Signals In Additive White Noise Research Articles

Generalized Multi-Lag Estimators (GMLE) for Polarimetric Weather Radar Observations

Observations of weather phenomenon by polarimetric pulsed-Doppler weather radars are employed worldwide to monitor impending severe storms, flash-floods, and other weather related public hazards. The basis for processing received meteorological signals from pulsed-radar waveforms relies on stochastic processes where the accurate estimation of radar variables from received signals in additive white noise is essential for meaningful interpretation of weather phenomena and algorithm-derived products. For polarimetric weather radars, these estimates are calculated from signal correlations in time and across the horizontal and vertical polarization channels. Conventional estimators only use 1 or 2 signal correlation time-lags and may not utilize all the available information intrinsic in the received signals. Weather-variable estimates could benefit from the use of all intrinsic characteristics in the received data; accordingly, more complex estimators use multiple lags to extract additional information. However, not all estimates are improved by the use of more lags; in fact, improvement in estimates depends on signal characteristics and requires that the additional correlation lags provide new information. In this article, we derive and examine general multi-lag estimators for reflectivity, differential reflectivity, polarimetric cross-correlation coefficient, and Doppler spectrum width. We compare the performance of these proposed estimators against conventional estimators using Monte-Carlo simulations on different meteorological signal characteristics to find estimators that can improve the quality of certain radar-variable estimates.

Weak signal detection: Condition for noise induced enhancement

For the detection of a weak known signal in additive white noise, a generalized correlation detector is considered. In the case of a large number of measurements, an asymptotic efficacy is analytically computed as a general measure of detection performance. The derivative of the efficacy with respect to the noise level is also analytically computed. Positivity of this derivative is the condition for enhancement of the detection performance by increasing the level of noise. The behavior of this derivative is analyzed in various important situations, especially showing when noise-enhanced detection is feasible and when it is not.

Weak-Periodic Stochastic Resonance in a Parallel Array of Static Nonlinearities

This paper studies the output-input signal-to-noise ratio (SNR) gain of an uncoupled parallel array of static, yet arbitrary, nonlinear elements for transmitting a weak periodic signal in additive white noise. In the small-signal limit, an explicit expression for the SNR gain is derived. It serves to prove that the SNR gain is always a monotonically increasing function of the array size for any given nonlinearity and noisy environment. It also determines the SNR gain maximized by the locally optimal nonlinearity as the upper bound of the SNR gain achieved by an array of static nonlinear elements. With locally optimal nonlinearity, it is demonstrated that stochastic resonance cannot occur, i.e. adding internal noise into the array never improves the SNR gain. However, in an array of suboptimal but easily implemented threshold nonlinearities, we show the feasibility of situations where stochastic resonance occurs, and also the possibility of the SNR gain exceeding unity for a wide range of input noise distributions.

Exploring weak-periodic-signal stochastic resonance in locally optimal processors with a Fisher information metric

For processing a weak periodic signal in additive white noise, a locally optimal processor (LOP) achieves the maximal output signal-to-noise ratio (SNR). In general, such a LOP is precisely determined by the noise probability density and also by the noise level. It is shown that the output–input SNR gain of a LOP is given by the Fisher information of a standardized noise distribution. Based on this connection, we find that an arbitrarily large SNR gain, for a LOP, can be achieved ranging from the minimal value of unity upwards. For stochastic resonance, when considering adding extra noise to the original signal, we here demonstrate via the appropriate Fisher information inequality that the updated LOP fully matched to the new noise, is unable to improve the output SNR above its original value with no extra noise. This result generalizes a proof that existed previously only for Gaussian noise. Furthermore, in the situation of non-adjustable processors, for instance when the structure of the LOP as prescribed by the noise probability density is not fully adaptable to the noise level, we show general conditions where stochastic resonance can be recovered, manifested by the possibility of adding extra noise to enhance the output SNR.

Fisher Information as a Metric of Locally Optimal Processing and Stochastic Resonance

The origins of Fisher information are in its use as a performance measure for parametric estimation. We augment this and show that the Fisher information can characterize the performance in several other significant signal processing operations. For processing of a weak signal in additive white noise, we demonstrate that the Fisher information determines (i) the maximum output signal-to-noise ratio for a periodic signal; (ii) the optimum asymptotic efficacy for signal detection; (iii) the best cross-correlation coefficient for signal transmission; and (iv) the minimum mean square error of an unbiased estimator. This unifying picture, via inequalities on the Fisher information, is used to establish conditions where improvement by noise through stochastic resonance is feasible or not.

