Signals In White Gaussian Noise Research Articles

Worst case Cramer-Rao bounds for parametric estimation of superimposed signals with applications

The problem of parameter estimation of superimposed signals in white Gaussian noise is considered. The effect of the correlation structure of the signals on the Cramer-Rao bounds is studied for both the single and multiple experiment cases. The best and worst conditions are found using various criteria. The results are applied to the example of parameter estimation of superimposed sinusoids, or plane-wave direction finding in white Gaussian noise, and best and worst conditions on the correlation structure and relative phase of the sinusoids are found. This provides useful information on the limits of the resolvability of sinusoid signals in time series analysis or of plane waves in array processing. The conditions are also useful for designing worst-case simulation studies of estimation algorithms, and for the design of minimax signal acquisition and estimation procedures, as demonstrated by an example.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Some contributions to a frequency location method due to He and Kedem

The author derives a useful information-theoretical technology from the tunable filter that automatically detects the frequency of an unknown signal in white Gaussian noise by adjusting its parameter to the correlation coefficient of its output. The tunable filter earlier proposed by S. He and B. Kedem (see ibid., vol.35, no.2, p.360-9, 1989) is known as the HK filter. The methodology sidesteps the white noise and normality assumptions. More importantly, it is shown that broad classes of linear filters can be made to implement the detection plan. In particular, bandpass filters can be expected to lock-in even more quickly and accurately than the HK filter. This analysis yields a formula for the error contraction at each recursion.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Upper bounds on the probability of error for M-ary orthogonal signaling in white Gaussian noise

The optimum detection of M orthogonal equiprobable equal-energy signals in additive white Gaussian noise is considered, and two upper bounds for the probability of error are derived. The behavior of these bounds is discussed and they are compared with previously known bounds for various values of signal-to-noise ratio and M. Some numerical results are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Flow velocity profile via time-domain correlation: error analysis and computer simulation

An ultrasonic human-blood-flow velocity profile measurement method using time-domain correlation of consecutive echo pairs has been developed. The time shift between a pair of range gated echoes is estimated by searching for the shift that results in the maximum correlation. The time shift indicates the distance a group of scatterers has moved, from which flow velocity is estimated. The basis for the computer simulations and error analyses of the scheme includes a band-passed white Gaussian noise signal model for an echo from a scattering medium, the estimate of flow velocity from both a single scatterer and multiple scatterers, and a derived precision estimation. The error analysis via computer simulation includes an evaluation of errors associated with the correlation method. For a uniform flow velocity profile, beamwidth modulation represents the greatest error source. However, for a nonuniform flow velocity profile, the jitter caused by a small flow velocity gradient can exceed the other error sources. A detailed computer simulation evaluated the interdependencies of window length, beam width, vessel diameter, and viewing angle on the estimation of flow velocity.

On the detection and classification of quadrature digital modulations in broad-band noise

Optimal and suboptimal decision rules for the detection of constant-envelope quadrature digital modulations in broadband noise are derived and analyzed. The effect of various stochastic models for the carrier phase is examined in detail, while no epoch or frequency uncertainty is assumed. The delay-and-multiply type of detector is considered. A new binary/quadrature phase shift keying (BPSK/QPSK) classifier is compared to the more traditional ad hoc techniques of a square-law classifier and a phase-based classifier (weighting on the phase histogram). The new classifier is derived by approximating the likelihood-ratio functionals of phase-modulated digital signals in white Gaussian noise, hence is named the quasi-log-likelihood ratio (qLLR) rule. It is shown analytically that its performance is significantly better than that of intuitively designed phase-based rules or the conventional square-law classifier. >

A time-frequency formulation of optimum detection

A time-frequency formulation is proposed for the optimum detection of Gaussian signals in white Gaussian noise. By choosing the Wigner-Ville distribution as the basic time-frequency tool, it is shown that the corresponding receivers generally take the form of a correlation between time-frequency structures, matching mathematical optimality with a physically meaningful interpretation. The case of low SNR is examined and various examples are considered: deterministic signal, Rayleigh fading signal, random jitter, and random time-varying channel. A general class of time-frequency receivers is proposed which admits as limiting cases different known structures, and its suboptimum performance is evaluated. Possible extensions to more elaborate situations (including parameter estimation) are mentioned. >

Wigner distribution method equivalent to dechirp method for detecting a chirp signal

It is shown that the Wigner distribution (WD) method of optimally detecting a chirp signal in white Gaussian noise [1] is equivalent mathematically to a dechirp method.

