Abstract:
Periodic nonuniform sampling has been considered in literature as an effective approach to reduce the sampling rate far below the Nyquist rate for sparse spectrum multiband signals. In the presence of non-ideality the sampling parameters play an important role on the quality of reconstructed signal. Also the average sampling ratio is directly dependent on the sampling parameters that they should be chosen for a minimum rate and complexity. In this paper we consider the effect of sampling parameters on the reconstruction error and the sampling ratio and suggest feasible approaches for achieving an optimal sampling and reconstruction.

Abstract:
Recent advances in optical systems make them ideal for undersampling multiband signals that have high bandwidths. In this paper we propose a new scheme for reconstructing multiband sparse signals using a small number of sampling channels. The scheme, which we call synchronous multirate sampling (SMRS), entails gathering samples synchronously at few different rates whose sum is significantly lower than the Nyquist sampling rate. The signals are reconstructed by solving a system of linear equations. We have demonstrated an accurate and robust reconstruction of signals using a small number of sampling channels that operate at relatively high rates. Sampling at higher rates increases the signal to noise ratio in samples. The SMRS scheme enables a significant reduction in the number of channels required when the sampling rate increases. We have demonstrated, using only three sampling channels, an accurate sampling and reconstruction of 4 real signals (8 bands). The matrices that are used to reconstruct the signals in the SMRS scheme also have low condition numbers. This indicates that the SMRS scheme is robust to noise in signals. The success of the SMRS scheme relies on the assumption that the sampled signals are sparse. As a result most of the sampled spectrum may be unaliased in at least one of the sampling channels. This is in contrast to multicoset sampling schemes in which an alias in one channel is equivalent to an alias in all channels. We have demonstrated that the SMRS scheme obtains similar performance using 3 sampling channels and a total sampling rate 8 times the Landau rate to an implementation of a multicoset sampling scheme that uses 6 sampling channels with a total sampling rate of 13 times the Landau rate.

Abstract:
Error estimation is given for a regularized Shannon's sampling formulae, which was found to be accurate and robust for numerically solving partial differential equations.

Abstract:
Many signal processing problems--such as analysis, compression, denoising, and reconstruction--can be facilitated by expressing the signal as a linear combination of atoms from a well-chosen dictionary. In this paper, we study possible dictionaries for representing the discrete vector one obtains when collecting a finite set of uniform samples from a multiband analog signal. By analyzing the spectrum of combined time- and multiband-limiting operations in the discrete-time domain, we conclude that the information level of the sampled multiband vectors is essentially equal to the time-frequency area. For representing these vectors, we consider a dictionary formed by concatenating a collection of modulated Discrete Prolate Spheroidal Sequences (DPSS's). We study the angle between the subspaces spanned by this dictionary and an optimal dictionary, and we conclude that the multiband modulated DPSS dictionary--which is simple to construct and more flexible than the optimal dictionary in practical applications--is nearly optimal for representing multiband sample vectors. We also show that the multiband modulated DPSS dictionary not only provides a very high degree of approximation accuracy in an MSE sense for multiband sample vectors (using a number of atoms comparable to the information level), but also that it can provide high-quality approximations of all sampled sinusoids within the bands of interest.

Abstract:
Filtering of multi-band bandlimited signals by means of a linear digital filter with one or more stopbands is explored. The main goal of the paper is to demonstrate that such a task can be accomplished using sampling rates lower than Landau rate, where the Landau rate is defined as the total bandwidth of the input signal. In order to reach such low rates Periodic Nonuniform Sampling is employed. We show that the proposed filtering method is most efficient when bandpass and multiband filtering is required. Necessary and sufficient conditions for filtering are derived, and an algorithm for designing PNS grids that allow sub-Landau sampling and filtering is proposed. Reconstruction systems are discussed and experimental examples are presented, which confirm the feasibility of the approach.

