The Least Mean Square $($LMS$)$ algorithm is one tool to filter out the noise so pilots can communicate clearly with one another and control towers. LMS is an adaptive algorithm, which uses a gradient$-$based method of steepest decent. The LMS algorithm uses the estimates of the gradient vector from the available data and incorporates an iterative procedure that makes successive corrections to the weight vector in the direction of the negative of the gradient vector which eventually leads to the minimum mean square error. Compared to other algorithms, the LMS algorithm is relatively simple in that it does not require correlation function calculation nor does it require matrix inversions. A component necessary for the LMS algorithm is the learning rate. The learning rate refers to the amount of change to the model during each step of the process.

Part A$-$Describe the example you chose to illustrate why the LMS algorithm will always converge to the best solution.

Part B$-$Detail why a different initialization of weights may lead to a different solution.

Part C$-$Explain the importance of a learning rate as it relates to your example.