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 Fourier coefficients can be determined from samples of @math using a scheme similar to that of Fig. with @math channels and modulators @math with @math . In this case, the input-output relation becomes @math , where @math is a vector of length @math and @math is a vector of Fourier coefficients of @math of length @math . In @cite_11 the authors proposed a more general scheme based on mixing the modulations @math with proper coefficients, resulting in periodic waveforms, before applying them to the signal. The corresponding samples are weighted superpositions of the Fourier coefficients @math . When the weights are properly chosen, @math can be recovered and therefore the time-delays and amplitudes as well. We incorporate the idea of mixing the channels into our sampling system in the follow-up paper, and show that under certain conditions on the Gabor frame, our generalized system can be used to sample signals of the form ). We note here, that the system of @cite_11 is inefficient for our signal model, since it reduces to the Fourier series method, which does not take sparsity in time into account.
@cite_11
2074380172
We consider minimal-rate sampling schemes for infinite streams of delayed and weighted versions of a known pulse shape. The minimal sampling rate for these parametric signals is referred to as the rate of innovation and is equal to the number of degrees of freedom per unit time. Although sampling of infinite pulse streams was treated in previous works, either the rate of innovation was not achieved, or the pulse shape was limited to Diracs. In this paper we propose a multichannel architecture for sampling pulse streams with arbitrary shape, operating at the rate of innovation. Our approach is based on modulating the input signal with a set of properly chosen waveforms, followed by a bank of integrators. This architecture is motivated by recent work on sub-Nyquist sampling of multiband signals. We show that the pulse stream can be recovered from the proposed minimal-rate samples using standard tools taken from spectral estimation in a stable way even at high rates of innovation. In addition, we address practical implementation issues, such as reduction of hardware complexity and immunity to failure in the sampling channels. The resulting scheme is flexible and exhibits better noise robustness than previous approaches.