Graphical Harmonic Analysis

With the Fourier transform begin{equation*}F(omega)=int^{+infty}_{-infty}f(t)text{exp}({-j}omega{t})dtend{equation*} it is not difficult to evaluate a good approximation of the envelope of F(?) simply by maximizing the integral in the second member of the preceding equation. The purpose of this document is to use the maximization technique for the graphical harmonic analysis of complex waveforms.     

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.     

Calculation of Magnetic Fields due to Line Currents

A modified form of the Biot-Savart law is derived which is of particular use in the calculation and/or graphical determination of the magnetic fields due to line currents. In a linear conductor carrying a uniform current, I, the magnetic intensity at an observation point is given by begin{equation*}H = frac{2I}{R}{sin}left(frac{alpha}{2}right)end{equation*} where ? is the viewing angle to the conductor extremities and R is the effective distance measured from the observation point to the conductor along the viewing angle bisector. Examples of the application of this formula are given. These illustrate the usefulness of the formula in simplifying EMC analyses involving circuit and system emanations.     

Synthesis of RC Bandpass Filters

For the first time a method of designing RC bandpass filters is presented. The method consists of two steps. The first step is a scheme to locate the necessary poles and zeros that are RC realizable to produce certain bandpass characteristics. The second step is the synthesis of RC networks to produce these poles and zeros. In the first step, the conformal transformati begin{equation*}s(z) = {left(frac{sn^{2}(z,k) - sn^{2}(alpha{K,k})}{sn^{2}(alpha{K,k})[1 - k^{2}sn^{2}(alpha{K,k})sn^{2}(z,k)]}right)}^{1/2}end{equation*} is used to map the complex frequency s plane into a rectangle in the z plane such that the passband becomes one side, and a part of the negative real axis becomes the opposite side of the rectangle. In the z plane, if poles are located along certain portions of the border and zeros in the interior of the rectangle, certain passband and stopband behavior can be achieved. Among the useful characteristics obtainable by this scheme, the following are three outstanding examples: 1) characteristics that are equal-ripple in the passband and monotonic in the stopband; 2) characteristics that are equal-ripple in the passband and have a number of transmission zeros in the stopband; and 3) characteristics with a maximum gain at the band center and monotonic elsewhere. The steepness of attenuation outside the passband can be altered by a change in the numbers of zeros at the origin and infinity.     

Spectral Components Analysis: A Bridge between Spectral Observations and Agrometeorological Crop Models

Spectral observations have been acknowledged to indicate general plant conditions over large areas but have yet to be exploited in connection with agrometeorological crop models. One reason is that it is not yet appreciated how periodic spectral observations of row-cropped and natural plant canopies, as expressed by vegetation indices (VI), can provide information on important crop model parameters, such as leaf area index (LAI) and absorbed photosynthetically active radiation (APAR). Two experiments were conducted under AgRISTARS sponsorship, one with cotton and one with spring wheat, specifically to determine the relationships for each term in the " spectral components analysis" identity begin{equation*} LAI/VI times APAR/LAI = APAR/VI.end{equation*} LAI and APAR could, indeed, be well estimated from vegetation indices such as normalized difference(ND) and perpendicular vegetation index (PVI)?apparently because of the close relation between the VI and amount of photosynthetically active tissue in the canopy. APAR and VI measurements are similarly affected by solar zenith angle (SZA), and LAI can be divided by cos SZA at the time of the VI and APAR measurements to achieve correspondence. APAR, ND, and PVI plotted against LAI all asymptote to limiting values in the same way yield does as LAI exceeds 5, further linking canopy development to yield capability. In summary, the spectral components analysis results presented add credence to the information conveyed by spectral canopy observations about plant development and yield, and establish a bridge between remote observations and agrometeorological crop modeling through the variables of mutual concern, LAI, biomass, and yield.     

