Complex Sinusoids Research Articles

Hanning’s Performance in 4253HT Smoother on Four Signals of Different Noise Levels

Smoothing is a method of data analysis that aims to provide well-defined patterns or signals by removing noise or unstructured patterns from data sets. To obtain clean and smooth data, Hanning is applied as one of the important components in smoothing. However, Hanning is not resistant to outliers. Therefore, this study aims to determine the best type of Hanning that is able to obtain the greatest performance of 4253HT smoother in signal recovery. Functions of Linear, Complex Sinusoidal, Custom Pulse Train, and Sawtooth signals corrupted with five levels of contaminated normal noise were used as signals in the smoothing process. All signals were applied to assess three different Hanning types, which were Tukey, Husain and Shitan. Besides, a Root Mean Square Error (RMSE) was used as an evaluator to determine and assess the performance of 4253HT smoother when utilizing an alternative Hanning. Based on the overall performance of 4253HT smoother, Husain Hanning presented the best outcome and worked most efficiently at all levels of noise except at the lowest noise (10%), which Tukey Hanning executed better. The findings of this study could benefit other researchers to decide the best Hanning to be used before performing forecasting and further analysis to improve the accuracy of predicting.

Fast and efficient K -D frequency estimation for multiple complex sinusoids

Fast and efficient <i>K</i> -D frequency estimation for multiple complex sinusoids

Harmonic retrieval using weighted lifted-structure low-rank matrix completion

In this paper, we investigate the problem of recovering the frequency components of a mixture of K complex sinusoids from a random subset of N equally-spaced time-domain samples. Because of the random subset, the samples are effectively non-uniform. Besides, the frequency values of each of the K complex sinusoids are assumed to vary continuously within a given range. For this problem, we propose a two-step strategy: (i) we first lift the incomplete set of uniform samples (unavailable samples are treated as missing data) into a structured matrix with missing entries, which is potentially low-rank; then (ii) we complete the matrix using a weighted nuclear minimization problem. We call the method a weighted lifted-structured (WLi) low-rank matrix recovery. Our approach can be applied to a range of matrix structures such as Hankel and double-Hankel, among others, and provides improvement over the unweighted existing schemes such as EMaC and DEMaC. We provide theoretical guarantees for the proposed method, as well as numerical simulations in both noiseless and noisy settings. Both the theoretical and the numerical results confirm the superiority of the proposed approach.

Frequency estimation of multiple complex sinusoids using noise suppressing predictive FIR filter

The paper presents a predictive filter based frequency estimation algorithm (PF-FE) for multiple complex sinusoids corrupted by additive white Gaussian noise. The proposed frequency estimator combines the noise suppression capability of an FIR filter with a least squares approach minimizing the sum of the squared errors of a prediction filter. The FIR filter coefficients are chosen such that the sinusoidal components are predicted with minimum distortion. Computer simulation results are given to contrast the performance of the proposed PF-FE method with six other state-of-the-art frequency-estimators and the Cramer-Rao lower bound. Furthermore, the paper provides a complexity analysis of the PF-FE method and compares it with those of the conventional MUltiple SIgnal Classification algorithm and some selected state-of-the-art methods from the literature. Performance of the proposed PF-FE method is validated with effective mean square error (EMSE) plots for model orders of three and five and under different observation lengths and signal-to-noise ratio values. Finally, the paper demonstrates how EMSE varies for the proposed method when the prediction filter length is increased from 20 to 75. After extensive simulations, it is observed that when filter length is larger than half the observation length the EMSE will start to degrade.

Prony Analysis of Left Ventricle Pressure and Volume.

