We study the local minima relocation of the fractionally spaced constant modulus algorithm (FSE-CMA) cost function in the presence of noise. Local minima move in a particular direction as the noise power increases and their number may be eventually reduced. In such cases the performance of FSE-CMA may fail to adequately reduce intersymbol interference (ISI), but achieve an approximated MMSE by reducing its equalizer noise gain under certain constraints. We analyze the mechanism of relocation of the FSE-CMA cost function local minima in terms of the auto-correlation matrix of the sub-channel convolution matrix and its eigenvectors