An Accurate Representation of the Complete Electromagnetic Field in the Vicinity of a Base-Driven Cylindrical Monopole

An accurate numerical representation of the electromagnetic field in the near zone of a cylindrical monopole oriented perpendicular to a highly conducting ground screen and driven at its base is needed for use in calibrating field strength measuring equipment. The fields H?(p,z), Ep(p,z), and Ez(p,z) are given by different integrals. The integrands are formed by multiplying the current distribution Iz(z) by certain derivations of K(p, z ? z') = e-j?R/R taken by hand, where ? is the radian wavenumber and begin{equation*}R = sqrt{(z - z^prime)^2 + p^2}.end{equation*}. Alternatively, the integrands may be constructed by multiplying K(p,z - z') by certain derivatives of Iz(z). The current Iz(z) is obtained by solving an integral equation with feedpoint correction employing a linear zoning technique. Generally speaking for a tubular monopole, the current may be obtained to any desired accuracy, and, of course, it is bounded at the driving point. The integrands are then formed, and the resulting integral expressions for the fields are evaluated using a digital computer. By this means it is felt that accurate numerical values of the fields H?(p,z), Ep(p,z), and Ez(p,z) in the vicinity of the monopole are found, excluding observation points near the feedpoint and end of the radiator. A brief discussion of the methodology employed in programming the Chang theory is presented.     

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.     

An algorithm for solving discrete-time Wiener-Hopf equations based upon Euclid's algorithm

An algorithm for solving a discrete-time Wiener-Hopf equation is presented based upon Euclid's algorithm. The discrete-time Wiener-Hopf equation is a system of linear inhomogeneous equations with a given Toeplitz matrix M, a given vector b, and an unknown vectorlambdasuch thatMlambda = b. The algorithm is able to find a solution of the discrete-time Wiener-Hopf equation for any type of Toeplitz matrices except for the all-zero matrix, while the Levinson algorithm and the Trench algorithm are not available when at least one of the principal submatrices of the Toeplitz matrixMis singular. The algorithm gives a solution, if one exists, even when the Toeplitz matrixMis singular, while the Brent-Gustavson-Yun algorithm only states that the Toeplitz matrixMis singular. The algorithm requiresO(t^{2})arithmetic operations fortunknowns, in the sense that the number of multiplications or divisions is directly proportional tot^{2}, like the Levinson and Trench algorithms. Furthermore, a faster algorithm is also presented based upon the half greatest common divisor algorithm, and hence it requiresO(t log^{2} t)arithmetic operations, like the Brent-Gustavson-Yun algorithm.     

On the fractional weight of distinct binaryn-tuples (Corresp.)

It is shown that the fractionpof ones in theMnpositions ofMdistinct binaryn-tuples satisfies the inequality begin{equation} h(p) geq (l/n) log_2 M end{equation} whereh(p) = - p log_2 p - (1 - p) log_2 (1 - p)is the binary entropy function. This inequality, which simplifies the derivation of the distance property of the Justesen codes, is proved using an elegant information-theoretic argument due to Kriz.     

An efficient solution of the congruencex^2 + ky^2 = mpmod{n}

The equation of the title arose in the proposed signature scheme of Ong-Schnorr-Shamir. The large integersn, kandmare given and we are asked to find any solutionx, y. It was believed that this task was of similar difficulty to that of factoring the modulusn;we show that, on the contrary, a solution can easily be found ifkandmare relatively prime ton. Under the assumption of the generalized Riemann hypothesis, a solution can be found by a probabilistic algorithm inO(log n)^{2}|loglog|k||)arithmetical steps onO(log n)-bit integers. The algorithm can be extended to solve the equationX^{2} + KY^{2} = M pmod{n}for quadratic integersK, M in {bf Z}[sqrt{d}]and to solve in integers the equationx^{3} + ky_{3} + k^{2}z^{3} - 3kxyz = m pmod{n}.     

Narrow-band systems and Gaussianity

The approach to Gaussianity of the outputy(t)of a narrow-band systemh(t)is investigated. It is assumed that the inputx(t)is ana-dependent process, in the sense that the random variablesx(t)andx(t + u)are independent foru > a. WithF(y)andG(y)the distribution functions ofy(t)and of a suitable normal process, a realistic boundBon the differenceF(y) -- G(y)is determined, and it is shown thatB rightarrow 0as the bandwidthomega_oof the system tends to zero. In the special case of the shot noise process begin{equation} y(t) = sum_i h(t - t_i) end{equation} it is shown that begin{equation} mid F(y) - G(y) mid < (omega_o/lambda) frac{1}{2} end{equation} wherelambda_iis the average density of the Poisson pointst_i.     

Bayes estimation with asymmetrical cost functions (Corresp.)

