system using the PSRP structurerequires less number of measurements to achieve the same performance of signalreconstruction. In addition, the latter is more robust to noise.
3. In order to reduce the complexity of the measuring structure and make itcompatible with the conventional sampling circuit, the direct sub-Nyquist randomsampling (DSRS) method is presented. In the proposed DSRS method, the randommeasurement of signal can be realized by adjusting the trigger time of the conventionalNyquist-based sampling circuit. For a conventional sampling circuit, besides theredesign of sampling clock, there is no need to change the other components to measurea signal based on compressed sensing theory. When the proposed DSRS structure isutilized in the measuring process in the framework of compressed sensing, the size,weight, energy consumption and cost of signal measuring will decrease greatly. Whenthe DSRS structure is employed in the data acquisition system for the echo signal basedon compressed sensing, under certain conditions, it can accomplish the low-ratemeasurement of signal, which is demonstrated by the simulation results.
4. In non-cooperative (or passive) environment, we research the method for acquiringgeneral UWB signal with high speed and high resolution. For general UWB signal, wefollow the conventional signal model in which the signal to be sampled is assumed to beband-limited. In order to get the high sampling rate and high resolution simultaneously,the sampling of the UWB signal in this paper is achieved by a sampling system with theparallel sampling structure based on random projection (PSRP). When M-channel PSRPstructure is used to sample signal, since the speed of ADC in channel is1/M of thespeed of the whole structure, the low-rate high-resolution ADC chip can be adopted, but the output of the PSRP structure is with properties of high rate and high resolution.Different from the data acquisition system based on compressed sensing, for theproposed system, no prior knowledge on signal besides the signal’s bandwidth is usedand the reduction of sample amount is not the purpose. So the number of themeasurement equals to the number of unknown Nyquist samples. Therefore the signalreconstruction can be accomplished by solving a linear equation, which can beimplemented very fast.
In order to make the channel number in the PSRP structure as small as possible andmake the complexity of the sampling system as low as possible, the signal is segmentedinto pieces with equal length, which will cause the segmentation error. In this paper, weprovide two strategies to suppress the segmentation error. One is software-basedanti-symmetric signal extension technique which will not bring any burden on hardwareimplement. However, the segmentation error cannot be eliminated completely. Theother is to preprocess the input signal with a high-rate sampling-and-hold circuit, whichcan eliminate the segmentation error completely. But the system cost will increase atsome extent due to the exact device. According to the practical situation and needs, thechoice of parallel sampling system using the ASE technique or the high rate S/H canmeet the different precision of high-rate data acquisition.
引文
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