Complex White Gaussian Noise Research Articles

Low complexity subspace approach for unbiased frequency estimation of a complex single-tone

We propose a single-tone frequency estimator of a one-dimensional complex signal in complex white Gaussian noise. The estimator is based on the subspace approach and the unitary transformation. Due to its low space and time-complexity, we name the estimator as Low complexity Unitary Principal-singular-vector Utilization for Model Analysis (LUPUMA). Regardless of the observation length, LUPUMA provides a uniform estimation variance over the whole frequency range, while achieving the lowest time-complexity among subspace methods. The proposed estimator asymptotically reaches the Cramér-Rao Lower Bound. For short observations, the signal-to-noise ratio threshold of LUPUMA corresponds to the threshold of the maximum likelihood estimator. The low space and time-complexity along with the stable and state-of-the-art estimation performance for short observations make LUPUMA an ideal candidate for applications with a limited number of signal samples, limited computational power, limited memory, and for applications that require rapid processing time (low latency).

Convergence in law for complex Gaussian multiplicative chaos in phase III

Gaussian multiplicative chaos (GMC) is informally defined as a random measure eγXdx where X is Gaussian field on Rd (or an open subset of it) whose correlation function is of the form K(x,y)=log1|y−x|+L(x,y), where L is a continuous function of x and y and γ=α+iβ is a complex parameter. In the present paper we consider the case γ∈PIII′, where PIII′:={α+iβ:α,γ∈R,|α|< d/2,α2+β2≥d}. We prove that if X is replaced by an approximation Xε obtained by convolution with a smooth kernel, then the random distribution eγXεdx, when properly rescaled, has an explicit nontrivial limit in law when ε goes to zero. This limit does not depend on the specific convolution kernel which is used to define Xε and can be described as a complex Gaussian white noise with a random intensity given by a real GMC associated with parameter 2α.

Autoregressive model of an underwater acoustic channel in the frequency domain

The underwater acoustic (UWA) channel model based on the multipath delay–amplitude or the angle of departure (AoD)–angle of arrival (AoA) require many parameters to describe a broadband underwater acoustic (UWA) channel. In this paper, an autoregressive (AR) model is developed to describe the channel in the frequency domain with few parameters. The AR model used here is with an order P, where P is also related to the number of main multipath clusters. Hence, only P + 1 parameters, which consist of the variance of the complex Gaussian white noise (CGWN) and the P pole values of the AR model, are required to describe UWA channel. Each pole value is corresponded to a multipath cluster in the channel, where its phase is correlated with the average multipath delay, and its absolute value is related to the multipath energy. A comprehensive analysis was performed to characterize the statistical distribution of each pole value and the CGWN variance. The results showed that the absolute value and phase of each pole satisfied the Weibull and gamma distributions, respectively, and the variance obeyed a lognormal distribution. Simulation channels were generated with different distributions of the AR model parameters, and the root mean square (RMS) delay obeyed a similar cumulative distribution function as the measured channel.

A Linear Model for Classification of Electronic Devices Using Harmonic Radar

A new approach to exploit nonlinear reradiation of electronic circuits under test (ECUT) for classification of electronic devices using harmonic radar is proposed in this article. Unlike prior approaches, we develop a linear model to relate the measurements to the unknown parameters representing the nonlinear characteristics of the ECUT. Each nonlinear circuit under test has a distinguishable response to a power-varying single-tone or two-tone incident wave. As a result, each ECUT possesses a unique unknown deterministic vector of parameters derived from the proposed linear model. We use maximum likelihood estimator in the presence of complex white Gaussian noise based on the newly constructed linear model. The statistical features of the normalized estimated vectors of parameters are employed as distinguishing factors in classification of different nonlinear electronic devices using K-nearest neighbors classification method. Simulation results show that the proposed method of linear model with power-swept signals and estimated vectors of parameters successfully classifies nonlinear devices.

Alternative representations for the probability of detection of non‐fluctuating targets

The authors derive a closed-form exact expression and a fast convergent series for the probability of phased-array detection of non-fluctuating targets embedded in complex white Gaussian noise. To be realistic, they assume that the amplitude of the target echoes and the noise power are unknown parameters. Their series representation achieved impressive savings in computational load and computation time when compared to the numerical evaluation of the existing integral-form solution.

