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In modern wireless communication systems, wide signal bandwidth is the most straightforward approach to accommodate high data rates. Wide signal bandwidth, on the other hand, introduces severe challenges to the power amplifier (PA) and digital predistortion (DPD) design in both performance and cost. Conventional DPD systems usually ignore the impact of the transmit low-pass filter (Tx LPF) bandwidth and assume the transmit bandwidth is sufficiently large. In wideband signal transmissions, the bandwidth of Tx LPF can become the system bottleneck, limiting DPDs compensation effects. Existing DPD studies mostly investigate the DPD with reduced feedback bandwidth. In this paper, we study the impact of Tx LPF bandwidth on the DPD performance. A full-band error minimization DPD based on direct learning structure is proposed. The DPD coefficients are estimated by minimizing the full-band error between the input signal and PA output signal in the frequency domain. Furthermore, we propose a weighted DPD with improved performance by introducing a weighting diagonal matrix to the error function. Compared to existing solutions, the weighted DPD achieves a good trade-off between the in-band distortion compensation and out-of-band spectral regrowth suppression. Simulations and experiments validate the effectiveness of the proposed DPD schemes. [ABSTRACT FROM AUTHOR]