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However, most of the CKAF applications involve the complex kernel least mean square (CKLMS) algorithms, which work in a pure complex or complexified reproducing kernel Hilbert space (RKHS). In this paper, we propose the generalized complex kernel affine projection (GCKAP) algorithms in the widely linear complex-valued RKHS (WL-RKHS). The proposed algorithms have two main notable features. One is that they provide a complete solution for both circular and non-circular complex nonlinear problems and show many performance improvements over the CKAP algorithms. The other is that the GCKAP algorithms inherit the simplicity of the CKLMS algorithm while reducing its gradient noise and boosting its convergence. The second-order statistical characteristics of WL-RKHS have also been developed. An augmented Gram matrix consists of a standard Gram matrix and a pseudo-Gram matrix. This decomposition provides more underlying information when the real and imaginary parts of the signal are correlated and learning is independent. In addition, some online sparsification criteria are compared comprehensively in the GCKAP algorithms, including the novelty criterion, the coherence criterion, and the angle criterion. Finally, two nonlinear channel equalization experiments with non-circular complex inputs are presented to illustrate the performance improvements of the proposed algorithms.<\/jats:p>","DOI":"10.1007\/s00034-021-01804-8","type":"journal-article","created":{"date-parts":[[2021,8,10]],"date-time":"2021-08-10T15:38:28Z","timestamp":1628609908000},"page":"831-850","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Generalized Complex Kernel Affine Projection Algorithms"],"prefix":"10.1007","volume":"41","author":[{"given":"Xiaojian","family":"Wang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5796-4754","authenticated-orcid":false,"given":"Ming","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jianxing","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wenchao","family":"Chen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anxue","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,8,10]]},"reference":[{"issue":"7","key":"1804_CR1","doi-asserted-by":"publisher","first-page":"2048","DOI":"10.1109\/TCSI.2014.2304665","volume":"61","author":"MZA Bhotto","year":"2014","unstructured":"M.Z.A. 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