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
$L$
paths with different angles of arrival (AoAs). It is assumed that Discrete-Time Fourier Transform (DTFT) beams (also known as analog beams with DFT beams as special cases) 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; the underlying signal is assumed to be constant during the projections. 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
$K$
needed to recover the
$L$
AoAs. Previous work on this problem showed (by applying compressed sensing theory) that
$K \approx L\mathcal {O}(\log (N/L))$
measurements are sufficient for recovering the AoAs, which grows with
$N$
. First, we show that necessary conditions for recovery are
$N \geq 2L$
and
$K \geq 2L$
. Second, by using properties of DTFT beam projections, we are able to show that if
$N \geq 2L$
then
$K \geq 3L$
arbitrary DTFT measurements suffice; hence, dependency on
$N$
is completely removed. Furthermore, if the DTFT beams are chosen to equal DFT beams with period
$N$
, then
$K \geq 2L$
beam measurements are enough, achieving sufficiency of the necessary conditions. 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 improved performance over conventional algorithms such as beamspace ESPRIT and compressed sensing.