Capacity pre-log of noncoherent SIMO channels via Hironaka's Theorem


Veniamin I. Morgenshtern, Erwin Riegler, Wei Yang, Giuseppe Durisi, Shaowei Lin, Bernd Sturmfels, and Helmut Bölcskei


IEEE Transactions on Information Theory, Vol. 59, No. 7, pp. 4213-4229, July 2013.

DOI: 10.1109/TIT.2013.2251394

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We find the capacity pre-log of a temporally correlated Rayleigh block-fading single-input multiple-output (SIMO) channel in the noncoherent setting. It is well known that for blocklength L and rank of the channel covariance matrix equal to Q, the capacity pre-log in the single-input single-output (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 for the blocklength L going to infinity, 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.


Fading, multiple-antenna channels, noncoherent capacity

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