Direct measurement of cardiac pressure-volume (PV) relationships is the gold standard for assessment of ventricular hemodynamics, but few innovations have been made to "multi-beat" PV analysis beyond traditional signal processing. The Prony method solves the signal recovery problem with a series of dampened exponentials or sinusoids. It achieves this by extracting the amplitude, frequency, dampening, and phase of each component. Since its inception, application of the Prony method to biologic and medical signal has demonstrated a relative degree of success, as a series of dampened complex sinusoids easily generalizes to multifaceted physiological processes. In cardiovascular physiology, the Prony analysis has been used to determine fatal arrythmia from electrocardiogram signals. However, application of the Prony method to simple left ventricular function based on pressure and volume analysis is absent. We have developed a new pipeline for analysis of pressure volume signals recorded from the left ventricle. We propose fitting pressure-volume data from cardiac catheterization to the Prony method for pole extraction and quantification of the transfer function. We implemented the Prony algorithm using open-source Python packages and analyzed the pressure and volume signals before and after severe hemorrhagic shock, and after resuscitation with stored blood. Each animal (n=6 per group) underwent a 50% hemorrhage to induce hypovolemic shock, which was maintained for 30min, and resuscitated with 3-week-old stored RBCs until 90% baseline blood pressure was achieved. Pressure-volume catheterization data used for Prony analysis were 1s in length, sampled at 1000Hz, and acquired at the time of hypovolemic shock, 15 and 30min after induction of hypovolemic shock, and 10, 30, and 60min after volume resuscitation. We next assessed the complex poles from both pressure and volume waveforms. To quantify deviation from the unit circle, which represents deviation from a Fourier series, we counted the number of poles at least 0.2 radial units away from it. We found a significant decrease in the number of poles after shock (p=0.0072vs. baseline) and after resuscitation (p=0.0091vs. baseline). No differences were observed in this metric pre and post volume resuscitation (p=0.2956). We next found a composite transfer function using the Prony fits between the pressure and volume waveforms and found differences in both the magnitude and phase Bode plots at baseline, during shock, and after resuscitation. In summary, our implementation of the Prony analysis shows meaningful physiologic differences after shock and resuscitation and allows for future applications to broader physiological and pathophysiological conditions.

Predictors for High Frequency Signals Based on Rational Polynomial Approximation of Periodic Exponentials

The paper presents linear integral predictors for continuous time high-frequency signals with a a finite spectrum gap. The predictors are based on approximation of a complex valued periodic exponential (complex sinusoid) by rational polynomials.

A Fast ISAR Imaging Method Based on Strategy Weighted CAMP Algorithm

Complex approximate message passing (CAMP) algorithm is a promising compressed sensing (CS) method for inverse synthetic aperture radar (ISAR) imaging with the merits of fast convergence and good reconstruction. To improve the imaging performance and accelerate the convergence, a new strategy weighted CAMP based ISAR imaging method has been proposed. Rather than solving a sequence of weighted <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${l}_{{1}}$ </tex-math></inline-formula> minimization problems, the weighted CAMP problem is solved directly by using a new weighted threshold, which not only ensures the convergence of the algorithm but also improves the convergence speed. To accelerate the convergence furtherly, a new point is chosen smartly in the scalar threshold function, which borrows the idea of fast iterative shrinkage-thresholding algorithm (FISTA). As weighting adaptively to the signal and the noise is more appropriate to maintain the target information and suppress the noise, random forests (RF) algorithm is employed to classify the signal and the noise, and strategy weighting is used based on the classification result. The proposed algorithm was applied into complex sinusoids, quasi-real, and real ISAR data. It has got the best imaging result among the algorithms of CAMP class and need less imaging time than CAMP and weighted CAMP (WCAMP). Therefore, the proposed algorithm is an efficient ISAR imaging method with good performance.

Limited memory predictors based on polynomial approximation of periodic exponentials

AbstractThe paper presents transfer functions for limited memory time‐invariant linear integral predictors for continuous time processes such that the corresponding predicting kernels have bounded support. It is shown that processes with exponentially decaying Fourier transforms are predictable with these predictors in some weak sense, meaning that convolution integrals over the future times can be approximated by causal convolutions over past times. For a given predicting horizon, the predictors are based on polynomial approximation of a periodic exponentials (complex sinusoid) in a weightedL2‐space.

