Sub-Nyquist Sampling: Bridging Theory and Practice

Sampling theory encompasses all aspects related to the conversion of continuous-time signals to discrete streams of numbers. The famous Shannon-Nyquist theorem has become a landmark in the development of digital signal processing. In modern applications, an increasingly number of functions is being pushed forward to sophisticated software algorithms, leaving only those delicate finely-tuned tasks for the circuit level. In this paper, we review sampling strategies which target reduction of the ADC rate below Nyquist. Our survey covers classic works from the early 50's of the previous century through recent publications from the past several years. The prime focus is bridging theory and practice, that is to pinpoint the potential of sub-Nyquist strategies to emerge from the math to the hardware. In that spirit, we integrate contemporary theoretical viewpoints, which study signal modeling in a union of subspaces, together with a taste of practical aspects, namely how the avant-garde modalities boil down to concrete signal processing systems. Our hope is that this presentation style will attract the interest of both researchers and engineers in the hope of promoting the sub-Nyquist premise into practical applications, and encouraging further research into this exciting new frontier.Comment: 48 pages, 18 figures, to appear in IEEE Signal Processing Magazin

Xampling: Signal Acquisition and Processing in Union of Subspaces

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. Analog compression that narrows down the input bandwidth prior to sampling with commercial devices. A nonlinear algorithm then detects the input subspace prior to conventional signal processing. A representative union model of spectrally-sparse signals serves as a test-case to study these Xampling functions. We adopt three metrics for the choice of analog compression: robustness to model mismatch, required hardware accuracy and software complexities. We conduct a comprehensive comparison between two sub-Nyquist acquisition strategies for spectrally-sparse signals, the random demodulator and the modulated wideband converter (MWC), in terms of these metrics and draw operative conclusions regarding the choice of analog compression. We then address lowrate signal processing and develop an algorithm for that purpose that enables convenient signal processing at sub-Nyquist rates from samples obtained by the MWC. We conclude by showing that a variety of other sampling approaches for different union classes fit nicely into our framework.Comment: 16 pages, 9 figures, submitted to IEEE for possible publicatio

Multichannel Sampling of Pulse Streams at the Rate of Innovation

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

Multifrequency Aperture-Synthesizing Microwave Radiometer System (MFASMR). Volume 1

Background material and a systems analysis of a multifrequency aperture - synthesizing microwave radiometer system is presented. It was found that the system does not exhibit high performance because much of the available thermal power is not used in the construction of the image and because the image that can be formed has a resolution of only ten lines. An analysis of image reconstruction is given. The system is compared with conventional aperture synthesis systems

Application of LSI to signal detection: The deltic DFPCC

The development of the DELTIC DFPCC serial mode signal processor is discussed. The processor is designed to detect in the presence of background noise a signal coded into the zero crossings of the waveform. The unique features of the DELTIC DFPCC include versatility in handling a variety of signals and relative simplicity in implementation. A theoretical performance model is presented which predicts the expected value of the output signal as a function of the input signal to noise ratio. Experimental results obtained with the prototype system, which was breadboarded with LSI, MSI and SSI components, are given. The device was compared with other LSI schemes for signal processing and it was concluded that the DELTIC DFPCC is simpler and in some cases more versatile than other systems. With established LSI technology, low frequency systems applicable to sonar and similar problems are feasible

Design and Development of an APD algorithm development board for Positron Emission Tomography

Using the PET scanner, three dimensional images of the human body with sufficient detail can be viewed which help physicians to visualize both normal metabolic functions and discover the chemical processes underlying physical abnormalities. Commercial PET scanners employ Photo Multiplier Tubes to detect the anti-matter annihilation photons and amplify the signals to a suitable level for digital sampling. Photomultiplier tubes provide extremely high sensitivity and exceptionally low noise compared to other photosensitive devices currently used to detect radiant energy in the ultraviolet, visible and near infrared regions. A combined magnetic resonance positron emission tomography (MR-PET) modality would require a solid-state photo detector due to the known gain/timing variation of PMTs with variable magnetic field. PET detector block designs have been described and implemented in the literature using APD photo detectors at moderate values of gain. The APD Algorithm Development Board is basically a signal processing board which receives the integrated APD analog signals and outputs a digital event packet composing of position and timing data for each detected photon. These digital event packets are digitally transmitted to a downstream module for comparison with opposing detectors to detect the coincidence photons fundamental to PET. The main functions are to process analog signals from the APDs to determine if an energy qualified gamma ray event has been detected, localize the crystal position and time of the event, and transmit the event information to the control interface, en route to a coincidence processor

Discrete multitone modulation with principal component filter banks

Discrete multitone (DMT) modulation is an attractive method for communication over a nonflat channel with possibly colored noise. The uniform discrete Fourier transform (DFT) filter bank and cosine modulated filter bank have in the past been used in this system because of low complexity. We show in this paper that principal component filter banks (PCFB) which are known to be optimal for data compression and denoising applications, are also optimal for a number of criteria in DMT modulation communication. For example, the PCFB of the effective channel noise power spectrum (noise psd weighted by the inverse of the channel gain) is optimal for DMT modulation in the sense of maximizing bit rate for fixed power and error probabilities. We also establish an optimality property of the PCFB when scalar prefilters and postfilters are used around the channel. The difference between the PCFB and a traditional filter bank such as the brickwall filter bank or DFT filter bank is significant for effective power spectra which depart considerably from monotonicity. The twisted pair channel with its bridged taps, next and fext noises, and AM interference, therefore appears to be a good candidate for the application of a PCFB. This is demonstrated with the help of numerical results for the case of the ADSL channel

Joint synchronization and calibration of multi-channel transform-domain charge sampling receivers

Transform-domain (TD) sampling is seen as a potential candidate for wideband and ultra-wideband high-performance receivers and is investigated in detail in this research. TD receivers expand the signal over a set of basis functions and operate on the digitized basis coefficients. This parallel digital signal processing relaxes the sampling requirements opening the doors to higher dynamic range and wider bandwidth in receivers. This research is focused on the implementation of a high performance multi-channel wideband receiver that is based on Frequency-domain (FD) sampling, a special case of TD sampling. To achieve high dynamic ranges in these receivers, it is critical that the digital post processing block matches the analog RF front end accurately. This accurate matching has to be ensured across several process variations, mismatches and o�sets that can be present in integrated circuit implementations. A unified model has been defined for the FD multi-channel receiver that contains all these imperfections and a joint synchronization and calibration technique, based on the Least-mean-squared (LMS) algorithm, is presented to track them. A maximum likelihood (ML) algorithm is used to estimate the frequency offset in carriers which is corrected prior to LMS calibration. Simulation results are provided to support these concepts. The sampling circuits in FD receivers are based on charge-sampling and a multi-channel charge-sampling receiver creates an inherent sinc filter-bank that has several advantages compared to the conventional analog filter banks used in other multi-channel receivers. It is shown that the sinc filter banks, besides reduced analog complexity, have very low computational complexity in data estimation which greatly reduces the digital power consumption of these filters. The digital complexity of data estimation in the sinc fiter bank is shown to be less than 1=10th of the complexity in analog filter banks