宽带信号采样若干关键技术研究
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摘要
现代通信与雷达系统的主流发展趋势是宽带化和软件化。有效的宽带信号采样与重构方法是顺应该发展趋势的必要基础，然而由于集成电路制造工艺和一些特殊应用条件的限制，基于传统采样结构和理论框架的采样系统越发难以满足实际需求。因此，研究高效采样结构和扩展的采样理论具有重要的理论意义和实用价值。
时间交替模数转换器（TIADC）是一种针对带宽有限的宽带信号进行采样的高效并行结构，而有限新息率（FRI）采样框架则是对一类非带限信号进行高效采样的理论基础，是基于空间映射概念对原始Shannon采样理论的扩展。本论文针对TIADC结构中存在的通道失配误差，以非线性混合滤波器组（NLFB）为基本模型，以二态Markov链、微分滤波器组和分数延时滤波器为主要工具，开展了自适应后端校准方法的研究。对于FRI采样框架，则重点对复杂脉冲波形情况下，系统设计过程中的内在约束关系，以及噪声环境下的重构稳健性问题进行了研究。本论文的主要研究成果概括如下：
1、针对TIADC通道间的直流偏置失配误差，本论文提出了基于数据随机化操作的自适应后端校准方法。该部分首先分析并验证了通道数据直接平均法的缺陷，然后根据缺陷产生的原因以及直流偏置误差引入的机制，在采样保持器和量化比较器之间引入了数据随机化操作。当随机序列均值为0且与输入信号不相关时，可以通过最佳线性估计算子得到通道直流偏置误差的无偏估计。鉴于最佳线性估计算子的运算复杂性，提出了采用直接平均算子作为次最佳估计，并证明了该算子与最佳线性估计算子的渐进一致性。为了便于物理实现，且能够提供更大的设计自由度，本论文用二态Markov链作为随机化操作序列的生成工具。理论分析和实验结果表明，通过增加采样值以及合理设置Markov链状态转移概率可以有效提高校准性能，而且该方法在系统同时存在静态增益失配误差和时间失配误差情况下依然有效。
2、针对TIADC通道间的静态增益和时间失配误差，本论文提出了基于微分滤波器组的全数字域自适应后端校准方法。该方法通过设计时序关系，使各个子通道与同一参考通道周期性地同时对输入信号进行采样，进而确立了参考通道和子通道采样值之间的泰勒级数关系；根据此关系，设计了基于微分滤波器组的自适应补偿结构，以实现同时对静态增益和时间失配误差进行校准。另外，本文在频域上分析了微分滤波器组与分数延时滤波器的等效关系，使所提出的校准方法能够简单地应用于DSP平台。仿真实验结果表明，在轻微过采样状态下，所提出的方法对点频信号和频带有限信号都能有效地校准失配误差；在已知信号所处Nyquist频带条件下，该方法对基带采样和带通采样的情况都能有效校准；校准性能与微分滤波器阶数及泰勒级数展开的长度有关，若要提高系统校准精度并降低过采样因子，需提高各个微分滤波器的阶数并适当选取泰勒展开的长度。
3、针对基于复杂脉冲波形的FRI信号，本论文推导了FRI采样核与输入信号脉冲波形之间的约束关系，在此基础上分别对基于复杂脉冲波形的周期FRI信号和有限长FRI信号给出了采样系统的设计方法。其一，对于周期FRI信号，需要采用sinc采样核或sinc函数加权求和（SoS）采样核，并通过对输入信号进行适当频移来满足约束关系；其二，对于有限长FRI信号，需要采用SoS采样核或指数再生采样核。当采用SoS采样核时，有限长FRI信号与采样核之间的约束关系与该采样核应用于周期FRI信号时的约束关系一致；当采用指数再生采样核时，约束关系由采样核参数确定，因此，对于不同的脉冲波形，可以通过合理设计采样核参数来满足约束关系。除此之外，本文详细分析了系统稳定性和物理实现等因素对采样核参数的约束，并给出了参数设计方法。实验仿真结果表明，本文设计方法能有效提高系统重构的性能。
4、针对FRI采样系统在噪声环境中的重构稳健性问题，本论文提出了两种解决办法，分别是改进的指数样条采样核和子空间白化预处理。在噪声环境中，需要用稳健的重构算法估计FRI信号参数，但已有重构算法都要求采样值中的噪声成分是加性白噪声。由于采样核的滤波作用，这一条件有时难以满足。因此，本文基于采样核对采样值的作用机理，并依据指数再生采样核固有的卷积性质，在频域对原始的指数样条函数重新进行设计，使其能够保持噪声原有的特性。对于改进采样核方法受限的情况，则可以通过子空间白化算法对数据进行预白化来保证重构算法的前提条件。仿真结果表明，两种方法都能有效改善系统重构性能，且相对而言，改进指数样条采样核方法效果更好，而子空间白化方法适用性更强。
The trend of modern communication and Radar systems are wideband andsoft-defined. Effective methods for sampling and reconstructing wideband signals areessentials to conform to the trend. However, the sampling systems based on traditionalschemes and theory become more and more difficult to meet requirements due to thelimitations of current IC processing and some especial application conditions. So,research of high efficient sampling schemes and general sampling theory are topics oftheoretical interest and also of great relevance to applications.
Time-Interleaved ADC is a kind of high efficient parallel sampling scheme aimingat wideband signals with limited bandwidth, while the framework of finite rate ofinnovation (FRI) is the basic theory to sample a kind of non-bandlimited signalseffectively. It is a generalization of standard Shannon’s sampling theorem from the viewpoint of space projection. Based on non-linear filter bank (NLFB) model, two stateMarkov chains, differential filter bank and fractional delay filter, this dissertationstudies adaptive backward calibration method for channel mismatch errors. For theframework of FRI, studies are conducted in this dissertation, mainly on the internalconstraint relationship between sampling system and complex pulse shapes, and on therobustness of reconstruction in noisy scenario.
