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Register a new userEfficient Sampling of Sparse Wideband Analog Signals
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Added by
Moshe Mishali
Publication date
2009
fields
Informatics Engineering
and research's language is
English
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Periodic nonuniform sampling is a known method to sample spectrally sparse signals below the Nyquist rate. This strategy relies on the implicit assumption that the individual samplers are exposed to the entire frequency range. This assumption becomes impractical for wideband sparse signals. The current paper proposes an alternative sampling stage that does not require a full-band front end. Instead, signals are captured with an analog front end that consists of a bank of multipliers and lowpass filters whose cutoff is much lower than the Nyquist rate. The problem of recovering the original signal from the low-rate samples can be studied within the framework of compressive sampling. An appropriate parameter selection ensures that the samples uniquely determine the analog input. Moreover, the analog input can be stably reconstructed with digital algorithms. Numerical experiments support the theoretical analysis.
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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 lowpass 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, realtime 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.
Advances of information-theoretic understanding of sparse sampling of continuous uncoded signals at sampling rates exceeding the Landau rate were reported in recent works. This work examines sparse sampling of coded signals at sub-Landau sampling rates. It is shown that with coded signals the Landau condition may be relaxed and the sampling rate required for signal reconstruction and for support detection can be lower than the effective bandwidth. Equivalently, the number of measurements in the corresponding sparse sensing problem can be smaller than the support size. Tight bounds on information rates and on signal and support detection performance are derived for the Gaussian sparsely sampled channel and for the frequency-sparse channel using the context of state dependent channels. Support detection results are verified by a simulation. When the system is high-dimensional the required SNR is shown to be finite but high and rising with decreasing sampling rate, in some practical applications it can be lowered by reducing the a-priory uncertainty about the support e.g. by concentrating the frequency support into a finite number of subbands.
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.
As technology grows, higher frequency signals are required to be processed in various applications. In order to digitize such signals, conventional analog to digital convertors are facing implementation challenges due to the higher sampling rates. Hence, lower sampling rates (i.e., sub-Nyquist) are considered to be cost efficient. A well-known approach is to consider sparse signals that have fewer nonzero frequency components compared to the highest frequency component. For the prior knowledge of the sparse positions, well-established methods already exist. However, there are applications where such information is not available. For such cases, a number of approaches have recently been proposed. In this paper, we propose several random sampling recovery algorithms which do not require any anti-aliasing filter. Moreover, we offer certain conditions under which these recovery techniques converge to the signal. Finally, we also confirm the performance of the above methods through extensive simulations.
We consider the problem of testing for the presence (or detection) of an unknown sparse signal in additive white noise. Given a fixed measurement budget, much smaller than the dimension of the signal, we consider the general problem of designing compressive measurements to maximize the measurement signal-to-noise ratio (SNR), as increasing SNR improves the detection performance in a large class of detectors. We use a lexicographic optimization approach, where the optimal measurement design for sparsity level $k$ is sought only among the set of measurement matrices that satisfy the optimality conditions for sparsity level k-1. We consider optimizing two different SNR criteria, namely a worst-case SNR measure, over all possible realizations of a k-sparse signal, and an average SNR measure with respect to a uniform distribution on the locations of the up to k nonzero entries in the signal. We establish connections between these two criteria and certain classes of tight frames. We constrain our measurement matrices to the class of tight frames to avoid coloring the noise covariance matrix. For the worst-case problem, we show that the optimal measurement matrix is a Grassmannian line packing for most---and a uniform tight frame for all---sparse signals. For the average SNR problem, we prove that the optimal measurement matrix is a uniform tight frame with minimum sum-coherence for most---and a tight frame for all---sparse signals.
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