  We show that a quantized large-scale system with unknown parameters and
training signals can be analyzed by examining an equivalent system with known
parameters by modifying the signal power and noise variance in a prescribed
manner. Applications to training in wireless communications and signal
processing are shown. In wireless communications, we show that the optimal
number of training signals can be significantly smaller than the number of
transmitting elements. Similar conclusions can be drawn when considering the
symbol error rate in signal processing applications, as long as the number of
receiving elements is large enough. We show that a linear analysis of training
in a quantized system can be accurate when the thermal noise is high or the
system is operating near its saturation rate.