The main contents of this dissertation are summarized as follows:
1. Focus on the offset mismatch error of TIADC, an adaptive backward calibrationmethod based on data-randomizing operation is proposed. First, this part analyzesand verifies the defect of direct data-averaging method, and then, based on thereason of defect and the mechanism of offset error, we introduce a randomizingoperator to the data between sample-and-hold (S/H) and quantizer. When therandom series is zero-mean and uncorrelated with input signal, an unbiasedestimation of the channel offset error can be got by the best linear unbiasedestimator. In the light of high computational complexity of best linear unbiasedestimator, a suboptimal estimator based on data-averaging is proposed. Theasymptotic equivalence between the data-averaging estimator and the best linearunbiased estimator is proved. In order to implement easily and to provide moredegrees of freedom for design, the two state Markov chain is used as the randomseries generator. Analysis and simulation results verify that the performance ofcalibration will be improved by increasing the number of samples and setting the transition probability of Markov chain, and that the proposed method is effectiveeven if gain mismatch errors and timing mismatch errors are exist.
2. Focus on the gain mismatch errors and timing mismatch errors, a digital adaptivebackward calibration method based on differential filter bank is proposed. First, bydesigning the sampling timing, each sub-channel is synchronized with the samereference channel periodically to sample input signals, and then, the samples of thereference channel can be expressed as the Taylor’s series of the samples ofsub-channels. According to this relationship, an adaptive calibration architecture isdesigned for both gain mismatch errors and timing mismatch errors. Therelationship between differential filter bank and fractional delay filter is analyzed,which provides a simple way to apply the proposed method to DSP platforms.Simulation results demonstrate that the proposed method is fit for both multi-tonesignals and bandlimited signals when they are slightly oversampled, and can workin both baseband and bandpass scenarios on the condition that we know the Nyquistband of input signal in advance. The performance of the proposed calibrationmethod is determined by the taps of differential filters and the length of Taylor’sexpansion. In order to improve precision of calibration and reduce the oversampling factor, the taps of differential filters must be increased, with the length ofTaylor expansion set properly.
