Wide-sense Stationary Signals Research Articles

Compressed covariance sensing for blade tip timing measurement

Due to harsh working conditions, rotating blades are prone to failures. Thus, it is urgently needed to monitor blade health. Blade tip timing (BTT) is a promising non-contact vibration technique for rotating blades owing to its high efficiency and long service time. However, due to severe undersampling, traditional spectrum analysis methods are ineffective in identifying characteristic parameters and recovering the power spectrum of vibrations for condition monitoring. To address this issue, this paper proposes a compressed covariance sensing-based BTT (CCS-BTT) method for power spectrum estimation. Specifically, we build a compressed covariance sensing representation model for BTT signal, where the covariance representation can be non-underdetermined owing to the Hermitian Toeplitz structure of the covariance matrix. Then, a universal covariance sampling pattern (UCSP) is derived to ensure the complete covariance information can be efficiently recovered without sparsity constraint. By utilizing the inherent periodicity, the derived UCSP significantly improves the efficiency of BTT measurement. Moreover, for practicality, we present a series of optimized layouts based on UCSP that cover almost all possible cases of probe numbers in BTT measurement. Finally, extensive simulations and experiments validate the effectiveness and performance of CCS-BTT. By shifting the perspective from the signal itself to its covariance, CCS-BTT opens many avenues for the development of BTT signal processing. In addition, the proposed method is expected to be used in deterministic compressive sampling of wide-sense stationary signals.

Frequency Estimation with Co-prime DFT Filter Bank and Half Nyquist Frequency Shift

When processing wide sense stationary (WSS) signals, the sampled data rate of conventional MN bands DFT filter bank is at least MN, while the co-prime DFT filter bank (co-prime DFTFB) only needs M+N, that sharply lowers the sampling rate. Here, M and N are co-prime pair of integers. However, practical FIR filter imperfections and correlation detection mode arouse spurious peaks in signal power spectrogram, that greatly limits the application of co-prime DFTFB. In this paper, we propose a modified co-prime DFTFB by adding a real valued multiplier to the original one, where M and N are both odd numbers. Together with original co-prime DFTFB, we get a 2MN bands filter bank. In the simulation section, two BPSK signals with random carrier frequencies are employed. The results of detection probability versus SNR curves through 500 Monte Carlo experiments verify the effectiveness of our method.

Correlation-Based Maximal Invariant Statistic for Spectrum Sensing

In this letter, spectrum sensing for wide-sense stationary signals with temporal correlation is studied. In order to perform uniformly most powerful invariant test (UMPIT) or locally most powerful invariant test (LMPIT), the ratio of the distributions of the maximum invariant statistics is derived in closed-form using Wijsman’s theorem. This ratio shows that UMPIT can be obtained when the signal-to-noise ratio (SNR) and the normalized auto-correlation matrix of the primary user signal are both known. In addition, the LMPIT with nominal SNR (LMPIT-NSNR) and the LMPIT with improved sample auto-correlation matrix (LMPIT-ISAC) are proposed based on this ratio. LMPIT-NSNR and LMPIT-ISAC are suitable for the case where the normalized auto-correlation matrix is known and the SNR is unknown, and for the case where both the normalized auto-correlation matrix and the SNR are unknown, respectively. Simulation results show that the performance of LMPIT-NSNR is robust to the fixed nominal SNR value and can approach the performance of UMPIT, and the LMPIT-ISAC performs better than the traditional covariance-based test.

Compressive Covariance Sensing-Based Power Spectrum Estimation of Real-Valued Signals Subject to Sub-Nyquist Sampling

In this work, an estimate of the power spectrum of a real-valued wide-sense stationary autoregressive signal is computed from sub-Nyquist or compressed measurements in additive white Gaussian noise. The problem is formulated using the concepts of compressive covariance sensing and Blackman-Tukey nonparametric spectrum estimation. Only the second-order statistics of the original signal, rather than the signal itself, need to be recovered from the compressed signal. This is achieved by solving the resulting overdetermined system of equations by application of least squares, thereby circumventing the need for applying the complicated ℓ 1 -minimization otherwise required for the reconstruction of the original signal. Moreover, the signal need not be spectrally sparse. A study of the performance of the power spectral estimator is conducted taking into account the properties of the different bases of the covariance subspace needed for compressive covariance sensing, as well as different linear sparse rulers by which compression is achieved. A method is proposed to benefit from the possible computational efficiency resulting from the use of the Fourier basis of the covariance subspace without considerably affecting the spectrum estimation performance.

