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The mean-squared-error criterion is widely used in the literature. However, there are applications where the squared-error is not the primary parameter affecting the performance of a system. In many communication systems, for instance, the information bits are carried over the phase of the transmitted signal. In this letter, we introduce a cost function that is based on both the error magnitude and the phase error. The criterion is useful for applications where the performance depends primarily on the phase of the estimated (recovered) signal. An adaptive filter is then developed using the proposed criterion with essentially the same complexity as the standard least mean squared (LMS) algorithm. The filter outperforms LMS specially in situations with fast channel variations. Bit error rate (BER) simulations for two communication systems using the proposed algorithm support the claims.