From Theory to Practice: Sub-Nyquist Sampling of Sparse Wideband Analog Signals

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.Comment: 17 pages, 12 figures, to appear in IEEE Journal of Selected Topics in Signal Processing, the special issue on Compressed Sensin

Novel Digital Alias-Free Signal Processing Approaches to FIR Filtering Estimation

This thesis aims at developing a new methodology of filtering continuous-time bandlimited signals and piecewise-continuous signals from their discrete-time samples. Unlike the existing state-of-the-art filters, my filters are not adversely affected by aliasing, allowing the designers to flexibly select the sampling rates of the processed signal to reach the required accuracy of signal filtering rather than meeting stiff and often demanding constraints imposed by the classical theory of digital signal processing (DSP). The impact of this thesis is cost reduction of alias-free sampling, filtering and other digital processing blocks, particularly when the processed signals have sparse and unknown spectral support. Novel approaches are proposed which can mitigate the negative effects of aliasing, thanks to the use of nonuniform random/pseudorandom sampling and processing algorithms. As such, the proposed approaches belong to the family of digital alias-free signal processing (DASP). Namely, three main approaches are considered: total random (ToRa), stratified (StSa) and antithetical stratified (AnSt) random sampling techniques. First, I introduce a finite impulse response (FIR) filter estimator for each of the three considered techniques. In addition, a generalised estimator that encompasses the three filter estimators is also proposed. Then, statistical properties of all estimators are investigated to assess their quality. Properties such as expected value, bias, variance, convergence rate, and consistency are all inspected and unveiled. Moreover, closed-form mathematical expression is devised for the variance of each single estimator. Furthermore, quality assessment of the proposed estimators is examined in two main cases related to the smoothness status of the filter convolution’s integrand function, \u1d454(\u1d461,\u1d70f)∶=\u1d465(\u1d70f)ℎ(\u1d461−\u1d70f), and its first two derivatives. The first main case is continuous and differentiable functions \u1d454(\u1d461,\u1d70f), \u1d454′(\u1d461,\u1d70f), and \u1d454′′(\u1d461,\u1d70f). Whereas in the second main case, I cover all possible instances where some/all of such functions are piecewise-continuous and involving a finite number of bounded discontinuities. Primarily obtained results prove that all considered filter estimators are unbiassed and consistent. Hence, variances of the estimators converge to zero after certain number of sample points. However, the convergence rate depends on the selected estimator and which case of smoothness is being considered. In the first case (i.e. continuous \u1d454(\u1d461,\u1d70f) and its derivatives), ToRa, StSa and AnSt filter estimators converge uniformly at rates of \u1d441−1, \u1d441−3, and \u1d441−5 respectively, where 2\u1d441 is the total number of sample points. More interestingly, in the second main case, the convergence rates of StSa and AnSt estimators are maintained even if there are some discontinuities in the first-order derivative (FOD) with respect to \u1d70f of \u1d454(\u1d461,\u1d70f) (for StSa estimator) or in the second-order derivative (SOD) with respect to \u1d70f of \u1d454(\u1d461,\u1d70f) (for AnSt). Whereas these rates drop to \u1d441−2 and \u1d441−4 (for StSa and AnSt, respectively) if the zero-order derivative (ZOD) (for StSa) and FOD (for AnSt) are piecewise-continuous. Finally, if the ZOD of \u1d454(\u1d461,\u1d70f) is piecewise-continuous, then the uniform convergence rate of the AnSt estimator further drops to \u1d441−2. For practical reasons, I also introduce the utilisation of the three estimators in a special situation where the input signal is pseudorandomly sampled from otherwise uniform and dense grid. An FIR filter model with an oversampled finite-duration impulse response, timely aligned with the grid, is proposed and meant to be stored in a lookup table of the implemented filter’s memory to save processing time. Then, a synchronised convolution sum operation is conducted to estimate the filter output. Finally, a new unequally spaced Lagrange interpolation-based rule is proposed. The so-called composite 3-nonuniform-sample (C3NS) rule is employed to estimate area under the curve (AUC) of an integrand function rather than the simple Rectangular rule. I then carry out comparisons for the convergence rates of different estimators based on the two interpolation rules. The proposed C3NS estimator outperforms other Rectangular rule estimators on the expense of higher computational complexity. Of course, this extra cost could only be justifiable for some specific applications where more accurate estimation is required

