Signal selection in communication and radar systems

A criterion for selecting a finite set of transmitter signals for a continuous communication channel is proposed. The "optimum" signal sets using this criterion are selected to maximize the minimum divergence between hypothesis pairs being tested at the receiver. The resulting signal sets have the property that the error probability using these signals is less than the error probability for any other choice of signals for some {em a priori} message statistics. The signal selection procedure may be applied without a knowledge of the {em a priori} message statistics and does not require an evaluation of error probabilities. Four examples of signal selection are included to illustrate the procedure.     

Likelihood ratios for sequential hypothesis testing on Markov sequences

A variety of likelihood ratios are derived for detecting Gauss-Markov and finite-state Markov sequences in additive Gaussian noise. The Bayesian recursions appropriate to related filtering problems are exploited, together with "known-form" likelihood ratios, to obtain the desired results. In the derivation of a discrete-time Gauss-Markov likelihood ratio, a "pure" causal estimator-correlator structure is sought and a "locally stable" state estimator is encountered that is of some interest in its own right. The likelihood ratio is "pure" in the sense that the locally stable estimator is used in precisely the same manner as the stored replica is used in known-form signal detection problems to form the likelihood ratio. Consequently, the likelihood ratio is devoid of the extra data-dependent term that arises whenever one uses least squares state estimators to form the likelihood ratio statistic. The locally stable estimator equalizes, within a constant related to the {em a priori} and {em a posteriori} filtering error covariances, the {em a priori} and {em a posteriori} filtering densities. Heuristically, the estimator is a compromise between the one-step predictor and the filtered estimator of a discrete-time Kalman filter. When the observation noise covariance is unknown, a generalization of the so-called unknown level problem, then a Wishart prior is assigned to the innovations covariance and an integral representation is obtained for the desired likelihood ratio. The representation suggests a parallel structure for approximating the likelihood ratio when the observation noise covariance is unknown. Finally, the likelihood ratio for detecting finite-state Markov sequences is derived to illustrate that in general no "pure" estimator-correlator structure can exist when the state-space is finite.     

The correlation function of Gaussian noise passed through nonlinear devices

This paper is concerned with the output autocorrelation functionR^{y}of Gaussian noise passed through a nonlinear device. An attempt is made to investigate in a systematic way the changes inR^{y}when certain mathematical manipulations are performed on some given device whose correlation function is known. These manipulations are the "elementary combinations and transformations" used in the theory of Fourier integrals, such as addition, differentiation, integration, shifting, etc. To each of these, the corresponding law governingR^{y}is established. The same laws are shown to hold for the envelope of signal plus noise for narrow-band noise with spectrum symmetric about signal frequency. Throughout the text and in the Appendix it is shown how the results can be used to establish unknown correlation function quickly with main emphasis on power-law devicesy = x^{m}withmeither an integer or half integer. Some interesting recurrence formulas are given. A second-order differential equation is derived which serves as an alternative means for calculatingR^{y}.     

On the distribution of positive-definite Gaussian quadratic forms

Quadratic signal processing is used in detection and estimation of random signals. To describe the performance of quadratic signal processing, the probability distribution of the output of the processor is needed. Only positive-definite Gaussian quadratic forms are considered. The quadratic form is diagonalized in terms of independent Gaussian variables and its mean, moment-generating function, and cumulants are computed; conditions are given for the quadratic form to bechi^{2}distributed and distributed like a sum of independent random variables having a Gamma distribution. A new method is proposed to approximate its probability distribution using an expansion in Laguerre polynomials for the central case and in generalizedchi^{2}distributions in the noncentral case. The series coefficients and bounds on truncation error are evaluated. Some applications in average power and power spectrum estimation and in detection illustrate our method.     

Optimal estimation for discrete time jump processes

Optimum estimates of nonobservable random variables or random processes which influence the rate functions of a discrete time jump process (D) are derived. The approach used is based on the {em a posteriori} probability of a nonobservable event expressed in terms of the {em a priori} probability of that event and of the sample function probability of the DTJP. Thus a general representation is obtained for optimum estimates, and recursive equations are derived for minimum mean-squared error (MMSE) estimates. In general, MMSE estimates are nonlinear functions of the observations. The problem of estimating the rate of a DTJP when the rate is a random variable with a beta probability density function and the jump amplitudes are binomially distributed is considered. It is shown that the MMSE estimates ale linear. The class of beta density functions is rather rich and explains why there are insignificant differences between optimum unconstrained and linear MMSE estimates in a variety of problems.     

