Detection Probabilities for Log-Normally Distributed Signals

The amplitude and power of a large family of radio signals are observed to have log-normal probability density functions. Among these are signals propagated through random inhomogeneous media, a notable example being low frequency atmospheric radio noise. Of greater importance are certain radar targets that have been observed to have essentially log-normal density functions. Both ships and space vehicles may fall into this category. Curves of probability of detection vs. signal-to-noise ratio for the case of log-normal signals in Gaussian noise have been computed and are presented in this paper. The curves apply for square-law detection with varying degrees of postdetection linear integration. Both fully correlated and completely uncorrelated fluctuating signals are considered. It is shown that for log-normal signal distributions having large variances, the probability of detection differs significantly from that obtained using curves based on an assumed Rayleigh signal distribution.     

Radiometric detection of spread-spectrum signals in noise ofuncertain power

The standard analysis of the radiometric detectability of a spread-spectrum signal assumes a background of stationary, white Gaussian noise whose power spectral density can be measured very accurately. This assumption yields a fairly high probability of interception, even for signals of short duration. By explicitly considering the effect of uncertain knowledge of the noise power density, it is demonstrated that detection of these signals by a wideband radiometer can be considerably more difficult in practice than is indicated by the standard result. Worst-case performance bounds are provided as a function of input signal-to-noise ratio (SNR), time-bandwidth (TW) product and peak-to-peak noise uncertainty. The results are illustrated graphically for a number of situations of interest. It is also shown that asymptotically, as the TW product becomes large, the SNR required for detection becomes a function of noise uncertainty only and is independent of the detection parameters and the observation interval     

Data-Adaptive Detection of a Weak Signal

Motivated by a form of the likelihood-ratio-tesf statistic for detection of a rank-one Gaussian signal in colored Gaussian noise, we apply our earlier technique for estimation of a low-rank signal to the problem of estimating and subtracting the waveform of a strong sinusoidal interference prior to detection of a weak sinusoidal signal. We consider the difficult case in which samples of data are taken over a short interval of time or space and the frequencies of the sinusoidal signal and sinusoidal interference are more closely spaced than the reciprocal of the extent of the aperture. The method can be applied to cases of nonsinusoidal and/or random signals and interference. The most important assumption is that when the samples of the interference are arranged in matrix form the matrix is approximately of low rank in the sense that, with high probability, the interference-only matrix can be well approximated by a matrix of low rank.     

Adaptive Array Behavior with Sinusoidal Envelope Modulated Interference

The behavior of a LMS (least mean square) adaptive array with modulated interference is described. An interference signal with sinusoidal, double-sideband, suppressed-carrier modulation is assumed. It is shown that such interference causes the array to modulate the desired signal envelope but not its phase. The amount of the desired signal modulation is determined as a function of signal arrival angles and powers and the modulation frequency of the interference. Such interference also causes the array output signal-to-interference-plus-noise ratio (SINR) to vary with time. However, it is shown that when the desired signal is a digital communication signal, the averaged bit error probability is essentially the same as for continuous wave (CW) interference.     

Interference Effects of Pseudo-Random Frequency-Hopping Signals

When a pseudo-random frequency-hopping signal is intercepted by a conventional receiver operating within the same frequency band, the interfering signal has the form of a pulse-amplitude modulated signal. Each pulse amplitude is dependent upon the hopping frequency and the selectivity characteristic of the victim receiver. The probability density function for the interfering pulse amplitude prior to demodulation is determined when the probability density function for the hopping frequency is uniform and the victim-receiver characteristic is 1) ideal flat bandpass, 2) single tuned, and 3) Gaussian shaped. It is shown that the average interfering pulse amplitude and interference power decrease as the frequency-hopping bandwidth increases with respect to the victim-receiver bandwidth. Fast Fourier transform computer techniques are used to obtain the probability density function of the interference amplitude in a Gaussian receiver when several (from 2 to 10) pseudo-random frequency-hopping systems are simultaneously using the same frequency band. The probability that the interference exceeds a prescribed threshold value is computed from the derived probability density functions. This probability may be used in signal-to-interference ratio calculations, to describe the capture effect, or to compute the expected number of clicks produced in an FM discriminator.     

Envelope Threshold Detection in a Class of Non-Gaussian Interference

This paper considers the detection of a sinusoidal or chirp signal imbedded in wideband FM interference (as might be generated by some types of active jamming), such that after pulse compression or other integration, the interference can be approximated by a sum of sinusoids of independent phase. The detection probability in such non-Gaussian noise is compared to that for Gaussian noise, with the Gaussian result approached, as required, in the limit that the number of sinusoids in the sum increases without bound. For detection using a comparison of the envelope with a threshold which yields a given false-alarm probability (CFAR detection), the detection probability is improved over the case of Gaussian noise, so that the usual approach basing the design on Gaussian noise would be conservative. Using a threshold determined from the envelope mean, the FM interference yields a lower false-alarm probability than for Gaussian noise, with detection probability only slightly degraded.     

