Counting solutions for the CDMA multiuser MAP demodulator

We evaluate the average number of locally minimal solutions for maximum-{\it a-posteriori} (MAP) demodulation in code-division multiple-access (CDMA) systems. For this purpose, we use a sophisticated method to investigate the ground state properties for the Sherrington-Kirkpatrick-type (i.e. fully connected) spin glasses established by Tanaka and Edwards in 1980. We derive the number of locally minimal solutions as a function of several parameters which specify the CDMA multiuser MAP demodulator. We also calculate the distribution function of the normalized-energies for the locally minimum states. We find that for a small number of chip intervals (or equivalently a large number of users) and large noise level at the base station, the number of local minimum solutions becomes larger than that of the SK model. This provides us with useful information about the computational complexity of the MAP demodulator.