This paper discusses the optimal design of linear-phase three-dimensional (3-D) FIR digital filters in the least-square sense. The design procedure is as follows. The linear-phase 3-D polynomial which is symmetric with respect to the origin is decomposed into four polynomials from the viewpoint of the symmetry (cubic symmetry, ω1 and ω2 axes skew symmetry, ω2 and ω3, axes skew symmetry and ω1 and ω3 axes skew symmetry). Each of those resulting polynomials is then used as the filter model. Next, the filter parameters are determined analytically so as to minimize the square errors between the actual magnitude response of the filter and the ideal magnitude response.
Finally, two numerical examples are given to demonstrate the validity of the theory. The proposed method allows us to reduce the computational complexity in the design since it is not necessary to use various optimization procedures or to calculate the inverse matrix. The filter with the desired response can be designed in a short time.
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