Complex Sinusoids Research Articles

Spectrum estimation using frequency shifting and decimation

Parametric spectral estimation techniques are widely used to estimate the parameters of sums of complex sinusoids corrupted by noise. In this work, the authors show that the numerical stability of the estimated frequencies not only depends on the size of the amplitudes associated with the real frequencies, but also to the distance among frequencies. Therefore, for closely spaced frequencies, the estimates are vulnerable to large deviate from their true values. To overcome this problem, they propose a strategy to artificially increase the frequency separation by downsampling the baseband equivalent of the noisy signal before applying a spectral estimation technique. This methodology significantly improves the estimation performance especially in the low signal to noise ratio regime. The performance of the technique is assessed in terms of the root mean square error and it is compared to results obtained in previous publications.

Closed-Form Estimator for Frequency Estimation of Complex Sinusoidal Signals in Multiplicative and Additive Noise

In this paper, a new method is proposed for estimating the frequency of a complex sinusoidal signal in multiplicative and additive noise. The proposed method is based on the linear prediction property of the signals’ autocorrelation function. This frequency estimation method is a closed-form algorithm which can directly estimate the frequency without peak searching. The performance of the proposed frequency estimator is investigated by computer simulations. Simulation results demonstrate that the performance of the proposed technique is reasonable and is close to Cramer–Rao bound.

Accurate Frequency Estimator of Sinusoid Based on Interpolation of FFT and DTFT

An accurate frequency estimator of complex sinusoid in additive white noise is proposed. It is based on interpolation of Fast Fourier Transform (FFT) and Discrete-Time Fourier Transform (DTFT). Zero-padding is firstly performed before the FFT of the sinusoid sampled data, and the coarse estimate is obtained by searching the discrete frequency index of the maximum FFT spectrum line. Then the fine estimate is obtained by employing the maximum FFT spectrum line and two DTFT sample values located on the left and right side of the maximum spectrum line. The correlation coefficients between the Fourier Transform of the noises on two arbitrarily spaced spectrum lines are derived, and the MSE calculation formula is derived in additive white noise background based on the correlation coefficients. Simulations results demonstrate that the proposed algorithm has lower MSE than the competing algorithms, and its signal-to-noise ratio (SNR) threshold is lower compared with Candan algorithm, AM algorithm and Djukanovic algorithms.

Optimum Combining for Coherent FFH/DS Spread Spectrum Receivers in the Presence of Multi-Tone Jammer

Fast Frequency Hopping/Direct Sequence (FFH/DS) hybrid spread spectrum is an effective technique of the communication network for the robot swarms operating in complex electromagnetic environment. It combines the advantages of both Frequency Hopping Spread Spectrum (FHSS) and Direct Sequence Spread Spectrum (DSSS), and also uses the diversity-combining in time and frequency domain to effectively reject unwanted jamming. Multi-Tone Interference (MTI) is one of the most deleterious jamming waveforms for frequency hopping systems, and consequently a severe threat for FFH/DS systems. In this article, we explore the effect of MTI on coherent FFH/DS, and we have shown that, if spectral sensing mechanism for MTI frequency and power monitoring is available, the MTI in FFH/DS reception is a set of complex sinusoid components with deterministic amplitudes and random phases, instead of the conventionally assumed Gaussian random variables. In light of this new finding, a novel combining method is proposed. It utilizes the spectral sensing information to adaptively compute the combining weights, following Maximum Signal to Interference plus Noise (MSINR) law. Its combining performance is analyzed and compared with other existing methods, and verified by simulations. It is shown that the proposed adaptive MSINR combining has satisfactory performance under both medium and strong MTI scenarios.

Low Complexity and High-Resolution Line Spectral Estimation Using Cyclic Minimization

The line spectral estimation problem has applications in radar, wireless communications, spectroscopy, and power electronics, among others. The signal is modeled as a sparse linear combination of complex sinusoids and the problem target is to estimate the number of sinusoids in the mixture, and respective parameters of each individual sinusoid, such as magnitude, phase, and frequency. In this work, we first introduce a novel formulation of the rank function, which involves the solution of a multi-convex optimization problem. Using the multi-convex reformulation of the rank function an over-parameterization of the line estimation problem is proposed together with a cyclic minimization procedure to obtain a solution. The cyclic method iterates between the optimization of the signal subspace and null-space and stops when the two are orthogonal. Every limit point of the sequence of iterates is shown to be a stationary point of the original problem. Numerical experiments show that only a small number of iterations is required for convergence. The signal subspace optimization is a semidefinite program (SDP) and the null-space optimization has a closed-form solution. The cost per iteration of a general-purpose interior point method (IPM) to solve the SDP is a quartic function of the problem dimension. It is shown that by exploiting the problem structure the cost per iteration of the IPM may be reduced to a cubic function of the problem dimension, comparable to the cost of the alternating direction method of multipliers (ADMM). However, contrarily to the latter, the former achieves the precision necessary for fine frequency localization. A large set of numerical experiments show the effectiveness of the proposed approach.

