Image Reconstruction Using Compressive Sensing Technique for Hardware Implementation

In this paper, we investigate sub-Nyquist sampling method, which can work at the sampling rate below the Nyquist rate. Nowadays, the sampling rate of the classical sub-Nyquist sampling is higher than discrete Compressed Sensing (CS), since the sparseness of the original signal is always destroyed by basis mismatch. We propose a new multi-channel co-prime sampling system along with Sparse Fourier Transform. The analog signal of each channel is firstly mixed by a series of exponential functions. Then, it is fed into integrators and commercial analog-to-digital converters (ADCs). It is noted that the frequency parameters of each subsystem are co-prime. Chinese remainder theorem (CRT) is utilized for the joint processing of all sets of samples and the reconstruction of original signals. Finally, the effects of the signal sparsity, the sampling rate and the signal to noise ratio are simulated in this system. In addition, the experimental results show that the proposed system can work at the rate lower than Random Demodulation RD), Modulated wideband converter (MWC) and their improved version.

An improved random equivalent sampling (RES) approach based on compressed sensing (CS) for repetitive ultrasonic signal detection is presented. The proposed system considers recovering ultrasonic signal with high equivalent sampling frequency from samples captured using analogue-to-digital converter (ADC) clocked at a rate much lower than Nyquist rate. A basis function is constructed to realise the ultrasonic signal sparse representation, which paves the way for applying CS theory to ultrasonic signal sub-Nyquist sampling. A sampling architecture is developed, which is applicable for ultrasonic compressive detection. The CS measurement matrix is constructed in the context of RES. Experimental results indicate that, with this sampling system, an ultrasonic signal sampled at ultra-low rate, but still can be recovered with overwhelming probability.

In practical compressive sensing (CS) communication system, Analog to Digital Converter (ADC) is a necessary component to convert analog signals into digital ones. However, nonlinear distortion in ADC is unavoidable and definitely affects the reception accuracy. Though recent works have studied various methods to combat the negative effect of ADC nonlinear distortion, few of them discuss the solution in communication system with CS. This paper studies the CS recovery method when compressive measurements suffer from ADC nonlinear distortion. A double-sparsity model is first formulated, where the original signal and the ADC nonlinear distortion are both sparse. Then, for the type of clipping ADC, we propose a corresponding algorithm based on the Alternating Direction Method of Multipliers (ADMM) strategy to solve the double-sparsity (DS) problem, named as DS-ADMM. For the type of self-reset (SR) ADC, we explore its essence of rounding operation to design an integer constraint based feedback updating (ICFU) strategy, and accordingly propose the DS-ADMM-ICFU recovery algorithm. Experiment results show that the DS-ADMM algorithm for the double-sparsity problem improves the recovery performance compared with the existing counterpart, and DS-ADMM-ICFU for SR ADC exhibits preferable advantage in typical communication systems.

There is a huge increase in data in terms of data conversion that effects the performance and complexity of the devices like analog-to-digital converters (ADC).The standard ADC uses the conventional Shannon-Nyquist Theorem which says that sampling frequency should be twice of the maximum frequency. These samples require very huge storage.Compressive sensing(CS) which gives solution to this problem. In CS we decrease the sampling rate much less than the Nyquist rate and reconstruct the original signal.There are different reconstruction algorithms evolved since from its origin each. In this paper, we are making a comparison between the performances of Orthogonal Matching Pursuit(OMP) and Regularised Orthogonal Matching Pursuit(ROMP) algorithms for image reconstruction.

The Compressive Sensing (CS) allows the acquisition of signals already compressed and the posterior reconstruction with much less number of measures than the minimum required by the Nyquist theorem. A subarea of CS which improves the performance at the reconstruction stage is named Model-Based CS. Some works have been developed within this subarea. However, most of them consider only the noise generated by sparse approximation, disregarding the noise generated by the quantization stage and its influence on efficiency and robustness of CS. The objective of this study is to investigate the influence of the noise generated by the quantization stage in Model-Based CS efficiency for images with different levels of sparsity and different distributions of coefficients in the frequency domain. In this work, the image acquisition stage is implemented using the partial Fourier matrix which results in a vector of measures. Then, different steps of uniform scalar quantization are added to this vector and the image reconstruction stage is performed using the Compressive Sampling Matching Pursuit (CoSaMP) on a quadtree model. PSNR and bits rate (BR) are then used to evaluate the efficiency of CoSaMP with quantization noise. The performance of this proposed Model-Based CS using different quantization steps were slightly better than other studies using the same model in terms of PSNR, but with the advantage of obtaining smaller values of bit rate (BR maior que 2 bpp).