Optimal invariant detection of a monochromatic plane wave with unknown amplitude, frequency, phase, direction of arrival and noise variance

This study deals with the problem of detecting a monochromatic plane wave with unknown amplitude, phase, temporal frequency and direction of arrival in complex white Gaussian noise with unknown variance. Depending on the natural invariance in scale, temporal modulation and spatial modulation of the problem, the uniformly most powerful invariant (UMPI) test is derived by using the statistical invariance principles. However, the UMPI test does not apply unless the signal-to-noise ratio (SNR) is known. However, it provides us with performance bound to evaluate any invariant test's performance when the SNR is unknown. Typically, the generalised likelihood ratio test (GLRT) and locally most powerful invariant (LMPI) test are derived as realisable suboptimal invariant tests with their performance comparison in different SNR through theoretical analysis. Computer simulation examples corroborate the authors analysis and indicate that the GLRT is close to the UMPI bound especially in the low probability-of-false-alarm (P FA) region of the receiver operating characteristic curve while the performance of the LMPI test is close to that of the UMPI test in the low SNR region.

Laplacian based non-local means denoising of MR images with Rician noise

Magnetic Resonance (MR) image is often corrupted with a complex white Gaussian noise (Rician noise) which is signal dependent. Considering the special characteristics of Rician noise, we carry out nonlocal means denoising on squared magnitude images and compensate the introduced bias. In this paper, we propose an algorithm which not only preserves the edges and fine structures but also performs efficient denoising. For this purpose we have used a Laplacian of Gaussian (LoG) filter in conjunction with a nonlocal means filter (NLM). Further, to enhance the edges and to accelerate the filtering process, only a few similar patches have been preselected on the basis of closeness in edge and inverted mean values. Experiments have been conducted on both simulated and clinical data sets. The qualitative and quantitative measures demonstrate the efficacy of the proposed method.

A new complex Duffing oscillator used in complex signal detection

In view of the fact that complex signals are often used in the digital processing of certain systems such as digital communication and radar systems, a new complex Duffing equation is proposed. In addition, the dynamical behaviors are analyzed. By calculating the maximal Lyapunov exponent and power spectrum, we prove that the proposed complex differential equation has a chaotic solution or a large-scale periodic one depending on different parameters. Based on the proposed equation, we present a complex chaotic oscillator detection system of the Duffing type. Such a dynamic system is sensitive to the initial conditions and highly immune to complex white Gaussian noise, so it can be used to detect a weak complex signal against a background of strong noise. Results of the Monte-Carlo simulation show that the proposed detection system can effectively detect complex single frequency signals and linear frequency modulation signals with a guaranteed low false alarm rate.

선형적 위상배열 코일구조의 시뮬레이션을 통한 민감도지도의 공간 해상도 및 필터링 변화에 따른 MR-SENSE 영상재구성 평가

Parallel imaging technique can provide several advantages for a multitude of MRI applications. Especially, in SENSE technique, sensitivity maps were always required in order to determine the reconstruction matrix, therefore, a number of difference approaches using sensitivity information from coils have been demonstrated to improve of image quality. Moreover, many filtering methods were proposed such as adaptive matched filter and nonlinear diffusion technique to optimize the suppression of background noise and to improve of image quality. In this study, we performed SENSE reconstruction using computer simulations to confirm the most suitable method for the feasibility of filtering effect and according to changing order of polynomial fit that were applied on variation of spatial resolution of sensitivity map. The image was obtained at 0.32T(Magfinder II, Genpia, Korea) MRI system using spin-echo pulse sequence(TR/TE = 500/20 ms, FOV = 300 mm, matrix = <TEX>$128{\times}128$</TEX>, thickness = 8 mm). For the simulation, obtained image was multiplied with four linear-array coil sensitivities which were formed of 2D-gaussian distribution and the image was complex white gaussian noise was added. Image processing was separated to apply two methods which were polynomial fitting and filtering according to spatial resolution of sensitivity map and each coil image was subsampled corresponding to reduction factor(r-factor) of 2 and 4. The results were compared to mean value of geomety factor(g-factor) and artifact power(AP) according to r-factor 2 and 4. Our results were represented while changing of spatial resolution of sensitivity map and r-factor, polynomial fit methods were represented the better results compared with general filtering methods. Although our result had limitation of computer simulation study instead of applying to experiment and coil geometric array such as linear, our method may be useful for determination of optimal sensitivity map in a linear coil array.

