# Channel Extrapolation in FDD Massive MIMO: Theoretical Analysis and Numerical Validation

## TL;DR

This paper analyzes the theoretical limits of frequency extrapolation for downlink channel estimation in FDD massive MIMO systems, demonstrating potential SNR gains and the impact of antenna configurations on extrapolation performance.

## Contribution

It derives a lower bound on the mean squared error for channel extrapolation, providing insights into the benefits of high-resolution estimation and antenna array configurations.

## Key findings

- SNR gain improves linearly with the number of antennas.
- Extrapolation range is limited in SISO but more promising in MIMO.
- Practical algorithms can approach the theoretical lower bound.

## Abstract

Downlink channel estimation in massive MIMO systems is well known to generate a large overhead in frequency division duplex (FDD) mode as the amount of training generally scales with the number of transmit antennas. Using instead an extrapolation of the channel from the measured uplink estimates to the downlink frequency band completely removes this overhead. In this paper, we investigate the theoretical limits of channel extrapolation in frequency. We highlight the advantage of basing the extrapolation on high-resolution channel estimation. A lower bound (LB) on the mean squared error (MSE) of the extrapolated channel is derived. A simplified LB is also proposed, giving physical intuition on the SNR gain and extrapolation range that can be expected in practice. The validity of the simplified LB relies on the assumption that the paths are well separated. The SNR gain then linearly improves with the number of receive antennas while the extrapolation performance penalty quadratically scales with the ratio of the frequency and the training bandwidth. The theoretical LB is numerically evaluated using a 3GPP channel model and we show that the LB can be reached by practical high-resolution parameter extraction algorithms. Our results show that there are strong limitations on the extrapolation range than can be expected in SISO systems while much more promising results can be obtained in the multiple-antenna setting as the paths can be more easily separated in the delay-angle domain.