@ARTICLE{Snopek_Kajetan_M._Quaternion_2025,
 author={Snopek, Kajetan M.},
 volume={vol. 71},
 number={No 2},
 journal={International Journal of Electronics and Telecommunications},
 pages={537-543},
 howpublished={online},
 year={2025},
 publisher={Polish Academy of Sciences Committee of Electronics and Telecommunications},
 abstract={ 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.},
 type={Article},
 title={Quaternion approach to the the oryof 2-D causal systems},
 URL={http://ochroma.man.poznan.pl/Content/135260/25-5103-Snopek-sk-new.pdf},
 doi={10.24425/ijet.2025.153602},
 keywords={causal/anti-causal signal, causal system, Gabor’sanalytic signal, quaternion analytic signal, quaternion Fouriertransform, dual symmetry, Pei’s formula, impulse response, frequency response},
}