Mixed-Signal Parallel Compressive Spectrum Sensing for Cognitive Radios

Wideband spectrum sensing for cognitive radios requires very demanding analog-to-digital conversion (ADC) speed and dynamic range. In this paper, a mixed-signal parallel compressive sensing architecture is developed to realize wideband spectrum sensing for cognitive radios at sub-Nqyuist rates by exploiting the sparsity in current frequency usage. Overlapping windowed integrators are used for analog basis expansion, that provides flexible filter nulls for clock leakage spur rejection. A low-speed experimental system, built with off-the-shelf components, is presented. The impact of circuit nonidealities is considered in detail, providing insight for a future integrated circuit implementation

Compressive sensing in electrical impedance tomography for breathing monitoring

Continuous functional thorax monitoring using EIT has been extensively researched. A limiting factor in high temporal resolution, three dimensional, and fast EIT is the handling of the volume of raw impedance data produced for transmission and storage. Owing to the periodicity of breathing that may be reflected in EIT boundary measurements, data dimensionality may be reduced efficiently at the time of sampling using compressed sensing techniques. Measurements using a 32-electrode 48-frame-per-second EIT system from 30 neonates were post-processed to simulate random demodulation acquisition method on 2000 frames for compression ratios (CRs) ranging from 2-100. Sparse reconstruction was performed by solving the basis pursuit problem using SPGL1 package. The global impedance data was used in the subsequent studies. The signal to noise ratio (SNR) for the entire frequency band (0 Hz - 24 Hz) and three local frequency bands were analysed. A breath detection algorithm was applied to traces and the subsequent error-rates were calculated while considering the outcome of the algorithm applied to a down-sampled and linearly interpolated version of the traces as the baseline. SNR degradation was proportional with CR. The mean degradation for 0 Hz - 8 Hz was below ~15 dB for all CRs. The error-rates in the outcome of the breath detection algorithm in the case of decompressed traces were lower than those of the associated down-sampled traces for CR≥25, corresponding to sub-Nyquist rate for breathing. For instance, the mean error-rate associated with CR = 50 was ~60% lower than that of the corresponding down-sampled traces. To the best of our knowledge, no other study has evaluated compressive sensing on boundary impedance data in EIT. While further research should be directed at optimising the acquisition and decompression techniques for this application, this contribution serves as the baseline for future efforts. [Abstract copyright: Creative Commons Attribution license.

Output-Only Vibration-Based Monitoring of Civil Infrastructure via Sub-Nyquist/Compressive Measurements Supporting Reduced Wireless Data Transmission

The consideration of wireless acceleration sensors is highly promising for cost-effective output-only system identification in the context of operational modal analysis (OMA) of large-scale civil structures as they alleviate the need for wiring. However, practical monitoring implementations for OMA using wireless units suffer a number of drawbacks related to wireless transmission of densely sampled acceleration time-series including the energy self-sustainability of the sensing nodes. In this work, two recently proposed approaches for output-only modal identification addressing the above issues through balancing monitoring accuracy with data transmission costs are comparatively studied and numerically assessed using field recorded acceleration datasets from two different structures: (i) an operating on-shore wind turbine, (ii) an open to traffic highway bridge. One approach utilizes non-uniform-in-time deterministic multi-coset sampling at sub-Nyquist rates to capture structural response acceleration time-series under ambient excitation assuming stationary signal conditions. In this approach, a power spectrum blind sampling technique is used to estimate the response acceleration power spectral density matrix from the low-rate sampled measurements and is coupled with the Frequency Domain Decomposition method of OMA. The other is a spectro-temporal compressive sensing approach which recovers response acceleration signals through time-series reconstruction in the time domain from sub-Nyquist non-uniform-in-time randomly sampled measurements. Prior knowledge of signal structure in the spectral domain is exploited through smart on-sensor operations and sensor/server communication. The benefits and limitations of the considered approaches are discussed and demonstrated by processing the field recorded datasets for different levels of signal compression and by estimating battery lifetime gains at a single sensor achieved by reduced data transmission. It is concluded that the two approaches are readily applicable in OMA of large-scale structures and can be used complementarily depending on the requirements of any particular acceleration monitoring campaign: time-series extraction for further interrogation vs. solely modal properties estimation