Sensing Fractional Power Spectrum of Nonstationary Signals with Coprime Filter Banks

The coprime discrete Fourier transform (DFT) filter banks provide an effective scheme of spectral sensing for wide-sense stationary (WSS) signals in case that the sampling rate is far lower than the Nyquist sampling rate. And the resolution of the coprime DFT filter banks in the Fourier domain (FD) is 2π/MN, where M and N are coprime. In this work, a digital fractional Fourier transform- (DFrFT-) based coprime filter banks spectrum sensing method is suggested. Our proposed method has the same sampling principle as the coprime DFT filter banks but has some advantages compared to the coprime DFT filter banks. Firstly, the fractional power spectrum of the chirp-stationary signals which are nonstationary in the FD can be sensed effectively by the coprime DFrFT filter banks because of the linear time-invariant (LTI) property of the proposed system in discrete-time Fourier domain (DTFD), while the coprime DFT filter banks can only sense the power spectrum of the WSS signals. Secondly, the coprime DFrFT filter banks improve the resolution from 2π/MN to 2π sin α/MN by combining the fractional filter banks theory with the coprime theory. Simulation results confirm the obtained analytical results.

Fast Compressed Power Spectrum Estimation: Toward a Practical Solution for Wideband Spectrum Sensing

There has been a growing interest in wideband spectrum sensing due to its applications in cognitive radios and electronic surveillance. To overcome the sampling rate bottleneck for wideband spectrum sensing, in this paper, we study the problem of compressed power spectrum estimation whose objective is to reconstruct the power spectrum of a wide-sense stationary signal based on sub-Nyquist samples. By exploring the sampling structure inherent in the multicoset sampling scheme, we develop a computationally efficient method for power spectrum reconstruction. An important advantage of our proposed method over existing compressed power spectrum estimation methods is that our proposed method, whose primary computational task consists of fast Fourier transform (FFT), has a very low computational complexity. Such a merit makes it possible to efficiently implement the proposed algorithm in a practical field-programmable gate array (FPGA)-based system for real-time wideband spectrum sensing. Our proposed method also provides a new perspective on the power spectrum recovery condition, which leads to a result similar to what was reported in prior works. Simulation results are presented to show the computational efficiency and the effectiveness of the proposed method.

First Order Statistic Based Fast Blind Calibration of Time Skews for Time-Interleaved ADCs

A full digital background method is proposed in this brief for timing mismatch calibrations of time-interleaved analog-to-digital converters (TIADCs) with wide sense stationary input signals. First, errors caused by timing mismatches are modeled as additive errors by applying the Taylor series approximation. Then, a mismatch estimation approach based on first-order statistics is proposed. In addition, computations are simplified significantly by using some valuable properties of the signals in TIADC channels. Further, a variable step-size iterative technique is presented to reduce steady-state errors. So the mismatch estimates are guaranteed to converge to actual values within three steps iterations. Theoretical analyses and experiments show that the proposed method has the advantages of low complexity and fast convergence over most other conventional methods. The proposed method is feasible for fast on-line timing mismatch calibrations of TIADCs.

Maximally stationary window design for overlap-add based random vibration synthesis

Abstract The accurate synthesis of realistic waveforms conforming to certain specifications is a fundamental step in random vibration testing. Since real-time implementation of digital signal processing systems for random vibration and noise synthesis necessarily operates frame by frame, the overlap-add (OLA) method, by which frames are windowed and overlapped, is widely used in practice to avoid artifacts at frame boundaries. When a wide-sense stationary random signal is desired, however, the OLA method presents a shortcoming, because the inherent periodicity of the frame-by-frame process unavoidably produces a cyclostationary signal, i.e., its statistics present an undesired periodic behavior. We analyze the impact of the window coefficients in the cyclostationarity properties of the synthetic process, and then present algorithms for window design with the goal of maximizing a measure of its stationarity, considering both second- and fourth-order statistical properties. The proposed designs are shown to significantly improve the stationarity properties when compared to commonly used windows.