3. For the FRI signals with complex pulse shape, the constraint relationship betweensampling kernels and input signals is deduced from the reconstructing mechanismof FRI framework, and then, the design method of sampling system for periodicFRI signals and finite length FRI signals is proposed. For the periodic FRI signals,sinc or sum of sincs (SoS) sampling kernels should be chose, and frequency shiftoperation can be used to meet the constraint. For the finite length FRI signals, weshould choose SoS or exponential reproducing sampling kernels. When SoSsampling kernels are used, the constraint between sampling kernels and inputsignals is the same as that of the SoS sampling kernels used for the periodic signals.When exponential reproducing sampling kernels are chose, the constraint isdetermined by the parameters of sampling kernels, so we can set properly theparameters of sampling kernels for different complex pulse shape to satisfy theconstraint. In addition, we also analyze the constraint from the consideration ofrobustness of system and practical implement, and then, give the setting method ofsampling kernels. The simulations results demonstrate that the performance ofreconstruction can be improved by our design methods.
4. Focus on the robustness of reconstruction of FRI sampling system in noisyscenarios, two solutions are given out, which are modified exponential splinesampling kernels and subspace prewhitening method. When noise exists, robustreconstructing algorithms must be used. However, these algorithms require that thenoise components in samples are additive white noise. Due to the filtering ofsampling kernels, the samples may not meet this requirement. So, based on themechanism of action of sampling kernel on samples and the inherent convolutionproperty of exponential reproducing sampling kernels, we modify the originalexponential spline in frequency domain to keep the characteristic of noisecomponents. For the scenarios that the modified exponential spline samplingkernels are restricted, we can prewhiten the noise components by subspacewhitening method before reconstruction to satisfy the prerequisite. The simulationresults verify the validity of the proposed solutions, and demonstrate that themodified exponential spline sampling kernel gets better performance, while theprewhitening method has more applicability.
时间交替模数转换器（TIADC）是一种针对带宽有限的宽带信号进行采样的高效并行结构，而有限新息率（FRI）采样框架则是对一类非带限信号进行高效采样的理论基础，是基于空间映射概念对原始Shannon采样理论的扩展。本论文针对TIADC结构中存在的通道失配误差，以非线性混合滤波器组（NLFB）为基本模型，以二态Markov链、微分滤波器组和分数延时滤波器为主要工具，开展了自适应后端校准方法的研究。对于FRI采样框架，则重点对复杂脉冲波形情况下，系统设计过程中的内在约束关系，以及噪声环境下的重构稳健性问题进行了研究。本论文的主要研究成果概括如下：
