  This paper revisited the problem of Private Information Retrieval (PIR),
where there are $N$ replicated non-communicating databases containing the same
$M$ messages and a user who wishes to retrieve one of the messages without
revealing the wanted message's index to the databases. However, we assume a
block-fading additive white Gaussian noise multiple access channel (AWGN MAC)
linking the user and the databases. Previous work \cite{shmuel2021private}
presented a joint channel-PIR scheme, utilizing the Compute and Forward
protocol, showing the potential of a joint channel-PIR scheme over a separated
one. This paper proposes an improved joint scheme tailored for the PIR problem
with $N$ databases over a block-fading AWGN. Unlike the C\&F protocol, our
scheme offers reduced computational complexity while improving the scaling laws
governing the achievable rate. Specifically, the achievable rate scales with
the number of databases $N$ and the power $P$ similarly to the channel capacity
without the privacy constraint and outperforms the C\&F-based approach.
Furthermore, the analysis demonstrates that the improved rate exhibits only a
finite gap from the unconstrained channel capacity -- one bit per second per Hz
as $N$ increases.