Numerical analysis of fringe patterns recorded in holographic interferometry using high-order ambiguity function

This letter introduces a new approach for the demodulation of fringe patterns recorded in holographic interferometry using high-order ambiguity function (HAF). The proposed approach is capable of retrieving the phase from a single fringe pattern. The main advantage of this approach is that it directly provides an estimation of the continuous phase distribution and thereby avoids the necessity of using a cumbersome 2D phase unwrapping procedure. This method first computes the discrete-time analytic signal of the recorded fringe pattern. Then, by modelling this analytic signal as a polynomial phase signal embedded in additive complex white Gaussian noise, a parametric estimation procedure based on HAF is employed to directly estimate the unwrapped phase distribution. Numerical simulations and experimental results demonstrate the potential of the proposed approach.

Tracking Analysis of Complex Adaptive IIR Notch Filter for a Linear Chirp Signal

This paper analyzes frequency tracking characteristics of a complex-coefficient adaptive infinite impulse response (IIR) notch filter with a simplified gradient-based algorithm. The input signal to the complex notch filter is a complex linear chirp embedded in a complex zero-mean white Gaussian noise. The analysis starts with derivation of a first-order real-coefficient difference equation with respect to steady-state instantaneous frequency tracking error. Closed-form expression for frequency tracking mean square error (MSE) is then derived from the difference equation. Lastly, closed-form expressions for optimum notch bandwidth coefficient and step size constant that minimize the frequency tracking MSE are derived. Computer simulations are presented to validate the analysis.

Random input problem for the nonlinear Schrödinger equation

We consider the random input problem for a nonlinear system modeled by the integrable one-dimensional self-focusing nonlinear Schrödinger equation (NLSE). We concentrate on the properties obtained from the direct scattering problem associated with the NLSE. We discuss some general issues regarding soliton creation from random input. We also study the averaged spectral density of random quasilinear waves generated in the NLSE channel for two models of the disordered input field profile. The first model is symmetric complex Gaussian white noise and the second one is a real dichotomous (telegraph) process. For the former model, the closed-form expression for the averaged spectral density is obtained, while for the dichotomous real input we present the small noise perturbative expansion for the same quantity. In the case of the dichotomous input, we also obtain the distribution of minimal pulse width required for a soliton generation. The obtained results can be applied to a multitude of problems including random nonlinear Fraunhoffer diffraction, transmission properties of randomly apodized long period Fiber Bragg gratings, and the propagation of incoherent pulses in optical fibers.

Autocorrelation-based algorithm for single-frequency estimation

A novel iterative algorithm for estimating the frequency of a single complex sinusoid in the complex white Gaussian noise is proposed. Inspired by the works of Fitz, L&R, and the ILP approaches, the algorithm is based on the repeated use of an autocorrelation-based frequency estimator. This approach is different from the iterative linear prediction (ILP) algorithm, yet produces very comparable performance. Like the ILP algorithm, the proposed estimator has the reduced threshold, and its performance is close to that of the ML estimation and uniform across the frequency range of - π to π . In addition, the proposed estimator demonstrates an asymptotic error variance of 0.14 dB above the CRB for N = 1024 and 0.62 dB for N = 48 , which is better than that of the ILP algorithm.

Estimating statistical properties of Eddy-current signals from steam generator tubes

We develop a model for characterizing amplitude and phase probability distributions of eddy-current signals and propose a maximum likelihood (ML) method for estimating the amplitude and phase distribution parameters from measurements corrupted by additive complex white Gaussian noise. The squared amplitudes and phases of the potential defect signals are modeled as independent, identically distributed (i.i.d.) random variables following gamma and von Mises distributions, respectively. Newton-Raphson iteration is utilized to compute the ML estimates of the unknown parameters. We also compute Crame/spl acute/r-Rao bounds (CRBs) for the unknown parameters and discuss initialization of the Newton-Raphson iteration. The proposed method is applied to analyze rotating-probe eddy-current data from steam-generator tube inspection in nuclear power plants. The obtained estimates can be utilized for maximum a posteriori (MAP) signal phase and amplitude estimation, as well as efficient feature extractors in a defect classification scheme. We present numerical examples with both real and simulated data to demonstrate the performance of the proposed methods.