Compressive Sensing of Multiband Spectrum towards Real-World Wideband Applications.

PhD Theses.Spectrum scarcity is a major challenge in wireless communication systems with their rapid evolutions towards more capacity and bandwidth. The fact that the real-world spectrum, as a nite resource, is sparsely utilized in certain bands spurs the proposal of spectrum sharing. In wideband scenarios, accurate real-time spectrum sensing, as an enabler of spectrum sharing, can become ine cient as it naturally requires the sampling rate of the analog-to-digital conversion to exceed the Nyquist rate, which is resourcecostly and energy-consuming. Compressive sensing techniques have been applied in wideband spectrum sensing to achieve sub-Nyquist-rate sampling of frequency sparse signals to alleviate such burdens. A major challenge of compressive spectrum sensing (CSS) is the complexity of the sparse recovery algorithm. Greedy algorithms achieve sparse recovery with low complexity but the required prior knowledge of the signal sparsity. A practical spectrum sparsity estimation scheme is proposed. Furthermore, the dimension of the sparse recovery problem is proposed to be reduced, which further reduces the complexity and achieves signal denoising that promotes recovery delity. The robust detection of incumbent radio is also a fundamental problem of CSS. To address the energy detection problem in CSS, the spectrum statistics of the recovered signals are investigated and a practical threshold adaption scheme for energy detection is proposed. Moreover, it is of particular interest to seek the challenges and opportunities to implement real-world CSS for systems with large bandwidth. Initial research on the practical issues towards the real-world realization of wideband CSS system based on the multicoset sampler architecture is presented. In all, this thesis provides insights into two critical challenges - low-complexity sparse recovery and robust energy detection - in the general CSS context, while also looks into some particular issues towards the real-world CSS implementation based on the i multicoset sampler

Compressive sensing in electrical impedance tomography for breathing monitoring

Continuous functional thorax monitoring using EIT has been extensively researched. A limiting factor in high temporal resolution, three dimensional, and fast EIT is the handling of the volume of raw impedance data produced for transmission and storage. Owing to the periodicity of breathing that may be reflected in EIT boundary measurements, data dimensionality may be reduced efficiently at the time of sampling using compressed sensing techniques. Measurements using a 32-electrode 48-frame-per-second EIT system from 30 neonates were post-processed to simulate random demodulation acquisition method on 2000 frames for compression ratios (CRs) ranging from 2-100. Sparse reconstruction was performed by solving the basis pursuit problem using SPGL1 package. The global impedance data was used in the subsequent studies. The signal to noise ratio (SNR) for the entire frequency band (0 Hz - 24 Hz) and three local frequency bands were analysed. A breath detection algorithm was applied to traces and the subsequent error-rates were calculated while considering the outcome of the algorithm applied to a down-sampled and linearly interpolated version of the traces as the baseline. SNR degradation was proportional with CR. The mean degradation for 0 Hz - 8 Hz was below ~15 dB for all CRs. The error-rates in the outcome of the breath detection algorithm in the case of decompressed traces were lower than those of the associated down-sampled traces for CR≥25, corresponding to sub-Nyquist rate for breathing. For instance, the mean error-rate associated with CR = 50 was ~60% lower than that of the corresponding down-sampled traces. To the best of our knowledge, no other study has evaluated compressive sensing on boundary impedance data in EIT. While further research should be directed at optimising the acquisition and decompression techniques for this application, this contribution serves as the baseline for future efforts. [Abstract copyright: Creative Commons Attribution license.