Detection of stochastic signals in the frequency domain

The optimum detector for a random signal, the estimator-correlator, is difficult to implement. If the power spectral density (PSD) of a continuous time signal is known, a locally optimum detector is available. It maximizes the deflection ratio (DR), a measure of the detector output signal-to-noise ratio (SNR). A discrete version of this detector is developed here, called the discrete-MDRD, which takes a weighted sum of the spectral components of the signal data as the detection statistic. Its derivation is applicable to nonwhite noise samples as well. A comparison of this new detector against three other common types, through their DR values and simulation results, reveals that the discrete-MDRD is near optimal at low SNRs. When the PSD of a signal is not known, a common test statistic is the peak of the PSD of the data. To reduce spectral variations, the PSD estimator first divides the data sequence into several segments and then forms the averaged PSD estimate. The segment length affects the DR values; the length that maximizes the DR is approximately the reciprocal of the signal bandwidth. Thus for unknown signal PSD, a detector that approaches the maximum DR is realizable from just the knowledge of the signal bandwidth, which is normally available. Examples and simulation results are provided to illustrate the properties and performance of the new detector     

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.     

Comments on, `Improved multilevel quantization for detection ofnarrowband signals' by C.W. Helstrom

It is shown that optimum quantization levels may be obtained by using C.W. Helstrom's approach of the above named work (see ibid., vol.AES-24, p.141-147, Mar. 1988) but maximizing exact detection probabilities instead of approximations to it if the moment generating function of the test statistic can be expressed in rational form. Adaptive levels may be obtained for the quantizers by cell averaging, leading to constant false alarm rate (CFAR) detectors     

Noncoherent Detection of Sinusoidal Signals in Sinusoidal Clutter

Probability density expressions associated with the noncoherent detection of a sinusoidal signal have been obtained. The signal is assumed to be imbedded in sinusoidal clutter at the same frequency and narrow-band Gaussian noise. The density expressions are shown to be a function of the signal-to-noise power ratio and the clutter-to-noise power ratio. The expressions have been numerically evaluated for a number of conditions, and the results under each reception hypothesis are presented graphically. Under large-sample conditions, the probability density for a multisample test statistic is shown to be Gaussian, and the probability of detection expression is written such that commonly available tabulated data can be utilized to determine the probabilities.     

Data quantization effects in CFAR signal detection

In this paper, we investigate data quantization effects in constant false alarm rate (CFAR) signal detection. Exponential distribution for the input data and uniform quantization are assumed for the CFAR detector analysis. Such assumptions are valid in the case of radar for a Swerling I target in Gaussian clutter plus noise and a receiver with analog square-law detection followed by analog-to-digital (A/D) conversion. False alarm and detection probabilities of the cell averaging (CA) and order statistic (OS) CFAR detectors operating on quantized observations are analytically determined. In homogeneous backgrounds with 15 dB clutter power fluctuations, we show analytically that a 12-bit uniform quantizer is sufficient to achieve false alarm rate invariance. Detector performance characteristics in nonhomogeneous backgrounds, due to regions of clutter power transitions and multiple interfering targets, are also presented and detailed comparisons are given     

Calculating the performance of linear and square-law detectors

A method is presented for calculating the performance of linear and square-law detectors in detection schemes that employ noncoherent integration. The method consists of transforming the coherent characteristic function, which is usually easy to obtain to a noncoherent moment generating function describing the test statistic of a linear or square-law detector. The method provides a single mathematical framework for many signal models (both classical and new) and can be implemented using standard numerical routines. Although the method is not always optimum in terms of computing speed for specific classical models, its common approach for all signal models makes it very efficient in term of learning and implementation times. Classical results as well as results for an extended set of target models consisting of an arbitrary number of constant amplitude random phase returns are presented to demonstrate the technique. It is shown for the signal parameters considered that the performance difference between the linear and square-law detectors is relatively insignificant     