The compressive sensing (CS) theory provides a new slight to the big-data problem led by the Shannon sampling theorem in rolling element bearings condition monitoring, where the measurement matrix of CS tends to be designed by the random matrix (RM) to preserve the integrity of signal roughly. However, when the signal to be analyzed is infected with strong noise, not only does the signal become insufficiently sparse, but the randomness of the measurement matrix will bring down the sensing efficiency, resulting in the loss of fault feature. Thus, a sensing-enhanced CS scheme based on a series of modes after VMD decomposition is proposed under this paper. The core of this scheme is as follows: (1) the principal mode of VMD with better sparsity replaces the raw signal for compressive sensing; (2) all these modes contain the time-frequency characteristics of the raw signal; (3) a new measurement matrix called mode-circulant matrix (MCM) is defined by circulating the mode matrix, and when the amount of samples is shrunk, the sensing efficiency can be enhanced greatly. Besides, considering the fault signal of rolling bearings under variable speed, there is a need to use order tracking to overcome the nonstationarity of the signal before applying CS theory. The analysis results of simulation and experiment prove that the VMD- and MCM-based CS can successfully extract the weak fault feature of rolling bearings with operating speed changing.

Optical communication systems, which operate at very high rates, are limited by the sampling rate bottleneck. The optical wideband regime exceeds analog to digital converters (ADCs) front-end bandwidth. Multi-channel sampling approaches, such as multicoset or interleaved ADCs, have been proposed tho sample the wideband signal using several channels. Each channel samples below the Nyquist rate such that the overall sampling rate is preserved. However, this scheme suffers from two practical limitations that make its implementation difficult. First, the inherent anti-aliasing filter of the samplers distorts the wideband signal. Second, it requires accurate time shifts on the order of the signal's Nyquist rate, which are challenging to maintain. In this work, we propose an alternative multi-channel sampling scheme, based on analog RF demodulation, where each channel aliases the spectrum using a periodic mixing function before integration and sampling. We show that intentionally using the inherent ADC filter to perform integration turns it to an advantage by increasing the signal to noise ratio (SNR). We demonstrate both theoretically and through numerical experiments that our system outperforms multicoset in terms of signal recovery and symbol estimation in the presence of both thermal and quantization noise.

Sampling rate required in the Nth Power Nonlinear Transformation (NPT) method is typically much greater than Nyquist rate, which causes heavy burden for the Analog to Digital Converter (ADC). Taking advantage of the sparse property of PSK signals' spectrum under NPT, we develop the NPT method for PSK signals with Sub-Nyquist rate samples. In this paper, combined the NPT method with Compressive Sensing (CS) theory, frequency spectrum reconstruction of the Nth power nonlinear transformation of PSK signals is presented, which can be further used for AMR and rough estimations of unknown carrier frequency and symbol rate.

Compressed sensing based multi-rate sub-Nyquist sampling system

The process of converting an analog, or continuous time, signal into a digital signal is known as analog‐to‐digital conversion (ADC). Many signals are inherently analog. Examples of analog signals obtained from a transducer abound. Other analog signals are inherently electrical, such as those obtained from biomedical electromyography or electroencephalography electrodes. In many cases, it is deemed desirable to process a signal by a microprocessor or computer to extract information, or to store the signal in some digital storage medium such as a hard drive, a floppy disk, or a CD ROM. If the original signal is in analog format, in such cases, it is necessary to convert the analog signal into an equivalent digital signal. This article describes the process of ADC. It introduces the topic and describes the operation of two of the most common methods of accomplishing ADC: the successive approximation ADC and the flash ADC. The sampling theorem is stated and proved, aliasing is described, and quantization noise is discussed. Oversampling is described, and the resultant improvements in ADC performance are documented. Then delta‐sigma ADCs are introduced and described. Conventional first‐order and second‐order delta‐sigma ADCs are described and evaluated. Performance comparisons among simple oversampling, first‐order delta‐sigma, and second‐order delta‐sigma ADCs are documented. The article concludes with an example of a higher order delta‐sigma ADC performance evaluation.

Random measurement matrices play a critical role in successful recovery with the compressive sensing (CS) framework. However, due to its randomly generated elements, these matrices require massive amounts of storage space to implement a random matrix in CS applications. To effectively reduce the storage space of the random measurement matrix for CS, we propose a random sampling approach for the CS framework based on the semi-tensor product (STP). The proposed approach generates a random measurement matrix, where the dimensions of the random measurement matrix are reduced to a quarter (or 1/16, 1/64, and even 1/256) of the number of dimensions, which are used for conventional CS. We then estimate the values of the sparse vector with a modified iteratively re-weighted least-squares (IRLS) algorithm. The results of numerical simulations showed that the proposed approach can reduce the storage space of a random matrix to at least a quarter while maintaining quality of reconstruction. All results confirmed that the proposed approach significantly influences the physical implementation of the CS in images, especially on embedded system and field programmable gate array (FPGA), where storage is limited.