A Real-Time Hardware Emulator for 3D Non-Stationary U2V Channels

Channel emulator is an important tool to evaluate communication system performance at the physical link, and network levels. In this paper, a new discrete 3D non-stationary geometry-based stochastic model (GBSM) for UAV to vehicle (U2V) channels is proposed, which considers 3D scattering space, 3D trajectory, and 3D antenna array. And a tailed channel emulator is developed on a field programmable gate array (FPGA) platform. All channel parameters, i.e., the power, delay, and phase are calculated by FPGA hardware for the first time instead of software or pre-storage method. Meanwhile, a Greedy CORDIC-based exponential calculation method for generating massive complex sinusoids is designed and implemented. The latency is reduced by 50% than traditional CORDIC method. By further utilizing the compact architecture with time division scheme, the hardware resource is significantly reduced from 16.51% to 7.55% for 16-bit data width. Meanwhile, the fixed-point output statistical properties are also derived for quantitatively validation. Finally, the U2V channel under the campus scenario is reproduced by the proposed emulator. The generated results demonstrate that the statistical properties are consistent well with the theoretical and ray tracing ones, which verifies the correctness of both proposed channel model and emulator.

Accurate frequency estimation of multiple complex and real sinusoids based on iterative interpolation

Frequency estimation of multiple complex and real sinusoids in noise has attracted wide attention in many fields. In this paper, two frequency domain parametric methods are proposed to estimate the frequencies of multiple complex and real sinusoids accurately. The first method for complex sinusoids adopts Discrete Fourier Transform (DFT) and interpolation algorithm to obtain the coarsely estimated frequencies, and then the interpolation is used iteratively to refine the estimated frequencies. In each step, the estimated frequency components are subtracted to reduce spectrum leakage. The second method for multiple real sinusoids, which is modified based on the first one, focuses on reducing the spectrum leakage caused by negative frequencies and then employs iterative interpolation to refine the estimated frequencies. Simulation results show that two proposed methods have lower mean square error (MSE) and signal-to-noise ratio (SNR) breakdown threshold than other frequency domain parametric methods, and are capable of achieving the Cramér-Rao lower bound (CRLB) for both multiple complex and real signals under a wide range of SNR.

Adaptive Sparse Cyclic Coordinate Descent for Sparse Frequency Estimation

The frequency estimation of multiple complex sinusoids in the presence of noise is important for many signal processing applications. As already discussed in the literature, this problem can be reformulated as a sparse representation problem. In this letter, such a formulation is derived and an algorithm based on sparse cyclic coordinate descent (SCCD) for estimating the frequency parameters is proposed. The algorithm adaptively reduces the size of the used frequency grid, which eases the computational burden. Simulation results revealed that the proposed algorithm achieves similar performance to the original formulation and the Root-multiple signal classification (MUSIC) algorithm in terms of the mean square error (MSE), with significantly less complexity.

An ISAR Imaging Method Based on Improved CAMP Algorithm

Compressed sensing (CS) provides a new idea for inverse synthetic aperture radar (ISAR) imaging. Complex approximate message passing (CAMP) algorithm, as a new CS reconstruction algorithm, has characteristics of fast convergence and good reconstruction, which is suitable for ISAR imaging. However, the off-grid problem is inevitable in ISAR imaging, which affects the reconstruction performance seriously and needs to be solved urgently. To solve the off-grid problem, a parametric CS ISAR imaging model is built in this paper, in which the off-grid is considered as an unknown parameter. To solve the joint optimization problem efficiently, first-order Taylor series approximation is used and the optimization problem is transformed into two least squares problems. To verify the effectiveness of the proposed method, it is applied into random sparse signals, complex sinusoids, quasi-real ISAR data and real ISAR data. Compared with the existing algorithms, the proposed method has got higher reconstruction signal-to-noise ratio (RSNR), smaller relative reconstruction error, smaller frequency position error and better imaging result. Therefore the proposed method is an effective CS ISAR imaging method which can solve the off-grid problem well.

A Reduced Complexity Tensor Approach for Order Selection and Frequency Estimation

This paper presents a reduced complexity tensor approach for order selection and subspace-based frequency estimation. The proposed covariance tensor based order selection algorithm, termed as CT-OS, uses singular values of the covariance tensor formed from the 1D noisy observations of multiple complex sinusoids. Experimental results show that the CT-OS algorithm is capable of providing accurate order selection under short observations and is robust under medium to high signal-to-noise ratios. The proposed covariance tensor based frequency estimator, termed as CT-FE, utilizes a singular vector matrix in the higher-order singular value decomposition of the covariance tensor. Experimental results show that the CT-FE outperforms the subspace alignment and separation algorithm (SAS-Est) and a recent two-stage order and frequency estimation algorithm. Furthermore, both theoretical analysis and experimental results demonstrate reduced computational complexity and time for the proposed CT-OS against the recent covariance tensor based order estimation algorithm CTB-OE. The CT-FE algorithm is also shown to enjoy reduced computational complexity and time when compared with the frequency estimator SAS-Est.

