Band-limited Signals Research Articles

Reconstruction of Signals From Level-Crossing Samples Using Implicit Information

This paper focuses on the recovery of bandlimited signals from level-crossing samples by exploiting not only the knowledge of when given levels are crossed by the input, but also the implicit information that the signal stays between neighboring levels in the time intervals between the level crossings. We propose to use the technique of projection onto convex sets (POCS) for perfect signal reconstruction from either the level crossings or the associated implicit information. Two POCS algorithms are proposed: iterative POCS and one-step POCS. While the one-step POCS is based on matrix inversion, the iterative projections can be implemented using a chain of standard circuit operations: a low-pass filter and a clipping circuit, respectively. The perfect signal recovery of the infinite projection iteration can be viewed as the completion of the nonperfect input reconstruction achieved in continuous-time digital signal processing from level crossings. The comparative analysis of simulation results for both iterative and one-step POCS algorithms show the importance of a good selection of the initial guess for the POCS reconstruction.

Adaptive decomposition of band-limited signals using the Fourier transform band-pass filter

Adaptive decomposition of band-limited signals using the Fourier transform band-pass filter

Finite Frequency $H_{\infty }$ Deconvolution With Application to Approximated Bandlimited Signal Recovery

This paper investigates a finite frequency ${H_{\infty }}$ deconvolution problem for single-input–single-output discrete-time systems. Motivated by the fact that many practical signals are located in a finite frequency range, this paper is focused on developing a design method for achieving better deconvolution performance over a finite frequency range. First, a finite frequency ${H_{\infty }}$ deconvolution problem is formulated such that the deconvolution specification is characterized by a modified $H_{\mathbf {\infty }}$ index associated with the corresponding finite frequency range. By an adapted generalized Kalman–Yakubovich–Popov lemma and an auxiliary vector approach, a matrix inequality condition is obtained for parameterizing a required deconvolution filter, based on which a simple iterative algorithm is then constructed for optimizing the filter performance. Multiobjective deconvolution is further discussed in the same framework. Compared with the traditional frequency weighting strategy, the proposed method can directly analyze and synthesize the deconvolution performance within finite frequency ranges, the advantage of which is demonstrated in a bandlimited signal recovery example.

ROAST: Rapid Orthogonal Approximate Slepian Transform

In this paper, we provide a Rapid Orthogonal Approximate Slepian Transform (ROAST) for the discrete vector that one obtains when collecting a finite set of uniform samples from a baseband analog signal. The ROAST offers an orthogonal projection which is an approximation to the orthogonal projection onto the leading discrete prolate spheroidal sequence (DPSS) vectors (also known as Slepian basis vectors). As such, the ROAST is guaranteed to accurately and compactly represent not only oversampled bandlimited signals but also the leading DPSS vectors themselves. Moreover, the subspace angle between the ROAST subspace and the corresponding DPSS subspace can be made arbitrarily small. The complexity of computing the representation of a signal using the ROAST is comparable to the FFT, which is much less than the complexity of using the DPSS basis vectors. We also give non-asymptotic results to guarantee that the proposed basis not only provides a very high degree of approximation accuracy in a mean squared error sense for bandlimited sample vectors, but also that it can provide high-quality approximations of all sampled sinusoids within the band of interest.

A New Ka-Band Doppler Radar in Robust and Precise Cardiopulmonary Remote Sensing

In this paper, a new Ka-band Doppler radar with an ability to remotely monitor respiration rate, heartbeat rate (HR), and heart rate variability (HRV) is presented. Exploiting the advantage of high sensitivity of Ka-band radar in sensing of chest-wall movement, we make more accurate reconstruction of cardiopulmonary activities from low-quality received signal. The advantage of using Ka-band radar is obtained by paying a penalty, which is harmonic interferences that make measuring of cardiopulmonary details infeasible in lots of real scenarios. In this paper, by using a more complete model of return signal, a new routine is introduced for the robust detection of breathing and HR that extremely alleviates the problems associated with harmonic interferences. Besides, a more suitable model is proposed, which can cover time-variability cardiorespiratory features. Taking advantage of this model and based on the unconditional orthonormal representation of band-limited signals, a filter with time-varying coefficients is proposed to extract HRV. This extraction is done without resorting to tracking HR trace that makes it more precise and robust than the conventional approach. The indicated filter enables us to suggest a novel approach to remove random body motion effects. By experimental data, the applicability of the new radar for a wide range of real world scenarios is investigated, and its advantages beside conventional approaches are proved practically in different aspects.

