This paper investigates the effect of sub-Nyquist sampling upon the capacity of an analog channel. The channel is assumed to be a linear time-invariant Gaussian channel, where perfect channel knowledge is available at both the transmitter and the receiver. We consider a general class of right-invertible time-preserving sampling methods which includes irregular nonuniform sampling, and characterize in closed form the channel capacity achievable by this class of sampling methods, under a sampling rate and power constraint. Our results indicate that the optimal sampling structures extract out the set of frequencies that exhibits the highest signal-to-noise ratio among all spectral sets of measure equal to the sampling rate. This can be attained through filterbank sampling with uniform sampling grid employed at each branch with possibly different rates, or through a single branch of modulation and filtering followed by uniform sampling. These results reveal that for a large class of channels, employing irregular nonuniform sampling sets, while are typi- cally complicated to realize in practice, does not provide capacity gain over uniform sampling sets with appropriate preprocessing. Our findings demonstrate that aliasing or scrambling of spectral components does not provide capacity gain in this scenario, which is in contrast to the benefits obtained from random mixing in spectrum-blind compressive sampling schemes.
However, for signals with certain structure, the Nyquist sampling rate may exceed that required for perfect signal reconstruction from the samples @cite_28 @cite_17 . For example, consider multiband signals, whose spectral contents reside within several subbands over a wide spectrum. If the spectral support is known, then the necessary sampling rate for the multiband signals is their spectral occupancy, termed the @cite_29 . Such signals admit perfect recovery when sampled at rates approaching the Landau rate, provided that the sampling sets are appropriately chosen (e.g. @cite_30 @cite_9 ). One type of sampling mechanism that can reconstruct multiband signals sampled at the Landau rate is a filter bank followed by sampling, studied in @cite_15 @cite_3 @cite_36 . Inspired by recent compressive sensing'' @cite_37 @cite_39 @cite_25 ideas, spectrum-blind sub-Nyquist sampling for multiband signals with random modulation has been developed @cite_13 as well.
@cite_30 @cite_37 @cite_28 @cite_36 @cite_9 @cite_29 @cite_3 @cite_39 @cite_15 @cite_13 @cite_25 @cite_17
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We study the problem of optimal sub-Nyquist sampling for perfect reconstruction of multiband signals. The signals are assumed to have a known spectral support spl Fscr that does not tile under translation. Such signals admit perfect reconstruction from periodic nonuniform sampling at rates approaching Landau's (1967) lower bound equal to the measure of spl Fscr . For signals with sparse spl Fscr , this rate can be much smaller than the Nyquist rate. Unfortunately the reduced sampling rates afforded by this scheme can be accompanied by increased error sensitivity. In a previous study, we derived bounds on the error due to mismodeling and sample additive noise. Adopting these bounds as performance measures, we consider the problems of optimizing the reconstruction sections of the system, choosing the optimal base sampling rate, and designing the nonuniform sampling pattern. We find that optimizing these parameters can improve system performance significantly. Furthermore, uniform sampling is optimal for signals with spl Fscr that tiles under translation. For signals with nontiling spl Fscr , which are not amenable to efficient uniform sampling, the results reveal increased error sensitivities with sub-Nyquist sampling. However, these can be controlled by optimal design, demonstrating the potential for practical multifold reductions in sampling rate. We address the problem of reconstructing a multiband signal from its sub-Nyquist pointwise samples, when the band locations are unknown. Our approach assumes an existing multi-coset sampling. To date, recovery methods for this sampling strategy ensure perfect reconstruction either when the band locations are known, or under strict restrictions on the possible spectral supports. In this paper, only the number of bands and their widths are assumed without any other limitations on the support. We describe how to choose the parameters of the multi-coset sampling so that a unique multiband signal matches the given samples. To recover the signal, the continuous reconstruction is replaced by a single finite-dimensional problem without the need for discretization. The resulting problem is studied within the framework of compressed sensing, and thus can be solved efficiently using known tractable algorithms from this emerging area. We also develop a theoretical lower bound on the average sampling rate required for blind signal reconstruction, which is twice the minimal rate of known-spectrum recovery. Our method ensures perfect reconstruction for a wide class of signals sampled at the minimal rate, and provides a first systematic study of compressed sensing in a truly analog setting. Numerical experiments are presented demonstrating blind sampling and reconstruction with minimal sampling rate. Conventional sub-Nyquist sampling methods for analog signals exploit prior information about the spectral support. In this paper, we consider the challenging problem of blind sub-Nyquist sampling of multiband signals, whose unknown frequency support occupies only a small portion of a wide spectrum. Our primary design goals are efficient hardware implementation and low computational load on the supporting digital processing. We propose a system, named the modulated wideband converter, which first multiplies the analog signal by a bank of periodic waveforms. The product is then low-pass filtered and sampled uniformly at a low rate, which is orders of magnitude smaller than Nyquist. Perfect recovery from the proposed samples is achieved under certain necessary and sufficient conditions. We also develop a digital architecture, which allows either reconstruction of the analog input, or processing of any band of interest at a low rate, that is, without interpolating to the high Nyquist rate. Numerical simulations demonstrate many engineering aspects: robustness to noise and mismodeling, potential hardware simplifications, real-time performance for signals with time-varying support and stability to quantization effects. We compare our system with two previous approaches: periodic nonuniform sampling, which is bandwidth limited by existing hardware devices, and the random demodulator, which is restricted to discrete multitone signals and has a high computational load. In the broader context of Nyquist sampling, our scheme has the potential to break through the bandwidth barrier of state-of-the-art analog conversion technologies such as interleaved converters. 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.