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In a recent paper, a simple least-squares (LS) based algorithm is introduced for unbiased parameter estimation of autoregressive (AR) signals observed in noise, under the assumption that the ratio between the driving source power and the corrupting noise variance is known. In the present paper, this LS based algorithm is modified with a more computationally efficient algorithmic structure. The mean convergence of the modified algorithm is then investigated. The issue of how the assumption of the known power ratio can be mitigated for practical applications is discussed, which leads to the development of an effective estimation algorithm for noisy AR signals. Theoretical results are validated through computer simulations.