Novel Sampling Formulas Associated with Quaternionic Prolate Spheroidal Wave functions

The Whittaker-Shannon-Kotel'nikov (WSK) sampling theorem provides a reconstruction formula for the bandlimited signals. In this paper, a novel kind of the WSK sampling theorem is established by using the theory of quaternion reproducing kernel Hilbert spaces. This generalization is employed to obtain the novel sampling formulas for the bandlimited quaternion-valued signals. A special case of our result is to show that the 2D generalized prolate spheroidal wave signals obtained by Slepian can be used to achieve a sampling series of cube-bandlimited signals. The solutions of energy concentration problems in quaternion Fourier transform are also investigated.

Joint angular and spectral estimation technique using nonlinear Kalman filters for cognitive radio

The joint estimation of direction of arrivals (DOA) and carrier frequencies of band-limited source signals is considered in this paper. A novel technique based on nonlinear Kalman filters is proposed for this joint angular and spectral estimation problem for cognitive radio (CR). Since sampling a wideband spectrum at Nyquist rate increases the analog-to-digital converter (ADC) requirements, we propose executing Kalman filter algorithm over a spatial state space model. Thus, one time sample is required and hardware complexity is reduced. Two types of nonlinear Kalman filters, extended Kalman filter (EKF) and unscented Kalman filter (UKF), are proposed. We consider their sub-optimal performance and show how to control their convergence. However, the proposed algorithms can detect a number of source signals limited to the number of elements in employing arrays.

Time-Varying Graph Signal Reconstruction

Signal processing on graphs is an emerging research field dealing with signals living on an irregular domain that is captured by a graph, and has been applied to sensor networks, machine learning, climate analysis, etc. Existing works on sampling and reconstruction of graph signals mainly studied static bandlimited signals. However, many real-world graph signals are time-varying, and they evolve smoothly, so instead of the signals themselves being bandlimited or smooth on graph, it is more reasonable that their temporal differences are smooth on graph. In this paper, a new batch reconstruction method of time-varying graph signals is proposed by exploiting the smoothness of the temporal difference signals, and the uniqueness of the solution to the corresponding optimization problem is theoretically analyzed. Furthermore, driven by practical applications faced with real-time requirements, huge size of data, lack of computing center, or communication difficulties between two nonneighboring vertices, an online distributed method is proposed by applying local properties of the temporal difference operator and the graph Laplacian matrix. Experiments on a variety of synthetic and real-world datasets demonstrate the excellent performance of the proposed methods.

Non-uniformly periodical sampling in phase-sensitive OTDR

The maximum detectable frequency response of phase-sensitive optical time domain reflectometry is limited by the sensing fiber length, which is a challenging issue in long-haul distributed vibration sensing. The vibration measurement in φ-OTDR system can be regarded as a discrete sampling process, and the sampling rate is the repetition rate of probe pulses. In this paper, we propose a method to enhance the detected frequency response range of φ-OTDR system by using non-uniformly periodical sampling, which is known for the ability of sub-Nyquist sampling for band-limited signals. The non-uniformly periodical probe pulses are injected into the sensing fiber to sample vibrations on the fiber. Experimental results show that the maximum detected frequency response is 2.3 kHz with ∼2397 Hz averaging frequency. The system can realize high frequency vibration measurement with low repetition rates of probe pulses in long-haul vibration measurement.

Theoretical Limits of Parameter Estimation Based on Quantized Data

Parameter estimation of band-limited periodic signals (sine and multisine waves) is a very common task in the field of measurement technology and control engineering. In the overwhelming majority of data acquisition and control systems the analog signals of the real world are sampled an quantized using analog-to-digital converters (ADCs). To estimate the parameters of the analog signal and the parameters of the quantizer from the same measurement record is an obvious need in these cases. The parameters of the recorded signal can be used to calculate the response of our system (e.g. signals of the actuators) while the parameters of the quantizer can be used to identify the transfer characteristic of the measurement channel. Maximum likelihood (ML) estimation of the quantizer and analog signal parameters has been developed to perform this task and to provide asymptotically unbiased and efficient estimators for the quantizer and signal parameters. This paper investigates the theoretical limits of this kind of estimation: provides the Cramér-Rao Lower Bound (CRLB) for the covariance of the achieved estimators and compares them to CRLB values obtained using less complex signal and channel models. This article also provides a comparison of the empirical covariance of estimator populations achieved different ways to the CRLB of estimation. The major tendencies are drawn and explanation for them is provided as well.