It is known that under certain restrictions on the posterior density and assigned cost function, the Bayes estimate of a random parameter is the conditional mean. The restrictions on the cost function are that it must be a symmetric convex upward function of the difference between the parameter and the estimate. In this correspondence, asymmetrical cost functions of the following form are examined: begin{equation} C(a, hat{a})= begin{cases} f_1(a- hat{a}),& a geq hat{a} \ f_2(hat{a}- a),& a < hat{a} end{cases} end{equation} wheref_1(cdot), f_2(cdot)are both twice-differentiable convex upward positive functions on[0, infty]that intersect the origin. It is shown that for posterior densities satisfying a certain symmetry condition, the biased Bayes estimate is a generalized median. Furthermore, for linear polynomial functionsf_1(cdot), f_2(cdot), the unbiased Bayes estimate is shown to be the conditional mean.     

Some cross correlation properties for distorted signals

For a nondecreasing distortion characteristicphi(cdot)and a given signalx(cdot), the "cross correlation" function defined byR_{phi} (tau) triangleq int_{-infty}^{infty} phi[x(t)]x(t - tau) dtis shown to satisfy the inequalityR_{phi}(tau) leq R_{phi}(0), for alltau, generalizing an earlier result of Richardson that requiredphi(cdot)to be continuous and strictly increasing. The methods of the paper also show that, under weak conditions, begin{equation} R_{phi,psi}(tau) triangleq int_{-infty}^{infty} phi[x(t)]psi[x(t - tau)] dt leq R_{phi,psi}(0) end{equation} whenpsiis strictly increasing andphiis nondecreasing. In the case of hounded signals (e.g., periodic functions), the appropriate cross correlation function is begin{equation} mathcal{R}_{phi,psi}(tau} triangleq lim_{T rightarrow infty} (2T)^{-l} int_{-T}^T phi[x(t)]psi[x(t - tau)] dt. end{equation} For this case it is shown thatmathcal{R}_{phi,psi} (tau) leq mathcal{R}_{phi,psi}(0)for any nondecreasing (or nonincreasing) distortion functionsphiandpsi. The result is then applied to generalize an inequality on correlation functions for periodic signals due to Prosser. Noise signals are treated and inequalities of a similar nature are obtained for ensemble-average cross correlation functions under suitable hypotheses on the statistical properties of the noise. Inequalities of this type are the basis of a well-known method of estimating the unknown time delay of an observed signal. The extension to nondecreasing discontinuous distortion functions allows the use of hard limiting or quantization to facilitate the cross correlation calculation.     

Asymptotics and Error Rate Bounds for M-ary DPSK

The symbol error probabilityP_{E}(M)forM-ary DPSK is shown to be bounded in terms of a recent asymptotic approximationP_{asym}(M)by the inequalitiesP_{asym}(M) < P_{E}(M) < 1.03P_{asym}(M);M geq 4, E_{b}/N_{0} geq 1whereE_{b}/N_{0}is the bit energy-to-noise spectral density ratio. Aside from the wide range of validity and the closeness of the lower and upper bounds, this result is striking in light of the often held view that such asymptotic approximations are primarily of value only in the limitE_{b}/N_{0} rightarrow infty; thus, one of the goals of this note is to demonstrate that asymptotic methods can lead to extremely good error rate approximations in lieu of the more traditional and more widely used bounding techniques. The results are also noted to be applicable in other similar situations which commonly occur.     

On the capacity of certain additive non-Gaussian channels

A model of an additive non-Gaussian noise channel with generalized average input energy constraint is considered. The asymptotic channel capacityC_{zeta}(S), for large signal-to-noise ratioS, is found under certain conditions on the entropyH_{ tilde{ zeta}}( zeta)of the measure induced in function space by the noise processzeta, relative to the measure induced bytilde{zeta}, where is a Gaussian process with the same covariance as that ofzeta. IfH_{ tilde{zeta}}( zeta) < inftyand the channel input signal is of dimensionM< infty, thenC_{ zeta}(S)= frac{1}{2}M ln(1 + S/M) + Q_{zeta}( M ) + {o}(1), where0 leq Q_{ zeta}( M ) leq H_{ tilde{ zeta}}( zeta). If the channel input signal is of infinite dimension andH_{ tilde{ zeta}}( zeta) rightarrow 0forS rightarrow infty, thenC_{ zeta}(S) = frac{1}{2}S+{o}(1).     

