DCT-compressive sampling of multifrequency sparse audio signals

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.

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.

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 differential frequency hopping (FH) wireless systems, data are expressed by frequency hopping pattern, and frequency detection involving the Fourier transform and widebands requires large numbers of Nyquist samples, which makes them difficult to implement. Due to the sparseness of FH signals in the frequency domain, compressive sampling (CS) techniques have been explored for the sampling and reconstruction of such signals. In CS, the number of measurements must satisfy $$M\ge {C}\cdot {\mu }^2(\varvec{\varPhi },\varvec{\varPsi })\cdot {K}\cdot {}log(N)$$M?C·μ2(?,?)·K·log(N), where $$C$$C and $$\mu (\varvec{\varPhi },\varvec{\varPsi })$$μ(?,?) are the constants, $$K$$K is the sparsity level, and $$N$$N is the dimension of the original signal. Because of the large number of channels in the operation band, many of these channels are often occupied by diverse frequency-fixed (FF) signals. For this reason, the band compression rate (BCR $$=N/M$$=N/M) may be greatly reduced. However, the FH signals only occupy a single frequency at a time. This means that $$K=1$$K=1, and the number of measurements can be far smaller than the number to restore the mixture of FH and FF signals. In this paper, a CS method that is not subject to the influence is presented, and the BCR can reach 1,000:6 when the FH signal is mixed with 50 interferences. The proposed algorithm covers the generation of the sampling matrix and signal reconstruction. Specifically, it maximizes the signal-to-noise ratio of the compressed signal with a white noise background. The new reconstruction algorithm was found to be 5 times faster than the OMP algorithm in the case of that $$K=4$$K=4, and it is not limited by the inequality given above.

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.

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 new technique that give an approach to reconstruct the signal with few numbers of observations or measurements. The CS is based on L1 -norm minimizations to find the sparse solutions and it is known as basis pursuit. In this investigation, CS scheme with different transforms is proposed. The three utilized transform techniques are: Discrete Fourier Transform(DFT), Discrete Cosine Transform(DCT) and Discrete Wavelet Transform(DWT) with daubechies1(DB1) and coiflets1(coif1) basis. The proposed system is tested by employing the following signals: Blocks, Heavy Sine, ‘Bumps’ and ‘Doppler’ which cover wide range of applications. The four testing signals are represented in sparse domain using different transforms. In order to threshold the coefficients of the signals in sparse domain, the universal threshold is utilized in the case of CS with FFT and DWT whereas, the universal threshold is modified to prune the DCT coefficients. The main aim of this study is to investigate the differences among CS with DFT, CS with DCT and CS with DWT, and consequently a suitable transform domain used with CS to be selected. The comparative study is established by assessing the performance of the proposed system using Root Mean Square Error (RMSE), output SNR, and the time required to reconstruct approximated signals. Simulation results have shown that the CS with DWT outperforms the CS with FFT and DCT. CS with DWT has achieved good RMSE values about (0.0014 to 3.359e-8) even when half of the signal elements are removed. CS with FFT and DCT enhanced the noisy Blocks and Bumps signals by 3dB and 1dB respectively, while it is failed to enhance noisy Heavy Sine and Doppler signals. CS with DWT of two basis and for single decomposition level have improved the noisy Blocks, Bumps, Heavy Sine and Doppler signals by 5dB, 4dB, 3dB, and 3dB respectively.

Purpose – The purpose of this paper is to present a compressed sensing (CS)-based sampling system for ultra-wide-band (UWB) signal. By exploiting the sparsity of signal, this new sampling system can sub-Nyquist sample a multiband UWB signal, whose unknown frequency support occupies only a small portion of a wide spectrum. Design/methodology/approach – A random Rademacher sequence is used to sense the signal in the frequency domain, and a matrix constructed by Hadamard basis is used to compress the signal. The probability of reconstruction is proved mathematically, and the reconstruction matrix is developed in the frequency domain. Findings – Simulation results indicate that, with an ultra-low sampling rate, the proposed system can capture and reconstruct sparse multiband UWB signals with high probability. For sparse multiband UWB signals, the proposed system has potential to break through the Shannon theorem. Originality/value – Different from the traditional sub-Nyquist techniques, the proposed sampling system not only breaks through the limitation of Shannon theorem but also avoids the barrier of input bandwidth of analog-to-digital converters (ADCs).

