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Quaternion approach to the theory of 2-D causal systems

Snopek Kajetana M.

International Journal of Electronics and Telecommunications

2025

Vol. 71, No. 2

537-543

The paper is devoted to the signal- and frequency characteristics of 2-D causal systems. Two different approaches to this problem are presented. First, the 2-D complex Fourier transformation of the 2-D causal impulse response 𝒉(𝒙𝟏, 𝒙𝟐) of a 2-D system defines the 2-D complex frequency response 𝑯(𝒇𝟏, 𝒇𝟐). The modulus |𝑯(𝒇𝟏, 𝒇2)| is the 2-D magnitude response and 𝐚𝐫𝐠 𝑯(𝒇𝟏, 𝒇2) is the 2-D phase response of a system. On the other hand, applying the Pei’s formula relating the 2-D complex Fourier transform with the 2-D right-sided quaternion Fourier transform we introduce a concept of the quaternion-valued frequency response 𝑯q(𝒇𝟏, 𝒇𝟏) of a 2-D causal system. We define the 2-D magnitude system response |𝑯q(𝒇𝟏, 𝒇2)| and three 2-D phase responses. These concepts constitute an original contribution of this paper. The theoretical aspects are illustrated with examples of magnitude and phase responses of a causal 2-D analog low-pass filter.