Non-searching coherent integration method for maneuvering target detection by using non-uniformly resampling technique

It is well known that, long-time integration improves the detection performance of radar system, which is however limited by the range migration (RM) and Doppler frequency migration (DFM) caused by target's high velocity and acceleration. To solve this problem, this paper proposed a three-dimensional modified cross correlation function (3DMCCF) method for maneuvering radar targets. In this method, we define a new cross correlation function under a designing frame based on non-uniformly resampling technique. By using the new function, the target's RM and DFM can be eliminated at the same time. The resulting signal is a three-dimensional complex sinusoid. Finally the integration is achieved efficiently by performing inverse fast Fourier transforms (IFFTs) and fast Fourier transforms (FFTs). The proposed method can work in low signal to noise ratio (SNR) environment and is applicable for multi-target scene. Compared with the generalized Radon Fourier transform (GRFT) and the Radon Lv's distribution (RLVD) algorithms, the proposed method is free from searching approach and has lower computational complexity. The effectiveness of the proposed method is demonstrated by simulations and real-data processing.

НИЖНІЙ ПОРІГ ВСШ ПРИ ШВИДКОМУ ОЦІНЮВАННІ КОМПЛЕКСНИХ СИНУСОЇД

A task of estimation of complex sinusoid frequency is considered. A particular but practically important case of low signal-to-noise ratio (SNR) is studied. The low SNR threshold, commonly overlooked in development of the rapid estimator of complex sinusoidals, is addressed. Signals of different length are considered and SNR is varied in wide limits. It is demonstrated that a simple interpolation with factor 2 lowers the SNR threshold by 1.5dB for the most complicated practical situations. Further interpolation does not bring any improvement. This allows proposing a compromise practical algorithm that provides accuracy close to the limit and is still very simple and fast.

An efficient algorithm for frequency estimation from cosine-sum windowed DFT coefficients

This paper presents an algorithm for estimating frequency of a complex sinusoid from three DFT coefficients. In particular, we consider the case where the signal is multiplied by a generic cosine-sum window for interference suppression. The algorithm is an interpolator that uses the peak sample in the DFT of the data and its two neighbors. The interpolator is based on a numerically efficient procedure which guarantees convergence to a solution. Performance of the proposed interpolator is shown to be nearly optimum and comparable to the best interpolator in the literature, while the computational burden required for the proposed algorithm is much lower than the generic numerical methods available. Degradation of performance due to interference is also investigated.

Fast and Efficient Sinusoidal Frequency Estimation by Using the DFT Coefficients

Fast and accurate frequency estimation of a complex sinusoidal plays an important role in many applications, including communications, radar, and sonar. This paper presents two methods for the frequency estimation of a complex sinusoidal in noisy environments. The proposed methods interpolate on shifted DFT coefficients to acquire the frequency estimates iteratively. The first method interpolates on the $q$ -shifted DFT coefficients of the signal, whose optimum iteration number is found to be a logarithmic function of the signal length. Furthermore, we also show that by appropriately selecting the value of the shift $q$ , the proposed method becomes asymptotically efficient. In the second method, we use hybrid half-shifted and $q$ -shifted DFT interpolators together, which converges in only two iterations. It is shown that both of the estimators are asymptotically unbiased with their mean square errors performing on the Cramer–Rao lower bound. Theoretical results are validated by extensive computer simulations.

Efficient and Accurate Detection and Frequency Estimation of Multiple Sinusoids

The method for detection of complex sinusoids in additive white Gaussian noise and estimation of their frequencies is proposed. It contains two stages: 1) sinusoid detection (model order estimation) and coarse frequency estimation, and 2) fine frequency estimation. The proposed method operates in the frequency domain, i.e., it uses the discrete Fourier transform (DFT) as the main tool. Sinusoid detection is performed so that a fixed probability of false alarm is provided (Neymann–Pearson criterion). For both coarse and fine frequency estimations, the three-point periodogram maximization approach is used. Simulations are carried out for variable signal-to-noise ratio, variable frequency displacement between the sinusoids and variable offset from the frequency grid. The proposed method meets the Cramer-Rao lower bound in frequency estimation and practically does not depend on the frequency displacement except for very small displacement values. In terms of model order estimation accuracy, it outperforms the state-of-the-art approaches. The most expensive operation in the method is the calculation of the DFT. Therefore, in terms of calculation complexity, the proposed method is on par with the most efficient algorithms for multiple frequency estimations.

