  In multiple antenna systems employing time-division duplexing, spatial
precoder design at the base station (BS) leverages channel state information
acquired through uplink pilot transmission, under the assumption of channel
reciprocity. Malicious eavesdroppers can start pilot spoofing attacks to alter
such design, so as to improve their eavesdropping performance in downlink. The
aim of this paper is to study the effects of pilot spoofing attacks on uplink
channel estimation, by assuming that the BS knows the angle of arrivals (AoAs)
of the legitimate channels. Specifically, after assessing the performance of
the simple least squares estimator (LSE), we consider more sophisticated
estimators, such as the maximum likelihood estimator (MLE) and different
versions of the minimum mean square error estimator (MMSEE), involving
different degrees of a priori information about the pilot spoofing attacks.
Theoretical analysis and numerical simulations are used to compare the
performance of such estimators. In particular, we analytically demonstrate that
the spoofing effects in the high signal-to-noise regime can be completely
suppressed, under certain conditions involving the AoAs of the legitimate and
spoofing channels. Moreover, we show that even an imperfect knowledge of the
AoAs and of the average transmission power of the spoofing signals allows the
MLE and MMSEE to achieve significant performance gains over the LSE.
