Quadratic Displacement Operators-Theory and Application to Millimeter-Wave Channel Tracking

First, a family of quadratic displacement operators based on group Fourier Transform has been proposed for joint distribution analysis. Second, considering the quadratic displacement operators, a novel millimeter-wave massive MIMO channel tracking has been proposed in Time-Angle (TA) plane. Due to a poor scattering environment, millimeter-wave communication suffers from an ill-conditioned channel matrix. Computationally effective compressed sensing has been proposed to estimate a few dominant paths. These methods rely on orthogonal pilot signals to estimate the channel reliably. However, applying compressed-sensing solutions to the continuous channel leads to significant pilot overhead. To reduce the pilot overhead, one needs to consider the channel dynamics, particularly the spectral overlap between the channel samples. Considering that channel spectral properties can be represented as a joint time-angle distribution, we propose a novel quadratic displacement operator in TA-plane. Considering group Fourier transform in TA-plane, we show that the subderivative of an angular parameter is the dual group of the original angular signal. For a group transform defined over the finite Abelian group, with respect to the finite-dimensional antenna arrays, the purposed time-angle representation matches the cover space of manifold defined in the virtual channel model recommended by A. Sayeed. In TA-plane, the quadratic displacement operator maps old time-angle coordinate onto a new coordinate by taking to account the local spectral properties of the channel samples. By applying the purposed quadratic displacement operator to the Saleh-Valenzuela channel model, the time evolution operator for directional millimeter-wave channel has been derived. Assuming independently identically distributed multipath contribution, we show that channel Wigner distribution is the direct sum of the cluster-wise Wigner distribution. Accordingly, we have shown that the continuous linear time-variant channel transfer function can be represented as the Hadamard product between Wigner distribution of the channel and Saleh-Valenzuela model. Considering the fact that the density matrix is a Weyl correspondence of Wigner distribution, a union of subspaces method has been employed to construct the density matrix for channels samples with slightly different states. Numerical results have been presented to evaluate the performance of the proposed channel tracking method.