Accurate Frequency Estimation of a Real Sinusoid by Three New Interpolators

Interpolators are widely employed in the frequency estimation of sinusoidal signals. However, for a real-valued sinusoidal signal, the existing interpolators do not consider the negative frequency component, which leads to a frequency estimation error floor. To solve this problem and further improve the estimation accuracy, three new interpolators are proposed in this paper. These three new interpolators consist of two initial interpolators and one fine interpolator. In the proposed algorithm, the initial interpolators consider both the positive and negative frequency components and are utilized first to eliminate this error floor and obtain an initial frequency estimate. Then, the fine interpolator for the frequency estimation of a complex sinusoid is exploited to improve the accuracy of the initial frequency estimate. The theoretical analysis demonstrates that the frequency estimation mean square error of the proposed algorithm is almost equal to the Cramer-Rao lower bound. Compared with the existing time-domain analysis algorithms, the proposed algorithm has better estimation performance, especially in the case of a low signal-to-noise ratio. Compared with the existing frequency-domain analysis algorithms, the proposed algorithm has lower computational complexity and a wider valid estimation range.

A Refinement to the Viterbi-Viterbi Carrier Phase Estimator and an Extension to the Case With a Wiener Carrier Phase Process

We provide a theoretical foundation for further analysis and optimization of the Mth-power (MP) carrier phase estimator for MPSK modulation. Also known as the Viterbi-Viterbi (VV) estimator, it is commonly used in practice because it leads to low-latency receiver implementations. The MP carrier phase estimator first raises the received noisy signal samples to the Mth-power to remove the unknown phase modulation, and then extracts the unknown carrier phase of the mid-symbol using a weighted sum of these modulation-wiped-off received signal samples over a symmetrical observation window. Our starting point is the single-term, complex exponential expression for a complex sinusoid received in complex, additive, white, Gaussian noise (AWGN), which leads to a great deal of simplicity in dealing with arbitrary powers of the noisy received signal sample when compared with the conventional approach of raising the sum of signal plus noise to higher powers. The single-exponential expression enables us to first optimize the weighting coefficients of the MP carrier phase estimator with respect to the statistics of the AWGN, in a manner much simpler than previous approaches. Then, it enables us to apply the linear minimum mean square error (LMMSE) criterion to optimize the MP estimator with respect to both the statistics of the AWGN and the carrier phase noise that we model here as a Wiener process. Although the LMMSE MP estimator is computationally intensive for online implementation, a much less complex version is suggested that can be efficiently implemented in real time. Extensive simulation results are presented to demonstrate the improved performance of the LMMSE MP estimator over the conventional MP estimator. By using a sufficiently long symmetrical observation window, the LMMSE estimator does not suffer from the block length effect, which leads to much performance gain over the VV/MP estimator especially at high signal-to-noise ratio (SNR) and high phase noise. A phase unwrapping algorithm is also presented for accurate unwrapping of the estimated carrier phase before it is used in data detection. The proposed LMMSE carrier phase estimator is suitable for implementing a coherent receiver at all SNRs.

Fast off grid compressed sensing ISAR imaging algorithm

Abstract To solve the off grid problem in compressed sensing (CS) based inverse synthetic aperture radar (ISAR) imaging, a fast and accurate algorithm has been proposed in the paper. By jointly estimating the off grid error and the sparse solution, off grid ISAR imaging is transformed into a joint optimization problem. Interestingly, it can be solved efficiently through two least squares problems based on first order Taylor approximation. When applied to complex sinusoids and quasi real ISAR data, the proposed algorithm has got better results than the conventional algorithm. Therefore, it is a promising off grid CS based ISAR imaging algorithm.

Robust Frequency Estimation in Symmetric $$\alpha $$ α -Stable Noise

The symmetric $$\alpha $$ -stable (S $$\alpha $$ S) noise is commonly encountered in a variety of applications such as wireless communications and image processing. In this work, combining the linear prediction property and $$\ell _p$$ -norm minimization, a robust frequency estimator is devised for a complex sinusoid in the presence of the S $$\alpha $$ S noise. The proposed algorithm, based on the $$\ell _p$$ -norm of the preprocessed linear prediction errors, can be regarded as the outlier-resistant version of the generalized weighted linear prediction frequency estimation approach. Computer simulations are conducted to contrast the performance of the proposed algorithm with three conventional frequency estimators. The results indicate that our method is robust to outliers and nearly optimal compared with Cramer–Rao lower bound.

