Capacity Pre-Log of Noncoherent SIMO Channels via Hironaka's Theorem
We find the capacity pre-log of a temporally correlated Rayleigh block-fading SIMO channel in the noncoherent setting. It is well known that for block-length L and rank of the channel covariance matrix equal to Q, the capacity pre-log in the SISO case is given by 1-Q/L. Here, Q/L can be interpreted...
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Zusammenfassung: | We find the capacity pre-log of a temporally correlated Rayleigh block-fading
SIMO channel in the noncoherent setting. It is well known that for block-length
L and rank of the channel covariance matrix equal to Q, the capacity pre-log in
the SISO case is given by 1-Q/L. Here, Q/L can be interpreted as the pre-log
penalty incurred by channel uncertainty. Our main result reveals that, by
adding only one receive antenna, this penalty can be reduced to 1/L and can,
hence, be made to vanish in the large-L limit, even if Q/L remains constant as
L goes to infinity. Intuitively, even though the SISO channels between the
transmit antenna and the two receive antennas are statistically independent,
the transmit signal induces enough statistical dependence between the
corresponding receive signals for the second receive antenna to be able to
resolve the uncertainty associated with the first receive antenna's channel and
thereby make the overall system appear coherent. The proof of our main theorem
is based on a deep result from algebraic geometry known as Hironaka's Theorem
on the Resolution of Singularities. |
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DOI: | 10.48550/arxiv.1204.2775 |