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A masking threshold constrained Kalman filter for speech enhancement is derived in the paper. A key step in a traditional Kalman filter requires minimizing an estimation error variance between a clean signal and its estimation. Our new method is to minimize the estimation error variance under the constraint that the energy of the estimation error is smaller than a masking threshold, computed from both time-domain forward masking and frequency-domain simultaneous masking properties of human auditory systems. The new Kalman filter provides a theoretical base for the application of the masking properties in Kalman filtering for speech enhancement. Due to the high computation cost of the proposed perceptually constrained Kalman filter, a perceptual post-filter concatenated with a standard Kalman filter is also proposed as a heuristic alternative for real-time implementation. The post-filter is constructed to make the estimation error obtained from the Kalman filter lower than the masking threshold. A wavelet Kalman filter with post-filtering is introduced to further reduce the computational load. Experimental results with colored noise show that the new constrained Kalman filter method produces the best performance when compared with other recent methods, and that the proposed heuristics with post-filtering can also produce a significant performance gain over other recent methods.