38 research outputs found

    Identification of Parametric Underspread Linear Systems and Super-Resolution Radar

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    Identification of time-varying linear systems, which introduce both time-shifts (delays) and frequency-shifts (Doppler-shifts), is a central task in many engineering applications. This paper studies the problem of identification of underspread linear systems (ULSs), whose responses lie within a unit-area region in the delay Doppler space, by probing them with a known input signal. It is shown that sufficiently-underspread parametric linear systems, described by a finite set of delays and Doppler-shifts, are identifiable from a single observation as long as the time bandwidth product of the input signal is proportional to the square of the total number of delay Doppler pairs in the system. In addition, an algorithm is developed that enables identification of parametric ULSs from an input train of pulses in polynomial time by exploiting recent results on sub-Nyquist sampling for time delay estimation and classical results on recovery of frequencies from a sum of complex exponentials. Finally, application of these results to super-resolution target detection using radar is discussed. Specifically, it is shown that the proposed procedure allows to distinguish between multiple targets with very close proximity in the delay Doppler space, resulting in a resolution that substantially exceeds that of standard matched-filtering based techniques without introducing leakage effects inherent in recently proposed compressed sensing-based radar methods.Comment: Revised version of a journal paper submitted to IEEE Trans. Signal Processing: 30 pages, 17 figure

    Density Criteria for the Identification of Linear Time-Varying Systems

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    This paper addresses the problem of identifying a linear time-varying (LTV) system characterized by a (possibly infinite) discrete set of delays and Doppler shifts. We prove that stable identifiability is possible if the upper uniform Beurling density of the delay-Doppler support set is strictly smaller than 1/2 and stable identifiability is impossible for densities strictly larger than 1/2. The proof of this density theorem reveals an interesting relation between LTV system identification and interpolation in the Bargmann-Fock space. Finally, we introduce a subspace method for solving the system identification problem at hand.Comment: IEEE International Symposium on Information Theory (ISIT), Hong Kong, China, June 201

    Super-Resolution Radar

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    In this paper we study the identification of a time-varying linear system from its response to a known input signal. More specifically, we consider systems whose response to the input signal is given by a weighted superposition of delayed and Doppler shifted versions of the input. This problem arises in a multitude of applications such as wireless communications and radar imaging. Due to practical constraints, the input signal has finite bandwidth B, and the received signal is observed over a finite time interval of length T only. This gives rise to a delay and Doppler resolution of 1/B and 1/T. We show that this resolution limit can be overcome, i.e., we can exactly recover the continuous delay-Doppler pairs and the corresponding attenuation factors, by solving a convex optimization problem. This result holds provided that the distance between the delay-Doppler pairs is at least 2.37/B in time or 2.37/T in frequency. Furthermore, this result allows the total number of delay-Doppler pairs to be linear up to a log-factor in BT, the dimensionality of the response of the system, and thereby the limit for identifiability. Stated differently, we show that we can estimate the time-frequency components of a signal that is S-sparse in the continuous dictionary of time-frequency shifts of a random window function, from a number of measurements, that is linear up to a log-factor in S.Comment: Revised versio

    Cornerstones of Sampling of Operator Theory

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    This paper reviews some results on the identifiability of classes of operators whose Kohn-Nirenberg symbols are band-limited (called band-limited operators), which we refer to as sampling of operators. We trace the motivation and history of the subject back to the original work of the third-named author in the late 1950s and early 1960s, and to the innovations in spread-spectrum communications that preceded that work. We give a brief overview of the NOMAC (Noise Modulation and Correlation) and Rake receivers, which were early implementations of spread-spectrum multi-path wireless communication systems. We examine in detail the original proof of the third-named author characterizing identifiability of channels in terms of the maximum time and Doppler spread of the channel, and do the same for the subsequent generalization of that work by Bello. The mathematical limitations inherent in the proofs of Bello and the third author are removed by using mathematical tools unavailable at the time. We survey more recent advances in sampling of operators and discuss the implications of the use of periodically-weighted delta-trains as identifiers for operator classes that satisfy Bello's criterion for identifiability, leading to new insights into the theory of finite-dimensional Gabor systems. We present novel results on operator sampling in higher dimensions, and review implications and generalizations of the results to stochastic operators, MIMO systems, and operators with unknown spreading domains

    Spatial Compressive Sensing for MIMO Radar

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    We study compressive sensing in the spatial domain to achieve target localization, specifically direction of arrival (DOA), using multiple-input multiple-output (MIMO) radar. A sparse localization framework is proposed for a MIMO array in which transmit and receive elements are placed at random. This allows for a dramatic reduction in the number of elements needed, while still attaining performance comparable to that of a filled (Nyquist) array. By leveraging properties of structured random matrices, we develop a bound on the coherence of the resulting measurement matrix, and obtain conditions under which the measurement matrix satisfies the so-called isotropy property. The coherence and isotropy concepts are used to establish uniform and non-uniform recovery guarantees within the proposed spatial compressive sensing framework. In particular, we show that non-uniform recovery is guaranteed if the product of the number of transmit and receive elements, MN (which is also the number of degrees of freedom), scales with K(log(G))^2, where K is the number of targets and G is proportional to the array aperture and determines the angle resolution. In contrast with a filled virtual MIMO array where the product MN scales linearly with G, the logarithmic dependence on G in the proposed framework supports the high-resolution provided by the virtual array aperture while using a small number of MIMO radar elements. In the numerical results we show that, in the proposed framework, compressive sensing recovery algorithms are capable of better performance than classical methods, such as beamforming and MUSIC.Comment: To appear in IEEE Transactions on Signal Processin
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