Identification of two dimensional complex sinusoids in white noise: a state-space frequency approach

This paper proposes a new frequency domain approach for identifying the parameters of two-dimensional complex sinusoids from a finite number of data, when the measurements are affected by additive and uncorrelated two-dimensional white noise. The new method extends in two dimensions a frequency identification procedure of complex sinusoids, originally developed for the one-dimensional case. The properties of the proposed method are analyzed by means of Monte Carlo simulations and its features are compared with those of other estimation algorithms. In particular the practical advantage of the method is highlighted. In fact the novel approach can operate just on a specified sub-area of the 2D spectrum. This area-selective feature allows a drastic reduction of the computational complexity, which is usually very high when standard time domain methods are used.

Flexible and Accurate Frequency Estimation for Complex Sinusoid Signal by Interpolation Using DFT Samples

A flexible and accurate frequency estimator is first proposed for frequency estimation of a complex sinusoid weighted with a rectangular window function in additive white Gaussian noise. This proposed frequency estimator can be operated in the application of an arbitrary length discrete Fourier transform where the original input data is padded with any zeroes, which makes it more flexible in practice. The proposed frequency estimator utilizes the maximum sample value and its two adjacent samples in the frequency domain to perform the fine frequency estimation with unbiased results obtained. Then a modified frequency estimator is proposed to estimate the frequency when the complex sinusoid signal is weighted with different nonrectangular window functions. Although the modified frequency estimator is nonanalytic and biased, it can still improve the estimation performance for certain applications. Simulation results demonstrate that both of the proposed frequency estimators are effective to achieve the high frequency estimation accuracy. And the root mean square errors of the proposed frequency estimators approach the Cramer-Rao bound when the signal-to-noise ratio is large enough to make the coarse frequency estimation work effectively.

2D-frequency domain identification of complex sinusoids in the presence of additive noise

This paper describes a new approach for identifying the parameters of two-dimensional complex sinusoids from a finite number of measurements, in presence of additive and uncorrelated two-dimensional white noise. The proposed approach is based on using frequency domain data. The new method extends to the two-dimensional (2D) case some recent results obtained with reference to the frequency ESPRIT algorithm. The properties of the proposed method are analyzed by means of Monte Carlo simulations and its features are compared with those of a classical time domain estimation algorithm. The practical advantages of the method are highlighted. In fact the novel approach can operate just on a specified sub-area of the 2D spectrum. This area-selective feature allows a drastic reduction of the computational complexity, which is usually very high when standard time domain methods are used.

Spectral Compressed Sensing via Projected Gradient Descent

Let $x\in\mathbb{C}^n$ be a spectrally sparse signal consisting of $r$ complex sinusoids with or without damping. We consider the spectral compressed sensing problem, which is about reconstructing $x$ from its partial revealed entries. By utilizing the low rank structure of the Hankel matrix corresponding to $x$, we develop a computationally efficient algorithm for this problem. The algorithm starts from an initial guess computed via one-step hard thresholding followed by projection, and then proceeds by applying projected gradient descent iterations to a non-convex functional. Based on the sampling with replacement model, we prove that $O(r^2\log(n))$ observed entries are sufficient for our algorithm to achieve the successful recovery of a spectrally sparse signal. Moreover, extensive empirical performance comparisons show that our algorithm is competitive with other state-of-the-art spectral compressed sensing algorithms in terms of phase transitions and overall computational time.

Subspace‐Based Method for Direction Finding of Multiple Partially Polarized Signals

We investigate the problem of direction finding for multiple Partially polarized (PP) electromagnetic signals, each of which possesses two spatial degrees of freedom. We use a uniform linear array of L sensors, where each consists of two co-located single-polarized elements. By exploiting the array geometry and its shift invariance property, we construct a set of data correlation sequences using the array output and its conjugate to transform the problem of direction finding to that of the estimation of complex sinusoid frequencies. The rank-2 signal correlation matrices of the PP signals become real-valued amplitudes of complex sinusoids, and thus, the rank-2K signal subspace of the array output transforms to the rank-K signal subspace of the constructed correlation sequences. We show that only L=K+1 sensors are needed to resolve K PP signals by solving the problem of estimation of sinusoid frequencies using subspace-based methods. The proposed method is also applicable to the scenario, where both Completely polarized (CP) and PP signals coexist, without the need of any prior information on the degree of polarization of the signal.

Nonlinear Fourier transform—towards the construction of nonlinear Fourier modes

We study a version of the nonlinear Fourier transform associated with ZS-AKNS systems. This version is suitable for the construction of nonlinear analogues of Fourier modes, and for the perturbation-theoretic study of their superposition. We provide an iterative scheme for computing the inverse of our transform. The relevant formulae are expressed in terms of Bell polynomials and functions related to them. In order to prove the validity of our iterative scheme, we show that our transform has the necessary analytic properties. We show that up to order three of the perturbation parameter, the nonlinear Fourier mode is a complex sinusoid modulated by the second Bernoulli polynomial. We describe an application of the nonlinear superposition of two modes to a problem of transmission through a nonlinear medium.