Signal-to-noise ratio estimation using higher-order moments

We consider the problem of estimation of the signal-to-noise ratio (SNR) of an unknown deterministic complex phase signal in additive complex white Gaussian noise. The phase of the signal is arbitrary and is not assumed to be known a priori unlike many SNR estimation methods that assume phase synchronization. We show that the moments of the complex sequences exhibit useful mean-ergodicity properties enabling a “method-of-moments” (MoM)-SNR estimator. The Cramer–Rao bounds (CRBs) on the signal power, noise variance and logarithmic-SNR are derived. We conduct experiments to study the efficiency of the SNR estimator. We show that the estimator exhibits finite sample super-efficiency/inefficiency and asymptotic efficiency, depending on the choice of the parameters. At 0 dB SNR, the mean square error in log-SNR estimation is approximately 2 dB 2 . The main feature of the MoM estimator is that it does not require the instantaneous phase/frequency of the signal, a priori. Infact, the SNR estimator can be used to track the instantaneous frequency (IF) of the phase signal. Using the adaptive pseudo-Wigner–Ville distribution technique, the IF estimation accuracy is the same as that obtained with perfect SNR knowledge and 8–10 dB better compared to the median-based SNR estimator.

On the distribution of poles of Padé approximants to the Z-transform of complex Gaussian white noise

In the application of Padé methods to signal processing a basic problem is to take into account the effect of measurement noise on the computed approximants. Qualitative deterministic noise models have been proposed which are consistent with experimental results. In this paper the Padé approximants to the Z -transform of a complex Gaussian discrete white noise process are considered. Properties of the condensed density of the Padé poles such as circular symmetry, asymptotic concentration on the unit circle and independence on the noise variance are proved. An analytic model of the condensed density of the Padé poles for all orders of the approximants is also computed. Some Monte Carlo simulations are provided.

Blind Multiuser Detection for Asynchronous MC-CDMA Systems Without Channel Estimation

This paper presents a blind adaptive decorrelating detector for asynchronous multicarrier code-division multiple-access (MC-CDMA) systems with Rayleigh-fading channels. This detector is derived by making use of the cross-correlation matrix between the consecutively received signals. The main attraction of the detection algorithm is its simplicity, since the detector can be implemented blindly without channel estimation, except for the synchronization of the desired user. To implement the proposed detector, a new adaptive subspace-tracking algorithm for any Hamiltonian matrices is developed, while the previously reported subspace-tracking methods can only estimate the subspace of a positive Hamiltonian matrix. Since a common drawback of the multiuser detector based on the subspace method is the existence of a phase factor that usually deteriorates the performance of the detector, a new method is developed to compensate for the phase of the proposed detector, based on the property of the complex white Gaussian noise. Results of extensive computer simulations show the performance improvement of the proposed detector.

Fast and accurate single frequency estimator

A new fast and accurate method for estimating the frequency of a complex sinusoid in complex white Gaussian noise is proposed. The proposed estimator has the lowest computational complexity among the estimators. Its performance and estimation range are similar to the Fitz and L&amp;R estimators, and its threshold is 20 log(m) less than the Kay estimator.

Optimal invariant detection of a sinusoid with unknown parameters

A uniformly most-powerful test does not exist for the problem of detecting a sinusoid of unknown amplitude, phase, and frequency in complex white Gaussian noise of unknown variance. The problem is shown to be invariant to scale and modulation transformations. Using invariance principles, it is found that the uniformly most-powerful-invariant (UMPI) test does not exist. A test is derived that is UMPI if it is given that the signal-to-noise ratio (SNR) is known. For SNR unknown, the derived test can be used as an upper performance bound for any invariant test. Suboptimal tests are typically implemented such as the generalized likelihood ratio and locally most powerful tests, which are also scale and modulation invariant. It is shown analytically that as SNR approaches zero, the locally most-powerful-invariant test performance approaches the UMPI performance bound, and as the SNR becomes large, the generalized likelihood ratio test performance approaches the performance of the UMPI bound. Computer simulation examples indicate that the generalized likelihood ratio test performance is close to the UMPI bound, particularly in the low-probability-of-false-alarm region of the receiver operating characteristic curve.

Modified Kay's method with improved frequency estimation

A modified frequency estimation method for use in the presence of additive complex white Gaussian noise is proposed. This method is basically an extension of Kay's method, but includes the use of autocorrelation functions. It retrieves the frequency value from the optimal linear combination of the differenced phases of the autocorrelation functions. It yields a considerably lower variance threshold than Kay's method while remaining unbiased and retaining the same frequency range.