Systems and methods for gain estimation and synchronization in adaptive filters

Digital pre-distortion is performed on a received signal using a set of pre-distortion coefficients to produce a digital pre-distorted signal. The digital pre-distorted signal is converted to an analog signal, which is amplified to produce a transmission output signal. The transmission output signal is converted to a digital feedback signal. A plurality of fractional delay filters is applied to the digital feedback signal to obtain a plurality of fractional delay compensated (FDC) candidates, and gain compensation is applied to each of the plurality of FDC candidates to obtain a plurality of gain and fractional delay compensated (XFT) candidates. The digital pre-distorted signal is used as a reference signal, and the XFT candidates and the reference signal are used to select a selected XFT candidate of the plurality of XFT candidates. The selected XFT candidate is used to generate the set of pre-distortion coefficients.