1、针对TIADC通道间的直流偏置失配误差，本论文提出了基于数据随机化操作的自适应后端校准方法。该部分首先分析并验证了通道数据直接平均法的缺陷，然后根据缺陷产生的原因以及直流偏置误差引入的机制，在采样保持器和量化比较器之间引入了数据随机化操作。当随机序列均值为0且与输入信号不相关时，可以通过最佳线性估计算子得到通道直流偏置误差的无偏估计。鉴于最佳线性估计算子的运算复杂性，提出了采用直接平均算子作为次最佳估计，并证明了该算子与最佳线性估计算子的渐进一致性。为了便于物理实现，且能够提供更大的设计自由度，本论文用二态Markov链作为随机化操作序列的生成工具。理论分析和实验结果表明，通过增加采样值以及合理设置Markov链状态转移概率可以有效提高校准性能，而且该方法在系统同时存在静态增益失配误差和时间失配误差情况下依然有效。
2、针对TIADC通道间的静态增益和时间失配误差，本论文提出了基于微分滤波器组的全数字域自适应后端校准方法。该方法通过设计时序关系，使各个子通道与同一参考通道周期性地同时对输入信号进行采样，进而确立了参考通道和子通道采样值之间的泰勒级数关系；根据此关系，设计了基于微分滤波器组的自适应补偿结构，以实现同时对静态增益和时间失配误差进行校准。另外，本文在频域上分析了微分滤波器组与分数延时滤波器的等效关系，使所提出的校准方法能够简单地应用于DSP平台。仿真实验结果表明，在轻微过采样状态下，所提出的方法对点频信号和频带有限信号都能有效地校准失配误差；在已知信号所处Nyquist频带条件下，该方法对基带采样和带通采样的情况都能有效校准；校准性能与微分滤波器阶数及泰勒级数展开的长度有关，若要提高系统校准精度并降低过采样因子，需提高各个微分滤波器的阶数并适当选取泰勒展开的长度。
3、针对基于复杂脉冲波形的FRI信号，本论文推导了FRI采样核与输入信号脉冲波形之间的约束关系，在此基础上分别对基于复杂脉冲波形的周期FRI信号和有限长FRI信号给出了采样系统的设计方法。其一，对于周期FRI信号，需要采用sinc采样核或sinc函数加权求和（SoS）采样核，并通过对输入信号进行适当频移来满足约束关系；其二，对于有限长FRI信号，需要采用SoS采样核或指数再生采样核。当采用SoS采样核时，有限长FRI信号与采样核之间的约束关系与该采样核应用于周期FRI信号时的约束关系一致；当采用指数再生采样核时，约束关系由采样核参数确定，因此，对于不同的脉冲波形，可以通过合理设计采样核参数来满足约束关系。除此之外，本文详细分析了系统稳定性和物理实现等因素对采样核参数的约束，并给出了参数设计方法。实验仿真结果表明，本文设计方法能有效提高系统重构的性能。
4、针对FRI采样系统在噪声环境中的重构稳健性问题，本论文提出了两种解决办法，分别是改进的指数样条采样核和子空间白化预处理。在噪声环境中，需要用稳健的重构算法估计FRI信号参数，但已有重构算法都要求采样值中的噪声成分是加性白噪声。由于采样核的滤波作用，这一条件有时难以满足。因此，本文基于采样核对采样值的作用机理，并依据指数再生采样核固有的卷积性质，在频域对原始的指数样条函数重新进行设计，使其能够保持噪声原有的特性。对于改进采样核方法受限的情况，则可以通过子空间白化算法对数据进行预白化来保证重构算法的前提条件。仿真结果表明，两种方法都能有效改善系统重构性能，且相对而言，改进指数样条采样核方法效果更好，而子空间白化方法适用性更强。
The trend of modern communication and Radar systems are wideband andsoft-defined. Effective methods for sampling and reconstructing wideband signals areessentials to conform to the trend. However, the sampling systems based on traditionalschemes and theory become more and more difficult to meet requirements due to thelimitations of current IC processing and some especial application conditions. So,research of high efficient sampling schemes and general sampling theory are topics oftheoretical interest and also of great relevance to applications.
Time-Interleaved ADC is a kind of high efficient parallel sampling scheme aimingat wideband signals with limited bandwidth, while the framework of finite rate ofinnovation (FRI) is the basic theory to sample a kind of non-bandlimited signalseffectively. It is a generalization of standard Shannon’s sampling theorem from the viewpoint of space projection. Based on non-linear filter bank (NLFB) model, two stateMarkov chains, differential filter bank and fractional delay filter, this dissertationstudies adaptive backward calibration method for channel mismatch errors. For theframework of FRI, studies are conducted in this dissertation, mainly on the internalconstraint relationship between sampling system and complex pulse shapes, and on therobustness of reconstruction in noisy scenario.
