Sensitivity Analysis of Finite-Lag Receivers

This paper presents the sensitivity analysis of a class of receivers called finite-lag receivers, introduced by the authors in [1] through [3]. Since these receivers are based on the use of fixed-lag smoothing techniques, algorithms for the calculation of large-scale and small-scale sensitivities of fixed-lag smoothing are derived using a state augmentation approach. Steady-state analysis of these algorithms shows that an explicit relation can be obtained between sensitivity coefficients of fixed-lag smoothing and filtering. The specific case of sensitivity to variations in the measurement (channel) noise is considered as an example. These results are applied to study the sensitivity performance of the finite-lag receivers for analog communication. It is shown, for example, that finite-lag receivers for AM signals, besides being superior in performance [1]-[3], [11], in terms of output SNR or error variance, are also much less sensitive to the additive noise power level, compared to zero-lag receivers.     

Applications of Mininum Variance Reduced-State Estimators

This paper presents an algorithm for a class of suitably constrained reduced-order filters which minimize the variance of the estimated variables. The algorithm generates both the filter gain history and the true estimation error covariance. The algorithm provides a quantitative criterion which can be used to measure the performance of any reduced-order estimator. Both continuous and discrete estimators are considered. Several examples are treated including an application of the technique to a hybrid navigation system of high order.     

Detection of Known Signals in Nonstationary Noise

The basic design of a nonlinear, time-invariant filter is postulated for detecting signal pulses of known shape imbedded in nonstationary noise. The noise is a sample function of a Gaussian random process whose statistics are approximately constant during the length of a signal pulse. The parameters of the filter are optimized to maximize the output signal-to-noise ratio (SNR). The resulting nonlinear filter has the interesting property of approximating the performance of an adaptive filter in that it weights each frequency band of each input pulse by a factor that depends on the instantaneous noise power spectrum present at that time. The SNR at the output of the nonlinear filter is compared to that at the output of a matched filter. The relative performance of the nonlinear system is good when the signal pulses have large time-bandwidth products and the instantaneous noise power spectrum is colored in the signal pass band.     

An Alternate Approach to the Prediction of Polynomial Signals in Noise from Discrete Data

When the problem of predicting the past, present, or future value of a polynomial signal or any of its derivatives is considered, where the signal is in white Gaussian noise, the standard approach has been to minimize mean-square-error with constraints by use of Lagrange multipliers. In this paper an alternate approach is described, using results of Rao and Bhattacharyya from the statistical literature, which reduce the specified prediction problem to a simple one requiring no formal minimizations and no use of Lagrange multipliers. It further has the advantage of yielding the covariances between estimates of the polynomial and its derivates. Useful engineering formulas for smoothing and prediction are developed in the main part of the paper. These include both filter and covariance expressions. A tutorial discussion of the theory is given in two appendixes.     

Precise navigation using adaptive FIR filtering and time domainspectral estimation

This paper introduces a new low cost, short range, positioning system based on adaptive finite impulse response (FIR) filtering and time domain spectral estimation. The system can determine absolute positions with a high degree of accuracy and is well suited for real time navigation. The approach is based upon signal processing techniques and a priori knowledge of the system transfer function. The first step is to measure the phase response of the linear transfer function and then using a FIR filter the time response of the system can be determined. The FIR filter computes the time response by performing a deconvolution between the measured phase response, and the complex conjugate of the transfer function. By correlating the known input impulse response with the output of the FIR filter, an error term is generated. The time delay of the system is determined by adjusting the FIR filter coefficients to minimize the error term. Simulated analysis of the system indicates a worst case error of ±16 cm     

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.     

Optimum Linear Estimation of Stochastic Signals in the Presence of Multiplicative Noise

This paper considers optimum (MMSE) linear recursive estimation of stochastic signals in the presence of multiplicative noise in addition to measurement noise. Often problems associated with phenomena such as fading or reflection of the transmitted signal at an ionospheric layer, and also situations involving sampling, gating, or amplitude modulation, can be cast into such formulation. The different kinds of estimation problems treated include one-stage prediction, filtering, and smoothing. Algorithms are presented for discrete time as well as for continuous time estimation.     

Logarithmic Detection with Postdetection Filtering

An analysis is presented of the performance of a logarithmic detector or followed by a low-pass filter, under the assumption of a large ratio of IF bandwidth to postdetection bandwidth. This allows for the probability of detection and similar quantities to be computed in a straightforward manner. The mean and variance of the filter output are found by first deriving the autocorrelation function of the logarithmic detector output and then obtaining the power spectral ral density. A useful engineering approximation is made for the variance which is valid for all signal-to-noise ratios.     

Tracking targets using adaptive Kalman filtering

A simple algorithm for estimating the unknown process noise variance of an otherwise known linear plant, using a Kalman filter is suggested. The process noise variance estimator is essentially dead beat, using the difference between the expected prediction error variance, computed in the Kalman filter, and the measured prediction error variance. The estimate is used to adapt the Kalman filter. The use of the adaptive filter is demonstrated in a simulated example in which a wildly maneuvering target is tracked     

Optimum Signal and Filter Design in Underwater Acoustic Echo Ranging Systems

Optimization of the filter, the signal, and the signal and filter jointly are studied in the sonar environment under noise and reverberation limited conditions. The maximization of the receiver output signal-to-interference ratio is used as a performance criterion with unit energy constraint on both signal and filter. In the filter design problem, the optimum filter function is the solution of a linear integral equation. The kernel of the integral equation is a function of the target and medium scattering functions and the reverberation distribution. In the signal design problem, a similar type of integral equation is obtained as in the filter optimization problem. In the joint signal and filter design problem, it is shown that the optimum signal and filter functions are the solutions to a pair of linear integral equations with the largest (SIR)O. Several examples are investigated for different mediums and reverberation distributions with the finite matrix approximation method. An interative technique is used to compute the joint optimization of signal and filter.     