Fisher-information condition for enhanced signal detection via stochastic resonance

Various situations where a signal is enhanced by noise through stochastic resonance are now known. This paper contributes to determining general conditions under which improvement by noise can be a priori decided as feasible or not. We focus on the detection of a known signal in additive white noise. Under the assumptions of a weak signal and a sufficiently large sample size, it is proved, with an inequality based on the Fisher information, that improvement by adding noise is never possible, generically, in these conditions. However, under less restrictive conditions, an example of signal detection is shown with favorable action of adding noise.

Measurement design for detecting sparse signals

We consider the problem of testing for the presence (or detection) of an unknown sparse signal in additive white noise. Given a fixed measurement budget, much smaller than the dimension of the signal, we consider the general problem of designing compressive measurements to maximize the measurement signal-to-noise ratio (SNR), as increasing SNR improves the detection performance in a large class of detectors. We use a lexicographic optimization approach, where the optimal measurement design for sparsity level k is sought only among the set of measurement matrices that satisfy the optimality conditions for sparsity level k−1. We consider optimizing two different SNR criteria, namely a worst-case SNR measure, over all possible realizations of a k-sparse signal, and an average SNR measure with respect to a uniform distribution on the locations of up to k nonzero entries in the signal. We establish connections between these two criteria and certain classes of tight frames. We constrain our measurement matrices to the class of tight frames to avoid coloring the noise covariance matrix. For the worst-case problem, we show that the optimal measurement matrix is a Grassmannian line packing for most–and a uniform tight frame for all–sparse signals. For the average SNR problem, we prove that the optimal measurement matrix is a uniform tight frame with minimum sum-coherence for most–and a tight frame for all–sparse signals.

A numerical solution for multichannel detection

A numerical approach to the vector signal detection problem is proposed by using an approximate series representation for vector complex-valued random processes. This technique allows us to provide computationally feasible suboptimum receivers for the problem of detecting an improper or proper complex signal in additive white noise. Its implementation requires the knowledge of the correlation matrix of the vector process involved. The performance of the technique devised is assessed by some communication examples.

Fast Tracking of a Real Sinusoid with Multiple Forgetting Factors

In this paper, a recursive Gauss-Newton (RGN) algorithm is first developed for adaptive tracking of the amplitude, frequency and phase of a real sinusoid signal in additive white noise. The derived algorithm is then simplified for computational complexity reduction as well as improved with the use of multiple forgetting factor (MFF) technique to provide a flexible way of keeping track of the parameters with different rates. The effectiveness of the simplified MFF-RGN scheme in sinusoidal parameter tracking is demonstrated via computer simulations.

Semiparametric estimation of the frequency of unknown periodic functions and its application to laser vibrometry signals

We propose a semiparametric approach to fundamental frequency estimation of an unknown periodic signal in additive white noise based on model selection. Our estimator maximizes a penalized version of the cumulated periodogram and is proved to be consistent and asymptotically efficient under very general conditions. When the number of observations is fixed, an implementation of this estimation method is proposed and illustrated on specific synthetic signals which arise in laser vibrometry. We extend this method for estimating the fundamental frequencies of two periodic functions having different fundamental frequencies when the data consist of their sum and additive white noise. We also compare the performances of our procedure with the so-called microdoppler technique, which is commonly used for laser vibrometry signals analysis. We show on simulated data that the penalized cumulated periodogram yields an accurate estimation of the frequencies at very low signal-to-noise ratios.

Nonlinear SNR amplification of harmonic signal in noise

The SNR of a harmonic signal in additive white noise is computed after transformation by an arbitrary memoryless nonlinearity. With a simple saturating nonlinearity having direct electronic implementation, an amplification of the SNR can be obtained, an outcome which is inaccessible with linear devices.

A quantitative analysis of SNR in the short-time Fourier transform domain for multicomponent signals

A quantitative analysis is given for the signal-to-noise ratio (SNR) in the short-time Fourier transform domain for multicomponent signals in additive white noise. It is shown that the SNR is increased on the order of O(N/K), where K is the number of components of a signal, N/T is the sampling rate, and T is the window size. The SNR increase rate is optimal for given K. For this result, the SNR definition is generalized, which is suitable for signals not only in the time domain but also in other domains. This theory is illustrated by one numerical example.

Iterative parameter estimation of multiple chirp signals

The authors propose an algorithm for estimating the parameters of multiple superimposed chirp signals in additive white noise. The algorithm is based on a novel iterative approach that significantly reduces the error propagation effect inherent in many existing techniques. Moreover, it allows the estimation over a wider range of phase parameter values while still maintaining a better estimation accuracy.

