Signals In White Gaussian Noise Research Articles

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.

Joint detection—Estimation of Gaussian signals in white Gaussian noise

Recent results on joint detection—estimation for discrete data are extended to the continuous case for Gaussian signals in white Gaussian noise. Stochastic differential equations are obtained describing the temporal evolution of the sufficient statistic for the minimum Bayes' risk detection and the optimal mean-square error estimate of the signal state vector. An approximate solution for the estimation equation is presented and the results are extended to the case of multishot joint detection estimation.

Detection of FSK signals using linear FM dispersion method

A new signal processing technique for the detection of FSK signals in white Gaussian noise is presented. The new feature of this technique is that the received FSK signal waveforms are first converted into new waveforms and the detection is carried out by using an appropriate matched filter matched to the new waveforms. The waveform conversion approach at the receiver is advantageous since the required matched filter for the new waveform is realizable more readily than that for the FSK waveforms.

An Optimum Phase Synchronizer in a Partially Coherent Receiver

The derivation and the statistical properties of the maximum a posteriori probability phase estimator of a sinusoidal signal in white Gaussian noise are considered. The probability density function of the phase estimate is developed. The estimator efficiency and performance as a phase synchronizer in a partially coherent receiver are calculated and compared with a first-order phase-locked loop phase estimator.

Realization of filter-squarer receivers (Corresp.)

Sufficient conditions for the realization of a causal filter-squarer receiver for the detection of a Gaussian signal in white Gaussian noise are established. Special consideration is given to a Gaussian signal that is the projection of a vector Gauss-Markov process. For this case, the causal filter-squarer receiver is shown to be related to the Kalman-Bucy filter for estimatng the underlying vector Gauss-Markov process.

Nonsingular detection and likelihood ratio for random signals in white Gaussian noise

This paper is concerned with the mathematical aspect of a detection problem (a random signal in white Gaussian noise). Specifically, we obtain a sufficient condition for nonsingular detection and derive a likelihood-ratio expression in terms of least-mean-square estimates. The problem itself is old, and the likelihood-ratio expression is also well known. The contribution of this paper is a relatively elementary and self-contained derivation of the likelihood-ratio expression as well as the nonsingularity condition.

Likelihood ratios for Gaussian processes

We give a comprehensive discussion of the structure of the likelihood ratio (LR) for discrimination between two Gaussian processes, one of which is white. Several more general problems can be reduced, usually by differentiation, to this form. We shall show that nonsingular detection problems of this form can always be interpreted as problems of the apparently more special "signal-in-noise" type, where the cross-covariance function of the signal and noise must be of a special "one-sided" form. Moreover, the LR for this equivalent problem can be written in the same form as that for known signals in white Gaussian noise, with the causal estimate of the signal process replacing the known signal. This single formula will be shown to be equivalent to a variety of other formulas, including all those previously known. The proofs are based on a resolvent identity and on a representation theorem for second-order processes, both of which have other applications. This paper also contains a discussion of the various stochastic integrals and infinite determinants that arise in Gaussian detection problems

Spectral Location of Rayleigh and "m" Fading Signals in White Gaussian Noise

The classical problem is considered of locating a fading sinusoidal signal known to be present in one of several frequency " cells," each of which contains additive white Gaussian noise. The signal fading is assumed to follow the popular Rayleigh distribution, but generalizations tions to non-Rayleigh fading are included in terms of the " in" distribution due to Nakagami. The channel observation time is allowed to be either predetermined or variable (corresponding, respectively, to " fixed sample size" and " sequential" reception), and the practically important situation of intermittent signal transmissions is also examined. Results are in the form of optimal and near-optimal receiver structures, and of measures of performance.

The efficiency of level tests in the detection of weak stochastic signals (Corresp.)

The efficiency of a class of test statistics for the detection of a stochastic signal in Gaussian and non-Gaussian white noise is examined. The tests are computationaUy simple, but not distribution free. In Gaussian noise these tests surpass two commonly used detection procedures.

