摘要
Delta算子是一种新的离散化方法。在处理高速采样过程时，Delta算子离散模型接近于原来的连续模型，所得结果也趋近于连续模型的相应结果，较好地解决了传统的移位算子方法在高速采样时引起的数值不稳定问题。
本论文研究Delta算子描述的系统参数估计与滤波问题。主要内容如下：
1)研究Delta算子的状态空间模型和传递函数模型描述，给出Z域和Delta域传递函数基于向量和矩阵的转换关系，仿真结果表明转换方法的有效性。
2)研究Delta算子自回归滑动平均模型(ARMA)参数估计，基于Delta算子矩阵求逆引理和最小二乘估计的一般格式，导出Delta算子ARMA模型的增广最小二乘估计和遗忘因子法。
3)研究Delta算子最小二乘格形滤波器(DLSL)，基于q算子格形滤波器的一般模型，应用Hilbert线性向量空间最小二乘正交投影的概念，推导Delta算子最小二乘(LS)格形滤波器的阶次和时间递推公式，给出Delta算子AR模型系数的格形算法。
Delta operator is a new discretization method. In processing high sampling data system, discrete time model using delta operator approaches corresponding continuous time model, and the results also tends to the corresponding ones of its continuous counterpart. As the sampling rates increase, the numerical instability problems caused by the conventional shift operator can be well conquered by the delta operator.
In this thesis, the problems of parameter estimation and filtering for delta operator systems are studied. The main contributions are as follows:
1) Both the state-space model and the transfer function model using delta operator are studied. Based on vectors and matrices, the transformation between Delta and Z domain transfer function is obtained. The simulation results show the validity of the proposed algorithm.
2) Parameter estimation of autoregression moving average model (ARMA) is developed. By the delta matrix inverse lemma and common form of delta operator recursive least squares algorithm, the delta operator extend least squares and forgetting factor algorithm for ARMA model are derived.
3) Delta least squares lattice filter is studied. Based on the least squares lattice filter by shift operator and the conception of least squares orthogonal projection in Hilbert linear vector space, the order and time update formulae are presented. In addition, the lattice algorithms of the coefficients for the delta operator AR model are also given.
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