Abstract:
The Random Demodulator (RD) and the Modulated Wideband Converter (MWC) are two recently proposed compressed sensing (CS) techniques for the acquisition of continuous-time spectrally-sparse signals. They extend the standard CS paradigm from sampling discrete, finite dimensional signals to sampling continuous and possibly infinite dimensional ones, and thus establish the ability to capture these signals at sub-Nyquist sampling rates. The RD and the MWC have remarkably similar structures (similar block diagrams), but their reconstruction algorithms and signal models strongly differ. To date, few results exist that compare these systems, and owing to the potential impacts they could have on spectral estimation in applications like electromagnetic scanning and cognitive radio, we more fully investigate their relationship in this paper. We show that the RD and the MWC are both based on the general concept of random filtering, but employ significantly different sampling functions. We also investigate system sensitivities (or robustness) to sparse signal model assumptions. Lastly, we show that "block convolution" is a fundamental aspect of the MWC, allowing it to successfully sample and reconstruct block-sparse (multiband) signals. Based on this concept, we propose a new acquisition system for continuous-time signals whose amplitudes are block sparse. The paper includes detailed time and frequency domain analyses of the RD and the MWC that differ, sometimes substantially, from published results.

Abstract:
In many applications of current interest, the observations are represented as a signal defined over a graph. The analysis of such signals requires the extension of standard signal processing tools. Building on the recently introduced Graph Fourier Transform, the first contribution of this paper is to provide an uncertainty principle for signals on graph. As a by-product of this theory, we show how to build a dictionary of maximally concentrated signals on vertex/frequency domains. Then, we establish a direct relation between uncertainty principle and sampling, which forms the basis for a sampling theorem for graph signals. Since samples location plays a key role in the performance of signal recovery algorithms, we suggest and compare a few alternative sampling strategies. Finally, we provide the conditions for perfect recovery of a useful signal corrupted by sparse noise, showing that this problem is also intrinsically related to vertex-frequency localization properties.

Abstract:
The reconstruction of an unknown continuously defined function from the samples of the responses of linear time-invariant (LTI) systems sampled by the th Nyquist rate is the aim of the generalized sampling. Papoulis (1977) provided an elegant solution for the case where is a band-limited function with finite energy and the sampling rate is equal to times cutoff frequency. In this paper, the scope of the Papoulis theory is extended to the case of bandpass signals. In the first part, a generalized sampling theorem (GST) for bandpass signals is presented. The second part deals with utilizing this theorem for signal recovery from nonuniform samples, and an efficient way of computing images of reconstructing functions for signal recovery is discussed.

Abstract:
In GHz-level wideband spectrum sensing, too high sampling rate exceeds the specifications of existed analog-to-digital converters (ADC) by sampling directly. Moreover, an accurate estimate of occupied spectrum band can improve spectrum utilization. Therefore, with modulated wideband converter (MWC), a wideband spectrum sensing approach based on multiband signal sampling and wavelet transform is proposed in this paper. With MWC, sub-bands signals are firstly obtained by low rate sampling; and then, an estimation method for noise power and threshold is proposed, such that the spectrum sensing for signal sub-bands can be carried out by energy detection; at last, the edge detection for signal sub-bands by using wavelet transform is used to obtain the exact locations of spectrum occupied by primary users. Simulation results show that the proposed wideband spectrum sensing approach is feasible and effective.

Abstract:
We study the problem of sampling k-bandlimited signals on graphs. We propose two sampling strategies that consist in selecting a small subset of nodes at random. The first strategy is non-adaptive, i.e., independent of the graph structure, and its performance depends on a parameter called the graph coherence. On the contrary, the second strategy is adaptive but yields optimal results. Indeed, no more than O(k log(k)) measurements are sufficient to ensure an accurate and stable recovery of all k-bandlimited signals. This second strategy is based on a careful choice of the sampling distribution, which can be estimated quickly. Then, we propose a computationally efficient decoder to reconstruct k-bandlimited signals from their samples. We prove that it yields accurate reconstructions and that it is also stable to noise. Finally, we conduct several experiments to test these techniques.