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This paper is concerned with the optimal filtering problem for discrete-time stochastic linear systems with multiple packet dropouts, where the number of consecutive packet dropouts is limited by a known upper bound. Without resorting to state augmentation, the system is converted to one with measurement delays and a moving average (MV) colored measurement noise. An unbiased optimal filter is developed in the linear least-mean-square sense. Its solution depends on the recursion of a Riccati equation and a Lyapunov equation. A numerical example shows the effectiveness of the proposed filter.