935 research outputs found
Time Delay Estimation from Low Rate Samples: A Union of Subspaces Approach
Time delay estimation arises in many applications in which a multipath medium
has to be identified from pulses transmitted through the channel. Various
approaches have been proposed in the literature to identify time delays
introduced by multipath environments. However, these methods either operate on
the analog received signal, or require high sampling rates in order to achieve
reasonable time resolution. In this paper, our goal is to develop a unified
approach to time delay estimation from low rate samples of the output of a
multipath channel. Our methods result in perfect recovery of the multipath
delays from samples of the channel output at the lowest possible rate, even in
the presence of overlapping transmitted pulses. This rate depends only on the
number of multipath components and the transmission rate, but not on the
bandwidth of the probing signal. In addition, our development allows for a
variety of different sampling methods. By properly manipulating the low-rate
samples, we show that the time delays can be recovered using the well-known
ESPRIT algorithm. Combining results from sampling theory with those obtained in
the context of direction of arrival estimation methods, we develop necessary
and sufficient conditions on the transmitted pulse and the sampling functions
in order to ensure perfect recovery of the channel parameters at the minimal
possible rate. Our results can be viewed in a broader context, as a sampling
theorem for analog signals defined over an infinite union of subspaces
Signal Recovery in Perturbed Fourier Compressed Sensing
In many applications in compressed sensing, the measurement matrix is a
Fourier matrix, i.e., it measures the Fourier transform of the underlying
signal at some specified `base' frequencies , where is the
number of measurements. However due to system calibration errors, the system
may measure the Fourier transform at frequencies
that are different from the base frequencies and where
are unknown. Ignoring perturbations of this nature can lead to major errors in
signal recovery. In this paper, we present a simple but effective alternating
minimization algorithm to recover the perturbations in the frequencies \emph{in
situ} with the signal, which we assume is sparse or compressible in some known
basis. In many cases, the perturbations can be expressed
in terms of a small number of unique parameters . We demonstrate that
in such cases, the method leads to excellent quality results that are several
times better than baseline algorithms (which are based on existing off-grid
methods in the recent literature on direction of arrival (DOA) estimation,
modified to suit the computational problem in this paper). Our results are also
robust to noise in the measurement values. We also provide theoretical results
for (1) the convergence of our algorithm, and (2) the uniqueness of its
solution under some restrictions.Comment: New theortical results about uniqueness and convergence now included.
More challenging experiments now include
- …