二维FIR数字滤波器优化设计理论与二维优化设计算法研究
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摘要
二维数字滤波器做为一种典型的多维数字系统已经被广泛应用于图像处理,声纳信号处理,雷达信号处理,地球物理信号处理等诸多方面。随着现代电子设备所处理数据量的快速增加,高效而精确地设计各种(尤其是高阶)二维数字滤波器在多维数字信号处理研究领域具有十分重要意义。与一维滤波器设计只涉及一元函数逼近不同,二维滤波器设计实质上是二元函数逼近问题,由于二元函数逼近理论尚没有一元函数逼近理论完善、成熟,一些有效的一维滤波器设计算法并不能扩展到二维滤波器设计中。即使能够扩展,扩展算法在设计二维滤波器时也会出现数值困难,这是由于二维滤波器设计中所需处理数据远多于一维情况。以上这些都造成了二维滤波器设计问题要比一维滤波器设计问题复杂得多。
高计算复杂性一直是二维滤波器设计中遇到的主要困难。近期,一些学者提出了二维优化算法来设计二维滤波器,与传统算法中把二维滤波器参数排成向量形式不同,这些二维算法处理数据时保持二维滤波器参数的原始矩阵形式,这有效提高了计算效率并节省了存储空间。但是,这些二维算法还存在这样或那样的缺点,应用受到限制。不过这些二维算法高效性表明如何充分利用二维滤波器参数成矩阵形式这一特性,将是发展高效稳定的二维滤波器设计技术的关键。本论文主要研究二维线性相位FIR数字滤波器的二维优化设计算法,旨在更加高效而精确地设计各种二维数字滤波器。
论文首先把二维线性相位FIR数字滤波器由一般到特殊分成三类：复二维线性相位FIR滤波器,中心(反)对称二维FIR滤波器和矩形(反)对称二维FIR滤波器。其中,矩形(反)对称二维FIR滤波器是中心(反)对称二维FIR滤波器的一种特殊形式,而中心(反)对称二维FIR滤波器则是复二维线性相位FIR滤波器的一种特殊形式。为了更好地描述所提出的算法,分别推导出这三类线性相位滤波器的相频特性和幅频特性函数,把其幅频特性表达成待求参数矩阵的函数,并给出了这些待求参数矩阵与滤波器的单位脉冲响应之间的对应关系。
在二维滤波器优化设计中,加权最小二乘(WLS)设计指标由于其简单性和较好的设计效果而被广泛使用。另外,许多滤波器设计问题,如：Minimax设计,最小lp范数设计,约束滤波器设计等都可以通过转化为一系列WLS设计子问题求解。故而,快速而稳定的二维滤波器WLS设计算法对进一步研究二维滤波器设计问题十分重要。论文首先研究了矩形(反)对称二维FIR滤波器的WLS优化设计问题。推导出此设计问题的最优性条件,以矩阵方程形式表达。首先针对四加权WLS设计问题,依据最优性条件,提出一种矩阵迭代算法和一种矩阵对角化设计算法,并证明了矩阵迭代算法的收敛性。而矩阵对角化设计算法的解是矩阵迭代算法的收敛极限。设计实例表明这两种算法不仅计算非常精确,更重要的是效率非常高,甚至接近无加权最小二乘方法。论文接着把矩阵对角化设计算法与迭代重加权最小二乘(IRLS)技术相结合,得到一种四加权二维IRLS算法,它通过迭代地调整各频带上的加权值来削减最大幅值逼近误差,同时,此算法计算效率也很高。
进一步,论文研究了任意加权情况下的矩形(反)对称二维FIR滤波器WLS设计问题。还是依据最优性条件,推导出三种矩阵迭代算法：矩阵迭代算法Ⅰ,Ⅱ,Ⅲ,和一种广义共轭梯度算法。在三种矩阵迭代算法中算法Ⅰ是基本算法,算法Ⅱ、Ⅲ是对算法Ⅰ的改进。用线性算子理论证明了矩阵迭代算法Ⅰ和Ⅱ的收敛性,而矩阵迭代算法Ⅲ在某些情况下可能不收敛。设计低阶滤波器时三种矩阵迭代算法设计效率相差不大,但当设计高阶滤波器时,一般来说算法Ⅱ,Ⅲ收敛更快。所提出的广义共轭梯度算法是传统共轭梯度算法在Hilbert内积空间中的扩展算法,它以矩阵为变量,并用Hilbert空间内积下的正交性代替传统共轭梯度算法中的共轭性。论文中用Hilbert空间内积理论证明了它可以在有限步内收敛。一般而言,广义共轭梯度算法在设计精度、设计时间等方面要优于三种矩阵迭代算法。这些算法都有各自的特点,论文通过计算复杂性分析和设计实例对各算法进行了详细分析,并和现有算法进行比较。仿真结果表明,本文所提出的算法在设计时间,设计精度和数值稳定性上比现有算法有很大提高。
进而,论文研究了任意加权情况下中心(反)对称二维FIR滤波器的WLS优化设计问题。首先建立此优化设计问题的数学模型,其目标函数中包含两个矩阵变量。令目标函数的导数为零,求出此优化问题的最优性条件,它是由包含两个矩阵变量的两个矩阵方程构成。根据最优性条件,把矩形(反)对称情况的矩阵迭代算法Ⅰ和Ⅱ进行适当的修改,扩展到中心(反)对称滤波器设计问题,并用线性算子理论证明扩展算法收敛。再通过在两个矩阵空间的Cartesian乘积空间上定义适当的内积,把矩形(反)对称二维FIR滤波器设计的广义共轭梯度算法扩展到中心(反)对称滤波器设计中,并用Hilbert空间内积理论证明扩展算法在有限步内收敛。仿真实例表明所得这些扩展算法能够非常有效且精确地设计中心(反)对称二维FIR滤波器,优于现有其他算法。
