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A Normalised Adaptive Amplitude Nonlinear Gradient Descent (NAANGD) algorithm for nonlinear Finite Impulse Response (FIR) filters is introduced. The FIR filter adapts its weights based upon a gradient descent type iteration and employs an adaptive multiplicative factor at the output of the activation function to overcome the problems encountered with previously introduced algorithms when the range of the desired signal exceeds the range of the nonlinear activation function. In this way, the proposed NAANGD reduces significantly the residual error, after convergence is attained, while due to the normalisation introduced in both update equations for the FIR filter weights and the multiplicative factor, it also increases the Convergence Rate (CR). Experimental results highlight these points and support the analysis.