Compressed sensing (CS) has gained much interests in the speech processing community and especially in compression. The success of this appealing paradigm relies, heavily, on the sparsity of speech signals in a given dictionary. Thanks to machine learning, it is possible to go beyond the limits of analytical dictionaries by designing learned dictionaries, which are more able to fit the nature of data. In this paper, we propose speech compression scheme based on CS using K-singular value decomposition (K-SVD) algorithm to learn a dictionary for sparse representation and iteratively reweighted least square (IRLS) algorithm for signal reconstruction from randomly generated measurements. Different sizes of dictionaries are trained and compared with discrete cosine transform (DCT). Signal quality results, measured with perceptual evaluation speech quality (PESQ) revealed that learned dictionary using K-SVD improves the performance of speech compressed sensing comparing to DCT.

Compressed Sensing (CS) is a promising method that recovers the sparse and compressible signals from severely under-sampled measurements. CS can be applied to wireless communication to enhance its capabilities. As this technology is proliferating, it is possible to explore its need and benefits for emerging applicationsCompressive Sensing for Wireless Communication provides: • A clear insight into the basics of compressed sensing • A thorough exploration of applying CS to audio, image and computer vision • Different dimensions of applying CS in Cognitive radio networks • CS in wireless sensor network for spatial compression and projection • Real world problems/projects that can be implemented and tested • Efficient methods to sample and reconstruct the images in resource constrained WMSN environmentThis book provides the details of CS and its associated applications in a thorough manner. It lays a direction for students and new engineers and prepares them for developing new tasks within the field of CS. It is an indispensable companion for practicing engineers who wish to learn about the emerging areas of interest.

Sparse gammatone signal model optimized for English speech does not match the human auditory filters

Compressive sampling (CS)is a emerging technique with many applications on signal processing field. It states that it is possible to reconstruct a signal from a number of samples below the well-known Nyquist limit. The success of the reconstruction depends on the capability of a frontend transform to represent the signal in a sparse way. In this paper, we propose the use of the discrete cosine transform (DCT) to preprocess a speech signal in order to obtain a sparse representation in the frequency domain, and thus, we show that the subsequent application of compressive sampling can represent vowels with less information than the Nyquist sampling theorem. The reader will find that the presented material differs from other speech processing techniques, as our results could be the basis for developing compression methods using the discrete cosine transform and compressive sampling. Both techniques, traditionally used for image compression, are now proposed for speech compression.

Variational Bayesian inference based robust multiple measurement sparse signal recovery

Currently, digital image watermarking is widely implemented as a tool for copyright protection. Along with technological developments, the technique of theft and manipulation of digital imagery increasingly sophisticated. So research on image watermarking should continue to grow for improved copyright security. This paper proposes the use of Discrete Cosine Transform (DCT) based on 8×8 sub-blocks to convert the original image from spatial to the frequency domain. The DCT transform result will produce AC and DC coefficients. Next, the DC coefficients are collected into a matrix to be transformed with DWT. The result of DWT transformation yield four subbands. Then the LL subband is selected for embedding copyright in the form of a binary image with a certain watermark strength value. The final step inverse the DWT-DCT transform to produce a watermarked image. From the results of this study, high imperceptibility values are evidenced by PSNR and MSE. Image watermark toughness is also measured by NC and is proven to be robust from various attacks mainly from JPEG compression attacks.

A compressive sensing (CS) based approach is applied in conjunction with an adaptive basis re-weighting procedure for multi-dimensional stochastic process power spectrum estimation. In particular, the problem of sampling gaps in stochastic process records, occurring for reasons such as sensor failures, data corruption, and bandwidth limitations, is addressed. Specifically, due to the fact that many stochastic process records such as wind, sea wave and earthquake excitations can be represented with relative sparsity in the frequency domain, a CS framework can be applied for power spectrum estimation. By relying on signal sparsity, and the assumption that multiple records are available upon which to produce a spectral estimate, it has been shown that a re-weighted CS approach succeeds in estimating power spectra with satisfactory accuracy. Of key importance in this paper is the extension from one-dimensional vector processes to a broader class of problems involving multidimensional stochastic fields. Numerical examples demonstrate the effectiveness of the approach when records are subjected to up to 75% missing data.