接下来论文对复二维线性相位FIR滤波器任意加权情况的WLS优化设计进行了研究。此优化设计问题数学模型的目标函数中包含了四个矩阵变量,从而求出的最优性条件是四个矩阵方程联立的矩阵方程组,方程组中包含四个矩阵变量。在四个矩阵空间的Cartesian乘积空间上定义适当的内积,并根据此内积定义把此前的广义共轭梯度算法扩展到复二维线性相位FIR滤波器设计中,最后仍用Hilbert空间内积理论证明算法在有限步内收敛。仿真实例说明所得算法能够非常有效且精确的设计复二维线性相位FIR滤波器。
论文最后研究了二维线性相位FIR数字滤波器在最小lp范数指标下的优化设计问题。最小lp范数指标能够有效的消除吉布斯效应,而且可以用来逼近Minimax设计。论文提出了一种基于任意加权WLS技术的二维IRLS算法用于设计最小lp范数指标下的二维线性相位FIR滤波器,包括矩形(反)对称、中心(反)对称和复二维线性相位FIR滤波器。这种二维IRLS算法是把经典IRLS技术与本论文提出的广义共轭梯度算法有效结合,并做适当地调整,使之更适用于最小lp范数二维滤波器设计问题。本论文广义共轭梯度算法的商效性保证了新二维IRLS算法能够快速收敛,最后的仿真结果也充分证实了这一点。
本论文提出的所有算法都利用了二维滤波器参数和二维滤波器频率采样点成矩阵形式这一特性,运算中保持其矩隈形式不变,都是二维优化算法。算法分析和设计实例都表明所提各算法较已有算法计算效率更高,占用计算机内存更小,能够有效且精确地设计WLS指标和最小lp范数指标下的各种(包括高阶)二维线性相位FIR数字滤波器。
As typical multi-dimensional digital systems, two-dimensional (2-D) digital filters have been widely applied in image processing, sonar and radar signal processing, geophysical signal processing and so on. Efficient design algorithms that are capable to fastly and accurately design various, especially high-order,2-D digital filters are becoming more and more important for the multi-dimensional digital signal processing due to the rapid increase of the amount of data processed by modern electronic equipments. Unlike the optimal design of one-dimensional (1-D) filters which is in fact a univariate function approximation problem, the design of2-D filters is essentially a bivariate function approximation problem. As a result, many efficient design algorithms for1-D filters cannot be extended to the2-D filter design problem due to the fact that the approximation theory of bivariate function is not as complete as that of univariate function. Though several algorithms can be extended to the design2-D filters, there are some inevitable numerical problems in these algorithms because of large amount of data required to be processed in the2-D case. The facts mentioned above make the design of2-D filters much more difficult than its1-D counterpart.