Design of Broadband Compressed Sampling Receiver Based on Concurrent Alternate Random Sequences

Modulated wideband converter (MWC) is a multi-branch sub-Nyquist sampling structure for processing the spare wideband signals. By mixing with a periodic pseudo-random bit sequence (PRBS), each branch of MWC compresses the information of the input signal into a narrow baseband which can be sampled at low speed. Based on the several sampling sequences, the input signals can be reconstructed by compressed sensing (CS) optimization algorithms. However, the classic MWC (C-MWC) still consumes a lot of hardware resources because of its complex structure. Especially when the spectrum is blind, the total sampling rate is required to be four times the Landau rate for joint sparse reconstruction. To solve this problem, this paper proposes alternate MWC (A-MWC). A-MWC replaces the periodic PRBSes with two sets of alternating PRBSes and uses a trigger to synchronize the PRBSes and the sample sequences. Based on a reasonable assumption that the input signals are the wide-sense stationary (WSS) signals which have the same power spectral density (PSD) in each time slice, this paper reduces the requirement the sampling rate for blind joint sparse reconstruction to twice Landau rate successfully. Both simulation and experiment prove that A-MWC reduces the number of branches required for blind reconstruction without reducing the quality of the reconstructed signal.

Transmission Strategies for Remote Estimation With an Energy Harvesting Sensor

We consider the remote estimation of a time-correlated signal using an energy harvesting (EH) sensor. The sensor observes the unknown signal and communicates its observations to a remote fusion center using an amplify-and-forward strategy. We consider the design of optimal power allocation strategies in order to minimize the mean-square error at the fusion center. Contrary to the traditional approaches, the degree of correlation between the signal values constitutes an important aspect of our formulation. We provide the optimal power allocation strategies for a number of illustrative scenarios. We show that the most majorized power allocation strategy, i.e., the power allocation as balanced as possible, is optimal for the cases of circularly wide-sense stationary (c.w.s.s.) signals with a static correlation coefficient and sampled low-pass c.w.s.s. signals for a static channel. We show that the optimal strategy can be characterized as a water-filling type solution for sampled low-pass c.w.s.s. signals for a fading channel. Motivated by the high-complexity of the numerical solution of the optimization problem, we propose low-complexity policies for the general scenario. Numerical evaluations illustrate the close performance of these low-complexity policies to that of the optimal policies and demonstrate the effect of the EH constraints and the degree of freedom of the signal.

Design of a Single Channel Modulated Wideband Converter for Wideband Spectrum Sensing: Theory, Architecture and Hardware Implementation.

In a cognitive radio sensor network (CRSN), wideband spectrum sensing devices which aims to effectively exploit temporarily vacant spectrum intervals as soon as possible are of great importance. However, the challenge of increasingly high signal frequency and wide bandwidth requires an extremely high sampling rate which may exceed today’s best analog-to-digital converters (ADCs) front-end bandwidth. Recently, the newly proposed architecture called modulated wideband converter (MWC), is an attractive analog compressed sensing technique that can highly reduce the sampling rate. However, the MWC has high hardware complexity owing to its parallel channel structure especially when the number of signals increases. In this paper, we propose a single channel modulated wideband converter (SCMWC) scheme for spectrum sensing of band-limited wide-sense stationary (WSS) signals. With one antenna or sensor, this scheme can save not only sampling rate but also hardware complexity. We then present a new, SCMWC based, single node CR prototype System, on which the spectrum sensing algorithm was tested. Experiments on our hardware prototype show that the proposed architecture leads to successful spectrum sensing. And the total sampling rate as well as hardware size is only one channel’s consumption of MWC.