The main contents of this dissertation are summarized as follows:
1. Focus on the offset mismatch error of TIADC, an adaptive backward calibrationmethod based on data-randomizing operation is proposed. First, this part analyzesand verifies the defect of direct data-averaging method, and then, based on thereason of defect and the mechanism of offset error, we introduce a randomizingoperator to the data between sample-and-hold (S/H) and quantizer. When therandom series is zero-mean and uncorrelated with input signal, an unbiasedestimation of the channel offset error can be got by the best linear unbiasedestimator. In the light of high computational complexity of best linear unbiasedestimator, a suboptimal estimator based on data-averaging is proposed. Theasymptotic equivalence between the data-averaging estimator and the best linearunbiased estimator is proved. In order to implement easily and to provide moredegrees of freedom for design, the two state Markov chain is used as the randomseries generator. Analysis and simulation results verify that the performance ofcalibration will be improved by increasing the number of samples and setting the transition probability of Markov chain, and that the proposed method is effectiveeven if gain mismatch errors and timing mismatch errors are exist.
2. Focus on the gain mismatch errors and timing mismatch errors, a digital adaptivebackward calibration method based on differential filter bank is proposed. First, bydesigning the sampling timing, each sub-channel is synchronized with the samereference channel periodically to sample input signals, and then, the samples of thereference channel can be expressed as the Taylor’s series of the samples ofsub-channels. According to this relationship, an adaptive calibration architecture isdesigned for both gain mismatch errors and timing mismatch errors. Therelationship between differential filter bank and fractional delay filter is analyzed,which provides a simple way to apply the proposed method to DSP platforms.Simulation results demonstrate that the proposed method is fit for both multi-tonesignals and bandlimited signals when they are slightly oversampled, and can workin both baseband and bandpass scenarios on the condition that we know the Nyquistband of input signal in advance. The performance of the proposed calibrationmethod is determined by the taps of differential filters and the length of Taylor’sexpansion. In order to improve precision of calibration and reduce the oversampling factor, the taps of differential filters must be increased, with the length ofTaylor expansion set properly.
3. For the FRI signals with complex pulse shape, the constraint relationship betweensampling kernels and input signals is deduced from the reconstructing mechanismof FRI framework, and then, the design method of sampling system for periodicFRI signals and finite length FRI signals is proposed. For the periodic FRI signals,sinc or sum of sincs (SoS) sampling kernels should be chose, and frequency shiftoperation can be used to meet the constraint. For the finite length FRI signals, weshould choose SoS or exponential reproducing sampling kernels. When SoSsampling kernels are used, the constraint between sampling kernels and inputsignals is the same as that of the SoS sampling kernels used for the periodic signals.When exponential reproducing sampling kernels are chose, the constraint isdetermined by the parameters of sampling kernels, so we can set properly theparameters of sampling kernels for different complex pulse shape to satisfy theconstraint. In addition, we also analyze the constraint from the consideration ofrobustness of system and practical implement, and then, give the setting method ofsampling kernels. The simulations results demonstrate that the performance ofreconstruction can be improved by our design methods.
4. Focus on the robustness of reconstruction of FRI sampling system in noisyscenarios, two solutions are given out, which are modified exponential splinesampling kernels and subspace prewhitening method. When noise exists, robustreconstructing algorithms must be used. However, these algorithms require that thenoise components in samples are additive white noise. Due to the filtering ofsampling kernels, the samples may not meet this requirement. So, based on themechanism of action of sampling kernel on samples and the inherent convolutionproperty of exponential reproducing sampling kernels, we modify the originalexponential spline in frequency domain to keep the characteristic of noisecomponents. For the scenarios that the modified exponential spline samplingkernels are restricted, we can prewhiten the noise components by subspacewhitening method before reconstruction to satisfy the prerequisite. The simulationresults verify the validity of the proposed solutions, and demonstrate that themodified exponential spline sampling kernel gets better performance, while theprewhitening method has more applicability.
引文
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