High computational complexity is the major difficulty encountered in the optimal design of2-D filters. Recently, a number of researchers proposed some two-dimension (2-D) based optimization algorithms for the2-D filters design. Unlike the conventional methods that rearrange the2-D filter coefficients into a vector, those2-D based algorithms deal with the filter coefficients in their nature matrix form, leading to a relatively high efficiency and a great saving of the memory space. However, the shortcomings of the existing2-D based algorithms have limited their applications. Nevertheless, high efficiency of those2-D based algorithms indicates that how to exploit the matrix form of the coefficients of2-D filters will be the key for developing efficient2-D filter design algorithms. In this dissertation, the efficient2-D based algorithms are investigated aiming to design linear-phase2-D FIR filters more efficiently and accurately.
Firstly,2-D linear-phase FIR filters are classifed into three categories:the complex2-D linear-phase FIR filter, the centro-(anti)symmetric2-D FIR filter and the quadrantally (anti)symmetric2-D FIR filter. It should be pointed out that the quadrantally (anti)symmetric2-D FIR filter is a special case of the centro-(anti)symmetric2-D FIR filter, and the centro-(anti)symmetric2-D FIR filter is a special case of the complex2-D linear-phase FIR filter. In order to describe the proposed algorithms more clearly, the magnitude responses are expressed as functions of coefficient matrices of the filters and the phase responses are derived for the three categories of filters, respectively. Furthermore, the relations btween the coefficient matrices and the impulse response of the filters are also given.
In the design of2-D filters, weighted least squares (WLS) criterion has been widely used due to its simplicity and relatively perfect design results. Moreover, many of filter design problems, such as the Minimax design, the least lp norm design, the constrained filters design and so on, can be solved by transforming the original design problem into a sequence of WLS design subproblems. Thus, the fast and numerically stable WLS algorithms for the design of2-D filters are of considerable important for further investigation of2-D filter design. In this dissertation, the WLS design problem of quadrantally (anti)symmetric2-D FIR filters is studied firstly. The optimality condition of such problem is obtained and expressed as a matrix equation. On the basis of the optimality condition, a matrix iterative algorithm and a matrix diagonalization based algorithm are proposed for the WLS design with four-valued weighting function. The convergence of the matrix iterative algorithm is established, while the solution obtained by the matrix diagonalization based algorithm is just the limite solution of the matrix iterative algorithm. Both of the algorithms are extremely computationally accurate, and at the same time, their computational efficiencies are very high and even comparable to that of unweighted least squares method. Consequently, the matrix diagonalization based algorithm is combined with the iterative reweighted least squares (IRLS) technique, resulting in a2-D based IRLS algorithm for four-valued weighting design. The2-D based IRLS algorithm can reduce the maximal magnitude error by iteratively adjusting the weights in different frequency bands, and is computationally very efficient as well.
Next, the WLS design problem with arbitrary weighting function is considered. Again according to the optimality condition, three matrix iterative algorithms:matrix iterative algorithm Ⅰ, Ⅱ, and Ⅲ, and a generalized conjugate gradient algorithm are presented. Among the three matrix iterative algorithms, the first one is the basic algorithm, and the other two are modified versions of the first. The convergences of the matrix iterative algorithm I, II are established by using linear operator theory, but the matrix iterative algorihtm III may fail to converge in some special situations. Generally speaking, the three matrix iterative algorithms have similar efficiencies for designing low-order filters, and the algorithm II and III converge more fast than the algorithm I for high-order cases. Using matrices as the variables, the generalized conjugate gradient algorithm is a generalization of the conventional conjugate gradient algorithm in Hilbert inner space by the way that the conjugacy in the conventional algorithm is replaced by the orthogonality under inner product defined on the Hilbert space. It is proved that the generalized conjugate gradient algorithm can converge in a finite number of steps by using the inner theory of Hilbert space. Generally, the generalied gradient conjugate algorithm is more efficient than the three matrix iterative algorithms. All those proposed algorithms have their respective characteristics and are analyzed and compared with existing algorihtms in detail along with the analyses of computational complexity through design examples. Simulation results show that those proposed algorithms provide significant impovements in terms of the design time, design accuracy and numerical stability compared to existing methods.
