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The linearly constrained minimum variance (LCMV) algorithm fails to capture the signal if coherent copies (multipath signals) impinge on the array at the same time with the signal. The constant modulus algorithm (CMA) and the multi-modulus algorithm (MMA) techniques solved this problem with the disadvantage of more complicated behavior and the violation of some constraints about null directions staled by the initial weights. This is the main point of interest of the new algorithms proposed in this paper. The proposed algorithms in this paper, namely, the linearly constrained CMA (LCCMA) and linearly constrained MMA (LCMMA) solved this problem and arc able to hold the initial constraints about deep nulls until the adaptation is finished. This property has been successfully used to improve the performance of the MMA algorithm to capture very high signal constellations in multipath and interference environment without having to use the more complicated generalized MMA (GMMA).