An Optimum Signal Detection Approach to the Joint ML Estimation of Timing Offset, Carrier Frequency and Phase Offset for Coherent Optical OFDM

Coherent orthogonal frequency-division multiplexing (OFDM) is one of the prime digital modulation techniques for present and future generations of wireless and optical communications. Accurate synchronization is the main obstacle to the implementation of a reliable OFDM receiver. We propose here a joint maximum likelihood (ML) estimator for the timing offset (TO), carrier frequency offset (CFO), and carrier phase offset (CPO) for coherent optical OFDM (CO-OFDM) motivated by the theory of ML signal detection. Our approach starts conceptually with the idea of first computing L replicas of the original OFDM spectrum, where L is a power of two and is assumed sufficiently large. This is done by padding (L-1)N zeros to the end of the N received noisy OFDM samples, where N is the number of OFDM subcarriers. By computing the LN-point discrete Fourier transform (DFT) of these LN time samples, we get L replicas of the DFT of the original OFDM spectrum, where each replica corresponds to the DFT for one hypothesized value of the CFO and the set of possible CFO values is {l/L} <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l=0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L-1</sup> . We select the most probable replica by choosing the one that is at the minimum Euclidean distance from the original OFDM spectrum, which leads to a matched-filtering (MF) operation in the frequency domain in either a blind or a data-aided manner. Building on this MF concept, we then develop a joint ML estimator of the TO and CPO for each hypothesized value of the CFO. The novelty here is that the TO and the CPO are estimated efficiently as the frequency and phase of a complex sinusoid observed in noise, via either a time-domain or a frequency-domain approach. The resulting joint CFO, CPO, and TO estimator is simpler than existing estimators, both conceptually and in implementation. A much simpler sequential approach in which we first decide on the CFO and then perform a joint TO and CPO estimation is also proposed. The performance loss of this sequential approach compared to the optimum joint approach is small at high signal-to-noise ratio (SNR). We obtain the performance of all our estimators via simulations, and show that they perform better when compared with the existing well-known estimators. Finally, we derive the Cramér-Rao lower bounds (CRLB) on the performance of our estimators, and show via simulations that our estimators for high SNR attain these performance lower bounds.

An FPGA based Real-Time Multi-Target Synthetic Aperture Radar Echoes Synthesizer

This paper proposes a technique for synthesizing multiple point target scatterer Synthetic Aperture Radar (SAR) echoesin real-time. Traditional approaches require high computation resources to calculate the complex SAR echoes due to its complex mathematical model. The proposed technique employs the low computation Direct Digital Synthesis (DDS) approach to generate these complex sinusoid echoes. The proposed Synthetic Aperture Radar Echoes Synthesizer(SAR-ES)is capable of synthesizing SAR echoes accurately in real-time and was built in a Field Programmable Gate Array (FPGA) platform. The system can be used as a testbed to validate and evaluate the performance of a real-time SAR processing algorithms/system prior to the actual flight mission. This could help in reducing the frequency of flight trials and to reduce the SAR system development risk especially for satellite-borne SAR system.

Fine Doppler Frequency Estimation of Windowed Complex Sinusoidal Signals for Radar Systems

This article presents a new fine Doppler frequency estimation method of complex sinusoidal signals for radar systems. The proposed method can work with nonrectangular window functions to suppress the interference signal caused by the undesired spectral leakage. This method is based on an interpolation curve that uses the sum and difference of two discrete Fourier transform (DFT) magnitudes, and can estimate an accurate frequency between the interpolation curve samples through precise linear approximation. The proposed method exhibits lower computational complexity than the conventional method, which compensates the bias of an arbitrary window function and exhibits high performance. Thus, the proposed method is applicable to radar systems requiring fast processing time. Furthermore, the simulation results show that the proposed method outperforms the conventional method for various window functions.

