Convexity in SISO blind equalization

A new class of blind equalization cost functions have been proposed (see. Shtrom, V. and Fan, H., IEEE Trans. on SP., vol.46, no.10, p.2674-83, 1998). They have the particularity of being unimodal, and some of them are sectionally convex in the combined system domain (CDS). Both properties seem very attractive for a cost function, though no proof (or specific definition) was given of the sectional convexity. We show, using a particular case of this class, that sectional convexity not only implies fixing a delay, but that it also requires to fix the value of the coefficient associated with such delay. We show that the CMA (constant modulus algorithm) criterion shares the same property, and that this property is difficult to maintain in actual algorithms.