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Download file# Sparse Channel Estimation From Discrete-Time Fourier Transform Beam Measurements

In this paper, we study channel estimation at a uniform linear array (ULA) with $N$ antennas, where the channel at the ULA is composed of $L$ scatterers incoming from $L$ different angles of arrival (AoAs). It is assumed that Discrete-Time Fourier Transform (DTFT) beams (also known as \emph{analog} beams) are applied at the ULA to project the incoming signal onto a single (or multiple) RF chain(s), after which the signal is sampled and measured in baseband domain. This measurement procedure arises in various communication systems, such as the receive beam sweeping phase in 5G NR, where DTFT beams are used due to their simple implementation as linear phase shifts on analog antennas. A fundamental question about this procedure is the number of DTFT measurements necessary to recover the $L$ AoAs. Previous work on this problem showed (by applying compressed sensing theory) that $L\mathcal{O}(\log(N/L))$ measurements are sufficient for recovering the AoAs, which grows large for large $N$. Instead, by using properties of DTFT beam projections, we are able to show that if $N \geq 2L$, then $3L$ arbitrary DTFT measurements are enough; hence, dependency on $N$ is completely removed. The main reason for this drastic improvement is that sparsity is found in a \emph{non-linear} mapping of unknowns to observations, while compressed sensing deals with \emph{linear} mappings of unknowns to observations. Furthermore, if the DTFT beams are centered at harmonic frequencies (resulting in DFT beams) with period $N$, then $2L$ beam measurements are enough. With these results, an AoA estimation algorithm is formulated which has enormous complexity savings compared to $L$-dimensional AoA search such as maximum likelihood (ML) estimation. Numerical simulations demonstrate the algorithm's superior performance over a conventional algorithm.