Multi-Band Covariance Interpolation with Applications in Massive MIMO

Abstract

In this paper, we study the problem of multi-band (frequency-variant) covariance interpolation with a particular emphasis towards massive MIMO applications. In a massive MIMO system, the communication between each BS with M 1 antennas and each single-antenna user occurs through a collection of scatterers in the environment, where the channel vector of each user at BS antennas consists in a weighted linear combination of the array responses of the scatterers, where each scatterer has its own angle of arrival (AoA) and complex channel gain. The array response at a given AoA depends on the wavelength of the incoming planar wave and is naturally frequency dependent. This results in a frequency-dependent distortion where the second order statistics, i.e., the covariance matrix, of the channel vectors varies with frequency. In this paper, we show that although this effect is generally negligible for a small number of antennas M , it results in a considerable distortion of the covariance matrix and especially its dominant signal subspace in the massive MIMO regime where M → ∞, and can generally incur a serious degradation of the performance especially in frequency division duplexing (FDD) massive MIMO systems where the uplink (UL) and the downlink (DL) communication occur over different frequency bands. We propose a novel UL-DL covariance interpolation technique that is able to recover the covariance matrix in the DL from an estimate of the covariance matrix in the UL under a mild reciprocity condition on the angular power spread function (PSF) of the users (this is in contrast with the UL-DL reciprocity of the instantaneous realization of the channel vectors which does not generally hold). We analyze the performance of our proposed scheme mathematically and prove its robustness under a sufficiently large spatial oversampling of the array. We also propose several simple off-the-shelf algorithms for UL-DL covariance interpolation and evaluate their performance via numerical simulations.Index Terms-UL-DL reciprocity, UL-DL reciprocity of the power spectral function, massive MIMO system.

Statistical Beamforming for FDD Downlink Massive MIMO via Spatial Information Extraction and Beam Selection

“…The CCM-based algorithms usually acquire the uplink CCM information via channel averaging and construct the downlink CCM from the uplink CCM. However, the transformation from the uplink CCM to the downlink CCM is non-trivial, for that the uplink and downlink communications occur over different frequency bands in the FDD massive MIMO systems [19].…”

Uplink-Downlink Channel Covariance Transformations and Precoding Design for FDD Massive MIMO

“…The ASF reciprocity was also assumed implicitly in several past works [9][10][11] by considering only ASFs with spike discrete components. Recent works [12][13][14] generalize these results to arbitrary ASFs consisting of both discrete spike and also continuous components, and show that such Uplink-Downlink covariance transformation (UDCT) is still feasible for sufficiently large number of BS antennas M although establishing the stability of UDCT is quite challenging when only a noisy estimate of the UL covariance matrix is available.…”

“…Let us consider a generic use and let us denote the ASF of this user by γpξq where ξ " sinpθq P r´1, 1s denotes a parametrization of the AoA θ P r´π 2 , π 2 s and where γpξq is density of the received signal power at the AoA ξ. The covariance matrix of the channel vector of this user at UL/DL carrier frequency f P tf ul , f dl u is given by [12] Σpf q "…”

“…and design an algorithm ALG that produces a structured ASF in A 0 whose UL covariance matrix matches x Σ ul under suitable metric. For example, [12,13] propose such algorithms using Non-Negative Least Squares (NNLS) and 2norm projections. However, none of these methods are able to capture the group-sparsity of the ASF in the AoA domain as illustrated in Fig.…”

“…1. For example, as we will illustrate via numerical simulations, the NNLS proposed in [12] is able to promote sparsity of ASF in the AoA domain but it does not yield necessarily group-sparse ASF. The 2 projection method [13], in contrast, is able to produce smooth ASFs but creates out-of-support components in the estimated ASF since 2 norm is inherently unable to promote sparsity.…”