Phase Analysis in Multiple-Sensor Receivers with High Signal-to-Noise Ratio

The ideal phase detector characteristic is analyzed for estimating the phase difference between two stochastic input signals. This can essentially be described as a correlation process formed by multiplying the two input signals and extracting the phase. High signal-to-noise ratio conditions are assumed to linearize the system with respect to the noise. The effects of the nonlinearity on the signal are handled in terms of a series expansion and by using low-pass filtering on the receiver output. The mean square error of the system is calculated for some typical parameters.     

Modeling and analysis for the GPS pseudo-range observable

In this paper, a digital system for the Global Positioning System (GPS) pseudo-range observable is modeled and analyzed theoretically. The observable is measured in a GPS receiver by accurately tracking the pseudorandom noise (PRN) code phase of the input GPS signal using a digital energy detector and a digital delay lock loop (DDLL). The following issues are presented: (1) mathematical modeling of the digital PRN code acquisition and tracking system, (2) the closed-form expression derivation for the detection and false-alarm probabilities of the acquisition process and for the variance of code phase tracking error, and (3) the linear and nonlinear performance analysis of the DDLL for optimizing the receiver structures and parameters with tradeoff between the tracking errors due to receiver dynamics and due to input noise     

Effect of Monopulse Signal Thresholding on Tracking Multiple Targets

The monopulse angle of arrival measured from two targets may wander far beyond the angular separation of the targets. In order to remove large errors, angle of arrival measurements are passed through a threshold detector and the measurements which do not satisfy the threshold are rejected. Thresholding has the tendency to move the mean indicated angle of arrival away from the stronger target and toward the power centroid of the targets, which may be undesirable in some applications. Expressions for the probability density function, the mean, and the variance of the in-phase angle of arrival are developed in this paper as a function of the threshold setting. Background noise and interfering signals are neglected in the analysis.     

Adjacent Channel Interference in a Binary Bandpass Communication System

The effect of adjacent channel interference on the probability of error in a binary bandpass communication system with an integrating and dumping detector is investigated. Narrowband filters are assumed in the receiver of the main signal and transmitters of both main and interfering signals. Plots of the probability of error as a function of signal to noise ratio in the main channel or as a function of carrier frequency difference between the main and interfering signals are presented, assuming that the filters are of the Butterworth type. These figures are helpful in the selection of minimal frequency spacing of adjacent channels.     

Noncoherent Detection of Split-Phase FSK by Multiple Predetection Filtering

When the cumulative drift in the center frequency of a binary split-phase FSK signal exceeds the peak deviation of the signal, a conventional noncoherent receiver (i.e., one provided with only two IF filters) may be unable to achieve the probability of error per bit which the designer desires. This limitation may be overcome if the receiver is provided with a bank of more than two contiguous filters (each followed by an envelope detector) tospan the total IF band the instantaneous IF signal might occupy. It is shown that the probability of error per bit for such a receiver is a function of 1) the ratio F of peak frequency deviation to peak frequency drift, 2) the number M of IF filter/detectors, and 3) the signal-to-noise ratio ? in the output of the filter containing the signal. It is further shown thatfor a given value of F an increase in M reduces the amount of transmitter power the communication system designer must provide to yield a given probability of error per bit.     

Analysis of a Signal Processor for an Antenna Array

A signal processor that provides ratio-squared predetection combining, has been investigated for application in an adaptive antenna array. The analysis and experimental data presented here pertain to the signal processing gain when the antenna array is illuminated by a coherent signal source and a partially coherent noise source. For a noise source which is coherent, the processing gain depends on relative strength of the signal and noise, relative directions of arrival, and the usual "array factor." The array exhibits capturing effects much as in an FM receiver. The effective antenna pattern is a superposition of two beams of different magnitudes, one directed to the signal source and the other to the noise source. When the noise is partially coherent, the behavior of the signal processor is quite complex. Analytical prediction and experimental simulation measurement on a four-channel system indicate that the partially coherent noise may be regarded as the source of an incoherent noise component plus a coherent noise component with the magnitude of the latter determined by the coherence coefficient for the noise source.     

Angle Estimation with Adaptive Arrays in External Noise Fields

A method of estimating the angle of arrival of a signal at an array of sensors in an external noise environment is outlined. The development is based on a maximum likelihood estimator and leads naturally to adaptive sum and difference beams which null the external noise sources. An algorithm for estimating angle of arrival, based on the outputs of adaptively distorted sum and differnce beams, is shown to perform well in the presence of sidelobe and/or main beam interference.     