On the approximation of the optimum detector by the energy detector in detection of colored Gaussian signals in noise

We demonstrate that the optimum detector of colored Gaussian signals in white Gaussian noise can be approximated, without serious degradation in detection performance, by the so-called energy detector which is optimum only when the signal is also white. Contrary to intuition, such an approximation is possible when the signal which is colored dominates the noise which is white. The demonstration is made analytically by deriving the optimum detection statistic and also numerically by simulating the detection performance of both detectors. A comparison of data processing complexity of the two detectors is also given.

Sequential Testing for Several Signals in Gaussian White Noise

Sequential Testing for Several Signals in Gaussian White Noise

The "Unity-lagged" short-time spectrum of a narrow-band Gaussian process

A form of the short-time cross-spectrum of a signal with its delayed self is considered. Its statistical and signal-characterizing properties are derived for the case of a narrow-band random Gaussian signal in white Gaussian noise.

Statistics of the maintained discharge of cat retinal ganglion cells.

Action potentials were recorded from single fibres in the optic tracts of anaesthetized cats. Continuous records were obtained at various levels of scotopic and mesopic retinal illumination. In some cases, the light intensity was modulated by a pseudorandom Gaussian white-noise signal. The maintained discharge of on-centre neurones increased while the maintained discharge of off-centre neurons decreased with increased illumination of the receptive field centre. For both cell types, the coefficient of variation declined with increased rate of discharge. There was minimal short-term dependency in the firing patterns, and it was unaffected by the level of retinal illumination. Virtually all of the structure revealed by the normalized autocovariance functions could be attributed to the shape of the interval distributions. The first few coefficients of the serial correlogram were slightly negative. The magnitude of this negativity was not related to illumination. Long-term dependency in the firing pattern was also quite small; the standard deviations of the mean rate in samples of about 1 sec duration were only slightly less than would be predicted from the interval distributions. This dependency tended to increase at higher retinal illuminations. Neural discharges elicited by Gaussian modulation of the light were strikingly different from those elicited by steady light. Modulation caused the first coefficient of the serial correlogram to become more positive, while the next several coefficients became more negative. A corresponding pattern could be seen in the normalized autocovariance functions, and in the differences between the normalized autocovariance and normalized autoconvolution. Long-term dependency also increased dramatically, such that the standard deviations of mean rate were about 60% of what would be expected given the interval distributions observed. These results place a number of constraints upon the ways in which intrinsic noise in the retina may enter the visual processing network. Two alternative models consistent with the data are presented.

Adaptive enhancement of finite bandwidth signals in white Gaussian noise

The steady-state behavior of the adaptive line enhancer (ALE) is analyzed for stationary inputs consisting of finite bandwidth signals embedded in a white Gaussian noise (WGN) background. Analytic expressions for the weights and output of the LMS adaptive filter are derived as functions of input signal bandwidth and SNR, as well as ALE length and bulk delay. The steady-state gain in broad-band SNR from input to output is derived as a function of these same four variables. For fixed ALE parameters and input SNR, it is shown that this gain increases as the input signal becomes narrower and approaches the sinusoidal limit. It is emphasized that because the correlation time of finite bandwidth signals is limited, excessively large values of the ALE bulk delay parameter result in diminished gain. Furthermore, there is an optimal filter length, whose value depends upon signal bandwidth and SNR, for which the broad-band gain is maximized. These results demonstrate the importance of including the effects of algorithm noise in analyzing the performance of real-time adaptive processors.