In the sequel, the WLS design of centro-(anti)symmetric2-D FIR filters with arbitrary weighting function is researched. The mathematical model of such optimization design problem is formulated with an objective function including two matrix variables, and by differentiating the objective function and setting the result to zero, the optimality condition is obtained, which is composed of two matrix equations in two matrix variables. On the basis of this optimality condition, the matrix iterative algorithms I and II are extended to the centro-(anti)symmetric filters design problem. Moreover, the convergences of these extended algorithms are proved by using the linear operator theory. Further, a porper inner product is defined on the Cartesian product space of two matrix spaces, thus the generalized conjugate gradient algorithm is extended to the centro-(anti)symmetric filters design problem and is proved by using the inner theory of Hilbert space that it can converge in a finite number of steps. Design examples illustrate that these extended algorithms are very efficent and far superior to existing algorithms.
In the following, the WLS design of complex2-D linear-phase FIR filters with arbitrary weighting function is studied. The objective function of this design problem is a matrix function of the four independent matrix variables. Thus, the obtained optimality condition consists of four matrix equations with respcet to four matrix variables. By defining a proper inner product on the Cartesian product space of four matrix spaces, the generalized conjugate gradient algorithm is extended for the WLS design of complex2-D linear-phase filters. Similarly, the convergence of the extended algorithm is proved by using the inner theory of Hilbert space. Simulation results demonstrate the good performance of the proposed algorithm in designing complex2-D linear-phase filters.
Finally, the least lp norm design of2-D linear-phase FIR filters is discussed, which can effectively eliminate the Gibbs'phenomenon appearing in the least squares design and can be used to approximate the Minimax design. Another2-D based IRLS algorithm with arbitrary weighted WLS technique as its iteration core is developed for designing2-D linear-phase filters in the least lp norm sense, including the quadrantally (anti)symmetric filter, the centro-(anti)symmetric filter and the complex2-D linear-phase filter. The new2-D based IRLS algorithm efficiently combines the genearalized conjugate gradient algorithm proposed in this dissertation with the conventional IRLS algorithm, as a result, it is especially well-suited for the least lp norm design of2-D FIR filters. In addition, high efficiency of the generalized conjugate gradient algorithm guarantees that the proposed IRLS algorithm can fast converge to the optimal filter, which fact has also been established by simulation results.
All of the algorithms proposed in this dissertation take advantage of the matrix forms of the2-D filter coefficients and the sampling frequencies, thereby are2-D based algorithms. Analyses of those algorithms along with large numbers of design examples indicate that they are computationally higher efficient, need smaller memory space compared with existing methods and can very efficiently design various, even high-order,2-D linear-phase filters in the WLS sense and the least lp norm sense.