Thermal Noise in Amplitude Comparison Monopulse Systems

In this paper we consider the problem of estimation of angle of arrival in an Amplitude Comparison Monopulse antenna arrangement with the explicit inclusion of internally generated thermal, i.e., receiver, noise as an interference to the desired measurement. A pulsed type radar is assumed, and an ideal (i.e., point) radar target is postulated. This latter restriction is made so that consideration of the effects of target scintillation, glint, or other external random phenomena can be excluded from our treatment of the problem. In this context, a maximum likelihood analysis is made to determine the form of the estimate of angle of arrival, and the probability density function (pdf) of this quantity is computed. The form of the estimate is found to be a ratio of Gaussian variables quite like that used in conventional monopulse signal processing. The pdf obtained for the estimate is believed to be new, and it serves to emphasize the bias and indeterminate variance effects associated with this type of nonlinear signal processing. Some useful approximations to the pdf are discussed, and a unit of precision for the estimate is defined.     

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.     

Spatial Correlation of Reverberation Noise in Active Sonar

A linear array of hydrophones is considered for detecting a signal echo from a stationary target in the presence of reverberation. The structure of the optimum (likelihood ratio) detector is compared with that of a beamformer-matched filter detector. The conditions causing an increase in the spatial noise correlation between two hydrophones are the conditions under which the optimum spatial detector performs significantly better than the beamforming detector. A study of the space-time correlation function of reverberation shows that 1) a decrease in scatterer angular spread (or a narrowing of the receiver directivity pattern) tends to increase the spatial correlation, 2) if the scatterer Doppler spread is much less than the signal carrier frequency and if the angular spread is uniform, it is still possible to get a high correlation if the intersensor distance is much smaller than the carrier wavelength. These conditions indicate situations where optimum techniques may be worthwhile.     

Waveform Design for Multistatic Radar Detection

We derive the optimal Neyman-Pearson (NP) detector and its performance, and then present a methodology for the design of the transmit signal for a multistatic radar receiver. The detector assumes a Swerling I extended target model as well as signal-dependent noise, i.e., clutter. It is shown that the NP detection performance does not immediately lead to an obvious signal design criterion so that as an alternative, a divergence criterion is proposed for signal design. A simple method for maximizing the divergence, termed the maximum marginal allocation algorithm, is presented and is guaranteed to find the global maximum. The overall approach is a generalization of previous work that determined the optimal detector and transmit signal for a monostatic radar.     

Monopulse Radar Angle Tracking Accuracy with Sum Channel Limiting

The variance of angle tracking error is found for an amplitude-comparison form of monopulse radar when the sum channel contains a limiter prior to the angle error detector. The error expression is valid for any shape of transmitted pulse and any duration of range tracking gate but does assume matched filters in signal processing channels. The procedures used are rigorous and an example of results is worked out for the special case of a rectangular transmitted pulse envelope. It is shown, for rectangular pulses, that achievable angle tracking error variance with sum channel limiting is not more than 2.22 dB larger than the theoretical minimum for any processor and not more than 1.29 dB larger than a similar signal processor that uses a "linear" angle error detector. Results apply for large single-pulse signal-to-noise ratio.     

Detection of Target Multiplicity Using Monopulse Quadrature Angle

The feasibility of using the indicated quadrature angle of arrival of a monopulse radar to discriminate a single target from multiple targets, separated in angle within a radar resolution cell, is investigated. The analysis is performed for steady (fixed) and Rayleigh fluctuating targets which cover a broad range of target characteristics. In both cases, the interfering signals due to noise and clutter in the sum and difference monopulse channels are assumed to be independent, zero-mean Gaussian processes. Detection and false alarm probabilities are evaluated analytically and the receiver operating characteristics are obtained for both fixed and fluctuating target cases. It is shown that multiple targets can be discriminated from a single target condition by integrating the indicated monopulse quadrature angle of arrival from several independent pulses. It is also shown that the probability of detecting multiple targets increases as the fluctuation in the target radar cross section decreases, approaching the fixed amplitude case in the limit.     

Performance of a Two-Sample Mann-Whitney Nonparametric Detector in a Radar Application

The detection of signals in an unknown, typically non-Gaussian noise environment, while attempting to maintain a constant false-alarm rate, is a common problem in radar and sonar. The raw receiver data is commonly processed initially by a bank of frequency filters. The further processing of the outputs from the filter bank by a two-sample Mann-Whitney detector is considered. When the noise statistics in all filters are identical, the Mann-Whitney detector is distribution free, i. e., the false-alarm probability may be prescribed in advance regardless of the precise form of the noise statistics. The primary purpose of this paper is to demonstrate the potential advantage of nonparametric detectors over conventional detectors. The signal detection performance of the Mann-Whitney detector is compared to that of an ordinary linear envelope detector plus integrator in the presence of Gaussian and several hypothetical forms of non-Gaussian noise. This comparison is made for both uniform and nonuniform distributions of noise power across the filter bank. Besides providing a much more constant false-alarm rate than the conventional detector, the Mann-Whitney detector's signal detection performance is found also to be much less sensitive to the form of the noise statistics. In one case, its detection sensitivity is found to be 11 dB better than that of the conventional detector. Even when the noise power density is made moderately nonuniform across the filter bank, the detection performance of the Mann-Whitney detector is found not to be significantly affected.     