Shannon’s Nyquist theorem has always dictated the conventional signal acquisition policies. Power system is not an exception to this. As per this theory, the sampling rate must be at least twice the maximum frequency present in the signal. Recently, compressive sampling (CS) theory has shown that the signals can be reconstructed from samples obtained at sub-Nyquist rate. Signal reconstruction in this theory is exact for “sparse signals” and is near exact for compressible signals provided certain conditions are satisfied. CS theory has already been applied in communication, medical imaging, MRI, radar imaging, remote sensing, computational biology, machine learning, geophysical data analysis, and so forth. CS is comparatively new in the area of computer based power system monitoring. In this paper, subareas of computer based power system monitoring where compressive sampling theory has been applied are reviewed. At first, an overview of CS is presented and then the relevant literature specific to power systems is discussed.

Time delay estimation (TDE) is a hot research topic in wireless location technology. Compressed sensing (CS) theory has been widely applied to image reconstruction and direction of arrival estimation since it was proposed in 2004. The sparse model can be constructed in time domain for estimating the time delay by using the CS theory. The measurement matrix plays a crucial role in the processing of signal reconstruction which is the core problem of CS theory. Therefore the research in the measurement matrix has becomes a hotspot in recent years. The existing measurement matrix is mainly divided into two categories, i.e., random measurement matrix and deterministic measurement matrix. The performance of random measurement matrix has bottlenecks. Firstly, because of the redundant measurement matrix data, the generation and storage of the random number put forward a high requirement for hardware. Secondly the random matrix can only satisfy the restricted isometry property in a statistical sense. The research of the deterministic measurement matrix is of great value under this background. The parity check matrix of low density parity check (LDPC) code has good performance in CS theory. However, the method of randomly selecting non-zero element position has a certain probability to generate a measurement matrix with a short loop structure during generating LDPC code measurement matrix. The robustness of the reconstruction performance decreases with the increase of iteration times. A novel quasi-cyclic CS algorithm based on progressive edge-growth is constructed to estimate the time delay. The purpose of this article is to deal with the need to store a large number of data in existing measurement matrix during time delay, by using the CS theory. The algorithm presented here can achieve TDE in a high precision. First, the theoretical bridge between CS and the maximum likelihood decoding is established. And the design criterion of measurement matrix based on the LDPC code is derived. The sparse measurement matrix with quasi-cyclic structure is constructed by introducing the idea of progressive edge-growth. Finally, the orthogonal matching pursuit algorithm is used to estimate the time delay. Furthermore, the computational complexity of the algorithm and the data storage of the measurement matrix are analyzed theoretically. Simulations show that the correct reconstruction probability of the proposed approach is higher than those of the Gauss random matrix and random LDPC matrix under the same dimension. Compared with the random LDPC matrix, the proposed method can improve performance at the expense of less complexity under the condition of the same data storage.

Electrocardiogram (ECG) is a vital signal which represents the state of health of astronauts in a manned spaceflight mission and hence must be continuously acquired and transmitted throughout the mission. Telemetry bandwidth is a premium resource in such applications. Traditional method of sampling at Nyquist rate is highly bandwidth inefficient. Hence it is advantageous to use Compressive sensing (CS) technique to optimize data at measurement point itself. Conventional CS techniques employ computationally intensive measurement matrices which are not hardware efficient for both acquisition as well as recovery. In this work, a hardware efficient scheme with use of a sparse binary measurement matrix is proposed. The signal is recovered from the compressive measurement using a constraint function based on inverted Laplace distribution function. Gradient decent method is used to recursively recover the original ECG signal from the compressive measurements in an efficient manner. Apart from this, a projection scheme is also proposed to minimize the recovery error. The proposed scheme was extensively evaluated with different ECG samples with different compression ratios. Finally, the proposed scheme was benchmarked with Approximated L0 norm based method and it is found to perform more efficiently in compressively sensing and recovery of ECG signals.

This paper presents a 12-bit successive approximation register (SAR) analog-to-digital converter (ADC) that uses a digital-to-analog converter (DAC) configurable window switching technique. By reusing the capacitors in the DAC, the proposed window switching scheme yields window boundaries to determine whether the input is located within the window, and thus avoid unnecessary capacitor switching. The proposed window SAR ADC improves the conversion efficiency and ADC linearity. A qualitative analysis of prior window switching schemes is presented to elaborate for various applications. A low-input capacitance of 1 pF was adopted to relax the input and reference buffers. A prototype ADC was implemented in 180-nm CMOS occupying an active area of 0.1 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . At 20 MS/s, it consumes a total power of 1.22 mW from a 1.5-V supply. The measured peak signal-to-noise and distortion ratio and spurious-free dynamic range were 61.7 and 79 dB, respectively. At the Nyquist rate, the measured effective number of bits (ENOB) was 9.53, equivalent to a figure-of-merit (FOM) of 83 fJ/conversion-step. In low-power mode (100 kS/s), it consumed a total power of 1.5μW from a 0.7-V supply. At the Nyquist rate, the measured ENOB was 9.82, equivalent to a FOM of 16.6 fJ/conversion step.