An information-theoretic proof of Burg's maximum entropy spectrum

It is known that the maximum entropy stationary Gaussian stochastic process, subject to a finite number of autocorrelation constraints, is the Gauss-Markov process of appropriate order. The associated spectrum is Burg's maximum entropy spectral density. We pose a somewhat broader entropy maximization problem, in which stationarity and normality are not assumed, and shift the burden of proof from the previous focus on the calculus of variations and time series techniques to a string of information-theoretic inequalities. This results in an elementary proof of greater generality.     

On the Binary DPSK Communication Systems in Correlated Gaussian Noise

Using the recently developed probability density function we obtain "directly" the error rate expressions for the binary differential phase-shift keyed (DPSK) systems when the noise values at the sampling instants in adjacent time slots are statistically dependent. Two cases are considered: One corresponding to equal SNR at each of the two sampling instants, and the other to unequal SNR's. The consideration of the former case, together with the assumption of unequal a priori symbol probabilities P0and P1results in an error rate expressionP_{E} =1/2[1 + (P_{0} - P_{1})rho] exp (-h^{2})where ρ is the noise correlation coefficient, and h²is the SNR. This expression shows clearly why PEis independent of noise correlation when the source symbols are equi-probable provided intersymbol interference is assumed absent. We then obtain the error probability expression in terms of unequal SNR's (at the two sampling instants) and the correlation coefficient ρ. Since intersymbol interference in a binary DPSK system gives rise to unequal SNR's, this expression provides a useful formula for estimating the system performance under such circumstances.     

Stationarizable random processes

The familiar notion of inducing stationarity into a cyclostationary process by random translation is extended through characterization of the class of all second-order continuous-parameter processes (with autocorrelation functions that possess a generalized Fourier transform) that are {em stationarizable} in the wide sense by random translation. This class includes the nested set of proper subclasses: {em almost cyclostationary} processes, {em quasi-cyclostationary} processes, and {em cyclostationary} processes. The random translations that induce stationarity are also characterized. The concept of stationarizability is extended to the concept of asymptotic stationarizability, and the class of {em asymptotically stationarizable} processes is characterized. These characterizations are employed to derive characterizations of optimum linear and nonlinear time-invariant filters for nonstationary processes. Relative to optimum time-varying filters, these time-invariant filters offer advantages of implementational simplicity and computational efficiency, but at the expense of increased filtering error which in some applications is quite modest. The uses of a random translation for inducing stationarity-of-order-n, for increasing the degree of local stationarity, and for inducing stationarity into discrete-parameter processes are briefly described.     

Correlation Entering New Fields with Real-Time Signal Analysis

Correlation analysis is a convenient technique for deyrtmining the spectral characteristics of a signal or the similarity of two different signals. One point of a correlation function is the long-term average of the product of two functions of time. The complete function is generated when the delay between the two time functions is varied. For example, if one voltage V1(t) and another voltage V2(t ? r), where r represents a finite and variable delay, are continuously multiplied together and the product fed into a low-pass filter, then the filter's output closely approximates the true mathematical correlation function. If V2; is identical to V1; in every respect except for the delay r, the result is the autocorrelation function. If V1; and V2; are totally different functions, then the result is the cross-correlation function. The outputs in both cases are functions of the delay time r. Mathematically for autocorrelation begin{equation*}C_{11}(r) = lim_{Trightarrowinfty}frac{1}{2T}int^T_{-T}V_1(t)V_1(t - r) dtend{equation*} for cross correlation begin{equation*} C_{12}(r) = lim_{Trightarrowinfty}frac{1}{2T}int^T_{-T}V_1(t)V_2(t - r) dt end{equation*} An instrument, therefore, that does this integrating process will show whether correlation exists between two signals and, if so, when maximum correlation takes place. In practice, the averaging process indicated in the above equations is performed only for a time longer than the longest period in signals f1(t) and f2(t). Autocorrelation is useful for the detection of an unknown periodic signal in the presence of noise or to measure some particular band of signal or noise frequencies.     

A First-Order Markov Model for Understanding Delta Modulation Noise Spectra

A first-order Markov process is used to model the sequence of quantization noise samples in delta modulation. An autocorrelation parameterCin the Markov model controls the shape of the noise spectrum, and asCdecreases from 1 to 0 and then to -1, the spectrum changes from a low-pass to a flat, and then to a high-pass characteristic. One can also use the Markov model to predict the so-called out-of-band noise rejection that is obtained when delta modulation is performed with an oversampled input, and the resulting quantization noise is lowpass filtered to the input band. The noise rejectionGis a function ofCas well as an oversampling factorFand an interesting asymptotic result is thatG=frac{1-C}{1+C} dot FifF gg frac{1+C}{1-C} dot frac{pi}{2}. Delta modulation literature has noted the importance of the specialGvalues,Fand2F. These correspond to autocorrelation values of 0 and -1/3.     

