  Reciprocity-based time-division duplex (TDD) Massive MIMO (multiple-input
multiple-output) systems utilize channel estimates obtained in the uplink to
perform precoding in the downlink. However, this method has been criticized of
breaking down, in the sense that the channel estimates are not good enough to
spatially separate multiple user terminals, at low uplink reference signal
signal-to-noise ratios, due to insufficient channel estimation quality.
Instead, codebook-based downlink precoding has been advocated for as an
alternative solution in order to bypass this problem. We analyze this problem
by considering a "grid-of-beams world" with a finite number of possible
downlink channel realizations. Assuming that the terminal accurately can detect
the downlink channel, we show that in the case where reciprocity holds,
carefully designing a mapping between the downlink channel and the uplink
reference signals will perform better than both the conventional TDD Massive
MIMO and frequency-division duplex (FDD) Massive MIMO approach. We derive
elegant metrics for designing this mapping, and further, we propose algorithms
that find good sequence mappings.
