In this paper, we present a detailed analysis of the Kalman filter for the application of speech enhancement and identify its shortcomings when the linear predictor model parameters are estimated from speech that has been corrupted with additive noise. We show that when only noise-corrupted speech is available, the poor performance of the Kalman filter may be attributed to the presence of large values in the Kalman gain during low speech energy regions, which cause a large degree of residual noise to be present in the output. These large Kalman gain values result from poor estimates of the LPCs due to the presence of additive noise. This paper presents the analysis and application of the Kalman gain trajectory as a useful indicator of Kalman filter performance, which can be used to motivate further methods of improvement. As an example, we analyse the previously-reported application of long and overlapped tapered windows using Kalman gain trajectories to explain the reduction and smoothing of residual noise in the enhanced output. In addition, we investigate further extensions, such as Dolph-Chebychev windowing and iterative LPC estimation. This modified Kalman filter was found to have improved on the conventional and iterative versions of the Kalman filter in both objective and subjective testing.