Some integrals involving theQ_Mfunction (Corresp.)

Some integrals are presented that can be expressed in terms of theQ_Mfunction, which is defined as begin{equation} Q_M(a,b) = int_b^{infty} dx x(x/a)^{M-1} exp (- frac{x^2 + a^2}{2}) I_{M-1}(ax), end{equation} whereI_{M-1}is the modified Bessel function of orderM-1. Some integrals of theQ_Mfunction are also evaluated.     

On theepsilon-entropy and the rate-distortion function of certain non-Gaussian processes

Letxi = {xi(t), 0 leq t leq T}be a process with covariance functionK(s,t)andE int_0^T xi^2(t) dt < infty. It is proved that for everyvarepsilon > 0thevarepsilon-entropyH_{varepsilon}(xi)satisfies begin{equation} H_{varepsilon}(xi_g) - mathcal{H}_{xi_g} (xi) leq H_{varepsilon}(xi) leq H_{varepsilon}(xi_g) end{equation} wherexi_gis a Gaussian process with the covarianeeK(s,t)andmathcal{H}_{xi_g}(xi)is the entropy of the measure induced byxi(in function space) with respect to that induced byxi_g. It is also shown that ifmathcal{H}_{xi_g}(xi) < inftythen, asvarepsilon rightarrow 0begin{equation} H_{varepsilon}(xi) = H_{varepsilon}(xi_g) - mathcal{H}_{xi_g}(xi) + o(1). end{equation} Furthermore, ff there exists a Gaussian processg = { g(t); 0 leq t leq T }such thatmathcal{H}_g(xi) < infty, then the ratio betweenH_{varepsilon}(xi)andH_{varepsilon}(g)goes to one asvarepsilongoes to zero. Similar results are given for the rate-distortion function, and some particular examples are worked out in detail. Some cases for whichmathcal_{xi_g}(xi) = inftyare discussed, and asymptotic bounds onH_{varepsilon}(xi), expressed in terms ofH_{varepsilon}(xi_g), are derived.     

On Parallelepiped-Shaped Passbands for Multidimensional Nonseparable ${bf M}$th-Band Low-Pass Filters

The “shape” of the desired frequency passband is an important consideration in the design of nonseparable multidimensional ($M$ -D) filters in $M$-D multirate systems. For $M$-D ${bf M}$th-band filters, the passband shape should be chosen such that the ${bf M}$th-band constraint is satisfied. The most commonly used shape of the passband for $M$-D ${bf M}$ th-band low-pass filters is the so-called symmetric parallelepiped (SPD) ${rm SPD}(pi {bf M}^{- {rm T}})$ . In this paper, we consider the more general parallelepiped passband ${rm SPD}(pi {bf L} ^{rm T})$, and derive conditions on $ {bf L} $ such that the ${bf M}$ th-band constraint is satisfied. This result gives some flexibility in designing $M$-D ${bf M}$th-band filters with parallelepiped shapes other than the commonly used case of $ {bf L} = {bf M}^{- 1}$. We present design examples of 2-D ${bf M}$th-band filters to illustrate this flexibility in the choice of $ {bf L} $.      

On the weight structure of Reed-Muller codes

The following theorem is proved. Letf(x_1,cdots, x_m)be a binary nonzero polynomial ofmvariables of degreenu. H the number of binarym-tuples(a_1,cdots, a_m)withf(a_1, cdots, a_m)= 1 is less than2^{m-nu+1}, thenfcan be reduced by an invertible affme transformation of its variables to one of the following forms. begin{equation} f = y_1 cdots y_{nu - mu} (y_{nu-mu+1} cdots y_{nu} + y_{nu+1} cdots y_{nu+mu}), end{equation} wherem geq nu+muandnu geq mu geq 3. begin{equation} f = y_1 cdots y_{nu-2}(y_{nu-1} y_{nu} + y_{nu+1} y_{nu+2} + cdots + y_{nu+2mu -3} y_{nu+2mu-2}), end{equation} This theorem completely characterizes the codewords of thenuth-order Reed-Muller code whose weights are less than twice the minimum weight and leads to the weight enumerators for those codewords. These weight formulas are extensions of Berlekamp and Sloane's results.     

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.