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.
The concept of using modulation waveforms is based on ideas presented in @cite_17 for a multiband model, which is Fourier dual to ours: the signals in @cite_17 are assumed to be sparse in frequency, while multipulse signals are sparse in time. More specifically, @cite_17 considers multiband signals whose Fourier transform is concentrated on @math frequency bands, and the width of each band is no greater than @math . The locations of the bands are unknown in advance. A low rate sampling scheme, called the modulated wideband converter (MWC), allowing recovery of such signals at the rate of @math was proposed in @cite_17 ; a hardware prototype appears in @cite_8 . This scheme consists of parallel channels where in each channel the input is modulated with a periodic waveform followed by a low-pass filter and low-rate uniform sampling. The main idea is that in each channel the spectrum of the signal is scrambled, such that a portion of the energy of all bands appears at baseband. Therefore, the input to the sampler contains a mixture of all the bands. Mixing of the frequency bands in @cite_17 is analogous to mixing the Gabor coefficients in our scheme.
@cite_8 @cite_17
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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. 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.