On the misbehavior of constant modulus equalizers for improper modulations

In this letter, the constant-modulus (CM) cost function is analyzed under the general assumptions that improper modulation schemes of practical interest are employed and the baseband equivalent of the channel impulse response is complex-valued. This study allows one to determine a broad family of undesired minima of the CM cost function, which do not lead to perfect symbol recovery in the absence of noise. The results developed herein generalize and subsume as a particular case existing studies of the CM cost function, which exclusively consider real-valued binary modulations.