Performance of Uniform and Sparse Non-Uniform Samplers In Presence of Modeling Errors: A Cramér-Rao Bound Based Study

This paper evaluates the performance of uniform and sparse nonuniform sampling techniques (namely nested and coprime sampling) for line spectrum estimation, in presence of nonideal conditions such as perturbation in sampling instants, and limited data for computing statistical averages. Coprime and nested sampling are well-known deterministic sampling techniques that operate at rates significantly lower than Nyquist, and yet allow perfect reconstruction of the spectra of wide sense stationary signals. However, theoretical guarantees for these samplers assume ideal conditions such as synchronous sampling, and ability to perfectly compute statistical expectations. This paper studies the performance of coprime and nested samplers when these assumptions are violated. Using a general grid-based signal model that applies to both spatial and temporal line spectrum estimation, the effect of perturbations in sampling instants is evaluated by deriving fundamental Cramer-Rao Bounds (CRB) for line spectrum estimation with perturbed samplers. For the first time, simplified expressions for the Fisher Information matrix for perturbed coprime and nested samplers are derived, which explicitly highlight the role of coarray. Even in presence of perturbations, it is possible to resolve O(M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) spectral lines under appropriate conditions on the size of the grid. The effect of finite data on the CRB is also studied, and necessary and sufficient conditions are derived to ensure that the CRB decreases monotonically to zero with the number of measurements, even when there are more sources than sensors. Finally, the theoretical results derived in this paper are supported by extensive numerical experiments.

Frequency‐domain wideband compressive spectrum sensing

The authors study the power spectrum density (PSD) estimation of wideband wide-sense stationary (WSS) signals with sub-Nyquist sampling rate. Owing to the large bandwidth, Nyquist rate sampling of such signals needs very high rate analogue-to-digital converters. It is important to note that PSD estimation does not necessarily require reconstruction of the original signal. Indeed, the power spectrum can be directly obtained from sub-Nyquist samples. In this study, a new method for reconstructing the power spectrum in the frequency domain for WSS signals is presented. The main idea is to divide the whole spectrum into N equal-length segments and calculate the average PSD in each segment using a frequency-domain representation of sub-Nyquist samples. In addition, the capability of the proposed method, as a detector of spectrum holes, is studied using receiver operating characteristic (ROC) curves. Then, the analysis of false alarm probability is provided. Simulation results for the additive white Gaussian noise channel and the slowly fading frequency selective channel show that the proposed method considerably outperforms available techniques.

Compressive wideband spectrum sensing based on single channel

The modulated wideband converter (MWC) is an attractive analogue compressed sensing technique that has been proposed in recent years. Regrettably, the MWC has high hardware complexity owing to its parallel channel structure. To reduce the complexity, a novel compressive wideband spectrum sensing scheme based on a single channel for band-limited wide-sense stationary (WSS) signals is proposed. In the proposed scheme, a special periodic waveform in the time domain is designed and a power spectrum sensing model is presented to detect the occupied frequency spectrum of such signals from sub-Nyquist samples of which the sample rate is only MWC's single-channel rate. Simulation results show that the proposed scheme has a perfect performance on wideband WSS signal spectrum sensing.

Joint multiband signal detection and cyclic spectrum estimation from compressive samples

This paper focuses on wide-sense stationary signal processing within a compressive sensing framework, proposing a new method of compressive sampling fast Fourier transform (FFT) accumulation method (CS-FAM). Depending on how it is applied, CS-FAM has one or two steps, allowing for versatility in multiband signal detection and parameter extraction. In the first step, the active sub-bands are detected using multiple measurement vectors (MMVs) and multiuser detection is achieved using a bandwidth constraint. In applications where it is required, such as in estimations of carrier frequency, symbol rate, or modulation format identification, the second step can be used to reconstruct the cyclic spectrums of each user individually. Based on the results of first step, parameter extraction is performed by searching for peaks in the cyclic spectrum rather than by the usual method of setting a threshold. Compared to other cyclic feature detection methods based on sub-Nyquist sampling, CS-FAM is low in complexity, allowing for practical implementation. Based on the results of the first step, parameter extraction from the cyclic spectrum is performed by searching for peaks rather than by setting a threshold. Although CS-FAM can only be employed for multiband signal detection, compared to other cyclic feature detection methods based on sub-Nyquist sampling, it is low in complexity, which makes practical implementation possible. Numerical simulations are presented to demonstrate the robustness of CS-FAM's multiband signal detection and the effectiveness of its cyclic spectrum estimation against both sampling rate reduction and noise uncertainty.

