We analyze a mobile wireless link comprising M transmitter and N receiver antennas operating in a Rayleigh flat-fading environment. The propagation coefficients between every pair of transmitter and receiver antennas are statistically independent and unknown; they remain constant for a coherence interval of T symbol periods, after which they change to new independent values which they maintain for another T symbol periods, and so on. Computing the link capacity, associated with channel coding over multiple fading intervals, requires an optimization over the joint density of T M complex transmitted signals. We prove that there is no point in making the number of transmitter antennas greater than the length of the coherence interval: the capacity for M> Tis equal to the capacity for M = T. Capacity is achieved when the T M transmitted signal matrix is equal to the product of two statistically independent matrices: a T T isotropically distributed unitary matrix times a certain T M random matrix that is diagonal, real, and nonnegative. This result enables us to determine capacity for many interesting cases. We conclude that, for a fixed number of antennas, as the length of the coherence interval increases, the capacity approaches the capacity obtained as if the receiver knew the propagation coefficients. Index Terms—Multi-element antenna arrays, wireless communications, space-time modulation 1

A stable recurrence for the incomplete

1.1. Background. L-functions and modular forms underlie much of twentieth century number theory and are connected to the practical applications of number theory in cryptography. The fundamental importance of these functions in mathematics is supported by the fact that two of the seven Clay Mathematics Million Dollar Millennium Problems [20] deal with properties of these functions, namely the