False Alarm Probability of a DWT-Based Estimation Algorithm

Noonan, J. P., and Marquis, D. A., False Alarm Probability of a DWT-Based Estimation Algorithm,Digital Signal Processing6 (1996), 155–159.Recently, the wavelet transform has been applied to the problem of detecting and estimating transient signals in additive white noise. The general procedure of these algorithms is to take a discrete wavelet transform (DWT) of the data, eliminate some of the DWT coefficients based on some criteria, and then inverse DWT the remaining coefficients to obtain the transient signal estimate. Most of the research has focused on the estimation performance of these algorithms. This research computes the false alarm probability on each DWT scale level for the algorithm described in (Noonanet al.,Digital Signal Processing 3, 1993, 89–96). The false alarm probability is the probability that one or more DWT coefficients on a difference level is nonzero.

A Novel Subspace-Based Approach to Parameter Estimation

The known eigendecomposition-based subspace techniques for parameter estimation of multiple signals in additive white noise or the directions of arrivals of wavefronts at an array of sensors use either the information generated from the noise subspace or the signal subspace. Here, we directly use information from both the noise and signal subspaces to estimate the parameters. Comparisons with other subspace techniques are made and an illustrative example is provided.

Enhancement of speech in additive, locally stationary and colored noise, using linear prediction

In this study, an algorithm is developed for enhancement of speech, degraded by additive, slightly time varying, colored, background noise. The procedure is based on modeling both the speech and noise using linear prediction. It is well-known that a clean speech can be successfully represented by an autoregressive (AR) model. If the background noise is white, then the AR signal and the additive white noise correspond to an autoregressive and moving average (ARMA) model where the moving average (MA) part represents the effect of the noise. By using this fact, it is suggested to model the speech and the additive noise as two linear predictive models. The additive colored noise is whitened by filtering the speech using the estimated noise model. Then, the filtered signal in additive white noise is modeled as an ARMA process where the MA part represents the effect of the additive white noise on speech. Finally, inverse filtering the signal in additive white noise by the MA part yields a speech signal with eliminated noise.

Estimation of instantaneous frequency using the discrete Wigner distribution

Analytical expressions for the performance of the discrete Wigner distribution (DWD) in estimating the instantaneous frequency of linear frequency modulated signals in additive white noise are derived and verified using simulation. It is shown that the DWD peak provides an optimal estimate at high input signal-to-noise ratios. The applicability of these results to the general case of nonlinear FM signals is discussed.

An entropy-based receiver for the detection of random signals and its application to radar

An entropy-based receiver is derived for detecting correlated signals in additive white noise, both having unknown statistics. The basic idea behind the new approach is that under a constant energy constraint, the entropy of a received process having a finite number of sample points attains its maximum when the process is white and decreases as it becomes correlated. When no signal is present, the test statistic is asymptotically a monotone function of a chi-square variable, which can be employed to determine the desired threshold for a constant false-alarm rate. Results based on simulated data and actual radar data are presented.

A Hankel-norm Approach to Approximate Recursive Estimation

Let {x(t)} and {y(t)} be multivariate processes which are weakly stationary and stationarily correlated We consider the problem of finding a approximate recursive low-dimensional filter of x(t), based on the observation of the past of y(t). We use a Hankel-norm approximation scheme to obtain a low order model. We also present some applications to Kalman filtering and to filtering of a signal in additive white noise.

The cause of line splitting in Burg maximum-entropy spectral analysis

Burg's algorithm for maximum-entropy autoregressive spectral estimation is analyzed for the case of one and two complex sinusoidal signals in additive white noise. In the latter case, two biases are found which can account for the line splitting and line shifting that occur in simulation studies when the SNR is very high. These biases vanish completely if the two complex sinusoids are in phase quadrature at the middle of the data record. If the data record contains an integral number of half-cycles of the difference frequency, the magnitude of the power spectral estimate will be biased, although the effects believed to cause splitting and shifting will be eliminated. For infinite SNR, the effect of these biases is to prevent perfect cancellation of the coherent signal by the correct-order Burg prediction-error filter (PEF). Then higher order PEF's can be based on the uncancelled signal. It is conjectured that the severity of splitting and shifting is closely related to the relative magnitude of the uncancelled signal. Results of simulation studies to verify this conjecture are presented. The analysis also shows that lengthening the observation interval by including more data recorded at the original sampling rate should reduce the magnitude of the biases, but that increasing the number of data by increasing the sampling rate, while maintaining the observation interval unchanged, is of little use.