Accelerated Structured Alternating Projections for Robust Spectrally Sparse Signal Recovery

Consider a spectrally sparse signal $\boldsymbol{x}$ that consists of $r$ complex sinusoids with or without damping. We study the robust recovery problem for the spectrally sparse signal under the fully observed setting, which is about recovering $\boldsymbol{x}$ and a sparse corruption vector $\boldsymbol{s}$ from their sum $\boldsymbol{z}=\boldsymbol{x}+\boldsymbol{s}$. In this paper, we exploit the low-rank property of the Hankel matrix formed by $\boldsymbol{x}$, and formulate the problem as the robust recovery of a corrupted low-rank Hankel matrix. We develop a highly efficient non-convex algorithm, coined Accelerated Structured Alternating Projections (ASAP). The high computational efficiency and low space complexity of ASAP are achieved by fast computations involving structured matrices, and a subspace projection method for accelerated low-rank approximation. Theoretical recovery guarantee with a linear convergence rate has been established for ASAP, under some mild assumptions on $\boldsymbol{x}$ and $\boldsymbol{s}$. Empirical performance comparisons on both synthetic and real-world data confirm the advantages of ASAP, in terms of computational efficiency and robustness aspects.

Multidimensional Spectral Super-Resolution With Prior Knowledge With Application to High Mobility Channel Estimation

The problem of estimating high-mobility channels is a special case of the general problem of recovering multi-dimensional (MD) complex sinusoids. This paper is concerned with estimation of multiple frequencies with prior knowledge from incomplete and/or noisy samples. Suppose that it is known a priori that the frequencies lie in some given intervals, we develop efficient super-resolution estimators by exploiting such prior knowledge based on frequency-selective (FS) atomic norm minimization. We study the MD Vandermonde decomposition of block Toeplitz matrices in which the frequencies are restricted to lie in given intervals. We then propose to solve the FS atomic norm minimization problems for the low-rank spectral tensor recovery by converting them into semidefinite programs based on the MD Vandermonde decomposition. We also develop fast solvers for solving these semidefinite programs via the alternating direction method of multipliers (ADMM), where each iteration involves a number of refinement steps to utilize the prior knowledge. Extensive simulation results are presented to illustrate the high performance of the proposed methods.

A low complexity model order and frequency estimation of multiple 2-D complex sinusoids

Model order and frequency estimation of multiple 2-D complex sinusoids in additive white Gaussian noise are addressed. Frequency estimation follows the coarse-fine search strategy. Coarse estimates, obtained by locating maxima of the 2-D discrete Fourier transform, are refined in a two-stage procedure. In both stages, frequency refinement is based on three-point periodogram maximization. In order to provide accurate model order estimation (MOE) for a wide signal-to-noise ratio (SNR) range, our approach combines two metrics for sinusoid detection in the 2-D frequency domain, one for low and the other for high SNR values. The proposed frequency estimation method attains the Cramér-Rao lower bound and it outperforms parametric methods in terms of the estimation accuracy and numerical efficiency. Compared with information criterion-based methods, the proposed MOE approach is numerically more efficient, it does not require estimation of noise variance and hence does not suffer from overestimation at high SNR.

Fast Rank-Revealing QR Factorization for Two-Dimensional Frequency Estimation

It is well known that the singular value decomposition (SVD), as the best rank-revealing factorization, furnishes the best rank- $k$ approximation to a dense matrix with expensive computational cost of $\mathcal O(M^{2}N+N^{2}M+\min (M,N)^{3})$ . Moreover, it is hard to implement in parallel, which challenges the memory storage in wireless communications data-driven system. In this letter, a fast rank-revealing technique, namely, bilateral random projections (BRP) with $\mathcal O(MN)$ operations, is exploited for two-dimensional (2-D) frequency estimation of a complex sinusoid in noisy environment. Based on the resulting data matrix, whereafter, two-stage QR factorization frequency estimation method with the weighted least squares (WLS) as solver, is proposed to reduce the computational complexity of those SVD-based frequency estimators. Simulation results demonstrate the efficiency of the proposed algorithm in comparison with several frequency estimation approaches and the Cramer-Rao lower bound (CRLB) as benchmark.