Unitary Precoding and Basis Dependency of MMSE Performance for Gaussian Erasure Channels

We consider the transmission of a Gaussian vector source over a multidimensional Gaussian channel where a random or a fixed subset of the channel outputs are erased. Within the setup where the only encoding operation allowed is a linear unitary transformation on the source, we investigate the minimum mean-square error (MMSE) performance, both in average, and also in terms of guarantees that hold with high probability as a function of the system parameters. Under the performance criterion of average MMSE, necessary conditions that should be satisfied by the optimal unitary encoders are established and explicit solutions for a class of settings are presented. For random sampling of signals that have a low number of degrees of freedom, we present MMSE bounds that hold with high probability. Our results illustrate how the spread of the eigenvalue distribution and the unitary transformation contribute to these performance guarantees. The performance of the discrete Fourier transform (DFT) is also investigated. As a benchmark, we investigate the equidistant sampling of circularly wide-sense stationary signals, and present the explicit error expression that quantifies the effects of the sampling rate and the eigenvalue distribution of the covariance matrix of the signal. These findings may be useful in understanding the geometric dependence of signal uncertainty in a stochastic process. In particular, unlike information theoretic measures such as entropy, we highlight the basis dependence of uncertainty in a signal with another perspective. The unitary encoding space restriction exhibits the most and least favorable signal bases for estimation.

Nested sparse sampling and co-prime sampling in sense-through-foliage target detection

This paper firstly introduces nested sampling and co-prime sampling, which were proposed recently, but have never been applied to real world target detection. We apply nested sampling and co-prime sampling to target detection in UWB radar sensor networks (RSN), based on a differential approach. The non-stationary UWB signal needs to be decomposed into several approximate wide sense stationary (WSS) signals so that nested sampling could be used in this situation. We also compare the performance of nested sampling and co-prime sampling against uniform under-sampling. The results show that in terms of good quality data and poor quality data, both nested sampling and co-prime sampling work better.

Mask operations in discrete fractional Fourier transform domains with nearly white real valued wide sense stationary output signals

For linear time invariant transforms, the multiplications in these transformed domains are referred to as filtering. On the other hand, the multiplications in linear time varying transformed domains are referred to as mask operations. Discrete fractional Fourier transforms (DFrFTs) are linear time varying transforms which map signals from the time domain to the rotated time frequency domains. In this paper, effects of the rotational angles of the DFrFTs on the output signals after applying the mask operations are studied. It is proved in this paper that if the rotational angles of the DFrFTs are not integer multiples of π, as well as they are not odd integer multiples of π2 when the signal lengths are odd, then there is only one degree of freedom for designing the mask coefficients. Otherwise, there are N degrees of freedom for designing the mask coefficients. Moreover, it is proved in this paper that satisfying the conditions for obtaining real valued output signals will automatically satisfy the conditions for obtaining wide sense stationary (WSS) output signals. Based on this result, designs of the mask coefficients are formulated as optimization problems with L1 norm nonconvex objective functions only subject to the conditions for obtaining real valued output signals. These constrained optimization problems are further reformulated to unconstrained optimization problems by a vector space approach. Finally, when there is only one degree of freedom for designing the mask coefficients, the globally optimal solutions of the unconstrained optimization problems are derived analytically. Computer numerical simulation results are presented for illustrations.

Condition Assessment in Hydraulically Noisy Pipeline Systems Using a Pressure Wave Splitting Method

Abstract For the past couple of decades, researchers have been investigating the use of hydraulic transients (pressure surge waves) to investigate the condition of a pipeline system, and detect anomalies such as leaks, blockages, or pipe wall condition. The use of controlled hydraulic transients, combined with signal processing methods on measured pressure responses, have had a demonstrated utility for pipeline condition assessment, largely owing to the sensitivity of the hydraulic transient waves to the said anomalies. This paper presents a simple extension to these existing methods by proposing a technique that does not require a controlled hydraulic transient input, but utilizes existing hydraulic noise (in the form of wide-sense stationary hydraulic signals) within the system of interest to perform the diagnosis.

Prediction of wide-sense stationary quaternion random signals

An efficient widely linear prediction algorithm is introduced for the class of wide-sense stationary quaternion signals. Specifically, using second order statistics information in the quaternion domain, a multivariate Durbin–Levison-like algorithm is derived. The proposed solution can be applied under a very general formulation of the problem, allowing for the estimation of a function of the quaternion signal which is observed through a system with both additive/multiplicative noises.