Adaptive filtering is an important topic in modern signal processing with the LMS algorithm being used widely in various applications. The wavelet transform provides good and in many times excellent results when used as a basic block transform in many systems such as electronic, communication, medical and even chemical systems. The work in this thesis proposes a new algorithm called the WPLMS algorithm, which combines the wavelet packet transform with the normal LMS adaptive filter systems to form wavelet packet transform adaptive filter systems. The two systems were compared on mathematical and simulation basis. Learning curves of adaptive channel equalization and adaptive channel estimation using WPLMS are compared with normal adaptive channel equalization and channel estimation using LMS only. The investigation of the Effects on the learning curve covered the following parameters: (i) The choice of the mother function (Haar, Daubechies, etc.); (ii) The number of the decomposition levels in the wavelet tree; (iii) The step size of the LMS algorithm; (iv) The signal to noise ratio (SNR = 30 dB and 40 dB); (v) The type of the telephone channel (chl, ch2 and ch3); (vi) Length of adaptation filter size. The results using this technique achieved good improvements in convergence time over the ordinary LMS algorithm. The change of mother function had little effect on the performance. It was found that the use of more than two deposition levels leads to divergence, while the increase in step size speed up the convergence. The improvement due to the WPLMS becomes more noticeable for higher levels of SNR. The proposed algorithm performs better in channel 1. For noisy channels the WPLMS requires larger filter sizes to converge. The improvement in convergence time in the case of adaptive channel estimation using the WPLMS nearly 42%, and in the case of adaptive channel equalization using the WPLMS it was nearly 33%. This demonstrates the validity and efficacy of the proposed technique. The simulation work was carried out on a PC with PI 233MHz Processor and programmed using the MATLAB simulation package version 5.3.