Compressive Sensing for Audio and Speech Signals

Cognitive Radio (CR) is one of the most promising techniques for optimizing the spectrum usage. However, the large amount of data of spectral information that must be processed to identify and assign spectral resources increases the channel assignment times, therefore worsening the quality of service for the devices using the spectrum. Compressive Sensing (CS) is a digital processing technique that allows the reconstruction of sparse or compressible signals using fewer samples than those required traditionally. This paper presents a model that addresses the Spectral Sensing problem in Cognitive Radio using Compressive Sensing as an effective way of decreasing the number of samples required in the sensing process. This model is based on Compressive Spectral Imaging (CSI) architectures where a centralized spectrum manager selects what power data must be delivered by the different wireless devices using binary patterns, and builds a multispectral data cube image with the geographical and spectral data power information. The results show that this multispectral data cube can be built with only a 50% of the samples generated by the devices and, therefore reducing the data traffic dramatically.

Compressed sensing (CS) is a technique to sample compressible signals below the Nyquist rate, whilst still allowing near optimal reconstruction of the signal. In this paper, we apply the iterative hard thresholding (IHT) algorithm for compressed sensing on the speech signal. The interested speech signal is transformed to the frequency domain using Discrete Fourier Transform (DCT) and then compressed sensing is applied to that signal. The compressed signal can be reconstructed using the recently introduced Iterative Hard Thresholding (IHT) algorithm and also by the tradditional \( \ell_{1} \) minimization (basic pursuit) for comparison. It is shown that the compressed sensing can provide better root mean square error (RMSE) than the tradition DCT compression method, given the same compression ratio.

Compressive sensing is a sub-Nyquist sampling technique for efficient signal acquisition and reconstruction of sparse or compressible signals. In order to account for the sparsity of the underlying signal of interest, it is common to use sparsifying priors such as Bernoulli–Gaussian-inverse Gamma (BGiG) and Gaussian-inverse Gamma (GiG) priors on the components of the signal. With the introduction of variational Bayesian inference, the sparse Bayesian learning (SBL) methods for solving the inverse problem of compressive sensing have received significant interest as the SBL methods become more efficient in terms of execution time. In this paper, we consider the sparse signal recovery problem using compressive sensing and the variational Bayesian (VB) inference framework. More specifically, we consider two widely used Bayesian models of BGiG and GiG for modeling the underlying sparse signal for this problem. Although these two models have been widely used for sparse recovery problems under various signal structures, the question of which model can outperform the other for sparse signal recovery under no specific structure has yet to be fully addressed under the VB inference setting. Here, we study these two models specifically under VB inference in detail, provide some motivating examples regarding the issues in signal reconstruction that may occur under each model, perform comparisons and provide suggestions on how to improve the performance of each model.

Compressed sensing (CS) has been applicably used in cognitive radio (CR) to sense wide-band spectrum, solving the wide-band sensing rate restriction problem. For a CR network which includes multiple nodes, it's considered that the received signals is high correlated. So in this paper, we focus on efficient compressed spectrum sensing reconstructions in CR network, proposing a probabilistic model called Cooperative Bayesian Compressed Spectrum Sensing (C-BCSS) algorithm. In the algorithm, node samples locally and a common sparse signal model is developed to the received signals where correlation exists. And then C-BCSS reconstruction is performed cooperatively at fusion center utilizing the information from all nodes. We compared C-BCSS with Multitask BCS and simulation results show that C-BCSS outperforms in condition of severe under-sampling and heavy noise.

In this paper, we propose to apply the discrete cosine transform (DCT) and the compressive sampling (CS) techniques to audio signal compression. Using spectral analysis and the properties of the DCT, we can treat audio signals as sparse signals in the frequency domain. This is especially true for sounds representing tones. Thus, we propose the use of DCT as a preprocessor in order to sparsely represent a signal in the frequency domain, combined with CS to obtain an efficient representation of the signals. We show that the subsequent application of CS represents our signals with less information than the well-known sampling theorem.

Compressive sensing (CS) is one of the effective data compression methods where a small number of measurements of a sparse signal is required to have an exact recovery. In Compressive sensing technique, the data is acquired and compressed at the same time. CS concept allows capturing only fewer sparse and compressible signal which contains only useful information that helps to recover a proper signal. In this paper, image compression is performed with compressive sensing concept. The different basis matrices and sensing matrices are considered which satisfy the Restricted Isometric Property (RIP) and Independent and Identically Distributed (IID). The hardware implementation of CS is covered to have real-time compression for various image-based applications. The performance is observed regarding SNR, compression ratio, and correlation. The reconstruction algorithms are implemented on MATLAB platform. The obtained results show satisfactory performance for CS based image compression.

