Automatic phase recovery properties of adaptive blind equalization algorithms

Assuming the intersymbol interference (ISI) has been removed by an adaptive equalizer which employs a suitable blind equalization algorithm, the output constellation may still be rotated by an arbitrary phase angle. These angles are the result of the existence of undesirable local minima in the phase parameter space. A simple technique to identify all such angles is presented. In addition, the stability of the rotated solutions and the range of phase angles that leads to such solutions can be determined too. Three blind algorithms, namely the multi-modulus algorithm (MMA), the reduced constellation algorithm (RCA), and the constant modulus algorithm (CMA) are analyzed using this approach. The immunity of the MMA towards such solutions is verified analytically while the susceptibility of the RCA towards them, especially the 45deg-solution, is verified through simulations. Our analysis also agrees with the well known fact that the CMA is indeed phase-blind.