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Taking above into account, we have proposed two efficient techniques for the design of 1-D and 2-D variable filters, which are based on the decomposition of the given 1-D and 2-D variable magnitude specifications [3, 4].
请考虑一下，我们假设用两种有效的方法来设计线形（1－D）的和平面（2－D）可变滤波器，这是根据给定的1－D与2－D可变幅度技术条件[3, 4]分解的。
The techniques are very computationally efficient because they solely need designing a set of 1-D constant filters and approximating several 1-D polynomials.
由于完全依赖设计一套1－D常量滤波器以及近似于数个1－D多项式，本技术是很有计算效率的。
In particular, the techniques can always guarantee the stability of the resulting variable filters.
特别是，本技术始终能够保证可变滤波器结果稳定性。
However, in the 2-D case, the variable filters designed by such a technique are constrained to only have quadrantally symmetrical magnitude characteristics.
然而，在平面条件下，用该技术设计的可变滤波器受到约束，仅仅只有象限幅度对称性。
That is to say, assume that is the given 2-D variable magnitude specification, where and are the normalized frequencies, and is a real valued vector whose elements are spectral parameters.
也就是说，设： 为给定的2－D可变技术条件，式中 与 为标准化的频率，并且为值实向量，其要素为光谱参数。
The technique [4] can only approximate the given 2-D variable magnitude specification in the case.
本技术[4]仅在这种情况下能够近似算出给定的2－D可变滤波器技术条件 。
= (1)
To circumvent this difficulty, this paper proposes a new method for designing 2-D variable digital filters with arbitrary magnitude characteristics.
围绕这一难题，本文提出了一项新方法，用于设计2－D可变数字滤波器，带有任意幅度性能。
In other words, the technique can approximate 2-D variable magnitude specifications even though
换言之，虽然 ≠ (2)，本技术能够近似计算出2－D可变幅度滤波器。
The technique is based on the decomposition of the given 2-D variable magnitude specifications, which is different from the one proposed in [4], and it only requires the normal 2-D constant filter designs and 1-D polynomial approximations.
本技术根据给定的1－D与2－D可变幅度技术条件分解的，但不同于[4]中提出技术方法，并且只需要公称2－D常量滤波器结构以及1－D多项式近似法。
Since 2-D constant filters are relatively easy to design by applying the existing design techniques and 1-D polynomials can be easily approximated by solving simultaneous linear equations, the technique significantly simplifies the original 2-D variable filter design problem.
由于，2－D常量滤波器用现有的设计技术以及1－D多项式也可用求出同步线形方程而相对简单，所以本技术相当地简化了原有的2－D可变滤波器设计的难题。
In particular, it should be emphasized that this technique can guarantee the stability of the resulting 2-D variable filters so long as the 2-D constant filters are designed to be stable. An example is given to illustrate the effectiveness of the proposed technique.
特别应当提出的是，本技术能够只要在2－D常量滤波器设计得稳定的话，就能保证取得的2－D可变滤波器结果的稳定性。文中详细说明了所提出的技术的效力。

2. Design problem formulation
2．设计问题的说明
In this section, we formulate the problem of designing 2-D variable digital filters with arbitrary magnitude responses based on the newly proposed outer product expansion of the given 2-D variable magnitude specifications.
在本节中，根据给定2－D可变幅度技术条件提出的外围设备，我们阐述了任意幅度反应2－D可变数字滤波器的设计。
Below, we first define an outer product expansion, then relate it with the 2-D variable filter design problem.
下面，我们首次对外围设备做了定义，然后将其同2－D可变滤波器

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