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

Ultra Wideband Impulse Radio Superregenerative Reception

Postprint (published version

Economical sampling of parametric signals

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (p. 107-115).This thesis proposes architectures and algorithms for digital acquisition of parametric signals. It furthermore provides bounds for the performance of these systems in the presence of noise. Our simple acquisition circuitry and low sampling rate enable accurate parameter estimation to be achieved economically. In present practice, sampling and estimation are not integrated: the sampling device does not take advantage of the parametric model, and the estimation assumes that noise in the data is signal-independent additive white Gaussian noise. We focus on estimating the timing information in signals that are linear combinations of scales and shifts of a known pulse. This signal model is well-known in a variety of disciplines such as ultra-wideband signaling, neurobiology, etc. The signal is completely determined by the amplitudes and shifts of the summands. The delays determine a subspace that contains the signals, so estimating the shifts is equivalent to subspace estimation. By contrast, conventional sampling theory yields a least-squares approximation to a signal from a fixed shift-invariant subspace of possible reconstructions. Conventional acquisition takes samples at a rate higher than twice the signal bandwidth.(cont.) Although this may be feasible, there is a trade-off between power, accuracy, and speed. Under the signal model of interest, when the pulses are very narrow, the number of parameters per unit time-the rate of innovation-is much lower than the Fourier bandwidth. There is thus potential for much lower sampling rate so long as nonlinear reconstruction algorithms are used. We present a new sampling scheme that takes simultaneous samples at the outputs of multiple channels. This new scheme can be implemented with simple circuitry and has a successive approximation property that can be used to detect undermodeling. In many regimes our algorithms provide better timing accuracy and resolution than conventional systems. Our new analytical and algorithmic techniques are applied to previously proposed systems, and it is shown that all the systems considered have super-resolution properties. Finally, we consider the same parameter estimation problem when the sampling instances are perturbed by signal-independent timing noise. We give an iterative algorithm that achieves accurate timing estimation by exploiting knowledge of the pulse shape.by Julius Kusuma.Ph.D