System identification is an exceptionally expansive topic and of remarkable significance in the discipline of signal processing and communication. Our goal in this paper is to show how simple adaptive FIR and IIR filters can be used in system modeling and demonstrating the application of adaptive system identification. The main objective of our research is to study the LMS algorithm and its improvement by the genetic search approach, namely, LMS-GA, to search the multi-modal error surface of the IIR filter to avoid local minima and finding the optimal weight vector when only measured or estimated data are available. Convergence analysis of the LMS algorithm in the case of coloured input signal, i.e., correlated input signal is demonstrated on adaptive FIR filter via power spectral density of the input signals and Fourier transform of the autocorrelation matrix of the input signal. Simulations have been carried out on adaptive filtering of FIR and IIR filters and tested on white and coloured input signals to validate the powerfulness of the genetic-based LMS algorithm.