The innovative concept of Compressed Sensing (CS) presents a breakthrough that enables the acquisition of sparse signals at much lower sampling rates compared to the conventional Nyquist rate. The scope of CS is not limited only to sparse signal but it is also applicable to compressible signals, such as multimedia signals including audio signals. Representing the random samples from CS process using finite-precision is a crucial problem in communication systems. In this paper, we focus on 1-bit quantized CS. We propose to take into account the perceptual CS model for audio compression, where the perceptual properties are taken into account. We propose two models, the first applies perceptual effect at the transmitter side. In the other model, a modified Binary Iterative Hard Thresholding (BIHT) is proposed to improve the performance of 1-bit compressed sensing by taking the perceptual properties of the received audio signal into account. The Mean Opinion Score (MOS) is used to compare the perceptual quality of the received signal for the proposed 1-bit perceptual CS algorithms. Simulation results show that a better performance is achieved using the proposed algorithms.

In this work, we study the problem of reconstructing a sparse signal from a limited number of its linear projections when the following knowledge is available. (1) We are given partial, and partly erroneous, knowledge of the signal’s support, denoted by T . (2) We are also given an erroneous estimate of the signal values on T . Alternatively, in recursive reconstruction applications, like real-time dynamic MRI, one can use the support estimate and the signal value estimate from the previous time instant. We presented algorithms by modifying Compressive Sensing (CS) using the partly erroneous support and also the erroneous signal estimate for both noiseless and noisy measurements. The idea of our proposed solution is to solve a convex relaxation of the following problem: find the signal that is sparsest outside the set T , while being “close enough” to signal estimate on T and satisfying the data constraint. We obtain sufficient conditions for exact reconstruction using modified-CS and regularized modified-BP. These are much weaker than those needed for CS when the size of the unknown part of the support is small compared to the support size. We also propose solutions modified-BPDN and regularized modified-BPDN for noisy measurements using the similar idea. We obtain the computable and tighter bounds without any sufficient conditions for the reconstruction error. Simulation comparisons for both sparse and compressible signals are shown. In this work, we also study the application of CS based approaches for blood oxygenation level dependent (BOLD) contrast functional MR imaging (fMRI). In particular, we show, via exhaustive experiments on actual MR scanner data for brain fMRI, that our recently proposed approach for recursive reconstruction of sparse signal sequences, modifiedCS-residual, outperforms other existing CS based approaches.

Compressive Sensing (CS) is a digital signal processing developed theory that encloses the signal sampling and compression, based on the sparsity characteristics of signal. This can decrease sampling rate, so reduce computational complexity of the system without degrading the performance of the system. This paper describes the theoretical frame and a few key technical, then illustrates the application of compressed sensing theory to wide-band cognitive radio signals. Spectrum sensing is a critical issue in wide-band Cognitive Radio (CR) networks as it faces hard challenges such as high hardware cost, complexity, sampling rate and processing speed. Thus, this paper shows that Compressive Sensing could be exploited in wide-band Cognitive Radio networks to solve the spectrum sensing problems mentioned above.

In the recent research, compressive sampling (CS) has received attention in the area of signal processing and wireless communications for the reconstruction of signals. CS aids in reducing the sampling rate of received signals thereby decreasing the processing time of analog-to-digital converters (ADC). The energy minimization is the key feature of CS. In this work, CS has been applied to spectrum sensing in cognitive radio networks (CRN). The primary user (PU) signal is optimally detected using the sparse representation of received signals. The received PU signal is compressed in the time domain to extract the minimum energy coefficients and then applied to sensing. Further, the signal is detected using energy detection technique and recovered using \(l_{1}\)-minimization algorithm. The detection performance for various compression rates is analyzed.

Compressive sensing (CS) based cooperative trans­missions is investigated in cognitive radio networks (CRN). Energy efficiency based optimal relay selection is implemented to determine the measurement numbers in CS. Homotopy re­construction method that solves a bound constrained quadratic programming (BCQP) problem is studied for signal sparse reconstruction at secondary user destination (SUD) in CRN. Based on the determined measurement numbers, binary phase shift keying (BPSK) modulation signal is sparsely sampled by analog-to-information converters (AICs) equipped at the selected secondary user relays (SURs). The corresponding measurement matrix is constructed in CS-based cooperative transmissions. Homotopy reconstruction method is implemented to obtain dynamic update of the solution for BCQP problem. The updating critical value of homotopy parameter and the updating solution of the reconstructed signals' autocorrelation vector can be both achieved via iteration. Simulation results are presented to validate the effectiveness of the proposed approach. It is shown that, homotopy method recovers original signal approximately in low signal-to-noise ratio (SNR) region. Compared with orthogonal matching pursuit (OMP) reconstruction approach, it provides a balanced solution considering normalized mean square error (NMSE) performance versus computational complexity. More­over, spectrum sensing overhead plays significant role in CS based cooperative transmissions.