Estimation of the maximum value of a signal in Gaussian white noise

Estimation of the maximum value of a signal in Gaussian white noise

Kalman filtering in two dimensions: Further results

The two-dimensional reduced update Kalman filter was recently introduced. The corresponding scalar filtering equations were derived for the case of estimating a Gaussian signal in white Gaussian noise and were shown to constitute a general nonsymmetric half-plane recursive filter. This paper extends the method to the deconvolution problem of image restoration. This paper also provides a more thorough treatment of the uniquely two-dimensional boundary condition problems. Numerical and subjective examples are presented.

Second-order output statistics of the adaptive line enhancer

The adaptive line enhancer (ALE) is an adaptive digital filter designed to suppress uncorrelated components of its input, while passing any narrow-band components with little attenuation. The purpose of this paper is to analyze the second-order output statistics of the ALE in steady-state operation, for input samples consisting of weak narrow-band signals in white Gaussian noise. The ALE output is shown to be the sum of two uncorrelated components, one arising from optimum finite-lag Wiener filtering of the narrow-band input components, and the other arising from the misadjustment error associated with the adaptation process. General expressions are given for the output auto-correlation function and power spectrum with arbitrary narrow-band input signals, and the case of a single sinusoid in white noise is worked out as an example. Finally, the significance of these results to practical applications of the ALE is mentioned.

A maximum likelihood ratio alternative to the multiple coherence statistic

The multiple coherence function has been suggested for application in multiple receiver detection of common narrowband acoustic signals. As an alternative the maximum likelihood ratio detection statistic is developed for multiple receiver detection of narrowband Gaussian signals in additive white Gaussian noise. For the assumptions of rectangular signal spectrums and sufficiently general nonlinear or random time varying propagation channel models, the maximum likelihood ratio statistic is shown to be equivalent |Σ|/Πtσij where Σ = [σij] is the cross-power spectral density matrix estimate within the signal spectral band. A Monte-Carlo computer simulation procedure is discussed. Receiver Operating Characteristic (ROC) curves resulting from the simulation are presented which suggest that in general the maximum likelihood ratio statistic is superior in detection capability to the multiple coherence statistic.

Fast time-invariant implementations of Gaussian signal detectors

A new implementation is presented for the optimum likelihood ratio detector for stationary Gaussian signals in white Gaussian noise that uses only two causal time-invariant filters. This solution also has the advantage that fast algorithms based on the Levinson and Chandrasekhar equations can he used for the determination of these time-invariant filters. By using a notion of "closeness to stationarity,' there is a natural extension of the above results for nonstationary signal processes.

Some zero‐crossing statistics of a signal consisting of band‐limited white Gaussian noise with a pure tone added to it

The number of zero crossings per second of a band‐limited white Gaussian noise signal x(t) is slightly greater than twice the midband frequency of the signal. This may be explained by considering the complex signal z(t) = x̂(t) + j x(t) where x̂(t) is the Hilbert transform of x(t). The vector z(t) rotates at a mean angular frequency that is equal to its mid‐band angular frequency, but it has a jitter superimposed upon it that; on occasion, causes it to rotate in the opposite direction. The extra crossings are produced by the vector rotating backwards just after it has crossed the real axis in the forward direction. It is the statistics of these backward crossings that have been studied when a pure signal has been added to the noise. [Work supported by ONR Code 436.]

An upper bound on the mean-square error for Bayesian parameter estimators (Corresp.)

The estimation of unknown parameters of deterministic time-varying signals contaminated with additive Gaussian white noise is considered, and an upper bound is obtained for the mean-square estimation error. As an example, the theory is applied to the synchronous demodulation of pulse frequency-modulated (PFM) signals in Gaussian white noise.

A likelihood ratio formula for two-dimensional random fields

This paper is concerned with the detection of a random signal in white Gaussian noise when both the signal and the noise are two-dimensional random fields. The principal result is the derivation of a recursive formula for the likelihood ratio relating it to certain conditional moments of the signal. It is also shown that, except for some relatively uninteresting cases, a simple exponential formula for the likelihood ratio, such as one has in one dimension, is not possible.