615 research outputs found
On Unlimited Sampling
Shannon's sampling theorem provides a link between the continuous and the
discrete realms stating that bandlimited signals are uniquely determined by its
values on a discrete set. This theorem is realized in practice using so called
analog--to--digital converters (ADCs). Unlike Shannon's sampling theorem, the
ADCs are limited in dynamic range. Whenever a signal exceeds some preset
threshold, the ADC saturates, resulting in aliasing due to clipping. The goal
of this paper is to analyze an alternative approach that does not suffer from
these problems. Our work is based on recent developments in ADC design, which
allow for ADCs that reset rather than to saturate, thus producing modulo
samples. An open problem that remains is: Given such modulo samples of a
bandlimited function as well as the dynamic range of the ADC, how can the
original signal be recovered and what are the sufficient conditions that
guarantee perfect recovery? In this paper, we prove such sufficiency conditions
and complement them with a stable recovery algorithm. Our results are not
limited to certain amplitude ranges, in fact even the same circuit architecture
allows for the recovery of arbitrary large amplitudes as long as some estimate
of the signal norm is available when recovering. Numerical experiments that
corroborate our theory indeed show that it is possible to perfectly recover
function that takes values that are orders of magnitude higher than the ADC's
threshold.Comment: 11 pages, 4 figures, copy of initial version to appear in Proceedings
of 12th International Conference on Sampling Theory and Applications (SampTA
Cornerstones of Sampling of Operator Theory
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
Signal and System Approximation from General Measurements
In this paper we analyze the behavior of system approximation processes for
stable linear time-invariant (LTI) systems and signals in the Paley-Wiener
space PW_\pi^1. We consider approximation processes, where the input signal is
not directly used to generate the system output, but instead a sequence of
numbers is used that is generated from the input signal by measurement
functionals. We consider classical sampling which corresponds to a pointwise
evaluation of the signal, as well as several more general measurement
functionals. We show that a stable system approximation is not possible for
pointwise sampling, because there exist signals and systems such that the
approximation process diverges. This remains true even with oversampling.
However, if more general measurement functionals are considered, a stable
approximation is possible if oversampling is used. Further, we show that
without oversampling we have divergence for a large class of practically
relevant measurement procedures.Comment: This paper will be published as part of the book "New Perspectives on
Approximation and Sampling Theory - Festschrift in honor of Paul Butzer's
85th birthday" in the Applied and Numerical Harmonic Analysis Series,
Birkhauser (Springer-Verlag). Parts of this work have been presented at the
IEEE International Conference on Acoustics, Speech, and Signal Processing
2014 (ICASSP 2014
Sampling from a system-theoretic viewpoint: Part II - Noncausal solutions
This paper puts to use concepts and tools introduced in Part I to address a wide spectrum of noncausal sampling and reconstruction problems. Particularly, we follow the system-theoretic paradigm by using systems as signal generators to account for available information and system norms (L2 and L∞) as performance measures. The proposed optimization-based approach recovers many known solutions, derived hitherto by different methods, as special cases under different assumptions about acquisition or reconstructing devices (e.g., polynomial and exponential cardinal splines for fixed samplers and the Sampling Theorem and its modifications in the case when both sampler and interpolator are design parameters). We also derive new results, such as versions of the Sampling Theorem for downsampling and reconstruction from noisy measurements, the continuous-time invariance of a wide class of optimal sampling-and-reconstruction circuits, etcetera
Oversampling PCM techniques and optimum noise shapers for quantizing a class of nonbandlimited signals
We consider the efficient quantization of a class of nonbandlimited signals, namely, the class of discrete-time signals that can be recovered from their decimated version. The signals are modeled as the output of a single FIR interpolation filter (single band model) or, more generally, as the sum of the outputs of L FIR interpolation filters (multiband model). These nonbandlimited signals are oversampled, and it is therefore reasonable to expect that we can reap the same benefits of well-known efficient A/D techniques that apply only to bandlimited signals. We first show that we can obtain a great reduction in the quantization noise variance due to the oversampled nature of the signals. We can achieve a substantial decrease in bit rate by appropriately decimating the signals and then quantizing them. To further increase the effective quantizer resolution, noise shaping is introduced by optimizing prefilters and postfilters around the quantizer. We start with a scalar time-invariant quantizer and study two important cases of linear time invariant (LTI) filters, namely, the case where the postfilter is the inverse of the prefilter and the more general case where the postfilter is independent from the prefilter. Closed form expressions for the optimum filters and average minimum mean square error are derived in each case for both the single band and multiband models. The class of noise shaping filters and quantizers is then enlarged to include linear periodically time varying (LPTV)M filters and periodically time-varying quantizers of period M. We study two special cases in great detail
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