On the Limitation of Linear MMSE Detection and (Generalized) Welch Bound Equality Signals
A. Kapur, Mahesh K. Varanasi · 2003
This paper addresses signal optimization for CDMA systems under linear minimum mean-squared error (MMSE) detection. In particular, generalized Welch-bound equality (WBE) signals, which have been found to be optimum for various measures/problems, are studied from a new perspec- tive. Specifically, we consider the asymptotic effective energy (AEE) criterion which characterizes bit error rate (BER) faith- fully in the high signal-to-noise ratio regime, and conservatively in low/medium regimes. Our analysis and numerical examples show that, with generalized WBE signals, at least one user has a BER that does not decay exponentially. In fact, in the equal- energy case, all users have an AEE that is provably equal to zero. Numerical simulations illustrate that, in general, the joint er- ror rate floors at very high values. We also prove for sufficient overload and conjecture for other overloaded cases that no signal set yields a joint error rate that decays exponentially under lin- ear MMSE detection, thereby highlighting a severe limitation of such detection. The results extend to the case where the received phases are unknown. Grant CCR-0112977. mance. Specifically, we prove that generalized WBE signals yield a joint error rate (JER), defined as the probability that at least one user is in error, that always floors for such detectors. The result is even stronger in the equal-energy case, wherein every user has a bit er- ror rate (BER) that floors. Moreover, we prove for the equal-energy case (and conjecture for the unequal-energy case) that any overloaded signal set yields a JER that floors. This is indicative of a severe lim- itation of linear MMSE detection, and further corroborates that non- linear detection is necessary to achieve reliable and spectrally effi- cient multiple access communication (11). Numerical examples not only confirm our results, but also illustrate that the JER of general- ized WBE signals typically floors at a very high value. Our analytical approach relies on the fact that BER is faithfully characterized by the asymptotic effective energy (AEE) in the high signal-to-noise ratio (SNR) regime (13), and conservatively in low/medium regimes. We show that bounds on the AEE of any user can be maximized by the direction of least interference, and that, extending this approach to joint signal design yields generalized WBE signals. However, their symmetric energy, defined as the smallest AEE (over all users), is al-