Optimum array detector for a weak signal in unknown noise

We study the design of constant false-alarm rate (CFAR) tests for detecting a rank-one signal in the presence of background Gaussian noise with unknown spatial covariance. We look at invariant tests, i.e., those tests whose performance is independent of the nuisance parameters, like the background noise covariance. Such tests are shown to have the desirable CFAR property. We characterize the class of all such tests by showing that any invariant decision statistic can be written as a function of two basic statistics which are in fact the adaptive matched filter (AMF) statistic and Kelly's generalized likelihood ratio statistic. Further, we establish an optimum test in the limit of low signal-to-noise ratio (SNR), the locally most powerful invariant (LMPI) test. We also derive the bound for the probability of detection of any invariant detector, at a fixed false-alarm rate, and compare the LMPI and the published detectors (Kelly and AMF) to it     

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.     

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.     

Leading Edge Estimation Errors

The leading edge estimator (LEE) of a pulse signal is defined as the instant at which a filtered version of the received noisy signal passes a preset threshold. A rigorous analysis for a rectangular pulse model of the signal results in an exact probability density function for the LEE, valid within the time interval of the leading edge of the filtered pulse. Possible occurrence of the threshold crossing outside of this interval is considered to be an anomalous estimate, since it leads to a gross error in comparison with the regular cases. It is found that the density function of the LEE error is asymmetrical and therefore biased, that the probability PA of anomalous estimation increases with the filter bandwidth, thus setting a well definable limit to the latter and that, for prespecified PA, the minimum bias and variance are proportional, respectively, to R-1 and R-2, minima being obtained by allowing for the largest bandwidth compatible with PA. On the other hand, for given bandwidth the variance decreases only as R-1. Here R is the signal-to-noise energy ratio. Results are presented in form of parameterized graphs.     

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.     

Asymptotically Optimum Sample Size for Quickest Detection

A method is presented for selecting the asymptotically optimum sample size M for detecting a sudden change in the statistics of an observed process. The test statistic is assumed to be a sum of M consecutive values of some single sample detector and the optimization criterion is to minimize the mean time to detection MD for a given mean time between false alarms MF. It is shown that for large MF and MD the solution can be expressed as a function of the single variable ?MF? (or alternatively ?MD?) where ? is a measure of the signal-to-noise ratio (SNR).     

Tracking the Frequency Translation of a Sum of Orthogonal Sinusoids

The maximum likelihood approach is used to derive a method for estimating and tracking the frequency translation of a signal consisting of a sum of orthogonal sinusoids corrupted by additive white noise. The likelihood function is reduced to an equivalent statistic expressed in terms of the squared magnitude of the finite Fourier transform of the received signal. A function that generates an error signal for a frequency translation tracking loop is derived, and a method of generating the error signal using the discrete Fourier transform (DFT) of the received signal weighted by a linear ramp is suggested. Two noise-free examples are presented.     

Weak Signal Detection in Non-Gaussian Noise of Unknown Level

An adaptive threshold detector to test for the presence of a weak signal in additive non-Gaussian noise of unknown level is discussed. The detector consists of a locally optimum detector, a noise level estimator, and a decision device. The detection threshold is made adaptive according to the information provided by the noise level estimator in order to keep a fixed false-alarm probability. Asymptotic performance characteristics are obtained indicating relationships among the basic system parameters such as the reference noise sample size and the underlying noise statistics. It is shown that, as the reference noise sample size is made sufficiently large, the adaptive threshold detector attains the performance of a corresponding locally optimum detector for detecting the weak signal were the noise level known.     

Calculating Detection Probabilities for Adaptive Thresholds

In calculating detection probabiities for radar and sonar systems it is usually assumed that the threshold required to yield a certain probability of false alarm is known. This is often not the case for real systems and therefore the threshold must be estimated using some measure related to the test statistic. This paper presents a calculation technique that handles estimated (adaptive) thresholds in a general framework that can be applied easily to many detection problems. False alarm and detection probabilities are calculated from the characteristic function of the noise or signal plus noise variate and the characteristic function of the threshold estimate. To illustrate the method the detection performance of overlapped discrete Fourier transforms (DFTs) is calculated for a narrowband Gaussian target signal.     

