A significant problem in communications is the generation of echoes. The echoes arise for a number of reasons, with the primary reason being an impedance mismatch. The impedance mismatch occurs when the two-wire network meets the four-wire network, this interface is known as the hybrid. This impedance mismatch causes some of the signal energy to be returned to the source as an echo . This can be seen in Figure 1. (All figures appear at the end of the report.)
Figure 1: schematic layout of the Echo canceller 
The delays between primary and echo signals are directly related to the transmission distance. For example, if a signal was sent to a satellite that redirected the signal back to another location on earth, that signal would have a very large time delay compared to a signal sent to a local switching station and back. Short delays (less than 50 ms) will not affect the quality of the signal as much as longer delays. Delays of this length are not noticed by the receiver and therefore are not considered an annoyance. However, these echoes may have an effect on data being transmitted through transmission lines . A sinusoid will be used as the input signal. The DSP board will create an echo of the sinusoid and add the echo to the original sinusoidal signal, thus creating a distorted version of the input signal. The DSP will then use LMS adaptive filtering to estimate the echo, and remove the echo from the distorted signal creating a reconstructed signal. The LMS algorithm seeks to minimize the excess mean-square error (MSE) between the echo signal and the estimated echo. The excess MSE refers to the LMS algorithm fluctuations about the adaptive filter coefficients after a large number of iterations . This thesis consists of designing and implementing an echo canceling system. The DSP chip is used to simulate the echo creating system and to implement the adaptive filtering system to cancel the echo in the distorted signal. Initially, the adaptive filter coefficients are far from the ideal numbers. After several iterations, the LMS algorithm will update these coefficients to converge on an optimal set of coefficients. Simulations of the LMS algorithm will be done in MATLAB to get approximate performance specifications before implementation. MSE plots that are called learning curves in the DSP field will be attained. The learning curves as well as the magnitude of the frequency response of the adaptive filter coefficients will be used to determine the performance of our system. Convergence of the adaptive filter coefficients and the similarity of the coefficient values in simulation and experimentation will also be of vital significance. The Least Mean Square (LMS) algorithm is a well-known adaptive estimation and prediction technique . It has been extensively studied in the literature and widely used in a variety of applications. The performance of the LMS algorithm is highly dependent on the selected convergence parameter μ and the signal condition. A larger convergence parameter value leads to faster convergence of the LMS algorithm, i.e., convergence of the filter coefficients to their optimal values. After coefficients converge to their optimal value, the convergence parameter should be small for better estimation accuracy.
In this thesis, the time-varying convergence parameter μn for the LMS algorithm in a white Gaussian noise environment is proposed. A general power decaying law has been studied, however, other time-varying laws could also be applicable. The main idea is to set the convergence parameter to a large value in the initial state in order to speed up the algorithm convergence. As time passes, the parameter will be adjusted to a smaller value so that the adaptive filter will have a smaller mean-squared error. The modified algorithm has been tested for noise reduction and estimation in linear frequency-modulated (LFM)...