On the detection of known binary signals in Gaussian noise of exponential covariance

The optimum test statistic for the detection of binary sure signals in stationary Gaussian noise takes a particularly simple form, that of a correlation integral, when the solution, denoted byq(t), of a given integral equation is well behaved(L_{2}). For the case of a rational noise spectrum, a solution of the integral equation can always be obtained if delta functions are admitted. However, it cannot be argued that the test statistic obtained by formally correlating the receiver input with aq(t)which is notL_{2}is optimum. In this paper, a rigorous derivation of the optimum test statistic for the case of exponentially correlated Gaussian noiseR(tau) = sigma^{2} e^{-alpha|tau|}is obtained. It is proved that for the correlation integral solution to yield the optimum test statistic whenq(t)is notL_{2}, it is sufficient that the binary signals have continuous third derivatives. Consideration is then given to the case where a, the bandwidth parameter of the exponentially correlated noise, is described statistically. The test statistic which is optimum in the Neyman-Pearson sense is formulated. Except for the fact that the receiver employsalpha_{infty}(which in general depends on the observed sample function) in place ofalpha, the operations of the optimum detector are unchanged by the uncertainty inalpha. It is then shown that almost all sample functions can be used to yield a perfect estimate ofalpha. Using this estimate ofalpha, a test statistic equivalent to the Neyman-Pearson statistic is obtained.     

Performance of anM-ary orthogonal communication system using stationary stochastic signals

This paper considers the performance of a communication system which transmits forTseconds the real part of a sample function of one ofMstationary complex Gaussian processes whose spectral densities are all frequency translations of the functionS_{xi (f). At the receiver white Gaussian noise of one-sided densityN_{0}is added. The center frequencies of the processes are assumed to be sufficiently separated that theMcovariance functions are orthogonal overT. Exponently tight bounds are obtained for the error probability of the maximum likelihood receiver. It is shown that the error probability approaches zero exponentially withTfor all ratesR = (ln M)/Tup toC= int_{-infty}^{infty} [S_{xi (f)/N_{0}] df - int_{- infty}^{infty} ln [1 + S_{xi}(f)/N_{0}] dfwhich is shown to be the channel capacity. Similar results are obtained for the case of stochastic signals with specular components.     

Complexity-based induction systems: Comparisons and convergence theorems

In 1964 the author proposed as an explication of {em a priori} probability the probability measure induced on output strings by a universal Turing machine with unidirectional output tape and a randomly coded unidirectional input tape. Levin has shown that iftilde{P}'_{M}(x)is an unnormalized form of this measure, andP(x)is any computable probability measure on strings,x, thentilde{P}'_{M}geqCP(x)whereCis a constant independent ofx. The corresponding result for the normalized form of this measure,P'_{M}, is directly derivable from Willis' probability measures on nonuniversal machines. If the conditional probabilities ofP'_{M}are used to approximate those ofP, then the expected value of the total squared error in these conditional probabilities is bounded by-(1/2) ln C. With this error criterion, and when used as the basis of a universal gambling scheme,P'_{M}is superior to Cover's measurebast. WhenHastequiv -log_{2} P'_{M}is used to define the entropy of a rmite sequence, the equationHast(x,y)= Hast(x)+H^{ast}_{x}(y)holds exactly, in contrast to Chaitin's entropy definition, which has a nonvanishing error term in this equation.     

Error performance of differentially coherent detection of binary DPSK data transmission on the hard-limiting satellite channel

The error performance of differentially coherent detection of a binary differential phase-shift keying (DPSK) system operating over a hard-limiting satellite channel is derived. The main objective is to show the extent of error rate degradation of a DPSK system when a power imbalance exists between the two symbol pulses that are used in a bit decision interval. Consideration is also given to the DPSK error rate performance for the special case of {em uncorrelated} uplink and {em correlated} downlink noises at the sampling instants in adjacent time slots. Error probabilities are given as functions of uplink signal-to-noise ratio (SNR) and downlink SNR with different levels of SNR imbalance and different downlink SNR and uplink SNR as parameters, respectively. Our numerical results show that 1) as long as the symbols are equiprobable, the error probability is not dependent upon the downlink noise correlation, regardless of whether there is a power imbalance; 2) error performance is definitely affected by the power imbalance for all cases of symbol distributions; and 3) the error probability does depend upon downlink noise correlation for all levels of power imbalance if the symbol probabilities are not equal.     