Dynamics of the Phase-Locked Loop without a Limiter

The general (nth order) phase-locked loop is analyzed, of which the amplitude is not constant. The input carrier signal is amplitude-modulated by wide-band stationary Gaussian noise, and the signal, superposed with the additive white stationary Gaussian noise, enters the nonlimited phase-locked loop. Under the above assumptions the loop can be shown to constitute an n-dimensional vector Markov process, so that the process satisfies the n-dimensional Fokker-Plank equation. The probability density function depends on the effective loop signal-to-noise ratio and the effective modulation power.     

Sonar Array Detection of Gaussian Signals in Gaussian Noise of Unknown Power

A statistical test is postulated for detecting, with an M-element hydrophone array, a Gaussian signal in spatially independent Gaussian noise of unknown power. The test is an extension of the uniformly-most-powerful (UMP) unbiased test for a two-element array. The output signal-to-noise ratio of the test is calculated and, for a large number of independent space-time samples, is shown to be no better than a mean-level detector (MLD). Receiver operating characteristic curves (ROC) for the MLD are computed and compared to the ROC curves for the optimum (Bayes) parametric detector. The input signal-to-noise power ratios required to provide a detection probability of 0.5 differ by less than 0.2 dB for a fifty-element array with wide variation in false-alarm probability and time-bandwidth product. This result suggests that both the extended bivariate UMP unbiased test and the MLD perform close to the unknown UMP unbiased test for independence of a multivariate Gaussian distribution.     

On Signal and Noise Level Estimation in a Coherent PCM Channel

Joint maximum likelihood estimators are presented for the signal amplitude and noise power density in a coherent PCM channel with white Gaussian noise and a correlation receiver. The estimates are based upon the correlation coefficient outputs of the receiver. From these estimators, an estimator for the quantity (received signal energy)/bit/,(noise power)/(unit bandwidth) upon which the error probabilities depend, is derived. This estimator is shown to be useful as 1) a point estimator for the signal-to-noise ratio for the higher values of this ratio (about 4 dB or greater), and 2) an easily calculated statistic upon which to base data acceptance or rejection criteria. The acceptance or rejection levels are obtained by the use of confidence interval curves in conjunction with word error probability data.     

Hard-Limiter Output Autocorrelation Function: Gaussian & Sinusoidal Input

An expression is derived for the autocorrelation function of the output of a hard limiter whose input is stationary Gaussian noise with zero mean plus independent random-phase sinusoidal signal. The output spectrum may then be evaluated. This spectrum is extremely useful in understanding the properties of a filter-limit-filter-detect signal processor whose signal input is an actual sinusoid, or when a sinusoid is used as a test signal.     

The Effect of Hard Limiting an Angle-Modulated Signal Plus Noise

The effect of hard limiting an angle-modulated signal plus narrow-band Gaussian noise is analyzed. Several examples are considered?sinusoidal angle modulation, Gaussian angle modulation, and biphase angle modulation. The general conclusion is that when a zonal band-pass filter is used, which rejects dc and second harmonics, an angle-modulated signal plus Gaussian noise provides the same output signal-to-noise ratio as shown by Davenport for a CW signal plus Gaussian noise. However, when a narrow bandpass filter is used, which has a bandwidth approximately equal to the input angle-modulated signal, an angle-modulated signal plus Gaussian noise has a better output signal-to-noise ratio than a CW signal plus Gaussian noise.     

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.     

A comparison of two adaptive detection schemes

Two schemes for adaptive detection are compared: Kelly's generalized likelihood ratio test (GLRT) and the mean level adaptive detector (MLAD). Detection performance, P_D, is predicted for the two schemes under the assumptions that the input noises are zero-mean complex Gaussian random variables that are temporally independent but spatially correlated; and the amplitude of the desired signal is Rayleigh distributed. P_D is computed as a function of the false alarm probability, the number of input channels, the number of independent samples per channel, and the matched filtered output signal-to-noise (S/N) power ratio. In this analysis the GLRT is shown to have better detection performance than the MLAD. The difference in detection performance increases as one uses fewer input samples. However, the required number of samples necessary to have only a 3 dB detection loss for both detection schemes is approximately the same. This is significant since for the present, the MLAD is considerably less complex to implement than the GLRT     

Synthesis Techniques for a Class of SSB-AM-PM Signals

This paper develops synthesis techniques for a particular type of single-sideband sinusoidal carrier which is phase modulated by a subcarrier. Mathematical expressions for signal efficiency, sensitivity of design to parameter variation, and ratio of peak to average power are derived and incorporated in a computer program. Given the desired power ratios for modulated signal components, the program solves for the corresponding modulation parameters and evaluates signal efficiency, design sensitivity, and peak to average power ratio. A sample signal design is presented for clarity.     

