Scattering of the two dimensional electromagnetic waves is studied by the infinite sequences of zeros arising on the complex plane, which just correspond to the null points of the far field pattern given as a function of the azimuthal angle θ. The convergent sequences of zeros around the point of infinity are evaluated when the scattering objects are assumed to be N-polygonal cylinders. Every edge condition can be satisfied if the locations of zeros are determined appropriately. The parameters, which allow us to calculate the exact positions of zeros, are given by the asymptotic analysis. It is also shown that there are N-directions of convergence, which tend to infinity. An illustrative example is presented.

Hypercomplex coefficient digital filters provide several attractive advantages such as compact realization with reduced system order, inherent parallelism. However, they also possess a drawback in that a multiplier requires a large amount of computations. This paper proposes a computationally efficient implementation of digital filters whose coefficient is a type of hypercomplex number; a bicomplex number. By decomposing a bicomplex multiplier into two parallel complex multipliers, we show that hypercomplex digital filters can be implemented as two parallel complex digital filters. The proposed implementation offers more than a 60% reduction in the count of real multipliers.

The mathematical theory of bicomplex electromagnetic waves in two-dimensional scattering and diffraction problems is developed. The Vekua's integral expression for the two-dimensional fields valid only in the closed source-free region is generalized into the radiating field. The boundary-value problems for scattering and diffraction are formulated in the bicomplex space. The complex function of a single variable, which obeys the Cauchy-Riemann relations and thus expresses low-frequency aspects of the near field at a wedge of the scatterer, is connected with the radiating field by an integral operator having a suitable kernel. The behaviors of this complex function in the whole space are discussed together with those of the far-zone field or the amplitude of angular spectrum. The Hilbert's factorization scheme is used to find out a linear transformation from the far-zone field to the bicomplex-valued function of a single variable. This transformation is shown to be unique. The new integral expression for the field scattered by a thin metallic strip is also obtained.

This correspondence reports novel computationally efficient algorithms for multiplication of bicomplex numbers, which belong to hypercomplex numbers. The proposed algorithms require less number of real multiplications than existing methods. Furthermore, they give more effective implementation when applied to constant coefficient digital filters.

It is shown from the Hilberts theory that if the real function Π(θ) has no zeros over the interval [0, 2π], it can be factorized into a product of the factor π+(θ) and its complex conjugate π-(θ)(=). This factorization is tested to decompose a real far-zone field pattern having zeros. To this end, the factorized factors are described in terms of bicomplex mathematics. In our bicomplex mathematics, the temporal imaginary unit "j" is newly defined to distinguish from the spatial imaginary unit i, both of which satisfy i2=-1 and j2=-1.

A bicomplex representation for time-harmonic electromagnetic fields appearing in scattering and diffraction problems is given using two imaginary units i and j. Fieldsolution integral-expressions obtained in the high-frequency and low-frequency limits are shown to provide the new relation between high-frequency diffraction and low-frequency scattering. Simple examples for direct scattering problems are illustrated. It may also be possible to characterize electric or magnetic currents induced on the obstacle in terms of geometrical optics far-fields. This paper outlines some algebraic rules of bicomplex mathematics for diffraction or scattering fields and describes mathematical evidence of the solutions. Major discussions on the relationship between high-frequency and low-frequency fields are relegated to the companion paper which will be published in another journal.