Quaternion approach to the the oryof 2-D causal systems - Kajetan M. Snopek (International Journal of Electronics and Telecommunications)
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 (, ) of a 2-D system defines the 2-D complex frequency response (, ). The modulus |(, )| is the 2-D magnitude response and (, ) 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 (, ) of a 2-D causal system. We define the 2-D magnitude system response |(, )| 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.