高计算复杂性一直是二维滤波器设计中遇到的主要困难。近期,一些学者提出了二维优化算法来设计二维滤波器,与传统算法中把二维滤波器参数排成向量形式不同,这些二维算法处理数据时保持二维滤波器参数的原始矩阵形式,这有效提高了计算效率并节省了存储空间。但是,这些二维算法还存在这样或那样的缺点,应用受到限制。不过这些二维算法高效性表明如何充分利用二维滤波器参数成矩阵形式这一特性,将是发展高效稳定的二维滤波器设计技术的关键。本论文主要研究二维线性相位FIR数字滤波器的二维优化设计算法,旨在更加高效而精确地设计各种二维数字滤波器。
论文首先把二维线性相位FIR数字滤波器由一般到特殊分成三类：复二维线性相位FIR滤波器,中心(反)对称二维FIR滤波器和矩形(反)对称二维FIR滤波器。其中,矩形(反)对称二维FIR滤波器是中心(反)对称二维FIR滤波器的一种特殊形式,而中心(反)对称二维FIR滤波器则是复二维线性相位FIR滤波器的一种特殊形式。为了更好地描述所提出的算法,分别推导出这三类线性相位滤波器的相频特性和幅频特性函数,把其幅频特性表达成待求参数矩阵的函数,并给出了这些待求参数矩阵与滤波器的单位脉冲响应之间的对应关系。
在二维滤波器优化设计中,加权最小二乘(WLS)设计指标由于其简单性和较好的设计效果而被广泛使用。另外,许多滤波器设计问题,如：Minimax设计,最小lp范数设计,约束滤波器设计等都可以通过转化为一系列WLS设计子问题求解。故而,快速而稳定的二维滤波器WLS设计算法对进一步研究二维滤波器设计问题十分重要。论文首先研究了矩形(反)对称二维FIR滤波器的WLS优化设计问题。推导出此设计问题的最优性条件,以矩阵方程形式表达。首先针对四加权WLS设计问题,依据最优性条件,提出一种矩阵迭代算法和一种矩阵对角化设计算法,并证明了矩阵迭代算法的收敛性。而矩阵对角化设计算法的解是矩阵迭代算法的收敛极限。设计实例表明这两种算法不仅计算非常精确,更重要的是效率非常高,甚至接近无加权最小二乘方法。论文接着把矩阵对角化设计算法与迭代重加权最小二乘(IRLS)技术相结合,得到一种四加权二维IRLS算法,它通过迭代地调整各频带上的加权值来削减最大幅值逼近误差,同时,此算法计算效率也很高。
进一步,论文研究了任意加权情况下的矩形(反)对称二维FIR滤波器WLS设计问题。还是依据最优性条件,推导出三种矩阵迭代算法：矩阵迭代算法Ⅰ,Ⅱ,Ⅲ,和一种广义共轭梯度算法。在三种矩阵迭代算法中算法Ⅰ是基本算法,算法Ⅱ、Ⅲ是对算法Ⅰ的改进。用线性算子理论证明了矩阵迭代算法Ⅰ和Ⅱ的收敛性,而矩阵迭代算法Ⅲ在某些情况下可能不收敛。设计低阶滤波器时三种矩阵迭代算法设计效率相差不大,但当设计高阶滤波器时,一般来说算法Ⅱ,Ⅲ收敛更快。所提出的广义共轭梯度算法是传统共轭梯度算法在Hilbert内积空间中的扩展算法,它以矩阵为变量,并用Hilbert空间内积下的正交性代替传统共轭梯度算法中的共轭性。论文中用Hilbert空间内积理论证明了它可以在有限步内收敛。一般而言,广义共轭梯度算法在设计精度、设计时间等方面要优于三种矩阵迭代算法。这些算法都有各自的特点,论文通过计算复杂性分析和设计实例对各算法进行了详细分析,并和现有算法进行比较。仿真结果表明,本文所提出的算法在设计时间,设计精度和数值稳定性上比现有算法有很大提高。
进而,论文研究了任意加权情况下中心(反)对称二维FIR滤波器的WLS优化设计问题。首先建立此优化设计问题的数学模型,其目标函数中包含两个矩阵变量。令目标函数的导数为零,求出此优化问题的最优性条件,它是由包含两个矩阵变量的两个矩阵方程构成。根据最优性条件,把矩形(反)对称情况的矩阵迭代算法Ⅰ和Ⅱ进行适当的修改,扩展到中心(反)对称滤波器设计问题,并用线性算子理论证明扩展算法收敛。再通过在两个矩阵空间的Cartesian乘积空间上定义适当的内积,把矩形(反)对称二维FIR滤波器设计的广义共轭梯度算法扩展到中心(反)对称滤波器设计中,并用Hilbert空间内积理论证明扩展算法在有限步内收敛。仿真实例表明所得这些扩展算法能够非常有效且精确地设计中心(反)对称二维FIR滤波器,优于现有其他算法。
接下来论文对复二维线性相位FIR滤波器任意加权情况的WLS优化设计进行了研究。此优化设计问题数学模型的目标函数中包含了四个矩阵变量,从而求出的最优性条件是四个矩阵方程联立的矩阵方程组,方程组中包含四个矩阵变量。在四个矩阵空间的Cartesian乘积空间上定义适当的内积,并根据此内积定义把此前的广义共轭梯度算法扩展到复二维线性相位FIR滤波器设计中,最后仍用Hilbert空间内积理论证明算法在有限步内收敛。仿真实例说明所得算法能够非常有效且精确的设计复二维线性相位FIR滤波器。
论文最后研究了二维线性相位FIR数字滤波器在最小lp范数指标下的优化设计问题。最小lp范数指标能够有效的消除吉布斯效应,而且可以用来逼近Minimax设计。论文提出了一种基于任意加权WLS技术的二维IRLS算法用于设计最小lp范数指标下的二维线性相位FIR滤波器,包括矩形(反)对称、中心(反)对称和复二维线性相位FIR滤波器。这种二维IRLS算法是把经典IRLS技术与本论文提出的广义共轭梯度算法有效结合,并做适当地调整,使之更适用于最小lp范数二维滤波器设计问题。本论文广义共轭梯度算法的商效性保证了新二维IRLS算法能够快速收敛,最后的仿真结果也充分证实了这一点。
本论文提出的所有算法都利用了二维滤波器参数和二维滤波器频率采样点成矩阵形式这一特性,运算中保持其矩隈形式不变,都是二维优化算法。算法分析和设计实例都表明所提各算法较已有算法计算效率更高,占用计算机内存更小,能够有效且精确地设计WLS指标和最小lp范数指标下的各种(包括高阶)二维线性相位FIR数字滤波器。
As typical multi-dimensional digital systems, two-dimensional (2-D) digital filters have been widely applied in image processing, sonar and radar signal processing, geophysical signal processing and so on. Efficient design algorithms that are capable to fastly and accurately design various, especially high-order,2-D digital filters are becoming more and more important for the multi-dimensional digital signal processing due to the rapid increase of the amount of data processed by modern electronic equipments. Unlike the optimal design of one-dimensional (1-D) filters which is in fact a univariate function approximation problem, the design of2-D filters is essentially a bivariate function approximation problem. As a result, many efficient design algorithms for1-D filters cannot be extended to the2-D filter design problem due to the fact that the approximation theory of bivariate function is not as complete as that of univariate function. Though several algorithms can be extended to the design2-D filters, there are some inevitable numerical problems in these algorithms because of large amount of data required to be processed in the2-D case. The facts mentioned above make the design of2-D filters much more difficult than its1-D counterpart.
