We develop sub-Nyquist sampling systems for analog signals comprised of several, possibly overlapping, finite duration pulses with unknown shapes and time positions. Efficient sampling schemes when either the pulse shape or the locations of the pulses are known have been previously developed. To the best of our knowledge, stable and low-rate sampling strategies for continuous signals that are superpositions of unknown pulses without knowledge of the pulse locations have not been derived. The goal in this paper is to fill this gap. We propose a multichannel scheme based on Gabor frames that exploits the sparsity of signals in time and enables sampling multipulse signals at sub-Nyquist rates. Moreover, if the signal is additionally essentially multiband, then the sampling scheme can be adapted to lower the sampling rate without knowing the band locations. We show that, with proper preprocessing, the necessary Gabor coefficients, can be recovered from the samples using standard methods of compressed sensing. In addition, we provide error estimates on the reconstruction and analyze the proposed architecture in the presence of noise.
Despite the similarity, there are some important differences between the systems. First, in the mixing stage we use time-limited and non-periodic waveforms, while the MWC relies on periodic functions. Second, following the mixing stage, we use an integrator in contrast to the low-pass filter in @cite_17 . These differences are due to the fact that we are interested in different quantities: content of the signal on time intervals in our work as opposed to frequency bands in @cite_17 . However, in both systems the mixing is used for the same purpose: to reduce the sampling rate relative to the Nyquist rate.
@cite_17
2054474363 2141116650 1993784401 2103898286
We address the challenging open problem of blindly separating time position varying mixtures, and attempt to separate the sources from such mixtures without having prior information about the sources or the mixing system. Unlike studies concerning instantaneous or convolutive mixtures, we assume that the mixing system (medium) is varying in time position. Attempts to solve this problem have mostly utilized, so far, online algorithms based on tracking the mixing system by methods previously developed for the instantaneous or convolutive mixtures. In contrast with these attempts, we develop a unified approach in the form of staged sparse component analysis (SSCA). Accordingly, we assume that the sources are either sparse or can be “sparsified.” In the first stage, we estimate the filters of the mixing system, based on the scatter plot of the sparse mixtures' data, using a proper clustering and curve surface fitting. In the second stage, the mixing system is inverted, yielding the estimated sources. We use the SSCA approach for solving three types of mixtures: time position varying instantaneous mixtures, single-path mixtures, and multipath mixtures. Real-life scenarios and simulated mixtures are used to demonstrate the performance of our approach. Wideband analog signals push contemporary analog-to-digital conversion (ADC) systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the band limit, although the locations of the frequencies may not be known a priori. For this type of sparse signal, other sampling strategies are possible. This paper describes a new type of data acquisition system, called a random demodulator, that is constructed from robust, readily available components. Let K denote the total number of frequencies in the signal, and let W denote its band limit in hertz. Simulations suggest that the random demodulator requires just O(K log(W K)) samples per second to stably reconstruct the signal. This sampling rate is exponentially lower than the Nyquist rate of W hertz. In contrast to Nyquist sampling, one must use nonlinear methods, such as convex programming, to recover the signal from the samples taken by the random demodulator. This paper provides a detailed theoretical analysis of the system's performance that supports the empirical observations. The authors present a sub-Nyquist analog-to-digital converter of wideband inputs. The circuit realises the recently proposed modulated wideband converter, which is a flexible platform for sampling signals according to their actual bandwidth occupation. The theoretical work enables, for example, a sub-Nyquist wideband communication receiver, which has no prior information on the transmitter carrier positions. The present design supports input signals with 2 GHz Nyquist rate and 120 MHz spectrum occupancy, with arbitrary transmission frequencies. The sampling rate is as low as 280 MHz. To the best of the authors' knowledge, this is the first reported hardware that performs sub-Nyquist sampling and reconstruction of wideband signals. The authors describe the various circuit design considerations, with an emphasis on the non-ordinary challenges the converter introduces: mixing a signal with a multiple set of sinusoids, rather than a single local oscillator, and generation of highly transient periodic waveforms, with transient intervals on the order of the Nyquist rate. Hardware experiments validate the design and demonstrate sub-Nyquist sampling and signal reconstruction. A noiseless ideal low-pass filter, followed by a limiter, is normally used as a binary data channel by sampling the output once per Nyquist interval. Detectors that sample more often encounter intersymbol interference, but can be used in ways that increase the information rate. A signaling system that achieves an information rate of 1.072 b Nyquist interval by allowing the detector to sample at times a half Nyquist interval apart is presented. This rate increase is small, but it shows that, in principle, oversampling does permit faster rates. Another signaling system that more closely resembles a digital data system is also presented; it has rate 1.050. By allowing a more irregular pattern of sampling times, the rate is increased to 1.090 (or 1.062 for the 'digital' system). >