Interference Mitigation Based on Bayesian Compressive Sensing for Wireless Localization Systems in Unlicensed Band

In wireless localization services, a wideband spectrum is required for high resolution ranging. For this purpose, unlicensed bands have received substantial interest for their potential to reduce deployment cost. However, in the unlicensed spectrum, narrowband interference is often present and distorts band-limited reference signals for channel impulse response (CIR) estimation that is a key component to determine the location of users. In this paper, we propose a new Bayesian compressive sensing (BCS) framework to estimate complex-valued targets and apply it to mitigate the impact of subband interference on CIR estimation accuracy. Our Bayesian approach estimates the CIR by maximizing the posterior probability of the CIR from frequency domain signals in which a portion of the signal is corrupted by the interference. Based on the BCS framework, we propose three interference mitigation techniques that utilize the information on interfered subbands differently. We demonstrate the superior performance of the proposed schemes by showing improved ranging error statistics using measured indoor channels in the 5.8 GHz band.

Bias-Variance Tradeoff of Graph Laplacian Regularizer

This paper presents a bias-variance tradeoff of graph Laplacian regularizer, which is widely used in graph signal processing and semi-supervised learning tasks. The scaling law of the optimal regularization parameter is specified in terms of the spectral graph properties and a novel signal-to-noise ratio parameter, which suggests selecting a mediocre regularization parameter is often suboptimal. The analysis is applied to three applications, including random, band-limited, and multiple-sampled graph signals. Experiments on synthetic and real-world graphs demonstrate near-optimal performance of the established analysis.

A Nonrecursive Approach for Zero Crossing Based Spectral Analysis

The spectral analysis of a band-limited signal can be achieved by means of a zero crossing approach without explicit analog-to-digital conversion. For this, a pure sinusoid is added to the signal to be analyzed. In this letter, it is shown that the cost for computing the spectrum can be significantly reduced and multiplicative recursions can be avoided if the frequency of the additional sinusoid is at least twice the band limit of the signal. In the limit case, for very small input signals, the computation method presented here is converging toward the computation of a discrete Fourier transform. It is demonstrated that the noise introduced by the quantization of the time axis is spectrally white.

A New Adaptive Subsample ETDE for Large‐Range Time‐Varying Delay Estimation Using Low‐Order Interpolators

We present a new adaptive subsample Explicit time delay estimator (ETDE) that is well suitable for determining and tracking the large-range time-varying delay between two noisy bandlimited signals only using low-order interpolators. Unlike conventional ETDE where a Fractional delay filter (FDF) is used, the new ETDE is based on a novel on-line interpolator which has the ability to optimally synthesize any continuously variable nonintegral sample delay. The theoretical derivations for our proposed on-line interpolator and the new ETDE algorithm are provided. Simulations are presented to evaluate the performance of the new estimator and to compare the performance with other estimators. Simulation results show that the new ETDE significantly outperforms FDF-based ETDE when the delay is large or varies in a large range, and the same filter orders are used.

A regularized two-dimensional sampling algorithm

Abstract In this paper, a regularized sampling algorithm for band-limited signals is presented in the two-dimensional case. The convergence of the regularized sampling algorithm is studied and compared with Shannon’s sampling theorem and the Tikhonov regularization method by some examples.

A derandomization approach to recovering bandlimited signals across a wide range of random sampling rates

Reconstructing bandlimited functions from random sampling is an important problem in signal processing. Strohmer and Vershynin obtained good results for this problem by using a randomized version of the Kaczmarz algorithm (RK) and assigning to every equation a probability weight proportional to the average distance of the sample from its two nearest neighbors. However, their results are valid only for moderate to high sampling rates; in practice, it may not always be possible to obtain many samples. Experiments show that the number of projections required by RK and other Kaczmarz variants rises seemingly exponentially when the equations/variables ratio (EVR) falls below 5. CGMN, which is a CG acceleration of Kaczmarz, provides very good results for low values of EVR and it is much better than CGNR and CGNE. A derandomization method, based on an extension of the bit-reversal permutation, is combined with the weights and shown to improve the performance of CGMN and the regular (cyclic) Kaczmarz, which even outperforms RK. A byproduct of our results is the finding that signals composed mainly of high-frequency components are easier to recover.

