On the existence of undesirable global minima of Godard equalizers

We consider the problem of global convergence of Godard (1980) equalizers in the special case of binary (2-PAM) input signals, when the channel impulse response is complex. We present a class of global minima of all Godard equalizers for this case, which do not correspond to settings free of intersymbol-interference (ISI). The equalizer output corresponding to these global minima appears as a four-point constellation in the complex plane, however it is easily shown that the decomposition in its real and imaginary part provides two ISI-free versions of the transmitted signal. In the case of multi-user constant modulus algorithms, the situation is somewhat more complicated: the real and imaginary parts of each equalizer output after convergence, may correspond to different user signals. These results can be extended to other types of real input signals.