Compressed Sensing (CS) is an emerging theory which can sample the sparse signal or compressible signal via sub-Nyquist sampling rate and reconstruct the original signal with small amount of measurements. Since speech signal is sparse in Discrete Cosine Transform (DCT) domain, a kind of adaptive Bayesian Compressed Sensing (BCS) based on speech signal is proposed in this paper. In the one hand, our proposed method exploits the difference of energy within different speech frame to allot measurements adaptively for each speech frame aim to promote the quality of recovery speech signal. In the other hand, the position information of sparse coefficient in each speech frame is also utilized by our proposed method to recover its neighboring speech frame for reducing the recovery time of speech signal. The experimental results prove that our proposed method is surely effective and practical.

Sparse signals can be reconstructed from a reduced set of signal samples using compressive sensing (CS) methods. The discrete cosine transform (DCT) can provide highly concentrated representations of audio signals. This property implies the DCT as a good sparsity domain for the audio signals. In this paper, the DCT is studied within the context of sparse audio signal processing using the CS theory and methods. The DCT coefficients of a sparse signal, calculated with a reduced set of available samples, can be modeled as random variables. It has been shown that the statistical properties of these variables are closely related to the unique reconstruction conditions. The main result of this paper is in an exact formula for the mean-square reconstruction error in the case of approximately sparse and nonsparse noisy signals reconstructed under the sparsity assumption. Based on the presented analysis, a simple and computationally efficient reconstruction algorithm is proposed. The presented theoretical concepts and the efficiency of the reconstruction algorithm are verified numerically, including examples with synthetic and recorded audio signals with unavailable or corrupted samples. Random disturbances and disturbances simulating clicks or inpainting in audio signals are considered. Statistical verification is done on a dataset with experimental signals. Results are compared with some classical and recent methods used in similar signal and disturbance scenarios.

As an alternative paradigm to the Shannon-Nyquist sampling theorem, compressive sensing enables sparse signals to be acquired by sub-Nyquist analog-to-digital converters thus may launch a revolution in signal collection, transmission and processing. In the practical compressive sensing applications, the sparse signal is always affected by noise and interference, and therefore the recovery performance reduces based on the conventional compressive sensing, especially in the low signal-to-noise scene, the sparse recovery is usually unavailable. In this paper, the influence of noise on recovery performance is analyzed, so as to provide the theoretical basis for the noise folding phenomenon in compressive sensing. From the analysis, we find that the expected noise gain in the random measure process is closely related to the row and column of the measurement matrix. However, only those columns corresponding to the support for the sparse signal contribute to the sparse vector. In the traditional Shannon-Nyquist sampling system, an antialiasing filter is applied before the sampling process, so as to filter the noise beyond the passband of interest. Inspired by the necessity of antialiasing filtering in bandpass signal sampling, we propose a selective measurement scheme, namely adapted compressive sensing, whose measurement matrix can be updated according to the noise information fed back by the processing center. The measurement matrix is specially designed, and the sensing matrix has directivity so that the signal noise can be suppressed. The measurement matrix senses only the spectrum of interest, where the sparse spectrum is most likely to lie. Moreover, we compare the recovery performance of the proposed adaptive scheme with those of the non-adaptive orthogonal matching pursuit algorithm, FOCal underdetermined system solver algorithm, and sarse Bayesian learning algorithm. Extensive numerical experiments show that the proposed scheme has a better improvement in the performance of the sparse signal recovery. From the viewpoint of implementation, the measurement noise should be taken into consideration in the system, and more efficient algorithms will be developed for source pre-estimation at lower signal-to-noise ratio.

The recently introduced Compressed Sensing (CS) theory explains how sparse or compressible signals can be reconstructed from far fewer samples than what was previously believed possible. The CS theory has attracted significant attention for applications such as Magnetic Resonance Imaging (MRI) where long acquisition times have been problematic. This is especially true for dynamic MRI applications where high spatio-temporal resolution is needed. For example, in cardiac cine MRI, it is desirable to acquire the whole cardiac volume within a single breath-hold in order to avoid artifacts due to respiratory motion. Conventional MRI techniques do not allow reconstruction of high resolution image sequences from such limited amount of data. Vaswani et al. recently proposed an extension of the CS framework to problems with partially known support (i.e. sparsity pattern). In their work, the problem of recursive reconstruction of time sequences of sparse signals was considered. Under the assumption that the support of the signal changes slowly over time, they proposed using the support of the previous frame as the "known" part of the support for the current frame. While this approach works well for image sequences with little or no motion, motion causes significant change in support between adjacent frames. In this paper, we illustrate how motion estimation and compensation techniques can be used to reconstruct more accurate estimates of support for image sequences with substantial motion (such as cardiac MRI). Experimental results using phantoms as well as real MRI data sets illustrate the improved performance of the proposed technique.