Noise Robust Formant Frequency Estimation Method Based on Spectral Model of Repeated Autocorrelation of Speech

In this paper, a noise robust formant frequency estimation scheme is developed based on a spectral model matching algorithm. Considering the vocal tract as an autoregressive system, a spectral model of repeated autocorrelation function RACF of band-limited speech signal is proposed. It is shown that because of the repeated autocorrelation operation on band-limited signal, the proposed model can exhibit prominent formant characteristics. First from given noisy speech observations, an adaptive band selection criterion is developed. Next, on each resulting band-limited noisy speech signal, a repeated autocorrelation operation is carried out, which not only reduces the effect of noise but also strengthens the dominant poles corresponding to the formant frequencies. Finally, spectrum of the RACF is computed and instead of direct spectral peak picking, a model fitting scheme is introduced to find out model parameters which lead to formant estimation. The proposed algorithm has been tested on natural vowels as well as some naturally spoken sentences in the presence of different environmental noises. It is found that the proposed scheme provides better formant estimation accuracy in comparison to some of the existing methods at low levels of signal-to-noise ratio.

Three-component Parallel and Synchronous Seismic Data Acquisition Based on Time-division Sampling System

To economize power dissipation and volume of the three-component parallel and synchronous seismic data acquisition system, under the inspiration of sampling rate conversion theory in the digital signal processing, a project of three-component parallel and synchronous seismic data acquisition method is proposed based on time-division sampling system, in which triple interpolation is implemented in the three sets of data acquired by time-division sampling. Then another interpolation or decimation is carried out to aim at converting sampling rate of the system into the target value. Results of the error analysis show that the accuracy of numerical calculation in this conversion process is superior to the resolution ratio of hardware system. the prominent dominant of this project is that power dissipation and volume of parallel and synchronous acquisition system applied to band-limited signal are optimized and performance of the whole system is improved.

Lower Bounds on Parameter Modulation–Estimation Under Bandwidth Constraints

The problem of modulating the value of a parameter onto a band-limited signal to be transmitted over a continuous-time, additive white Gaussian noise (AWGN) channel, and then estimating this parameter at the receiver, is considered. The performance is measured by the mean power- $\alpha $ error (MP $\alpha \text{E}$ ), which is defined as the worst case $\alpha $ th-order moment of the absolute estimation error. The optimal exponential decay rate of the MP $\alpha \text{E}$ as a function of the transmission time is investigated. Two upper (converse) bounds on the MP $\alpha \text{E}$ exponent are derived, on the basis of known bounds for the AWGN channel of inputs with unlimited bandwidth. The bounds are computed for typical values of the error moment and the signal-to-noise ratio (SNR), and the SNR asymptotics of the different bounds are analyzed. The new bounds are compared with known converse and achievability bounds, which were derived from channel coding considerations.

Prolate spheroidal wave functions associated with the quaternionic Fourier transform

One of the fundamental problems in communications is finding the energy distribution of signals in time and frequency domains. It should therefore be of great interest to find the quaternionic signal whose time‐frequency energy distribution is most concentrated in a given time‐frequency domain. The present paper finds a new kind of quaternionic signals whose energy concentration is maximal in both time and frequency under the quaternionic Fourier transform. The new signals are a generalization of the classical prolate spheroidal wave functions to a quaternionic space, which are called the quaternionic prolate spheroidal wave functions. The purpose of this paper is to present the definition and fundamental properties of the quaternionic prolate spheroidal wave functions and to show that they can reach the extreme case within the energy concentration problem both from the theoretical and experimental description. The superiority of the proposed results can be widely applied to the application of 4D valued problems. In particular, these functions are shown as an effective method for bandlimited quaternionic signals relying on the extrapolation problem. Copyright © 2017 John Wiley & Sons, Ltd.