On MMSE Crossing Properties and Implications in Parallel Vector Gaussian Channels

The scalar additive Gaussian noise channel has the "single crossing point" property between the minimum mean square error (MMSE) in the estimation of the input given the channel output, assuming a Gaussian input to the channel, and the MMSE assuming an arbitrary input. This paper extends the result to the parallel vector additive Gaussian channel in three phases. 1) The channel matrix is the identity matrix, and we limit the Gaussian input to a vector of Gaussian i.i.d. elements. The "single crossing point" property is with respect to the signal-to-noise ratio (as in the scalar case). 2) The channel matrix is arbitrary, and the Gaussian input is limited to an independent Gaussian input. A "single crossing point" property is derived for each diagonal element of the MMSE matrix. 3) The Gaussian input is allowed to be an arbitrary Gaussian random vector. A "single crossing point" property is derived for each eigenvalue of the difference matrix between the two MMSE matrices. These three extensions are then translated to new information theoretic properties on the mutual information, using the I-MMSE relationship, a fundamental relationship between estimation theory and information theory revealed by Guo and coworkers. The results of the last phase are also translated to a new property of Fisher information. Finally, the applicability of all three extensions on information theoretic problems is demonstrated through a proof of a special case of Shannon's vector entropy power inequality, a converse proof of the capacity region of the parallel degraded broadcast channel (BC) under an input per-antenna power constraint and under an input covariance constraint, and a converse proof of the capacity region of the compound parallel degraded BC under an input covariance constraint.