  The current evolution towards a massive number of antennas and a large
variety of transceiver architectures forces to revisit the conventional
techniques used to improve the fundamental power amplifier (PA)
linearity-efficiency trade-off. Most of the digital linearization techniques
rely on PA measurements using a dedicated feedback receiver. However, in modern
systems with large amount of RF chains and high carrier frequency, dedicated
receiver per RF chain is costly and complex to implement. This issue can be
addressed by measuring PAs over the air, but in that case, this extra
signalling is sharing resources with the actual data transmission. In this
paper, we look at the problem from an estimation theory point of view so as to
minimize pilot overhead while optimizing estimation performance. We show that
conventional results in the mathematical statistics community can be used. We
find the least squares (LS) optimal training design, minimizing the maximal
mean squared error (MSE) of the reconstructed PA response over its whole input
range. As compared to uniform training, simulations demonstrate a factor 10
reduction of the maximal MSE for a L = 7 PA polynomial order. Using prior
information, the LMMSE estimator can achieve an additional gain of a factor up
to 300 at low signal-to-noise ratio (SNR).
