This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS 1 Adaptive Nonlinear Equalization of a Tunable Bandpass Filter Nicholas Peccarelli , Student Member, IEEE, and Caleb Fulton , Senior Member, IEEE Abstract— The mitigation of third-order intermodulation dis- tortion produced by a nonlinear tunable bandpass filter (BPF) in a digital receiver channel is demonstrated. An iterative solution is proposed, in the use of the least-mean-square algorithm, as an effective way to allow the correction coefficients to adapt to the changes in the system characteristics when the center frequency is changed. Results are shown using a two-pole, varactor-loaded BPF, and an Analog Devices AD9371 transceiver. Index Terms— Adaptive signal processing, least mean square, nonlinear distortion and intermodulation distortion (IMD), nonlinear filters. I. I NTRODUCTION R ECENTLY, there has been a growing interest in fully digital, low-cost arrays, for a variety of applications [1]–[5]. These systems are also sometimes making use of tun- able components. For size and cost reasons, these are typically either based on integrated technologies or are otherwise lim- ited in linearity, leading to intermodulation distortion (IMD)— a threat that is exacerbated by the wide-angle response of each element in a digital array [1], [2]. Digital postdistortion, or nonlinear equalization (NLEQ), has been increasingly researched in recent years [1], [6], [7]. Such corrections often focus on the main cause of receiver nonlinearities, such as the low-noise amplifier (LNA), mixer, or baseband stages. With a few small exceptions, corrections for tunable filter nonlinearities have not been widely explored [8], [9] (see Fig. 1). Low-cost tunable filters typically use varactor diodes or other integrated approaches (as opposed to mechanical or other exotic techniques) and, since they are often placed immediately after the LNA stage, must have inherently higher input third-order intercept points than the LNA, lest they contribute significantly to IMD. Iterative NLEQ approaches are quite natural to correct for IMD in these tunable devices, since both the filtering characteristics and nonlinear behavior of the filter can change greatly with center frequency, temperature, and time. The least-mean-square (LMS) algorithm is a convenient iterative algorithm for such purposes, with the proper basis choice. Most of the previous work, with few exceptions, has used a static power series as a basis, which was commonly used in digital predistortion. But the receiver channel, compared to that of the transmitter, usually contains a bandpass filter (BPF), Manuscript received November 19, 2018; accepted December 10, 2018. This work was supported by the Defense Advanced Research Projects Agency under Grant D15A00090. (Corresponding author: Nicholas Peccarelli.) The authors are with the Advanced Radar Research Center, College of Electrical and Computer Engineering, University of Oklahoma, Norman, OK 73019 USA (e-mail: peccarelli@ou.edu; fulton@ou.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LMWC.2018.2887387 Fig. 1. Example of a nonlinear receiver front end. Red: nonlinear components that are typically accounted for. Blue: nonlinear tunable BPF in question. Fig. 2. Example of an LMS-based NLEQ correction of a compressed receiver front end, trained by an auxiliary channel. which can greatly add frequency dependence, requiring the use of memory terms for equalization. A memory polynomial (MP) may not be optimal, but it has shown to be effective for a range of linearity and equalization purposes in previous work [10]. An example of the desired result is summarized in Fig. 2. This letter demonstrates these concepts with a varactor-tuned S-band filter in front of a modern digital transceiver for digital array applications. Section II summarizes the approach, and Section III shows measured results with the filter. It is demonstrated that the NLEQ coefficients that result from training at one frequency must be updated for use at another frequency, due to changes in the filter characteristics, and that the iterative LMS algorithm is an effective choice for facilitating these updates. II. LEAST-MEAN-SQUARE ALGORITHM The LMS algorithm iteratively minimizes the mean-square error (MSE) between the desired signal d (n) and the corrected signal y (n), given by MSE = 1 N N  n=1 [d (n) - y (n)] 2 . (1) In this letter, an auxiliary receiver with a higher noise floor is used to provide the reference signal for one of what would 1531-1309 © 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.