This paper proposes a new adaptive predistortion-postdistortion scheme based on a recurrent neural network to reduce nonlinear distortion introduced by a high power amplifier in the amplitude and phase of received Quadrature Phase Shift Keying (QPSK) signals in a digital microwave system. The recurrent neural network structure is inspired by the model proposed by Williams and Zipser, with a modified backpropagation algorithm. The input signal is processed by a nonlinear predistorter which reduces the warping effect. The received output signal is passed through a postdistorter to compensate for the warping and clustering effects produced by an amplifier. The proposed scheme yields a significant improvement when it is compared to the system without predistortion-postdistortion, performance is evaluated in terms of the bit error rate and output signal constellation.