A general likelihood-ratio formula for random signals in Gaussian noise

It is shown that the likelihood ratio for the detection of a random, not necessarily Gaussian, signal in additive white Gaussian noise has the same form as that for a known signal in white Gaussian noise. The role of the known signal is played by the casual least-squares estimate of the signal from the observations. However, the "correlation" integral has to be interpreted in a special sense as an Itô stochastic integral. It will be shown that the formula includes all known explicit formulas for signals in white Gaussian noise. However, and more important, the formula suggests an "estimator-correlator" philosophy for engineering approximation of the optimum receiver. Some extensions of the above result are also discussed, e.g., additive finite-variance, not necessarily Gaussian, noise plus a white Gaussian noise component. Purely colored Gaussian noise can be treated if whitening filters can be specified. The analog implementation of Itô integrals is briefly discussed. The proofs of the formulas are based on the concept of an innovation process, which has been useful in certain related problems of linear and nonlinear least-squares estimation, and on the concept of covariance factorization.

Energy detection of unknown deterministic signals

By using Shannon's sampling formula, the problem of the detection of a deterministic signal in white Gaussian noise, by means of an energy-measuring device, reduces to the consideration of the sum of the squares of statistically independent Gaussian variates. When the signal is absent, the decision statistic has a central chi-square distribution with the number of degrees of freedom equal to twice the time-bandwidth product of the input. When the signal is present, the decision statistic has a noncentral chi-square distribution with the same number of degrees of freedom and a noncentrality parameter λ equal to the ratio of signal energy to two-sided noise spectral density. Since the noncentral chi-square distribution has not been tabulated extensively enough for our purpose, an approximate form was used. This form replaces the noncentral chi-square with a modified chi-square whose degrees of freedom and threshold are determined by the noncentrality parameter and the previous degrees of freedom. Sets of receiver operating characteristic (ROC) curves are drawn for several time-bandwidth products, as well as an extended nomogram of the chi-square cumulative probability which can be used for rapid calculation of false alarm and detection probabilities. Related work in energy detection by J. I. Marcum and E. L Kaplan is discussed.

The likelihood ratio filter for the detection of Gaussian signals in white noise

This paper derives a solution to the equation \rho (u, s) = N_{0}h(s, u) + \int_{0}^{T} \rho (t, s)h(t, u) dt , for the case in which the kernel is the covariance function of a stationary random process whose spectral density is a rational function of frequency.

Crest and Extremal Statistics of a Square-Law-Derived Random Process

Theoretical analysis of the crest and extremal statistics of a square-law-derived random process is presented. The differences between the crest and the extremal statistics, particularly the bandwidth dependence of the latter, are illustrated for the Gaussian-derived case. The theory is applied to the statistical response of a uniform bar in tension. Particular attention is paid to the crest distributions which deviate from the integrated Rayleigh distributions in a predictable way at large amplitudes of vibration. Experimental distributions obtained from the strain response of a uniform bar forced in its fundamental mode by band-limited Gaussian white-noise signals are in excellent agreement with calculated values.

The detectability of fading radiotelegraph signals in noise

An ideal receiver for binary synchronous telegraphy is postulated; this receiver is defined as one that interprets each element of a received signal with the minimum probability of error. The relations between error liability and signal/noise ratio are determined for steady signals and for Rayleigh-fading signals in white Gaussian noise, using up to four diversity branches. The noisy-signal performance of a practical receiver can be described in terms of the amount by which it falls short of the ideal, and it is proposed that this measure of imperfection, expressed as an energy ratio, be known as the ‘demodulation factor’.The analysis leads to a mathematical specification for the ideal diversity receiver, and this provides a starting-point for the design of practical receivers. The outputs of diversity branches are combined, weighted according to signal energy, instead of the largest output being selected. Very deep fading proves to be of little importance. It is found that the use of a 7-unit error-detecting code in place of an unprotected 5-unit code is roughly equivalent to doubling the number of diversity branches. Cross-correlation between the mark and space signals in two-tone or frequency-shift systems is found to be insignificant under the conditions ordinarily encountered in long-distance radiotelegraphy.Although it is primarily concerned with high-frequency radiotelegraphy, the study may prove useful in other fields, including that of microwave pulse communication.