High computational complexity is the major difficulty encountered in the optimal design of2-D filters. Recently, a number of researchers proposed some two-dimension (2-D) based optimization algorithms for the2-D filters design. Unlike the conventional methods that rearrange the2-D filter coefficients into a vector, those2-D based algorithms deal with the filter coefficients in their nature matrix form, leading to a relatively high efficiency and a great saving of the memory space. However, the shortcomings of the existing2-D based algorithms have limited their applications. Nevertheless, high efficiency of those2-D based algorithms indicates that how to exploit the matrix form of the coefficients of2-D filters will be the key for developing efficient2-D filter design algorithms. In this dissertation, the efficient2-D based algorithms are investigated aiming to design linear-phase2-D FIR filters more efficiently and accurately.
Firstly,2-D linear-phase FIR filters are classifed into three categories:the complex2-D linear-phase FIR filter, the centro-(anti)symmetric2-D FIR filter and the quadrantally (anti)symmetric2-D FIR filter. It should be pointed out that the quadrantally (anti)symmetric2-D FIR filter is a special case of the centro-(anti)symmetric2-D FIR filter, and the centro-(anti)symmetric2-D FIR filter is a special case of the complex2-D linear-phase FIR filter. In order to describe the proposed algorithms more clearly, the magnitude responses are expressed as functions of coefficient matrices of the filters and the phase responses are derived for the three categories of filters, respectively. Furthermore, the relations btween the coefficient matrices and the impulse response of the filters are also given.
In the design of2-D filters, weighted least squares (WLS) criterion has been widely used due to its simplicity and relatively perfect design results. Moreover, many of filter design problems, such as the Minimax design, the least lp norm design, the constrained filters design and so on, can be solved by transforming the original design problem into a sequence of WLS design subproblems. Thus, the fast and numerically stable WLS algorithms for the design of2-D filters are of considerable important for further investigation of2-D filter design. In this dissertation, the WLS design problem of quadrantally (anti)symmetric2-D FIR filters is studied firstly. The optimality condition of such problem is obtained and expressed as a matrix equation. On the basis of the optimality condition, a matrix iterative algorithm and a matrix diagonalization based algorithm are proposed for the WLS design with four-valued weighting function. The convergence of the matrix iterative algorithm is established, while the solution obtained by the matrix diagonalization based algorithm is just the limite solution of the matrix iterative algorithm. Both of the algorithms are extremely computationally accurate, and at the same time, their computational efficiencies are very high and even comparable to that of unweighted least squares method. Consequently, the matrix diagonalization based algorithm is combined with the iterative reweighted least squares (IRLS) technique, resulting in a2-D based IRLS algorithm for four-valued weighting design. The2-D based IRLS algorithm can reduce the maximal magnitude error by iteratively adjusting the weights in different frequency bands, and is computationally very efficient as well.
