Wednesday, October 20, 2010

CS: Sub-Nyquist Sampling of Short Pulses, Xampling and Traditional Compressive Sensing

Here is an interesting paper, after the hardware implementation of Xampling in the Modulated WideBand Converter here is some more on the subject of Xampling that seems to follow the generalized sampling theory and provide sub-nyquist sampling results.

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 a superposition of unknown pulses without knowledge of the pulse locations have not been derived. The goal in this two-part 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. Our approach is based on modulating the input signal in each channel with a properly chosen waveform, followed by an integrator. We show that, with proper preprocessing, the Gabor coefficients necessary for almost perfect reconstruct of the input signal, 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 resulting scheme is flexible and exhibits good noise robustness. The first part in this series is focused on the basic underlying principles. The second part generalizes the present sampling system in several directions. In particular, we consider practical implementations from a hardware perspective and extend the architecture to efficiently sample radar-like signals that are sparse in both time and frequency.
My Arxiv filter now include the Xampling keyword. In the following paper: Xampling: Analog Data Compression by Moshe Mishali and Yonina Eldar there is simple parallel between Xampling and compressive Sensing. The abstract of the paper reads:
We introduce Xampling, a design methodology for analog compressed sensing in which we sample analog bandlimited signals at rates far lower than Nyquist, without loss of information. This allows compression together with the sampling stage. The main principles underlying this framework are the ability to capture a broad signal model, low sampling rate, e cient analog and digital implementation and lowrate baseband processing. In order to break through the Nyquist barrier so as to compress the signals in the sampling process, one has to combine classic methods from sampling theory together with recent developments in compressed sensing. We show that previous attempts at sub-Nyquist sampling su er from analog implementation issues, large computational loads, and have no baseband processing capabilities. We then introduce the modulated wideband converter which can satisfy all the Xampling desiderata. We also demonstrate a board implementation of our converter which exhibits sub-Nyquist sampling in practice.

 From that paper, I note the following comparison with traditional Compressed Sensing:
....Exploiting sparsity for rate reduction has been explored in the context of CS, which treats underdetermined sparse recovery. The signal model assumes a vector x of nite length n, which has only a few nonzero entries. Sensing is carried out by computing the linear projection y = Ax, with A having far fewer rows than columns. Results from this fi eld [11, 12] show that under suitable conditions, the linear sensing is stably invertible, even when the length of y is proportional to the number of nonzeros in x, rather than the ambient dimension n. However, it is not straightforward to generalize the discrete CS formulation to analog signals. The difficulty can be noticed immediately in the signal model. Sparsity is de ned in CS by counting the number of nonzeros in x, while analog sparsity of x(t) involves an uncountable number of zeros and nonzeros. Naive extensions of CS-type algorithms for sparse recovery to in nite dimensions leads to unde ned or difficult problems [14]. Discretization methods result in very large scale CS systems, which impose a severe burden on the digital processing units.
The CS paradigm aims at avoiding high-rate redundant sampling. The discrete CS framework [11,12] initiated a long line of highly in uential works. However, it still remains puzzling from the analog sampling viewpoint; sensing by y = Ax implicity assumes that x is the Nyquist rate samples of some continuous signal x(t) on a specific time-interval. We propose Xampling as a general framework within various solutions for analog signals can be compared according to the four rules (X1)-(X4).....

Version 2 also came out Xampling: Signal Acquisition and Processing in Union of Subspaces by Moshe Mishali, Yonina C. Eldar, Asaf Elron. The abstract reads:
We introduce Xampling, a unified framework for signal acquisition and processing of signals in a union of subspaces. The main functions of this framework are two. An analog projection narrows down the input bandwidth prior to sampling with commercial devices. A nonlinear algorithm detects the input subspace prior to conventional signal processing. We build the framework bottom-up, from the application layer to the abstract setting, in three steps. First, we study Xampling for lowrate signal acquisition by conducting a thorough comparative study between two sub-Nyquist acquisition strategies, the random demodulator and the modulated wideband converter (MWC), in terms of model robustness and hardware and software complexities. Second, we develop an algorithm that enables convenient signal processing at sub-Nyquist rates from samples obtained by the MWC. Third, with the intuition we gain from the previous stages, we describe the line of reasoning underlying the proposed Xampling framework. In addition, we show that a variety of sampling approaches for union classes fit nicely into our framework, supporting the generality of Xampling.

Image Credit: NASA/JPL/Space Science Institute, N00164548.jpg was taken on September 25, 2010 and received on Earth September 26, 2010. The camera was pointing toward TITAN at approximately 246,162 kilometers away, and the image was taken using the CL1 and CB3 filters.

No comments: