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.
One interesting fact discovered in @cite_35 is the non-monotonicity of capacity with sampling rate under filter- and modulation-bank sampling, assuming an equal sampling rate per branch for a given number of branches. This indicates that more sophisticated sampling techniques, adaptive to the channel response and the sampling rate, are needed to maximize capacity under sub-Nyquist rate constraints, including both uniform and nonuniform sampling. However, none of the aforementioned work has investigated the question as to which sampling method can best exploit channel structure, thereby maximizing sampled capacity under a given sampling rate constraint. Although several classes of sampling methods were shown in @cite_35 to have closed-form capacity solutions, the capacity limits might not even exist for general sampling methods. This raises the question as to whether there exists a capacity upper bound over a general class of sub-Nyquist sampling systems beyond the classes we discussed in @cite_35 and, if so, when the bound is achievable. That is the question we investigate herein.
@cite_35
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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 explore two fundamental questions at the intersection of sampling theory and information theory: how channel capacity is affected by sampling below the channel's Nyquist rate, and what sub-Nyquist sampling strategy should be employed to maximize capacity. In particular, we derive the capacity of sampled analog channels for three prevalent sampling strategies: sampling with filtering, sampling with filter banks, and sampling with modulation and filter banks. These sampling mechanisms subsume most nonuniform sampling techniques applied in practice. Our analyses illuminate interesting connections between undersampled channels and multiple-input multiple-output channels. The optimal sampling structures are shown to extract out the frequencies with the highest SNR from each aliased frequency set, while suppressing aliasing and out-of-band noise. We also highlight connections between undersampled channel capacity and minimum mean-squared error (MSE) estimation from sampled data. In particular, we show that the filters maximizing capacity and the ones minimizing MSE are equivalent under both filtering and filter-bank sampling strategies. These results demonstrate the effect upon channel capacity of sub-Nyquist sampling techniques, and characterize the tradeoff between information rate and sampling rate. A constructive solution of the irregular sampling problem for band- limited functions is given. We show how a band-limited function can be com- pletely reconstructed from any random sampling set whose density is higher than the Nyquist rate, and give precise estimates for the speed of convergence of this iteration method. Variations of this algorithm allow for irregular sampling with derivatives, reconstruction of band-limited functions from local averages, and irregular sampling of multivariate band-limited functions. In the irregular sampling problem one is asked whether and how a band- limited function can be completely reconstructed from its irregularly sam- pled values f(xi). This has many applications in signal and image processing, seismology, meteorology, medical imaging, etc. Finding constructive solutions of this problem has received considerable attention among mathematicians and engineers. The mathematical literature provides several uniqueness results (1, 2, 17, 18, 19). It is now part of the folklore that for stable sampling the sampling rate must be at least the Nyquist rate (18). These results, as deep as they are, have had little impact for the applied sciences, because they were not constructive. If the sampling set is just a perturbation of the regular oversampling, then a reconstruction method has been obtained in a seminal paper by Duffin and Schaeffer (6) (see also (29)): if for some L > 0, a > 0, and o > 0 the sampling points xk , k e Z , satisfy (a) - ok a, k ^ I, then the norm equivalence A iR (x) ) with w < n o. This norm equivalence implies that it is possible to reconstruct through an iterative procedure, the so-called frame method. Most of the later work on constructive methods consists of variations of this method (3, 21, 22, 26). The above conditions on the sampling set exclude random irregular sampling sets, e.g., sets with regions of higher sampling density. A partial, but undesirable remedy, to handle highly irregular sampling sets, would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration. 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.