Compensation for the nonlinear systems represented by polynomials involves polynomial inverse. In this paper, a new algorithm is proposed that gives the baseband polynomial inverse with a limited order. The algorithm employs orthogonal basis that is predetermined from the distribution of input signal and finds the coefficients of the inverse polynomial to minimize the mean square error. Compared with the well established p-th order inverse method, the proposed method can suppress the distortions better including higher order distortions. It is also extended to obtain memory polynomial inverse through a feedback-configured structure. Both numerical simulations and experimental results demonstrate that the proposed algorithm can provide good performance for compensating the nonlinear systems represented by baseband polynomials.

In many practical situations in NN learning, training examples tend to be supplied one by one. In such situations, incremental learning seems more natural than batch learning in view of the learning methods of human beings. In this paper, we propose an incremental learning method in neural networks under the projection learning criterion. Although projection learning is a linear learning method, achieving the above goal is not straightforward since it involves redundant expressions of functions with over-complete bases, which is essentially related to pseudo biorthogonal bases (or frames). The proposed method provides exactly the same learning result as that obtained by batch learning. It is theoretically shown that the proposed method is more efficient in computation than batch learning.

The purpose of this paper is to deal with the problem of recovering a signal from its noisy version. One example is to restore old images degraded by noise. The recovery solution is given within the framework of series expansion and we shall show that for the general case the recovery functions have to be elements of an extended pseudo biorthogonal basis (EPBOB) in order to suppress efficiently the corruption noise. After we discuss the different situations of noise, we provide some methods to construct the optimal EPBOB in order to deal with these situations.

As a generalization of the concept of pseudo-biorthogonal bases (PBOB), we already presented in Ref. [3] the theory of the so-called extended pseudo-biorthogonal bases (EPBOB). We introduce in this paper two special types of EPBOB called EPBOB's of type O and of type L. This paper discusses characterizations, construction methods, inherent properties, and mutual relations of these types of EPBOB.

This paper introduces the concept of an extended pseudo-biorthogonal basis" (EPBOB), which is a generalization of the concepts of an orthonormal (OB), a biorthonormal (BOB), a pseudo-orthogonal (POB), and a pseudo-biorthogonal (PBOB) bases. Let HN be a subspace of a Hilbert space H. The concept of EPBOB says that we can always construct a set of 2M (MN) elements of H but not necessarily all in HN such that like BOB any element f in HN can be expressed by fMΣm=1(f,φ*m)φm. For a better understanding and a wide application of EPBOB, this paper provides their characterization and shows how they preserve the formalism of BOB. It also shows how to construct them.