Improving the energy efficiency of cellular communication networks is necessary for future green and sustainable wireless communications. The advent of Fifth Generation (5G) communications has introduced new communication scenarios which have augmented this issue; large scale networks of small cell battery powered nodes, massive Multiple Input Multiple Output (MIMO) transmission consisting of large multiples of transmitter elements per base station and beam steering antenna arrays for high frequency communications. Power Amplifiers (PA) are a major cause for power inefficiency in Radio Frequency (RF) transmitters. An trade-off in PA design exists between efficiency and linearity. To satisfy both high efficiency and high linearity, an external linearisation technique can be applied to a PA. Digital Pre-Distortion (DPD) has been widely demonstrated in industry and academia to perform linearisation. Some issues concerning DPD are related to implementation efficiency, robust function estimation and the complexity of DPD solutions. This dissertation contributes to the optimised design of pre-distorters in hardware, the improvement of the DPD function estimation process in terms of accuracy and robustness, and the dimension reduction of DPD functions. These contributions are accomplished with modern 5G transmission scenarios in mind. Four novel contributions (B-E) are presented in this thesis, directly related to publications listed on page xi. These contributions were focused on the aspects of DPD function implementation, minimisation and estimation. Two novel contributions were made related to the aspect of implementation. These focused on the design of polynomial pre-distorters. They present optimised pre-distorter designs based on whether there is prior knowledge on the exact structure or not. The first proposed method (B) optimises a pre-distorter with regards to hardware efficiency and latency for a given predistorter structure. It achieves the best compromise of both compared to existing methods. The second proposed method (D) is focused on hardware efficiency and can optimise predistorters without prior knowledge of the polynomial structure. The remaining two contributions are related to the minimisation and estimation aspects of a DPD function. In this thesis contributions were focused on minimised sets of Least Squares (LS) estimated DPD parameters. The first of these proposed methods (C) gauges the influence of training samples on the fitted function. Samples with an abnormally large influence relative to the other training samples can be deemed outlier data points and excluded from a repeated regression. Experimental results while using the method (E) showed improved linearisation performance while using a reduced training dataset. The second of these proposed methods (E) extends the concept of (C) to analysing the influence each estimated parameter has on the fitted function. From a scale of 0 to 1 the linear independence of each individual basis function is gauged, which reflects its relative importance in the regression. Not only can this then be used to prune the least relevant features, it also offers a simple and informative level of interpretability not found among other feature selection methods. Experimental results, using measured PA output signals, in which feature selection was performed with the proposed method (E) showed comparable linearisation performance with reduced pre-distorter complexity to the LS method.