The original Japanese edition of this book, published by Saiensu-sha, Japan, in March 2005, has fortunately acquired a favorable reputation. I am grateful to the readers for their kind feedbacks, many of which are included in this edition. I hope this English publication attracts readers in wider areas, and evokes valuable feedbacks furthermore. In the months after the Japanese publication, researches on the compl- valued neural networks have kept evolution in respective directions. There are some plans of special sessions in international conferences and special issues in journals. The bibliography has been slightly modi?ed to include the special sessions and latest journal publications. On the other hand, references written in Japanese on domestic circumstances have been omitted. Besides, Fig.1.1 has been added, and Fig.2.1 has been modi?ed, which are related to the Special Issue on Complex-Valued Neural Networks, The Journal of the IEICE, 87 (6), June 2004 (in Japanese), so that even readers not having glancedtheissuecanobtainclearconcepts.Withthesemodi?cations, Iexpect a higher appeal in this English edition. I am very much obliged to Dr. Thomas Ditzinger, Engineering Editor, Springer-Verlag, for his continuous help in publication. Tokyo, Japan Akira Hirose April 2006 Preface The studies on complex-valued neural networks have recently been evolving in various directions. The pioneering areas include electromagnetic-wave and lightwave sensing and imaging, independent component analysis in blind s- aration, blur restoration in image processing, and so on. Developing appli- tionsinvolveadaptivequantumdevices, quantumcomputation, socialsystems related to periodicity and oscillation, and so forth.