Gaussian arbitrarily varying channels

The {em arbitrarily varying channel} (AVC) can be interpreted as a model of a channel jammed by an intelligent and unpredictable adversary. We investigate the asymptotic reliability of optimal random block codes on Gaussian arbitrarily varying channels (GAVC's). A GAVC is a discrete-time memoryless Gaussian channel with input power constraintP_{T}and noise powerN_{e}, which is further corrupted by an additive "jamming signal." The statistics of this signal are unknown and may be arbitrary, except that they are subject to a power constraintP_{J}. We distinguish between two types of power constraints: {em peak} and {em average.} For peak constraints on the input power and the jamming power we show that the GAVC has a random coding capacity. For the remaining cases in which either the transmitter or the jammer or both are subject to average power constraints, no capacities exist and onlylambda-capacities are found. The asymptotic error probability suffered by optimal random codes in these cases is determined. Our results suggest that if the jammer is subject only to an average power constraint, reliable communication is impossible at any positive code rate.     

An adaptive digital filter for improving the signal-to-noise ratio of a signal with additive noise (Corresp.)

An adaptive digital filtering scheme is developed to deal with the problem of improving the signal-to-noise ratio of a signal corrupted by noise when only general {em a priori} assumptions regarding the signal and the noise are possible. Specifically, the noise is assumed to be white zero-mean and uncorrelated; while the signal is considered to be band-limited, possibly with slowly varying spectrum. The proposed adaptive digital filtering scheme is based upon a class of variable wave digital filters. Adaptation of the digital filter multipliers is accomplished through the use of an identification procedure based on an adaptive spectral estimation method.     

Estimation of a quadrature cross- correlation detector by analog, polarity-coincidence, and relay methods (Corresp.)

The variance of the output of a cross-correlation detector, which is called a quadrature cross-correlation detector, is estimated. In this type of detector two zero-mean Gaussian quadrature processesalpha(t)andbeta(t)of a complex process(alpha(t) -j beta(t))are cross correlated. This cross-correlation functionR_{A} (tau)is estimated when neither of the two processes is distorted (the analog method), when both processes are distorted by a signum function before being cross correlated (the polarity coincidence method), and when one of the two processes is distorted by either a signum function or by a "comparator logic" function (the relay method). These quadrature cross-correlation detectors then are compared on the basis of output signal-to-noise ratio (s/n) and the clipping and relay losses are computed for two test quadrature processes of an Edgeworth-expansion-approximated power spectrum. SinceR_{A} (0)is zero, the four corresponding differential estimators, such as(R_{A} (tau) - R_{A} (0))are also estimated and are compared on the basis of s/n. For these differential estimators, the clipping and relay losses are computed for the two test processes. In all cases the exact expressions for the s/n are derived as a function oftau. Some applications of these correlation detectors are outlined. The mathematical techniques employed here are thought to have potential usefulness for related problems in statistical communication theory and signal processing.     

A new approach to the problem of wave fluctuations in localized smoothly varying turbulence

In the past, smoothly varying turbulence has been studied by changing the structure constant to the functionC_{n}^{2}(bar{r}). The purpose of this paper is to show that this approach is insufficient, and that a random process developed by Silverman can be used to describe the wave fluctuations in localized smoothly varying turbulence. The localized turbulence is characterized by a correlation function which is a product of a function of the average coordinate and a function of the difference coordinate. The corresponding spectrum is also given by a product of a function of the difference wavenumber and a function of the average wavenumber. They are related to each other through two Fourier transform pairs. Making use of the preceding representations, the fluctuations of a wave propagating through such a turbulence can be given either by the integrals with respect to the two wavenumbers or by a convolution integral of the structure constantC_{n}^{2}(bar{r}) and a function involving the outer scale of the turbulenceL_{0}. It is shown that for a plane wave case, if the distanceLis within (L_{0}^{2}/lambda), then the usual formula given by Tatarski is valid. But if the distance is betweenL_{0}^{2}/lambdaand(bL_{0})/lambdawherebis the total transverse size of the turbulence, the variance of the wave is nearly constant, and ifL gg (bL_{0})/lambda, the variance decays asL^{-2}. Similar conclusions are shown for a spherical wave case. Some examples are shown illustrating the effectiveness of this method.     

Information and quantum measurement

Given a finite number of quantum states with {em a priori} probabilities, the positive operator-valued measure that maximizes the Shannon mutual information is investigated. The group covariant case is examined in detail.     

周期为4v(几乎)平衡理想二进制序列构造研究

具有理想自相关特性的序列在无线通信、雷达以及密码学中具有重要的作用.因此为了扩展更多可应用于通信系统的理想序列,该文基于2阶分圆类和中国剩余定理,提出3类新的周期为T=4v(v是奇素数)平衡或几乎平衡理想二进制序列构造方法.构造所得序列的周期自相关函数满足:当v≡3(mod4)时,序列的周期自相关函数旁瓣值取值集合为{...