Next, the WLS design problem with arbitrary weighting function is considered. Again according to the optimality condition, three matrix iterative algorithms:matrix iterative algorithm Ⅰ, Ⅱ, and Ⅲ, and a generalized conjugate gradient algorithm are presented. Among the three matrix iterative algorithms, the first one is the basic algorithm, and the other two are modified versions of the first. The convergences of the matrix iterative algorithm I, II are established by using linear operator theory, but the matrix iterative algorihtm III may fail to converge in some special situations. Generally speaking, the three matrix iterative algorithms have similar efficiencies for designing low-order filters, and the algorithm II and III converge more fast than the algorithm I for high-order cases. Using matrices as the variables, the generalized conjugate gradient algorithm is a generalization of the conventional conjugate gradient algorithm in Hilbert inner space by the way that the conjugacy in the conventional algorithm is replaced by the orthogonality under inner product defined on the Hilbert space. It is proved that the generalized conjugate gradient algorithm can converge in a finite number of steps by using the inner theory of Hilbert space. Generally, the generalied gradient conjugate algorithm is more efficient than the three matrix iterative algorithms. All those proposed algorithms have their respective characteristics and are analyzed and compared with existing algorihtms in detail along with the analyses of computational complexity through design examples. Simulation results show that those proposed algorithms provide significant impovements in terms of the design time, design accuracy and numerical stability compared to existing methods.
In the sequel, the WLS design of centro-(anti)symmetric2-D FIR filters with arbitrary weighting function is researched. The mathematical model of such optimization design problem is formulated with an objective function including two matrix variables, and by differentiating the objective function and setting the result to zero, the optimality condition is obtained, which is composed of two matrix equations in two matrix variables. On the basis of this optimality condition, the matrix iterative algorithms I and II are extended to the centro-(anti)symmetric filters design problem. Moreover, the convergences of these extended algorithms are proved by using the linear operator theory. Further, a porper inner product is defined on the Cartesian product space of two matrix spaces, thus the generalized conjugate gradient algorithm is extended to the centro-(anti)symmetric filters design problem and is proved by using the inner theory of Hilbert space that it can converge in a finite number of steps. Design examples illustrate that these extended algorithms are very efficent and far superior to existing algorithms.
In the following, the WLS design of complex2-D linear-phase FIR filters with arbitrary weighting function is studied. The objective function of this design problem is a matrix function of the four independent matrix variables. Thus, the obtained optimality condition consists of four matrix equations with respcet to four matrix variables. By defining a proper inner product on the Cartesian product space of four matrix spaces, the generalized conjugate gradient algorithm is extended for the WLS design of complex2-D linear-phase filters. Similarly, the convergence of the extended algorithm is proved by using the inner theory of Hilbert space. Simulation results demonstrate the good performance of the proposed algorithm in designing complex2-D linear-phase filters.
Finally, the least lp norm design of2-D linear-phase FIR filters is discussed, which can effectively eliminate the Gibbs'phenomenon appearing in the least squares design and can be used to approximate the Minimax design. Another2-D based IRLS algorithm with arbitrary weighted WLS technique as its iteration core is developed for designing2-D linear-phase filters in the least lp norm sense, including the quadrantally (anti)symmetric filter, the centro-(anti)symmetric filter and the complex2-D linear-phase filter. The new2-D based IRLS algorithm efficiently combines the genearalized conjugate gradient algorithm proposed in this dissertation with the conventional IRLS algorithm, as a result, it is especially well-suited for the least lp norm design of2-D FIR filters. In addition, high efficiency of the generalized conjugate gradient algorithm guarantees that the proposed IRLS algorithm can fast converge to the optimal filter, which fact has also been established by simulation results.
All of the algorithms proposed in this dissertation take advantage of the matrix forms of the2-D filter coefficients and the sampling frequencies, thereby are2-D based algorithms. Analyses of those algorithms along with large numbers of design examples indicate that they are computationally higher efficient, need smaller memory space compared with existing methods and can very efficiently design various, even high-order,2-D linear-phase filters in the WLS sense and the least lp norm sense.
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