Approximate Maximum Likelihood Algorithm for Moving Source Localization Using TDOA and FDOA Measurements

 A closed-form approximate maximum likelihood (AML) algorithm for estimating the position and velocity of a moving source is proposed by utilizing the time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements of a signal received at a number of receivers. The maximum likelihood (ML) technique is a powerful tool to solve this problem. But a direct approach that uses the ML estimator to solve the localization problem is exhaustive search in the solution space, and it is very computationally expensive, and prohibits real-time processing. On the basis of ML function, a closed-form approximate solution to the ML equations can be obtained, which can allow real-time implementation as well as global convergence. Simulation results show that the proposed estimator achieves better performance than the two-step weighted least squares (WLS) approach, which makes it possible to attain the Cram閞-Rao lower bound (CRLB) at a sufficiently high noise level before the threshold effect occurs.     

三角波信号沿线性度的统计分析评价方法

介绍了三角波信号沿线性度参数的一种统计分析评价方法 ;在均匀分布的条件下 ,通过使用波形测量手段和直方图统计分析方法 ,对三角波信号的幅度分布直方图、沿线性度等指标进行了评价 ,详细讨论了方法的实现过程以及有关技术问题 ,并对相应参数指标进行了误差分析 ;实验验证结果表明了该方法的有效性和实用性 ,该方法可应用到三角波信号源的性能指标评价中。     

Sensorless synchronous reluctance drive with standstill starting

This paper presents a novel sensorless synchronous reluctance drive system. Based on the dynamic model of the synchronous reluctance motor (SRM), a new rotor position estimator is proposed. This estimator is only related to the rate change of the stator currents. It is not related to the parameters, speed, voltage, and external load of the motor. As a result, the estimator is simple and robust. Based on the proposed estimator, a sensorless synchronous reluctance drive has been implemented. This drive system can start from standstill and accelerate to a rated speed; the traditional open-loop starting method, therefore, is unnecessary. A digital signal processor, TMS-320-C30, is used to implement the estimating and control algorithms. Experimental results show that the proposed system works well. The adjustable speed range can be from 1 r/min to 1800 r/min. Moreover, by adding the position-loop controller the proposed system can perform as a position control drive as well. Several experimental results validate the theoretical analysis.     

Coarse quantization for data compression in coherent location systems

When emitter location systems measure time-difference-of-arrival (TDOA) and differential Doppler (DD) by coherently cross-correlating the signal pairs, data compression techniques are needed to facilitate data transfer of one of the signals to the receiving site of the other signal. Two block-adaptive quantization schemes are analyzed here to determine their impact on the signal-to-noise ratio (SNR) of the quantized signal as well as on the post-correlation SNR. Comparisons are made between two approaches: quantization of the real/imaginary (R/I) components or the magnitude/phase (M/P) components. For the M/P approach, a rule is derived for optimally allocating the bits between the magnitude and phase. The M/P approach provides better post-quantization/precorrelation SNR for most signals; however, when the SNR of the signal not being quantized is small, the post-correlation SNR can be largely unaffected by the quantization. In that case, there is little difference between R/I and M/P, even under the most favorable scenario for M/P.     

Optimum FFT-based frequency acquisition with application toCOSPAS-SARSAT

In the case of a single sinusoid or multiple well-separated sinusoids, a coarse estimator consisting of a windowed Fourier transform followed by a fine estimator which is an interpolator is a good approximation to an optimal frequency acquisition and measurement algorithm. The design tradeoffs are described. It is shown that for the fine-frequency estimator a good method is to fit a Gaussian function to the fast-Fourier-transform (FFT) peak and its two neighbors. This method achieves a frequency standard deviation and a bias in the order of only a few percent of a bin. In the case of short-time stationarity, for a moderate number of averages and for an adaptive threshold detector, only between 0.5 and 1 dB is lost when averaging is traded off for FFT length, in contrast to the asymptotic result of 1.5 dB. The COSPAS-SARSAT satellite system for emergency detection and localization is used to illustrate the concepts. The algorithm is analyzed theoretically, and good agreement is found with test results