IEICE TRANS. FUNDAMENTALS, VOL.E96–A, NO.3 MARCH 2013 675 PAPER An Algorithm for Obtaining the Inverse for a Given Polynomial in Baseband Yuelin MA †a) , Student Member, Yasushi YAMAO † , and Yoshihiko AKAIWA † , Fellows SUMMARY 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 dis- tribution 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 distor- tions better including higher order distortions. It is also extended to ob- tain 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. key words: baseband polynomial, orthogonal basis, nonlinearity compen- sation, polynomial inverse, predistortion 1. Introduction In recent years, a lot of researches have been carried out on behavioral modeling and compensation of nonlinear sys- tems. Especially, nonlinear system modeling and compen- sation of the radio frequency (RF) power amplifiers (PAs) are important subjects from the viewpoint of practical use. As an effective technique to compensate for the nonlinear distortion in PAs, digital predistortion (DPD), has attracted much attention. It essentially involves inverse modeling of nonlinear distortion of the PA, either directly or indirectly. Most of the predistortion architectures for PA linearization are based on indirect learning which is more flexible and robust than the direct learning architecture [1]. As noticed by some researchers, however, this method suffers from the noisy output of the PA [2] and thus more complicated archi- tectures can be found in the literature in order to overcome this issue. In [2] and [3], the PA is first modeled using a poly- nomial model and the predistortion function is calculated based on the inverse of this polynomial. Such scheme is shown in Fig.1, which is also known as “PA modeling and inverse”. As observed from the figure, it basically requires an adaptive algorithm or a least square fitting at both the PA modeling and the inverse estimation based on the observa- tions from the PA, which result in remarkably high compu- tational load. In order to mitigate this high computational Manuscript received June 20, 2012. Manuscript revised November 5, 2012. † The authors are with the Advanced Wireless Communica- tion Research Center of the University of Electro Communications, Chofu-shi, 182-8585 Japan. a) E-mail: yuelinma1985@gmail.com DOI: 10.1587/transfun.E96.A.675 Fig. 1 Digital predistortion based on PA modeling and inverse. load, more efficient methods for inverting the given polyno- mial model need to be developed. There are several techniques for inverting a given poly- nomial. Schetzen [4] proposed “ pth-order inverse” method to invert a Volterra series which is known as the most general polynomial for representing weak nonlinear systems with memory. In order to increase the speed and decrease the complexity for constructing the Volterra kernels, Sarti et al. proposed the recursive technology for the pth-order inverse [5]. The limitation of the pth-order inverse is that it can only remove the nonlinear distortion introduced by the terms up to pth-order and leaves residual higher order distortion. It results in ineffective inverses in some circumstances. Tsim- binos et al. [6] derived the method to invert the power series based on Chebyshev polynomial and Hermite polynomial, which provides superior performance to that of pth-order inverse. A common issue for all the aforementioned tech- niques is that they were developed for real bandpass signal. In [7], the baseband polynomial with even-order terms [8] has been considered, and a novel algorithm was pro- posed that gives the baseband polynomial inverse with a limited order. The algorithm employs orthogonal basis that is predetermined from the distribution of input signal. The major difference of the that method from the conventional methods is that it focuses on minimizing the mean square error, while conventional works aims to remove the higher distortions of pth-order or above, not to minimize the whole distortion power. This paper is an extension of our previous work [7]. Here, we take deeper insights into that algorithm, and also extensions are made to adapt it to the memory case. The algorithm is based on the assumption that the non- linear system can be represented by a baseband polynomial. When a stationary input signal and its probability distribu- tion are given, the algorithm can find the coefficients of the Copyright c  2013 The Institute of Electronics, Information and Communication Engineers