Matched subspace CFAR detection of hovering helicopters

A constant false alarm rate (CFAR) strategy for detecting a Gaussian distributed random signal against correlated non-Gaussian clutter is developed. The proposed algorithm is based on Scharf's matched subspace detector (MSD) and has the CFAR property with respect to the clutter amplitude probability density function (apdf), provided that the clutter distribution belongs to the compound-Gaussian family and the clutter covariance matrix is known to within a scale factor. Analytical expressions of false alarm and detection probabilities are derived. An application to the problem of detecting hovering helicopters against vegetated ground clutter is reported     

Energy Detection of Unknown Deterministic Signals in the Presence of Jamming

Urkowitz [1]has discussed the detection of a deterministic signal of unknown structure in the presence of flat, band-limited, Gaussian noise of known power density. That analysis is extended here to the case where jamming or other conditions preclude knowledge of the noise power density. The chi-square statistic of Urkowitz is replaced with Fisher's variance-ratio statistic, using a separate set of noise samples to estimate the unknown noise power density. Curves are given to show the additional degradation of perform over that due to ignorance of the signal structure, caused by ignorance of the noise power density.     

Analysis of an adaptive CFAR detector in non-Gaussian interference

Coherent signal detection in non-Gaussian interference is presently of interest in adaptive array applications. Conventional array detection algorithms inherently model the interference with a multivariate Gaussian random vector. However, non-Gaussian interference models are also under investigation for applications where the Gaussian assumption may not be appropriate. We analyze the performance of an adaptive array receiver for signal detection in interference modeled with a non-Gaussian distribution referred to as a spherically invariant random vector (SIRV). We first motivate this interference model with results from radar clutter measurements collected in the Mountain Top Program. Then we develop analytical expressions for the probability of false alarm and the probability of detection for the adaptive array receiver. Our analysis shows that the receiver has constant false alarm rate (CFAR) performance with respect to all the interference parameters. Some illustrative examples are included that compare the detection performance of this CFAR receiver with a receiver that has prior knowledge of the interference parameters     

An Upper Bound on Detection Probability for Fluctuating Signals

In the theory of signal detectability, the signal-to-noise ratio (SNR), defined as the quotient of the average received signal energy and the spectral density of the white Gaussian noise, is a fundamental parameter. For a signal which is exactly known, or known except for a random phase, this ratio uniquely defines the detection performance which can be achieved with a matched filter receiver. However, when the signal amplitude is a random parameter, the detection performance is changed and must be determined from the probability density function (pdf) of the amplitude. Relative to the case of a constant signal amplitude, such signal amplitude fluctuation usually degrades performance when a high probability of detection (Pd) is required, but improves performance at low values of Pd; the corresponding change in the required SNR is the so-called signal fluctuation loss Lf. Thus, since Lf in some cases represents an improvement in performance for low values of Pd, a question of at least theoretical interest is: how large might this improvement be, when the class of all signal amplitude pdf's is considered. The solution, presented here, results in a lower bound on the signal fluctuation loss Lf as a function of Pd, or equivalently an upper bound on Pd as a function of SNR. The corresponding most favorable pdf was determined using the Lagrange multiplier technique and results of a numerical maximization are included to provide insight into the general properties of the solution.     

Spatially distributed target detection in non-Gaussian clutter

Two detection schemes for the detection of a spatially distributed, Doppler-shifted target in non-Gaussian clutter are developed. The non-Gaussian clutter is modeled as a spherically invariant random vector (SIRV) distribution. For the first detector, called the non-scatterer density dependent generalized likelihood ratio test (NSDD-GLRT), the detector takes the form of a sum of logarithms of identical functions of data from each individual range cell. It is shown under the clutter only hypothesis, that the detection statistic has the chi-square distribution so that the detector threshold is easily calculated for a given probability of false alarm P_F. The detection probability P_D is shown to be only a function of the signal-to-clutter power ratio (S/C)_opt of the matched filter, the number of pulses N, the number of target range resolution cells J, the spikiness of the clutter determined by a parameter of an assumed underlying mixing distribution, and P_F. For representative examples, it is shown that as N, J, or the clutter spikiness increases, detection performance improves. A second detector is developed which incorporates a priori knowledge of the spatial scatterer density. This detector is called the scatterer density dependent GLRT (SDD-GLRT) and is shown for a representative case to improve significantly the detection performance of a sparsely distributed target relative to the performance of the NSDD-GLRT and to be robust for a moderate mismatch of the expected number of scatterers. For both the NSDD-GLRT and SDD-GLRT, the detectors have the constant false-alarm rate (CFAR) property that P_F is independent of the underlying mixing distribution of the clutter, the clutter covariance matrix, and the steering vector of the desired signal     

Coherent detection in Laplace noise

The discrete-time detection of a time-varying, additive signal in independent Laplace noise is considered. Previous efforts in this area have been restricted to the constant signal, and identically distributed noise case. Theoretical (closed form) expressions for the false alarm and detection probabilities are developed for both the Neyman-Pearson optimal detector and the classical matched filter detector. Comparisons between the two detectors are made which illustrate the effects of signal-to-noise power ratio and sample size for certain false alarm and detection probability constraints. In view of the fact that the optimal Laplace detector is not UMP, we also investigate the effect of signal amplitude mismatch