Energy Detection of a Random Process in Colored Gaussian Noise

A likelihood receiver for a Gaussian random signal process in colored Gaussian noise is realized with a quadratic form of a finite-duration sample of the input process. Such a receiver may be called a "filtered energy detector." The output statistic is compared with a threshold and if the threshold is exceeded, a signal is said to be present. False alarm and detection probabilities may be estimated if tabulated distributions can be fitted to the actual distributions of the test statistic which are unknown. Gamma distributions were fitted to the conditional probability densities of the output statistic by equating means and variances, formulas for which are derived assuming a large observation interval. A numerical example is given for the case in which the noise and signal processes have spectral densities of the same shape or are flat. The optimum filter turns out to be a band-limited noise whitener. The factors governing false alarm and detection probabilities are the filter bandwidth, the sample duration, and the signal level compared to the noise. Two sets of receiver operating characteristic curves are presented to complete the example.     

Numerical Results on the Convergence of Relative Efficiencies

This paper considers the detection of a known constant signal in an additive non-Gaussian noise under the assumptions of discrete time and statistically independent noise samples. The objective is to determine how large sample size must be before the easily computed asymptotic relative efficiency becomes a valid measure of performance. The exact small-sample error probabilities are calculated for a Neyman-Pearson optimal nonlinear detector consisting of a zeromemory nonlinearity followed by summation and threshold comparison. "Large-tailed" noise having a double exponential distribution is used as an example. The exact distribution of the test statistics for a linear detector and for the Neyman-Pearson optimal detector are calculated. Then the relative efficiency of the Neyman-Pearson optimal detector, as compared to a linear detector, is computed in order to study the rate of approach of the relative efficiency to its asymptotic value.     

Detection in Laplace Noise

The discrete time detection of a known constant signal in white stationary Laplace noise is considered. Exact expressions describing the performance of both the Neyman-Pearson optimal detector and the suboptimal linear detector are presented. Also, graphs of the receiver operating characteristics are given. The actual performance of the Neyman-Pearson optimal detector is compared to that predicted by a Gaussian approximation to the distribution of the test statistic.     

Phase-Locked Loop Behavior near Threshold

The phase-locked loop behavior is analyzed following the quasilinearization Booton's method. When the loop is locked on an unmodulated input signal with a static phase error, the phase detector nonlinearity produces an interaction between the static phase error and the voltage-controlled-oscillator (VCO) noise phase fluctuations. Formulas allowing one to compute the static phase error increase and the VCO phase variance increase are derived. When the input signal is phase modulated, there is an interaction between the static phase error, the VCO noise phase fluctuations, and the input signal phase modulation. Formulas are obtained that allow one to compute the loop loss of performances (static phase error increase and VCO phase variance increase) and the coherent phase demodulator output signal-to-noise ratio decrease. Finally, a slight modification to Booton's procedure is proposed, leading to results in better agreement with experimental data.     

Detection Performance of the Cell Averaging LOG/CFAR Receiver

The cell averaging LOG/CFAR receiver is a special implementation of a constant-false-alarm-rate (CFAR) receiver in which the noise level estimate is derived from a set of contiguous time samples of the output of a logarithmic (LOG) detector as obtained from a tapped delay line. This CFAR receiver is capable of operating over a larger dynamic range of noise levels than a conventional cell averaging CFAR receiver, but with somewhat poorer detectability. The performance in stationary Gaussian noise of the cell averaging LOG/CFAR receiver with no post-detection integration is determined in this paper. For a small number of reference noise samples, results were obtained by a Monte Carlo simulation using the technique of importance sampling. For a large number of reference noise samples, a second moment analysis gave the desired results. Both these results can be summarized in the following simple formula, NLOG = 1.65NLIN - 0.65, which relates the number of reference samples required by each of the two receivers for equivalent performance. Thus, for the cell averaging LOG/CFAR receiver to give the same detection performance as the conventional cell averaging CFAR receiver, the number of reference noise samples has to be increased by up to 65 percent.     

Performance analysis of GPS carrier phase observable

The accuracy analysis of Global Positioning System (GPS) carrier phase observable measured by a digital GPS receiver is presented. A digital phase-locked loop (DPLL) is modeled to extract the carrier phase of the received signal after a pseudorandom noise (PRN) code synchronization system despreads the received PRN coded signal. Based on phase noise characteristics of the input signal, the following performance of the first, second, and third-order DPLLs is analyzed mathematically: (1) loop stability and transient process; (2) steady-state probability density function (pdf), mean and variance of phase tracking error; (3) carrier phase acquisition performance; and (4) mean time to the first cycle-slipping. The theoretical analysis is verified by Monte Carlo computer simulations. The analysis of the dependency of the phase input noise and receiver design parameters provides with an important reference in designing the carrier phase synchronization system for high accuracy GPS positioning