Control-Loop Noise in Adaptive Array Antennas

Adaptive array receiving antennas can be designed to sense the external noise field and to optimize the array illumination function. A substantial improvement in signal-to-noise ratio can be obtained with adaptive arrays when the external noise field is nonuniformly distributed in angle. The external noise process may be time varying and contain both discrete sources and continuously distributed sources. Two adaptive array implementations which maximize the signal-to-noise ratio are described in this paper. Expressions are derived for control-loop noise, i.e., the variance of the array element weights, and for the additional noise in the array output due to this element weight noise. It is shown that both the element weight noise and the array convergence rate are determined by the eigenvalues of the noise covariance matrix.     

Novel Nonlinear Filtering & Prediction Method for Maneuvering Target Tracking

A new nonlinear filtering and prediction (NFP) algorithm with input es?imation is proposed for maneuvering target tracking. In the proposed method, the acceleration level is determined by a decision process, where a least squares (LS) estimator plays a major role in detecting target maneuvering within a sliding window. We first illustrate that the optimal solution to minimize the mean squared error (MSE) must consider a trade-off between the bias and error variance. For the application of target tracking, we then derive the MSE of target positions in a closed form by using orthogonal space decompositions. Then we discuss the NFP estimator, and evaluate how well the approach potentially works in the case of a set of given system parameters. Comparing with the traditional unbiased minimum variance filter (UMVF), Kalman filter, and interactive multiple model (IMM) algorithms, numerical results show that the newly proposed NFP method performs comparable or better in all scenarios with significantly less computational requirements.     

The Performance of Analog Correlators for Several Integrator Functions

A measure of analog correlator performance is considered and interpreted reted as an output signl-to-noise ratio. This quantity is shown to depend on the fourth-order expectation of the input signals and the characteristics of the integrating system. Based on the output signal-to-noise ratio, figures of merit are established for correlators utilizing an ideal low-pass filter, an RC filter, and a finite time integrator. These figures of merit indicate to what degree the correlators reject noise components. For jointly Gaussian inputs, the variation of the output signal-to-noise ratio with the input signal-to-noise ratio is shown to be independent of the integrating system. Finally, a graphical comparison indicates the characteristics of the three systems for several different parameters. This comparison shows that the finite time integrator is superior to the other two systems considered.     

Steady state analysis for discrete tracking filters

An analysis of a discrete constant-gain α-β-γ tracking filter is presented. Steady-state tracking filter gains are determined without solving a discrete matrix Ricatti equation. Explicit closed-form gain solutions are found as a function of the roots to a cubic equation. Gain variations as a function of process noise, measurement noise, and sampling time are investigated     

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.     

Reconciling steady-state Kalman and alpha-beta filter design

The deterministic design of the alpha-beta filter and the stochastic design of its Kalman counterpart are placed on a common basis. The first step is to find the continuous-time filter architecture which transforms into the alpha-beta discrete filter via the method of impulse invariance. This yields relations between filter bandwidth and damping ratio and the coefficients, α and β. In the Kalman case, these same coefficients are related to a defined stochastic signal-to-noise ratio and to a defined normalized tracking error variance. These latter relations are obtained from a closed-form, unique, positive-definite solution to the matrix Riccati equation for the tracking error covariance. A nomograph is given that relates the stochastic and deterministic designs     

Some Matched Filter Configurations for Infrared Systems

An analysis of the output of three alternative matched filter configurations in an infrared scanning system model is presented. The sensor is corrupted by thermal noise, generation-recombination noise, photon noise, and modulation noise, the latter providing an extreme discoloration in the signal passband. Expressions for the signal voltage density spectrum, signal pulse shape, noise power spectrum, and average noise power at the matched filter output are derived where the integral evaluations attendant to these derivations do not appear elsewhere in the literature. The paper also provides graphical displays of the signal-to-noise power ratio at the filter output versus various system parameters, noise power spectrum out of the matched filter versus ?, and the signal pulse shape out of the filter versus time. Also included are discussions of practically realizable approximations to the matched filters and curve fitting techniques for the signal pulse shape function.     

An optimal reduced-order stochastic observer-estimator

An optimal reduced-order observer-estimator (filter) is developed which can provide a full-dimensional vector of state estimates for systems where the dimension of the measurement vector is smaller than that of the state vector and none of the measurements are noise free. The reduced-order filter consists of two subfilters each of which provides a subset of the optimal estimate. A two-step L-K transformation is employed to minimize the estimate error variance of each subfilter. The optimal reduced-order filter developed is computationally efficient     

相关系数平稳序列的估计与识别

本文提出了通过观测序列确定不可直接测得的相关系数平稳序列模型参数的方法,详细讨论了相关系数AR(p),MA(q)和ARMA(p,q)序列状态方程的参数估计,为非平稳序列的滤波、预测和平滑提供了依据。文中还给出了观测系统参数的识别方法,与传统方法相比,本文方法考虑了测量噪声的均值和方差随时间变化情况,具有更高的识别精度。