Time Delay Estimation from Low Rate Samples: A Union of Subspaces Approach (0905.2429v3)

Abstract: 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.

• The paper achieves perfect recovery of time delays in multipath environments using sub-Nyquist sampling rates by treating signals as a union of subspaces.
• The proposed methodology employs the ESPRIT algorithm within a subspace framework and uses multichannel filters, enabling robust channel identification even with overlapping pulses.
• This approach allows for more efficient hardware implementations by reducing required sampling rates and extends compressed sensing into continuous signals, with potential applications in wireless communication and radar.

Time Delay Estimation from Low Rate Samples: A Union of Subspaces Approach

This paper addresses the problem of estimating time delays in multipath environments from low-rate samples, expanding upon existing methodologies that often require sampling at the Nyquist rate or operate on the analog signal. The research aims to achieve perfect recovery of time delays using sub-Nyquist sampling rates, which are determined by the number of multipath components and transmission rate, but remain independent of the pulse bandwidth. This approach holds significant potential for practical implementations, particularly in settings with a limited number of propagation paths or high-bandwidth pulse requirements.

The paper proposes a sampling paradigm within the context of signals lying in a union of subspaces, treating the multipath signal samples via the ESPRIT algorithm, a subspace method similarly employed in direction of arrival (DOA) estimations. This formulation leads to a robust mechanism for channel identification, even when dealing with overlapping pulse scenarios; further, it illustrates conditions under which time delay recovery from minimal sampling is feasible. By focusing on formulated delayed subspace models, the authors present efficient sampling schemes that utilize multichannel filters.

Theoretical foundations for this approach hinge upon the transformation of a parametric estimation problem into the subspace framework within sampling theory. Their formulation involves a bank of $p$ sampling functions, which when appropriately manipulated, yield conditions for estimation. Prior approaches often neglect sampling implications or require a prohibitively high sampling rate; this paper introduces the notion that sufficient knowledge of multipath parameters renders it possible to sample below the Nyquist threshold.

The implications of this work are manifold. Practically, reducing the required sampling rate allows for more efficient hardware implementations, such as lower power-consuming ADCs, without sacrificing the precision necessary for channel estimation. Theoretically, the paper extends the domain of compressed sensing into the continuous regime of a union of subspaces, offering new directions in sampling theory. As this research can lower barriers inherent in high-bandwidth or multi-channel systems, it suggests future advancements in systems such as wireless communication or radar technologies.

Future directions may explore integrating this sampling paradigm with compressed sensing methodologies or adaptive filtering arrangements to address the dynamic nature of signal environments. Another potential avenue could investigate the robustness of this approach under varying noise and interference conditions, ensuring practical deployment. Furthermore, there might be a fruitful cross-disciplinary application of union-based subspace sampling in other domains of signal processing, perhaps in imaging or acoustics, where capturing the essence of signal scenes without full fidelity sampling becomes invaluable.