This book includes the research progress and achievement during my Ph. D. study period. It results in the publication of more than 10 papers in international journals and conferences. These journals are IEEE Transactions on Circuits and Systems-II, Electronics Letters and International Journal of Electronics. The content of this book covers the characteristic investigation of the pathological elements, the filter and oscillator synthesis and transform using pathological elements and impedance simulated circuit synthesis with mathematic approach. One can step into the research realm easily with basic electronic and mathematic background knowledge. These research experiences are conductive to publishing further research results in other journals, such as Circuits, Systems, and Signal Processing, Microelectronics Journal, Analog Integrated Circuits and Signal Processing, Applied Sciences, International Journal of Electronics and Communications (AEU), and International Journal of Circuits, Systems and Computers. They have been attained by my foreign and native doctoral students and me.