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

Sub-Nyquist Sampling: Bridging Theory and Practice

Sampling theory encompasses all aspects related to the conversion of continuous-time signals to discrete streams of numbers. The famous Shannon-Nyquist theorem has become a landmark in the development of digital signal processing. In modern applications, an increasingly number of functions is being pushed forward to sophisticated software algorithms, leaving only those delicate finely-tuned tasks for the circuit level. In this paper, we review sampling strategies which target reduction of the ADC rate below Nyquist. Our survey covers classic works from the early 50's of the previous century through recent publications from the past several years. The prime focus is bridging theory and practice, that is to pinpoint the potential of sub-Nyquist strategies to emerge from the math to the hardware. In that spirit, we integrate contemporary theoretical viewpoints, which study signal modeling in a union of subspaces, together with a taste of practical aspects, namely how the avant-garde modalities boil down to concrete signal processing systems. Our hope is that this presentation style will attract the interest of both researchers and engineers in the hope of promoting the sub-Nyquist premise into practical applications, and encouraging further research into this exciting new frontier.Comment: 48 pages, 18 figures, to appear in IEEE Signal Processing Magazin

Semiparametric Estimation of Fractional Cointegrating Subspaces

We consider a common components model for multivariate fractional cointegration, in which the s>=1 components have different memory parameters. The cointegrating rank is allowed to exceed 1. The true cointegrating vectors can be decomposed into orthogonal fractional cointegrating subspaces such that vectors from distinct subspaces yield cointegrating errors with distinct memory parameters, denoted by d_k for k=1,...,s. We estimate each cointegrating subsspace separately using appropriate sets of eigenvectors of an averaged periodogram matrix of tapered, differenced observations. The averaging uses the first m Fourier frequencies, with m fixed. We will show that any vector in the k'th estimated coingetraging subspace is, with high probability, close to the k'th true cointegrating subspace, in the sense that the angle between the estimated cointegrating vector and the true cointegrating subspace converges in probability to zero. The angle is O_p(n^{- \alpha_k}), where n is the sample size and \alpha_k is the shortest distance between the memory parameters corresponding to the given and adjacent subspaces. We show that the cointegrating residuals corresponding to an estimated cointegrating vector can be used to obtain a consistent and asymptotically normal estimate of the memory parameter for the given cointegrating subspace, using a univariate Gaussian semiparametric estimator with a bandwidth that tends to \infty more slowly than n. We also show how these memory parameter estimates can be used to test for fractional cointegration and to consistently identify the cointegrating subspaces.Fractional Cointegration; Long Memory; Tapering; Periodogram