On the quality factor of strip and line source antennas and its relationship to superdirectivity ratio

Exact formulas are derived for the quality factorQof strip and line source antennas. Contrary to popular opinion, none of them is equal to Taylor's superdirectivity ratiogammaor togamma - 1. But in the case ofE-plane strip sources (the complement of the type of strip source treated by Woodward and Lawson) the value ofQis precisely equal togamma_{alpha}^{-1/2}- 1, wheregamma_{alpha}^{beta}is a generalized superdirectivity ratio that reduces to Taylor'sgammawhen the edge exponentalphaand the pattern weighting exponentbetaare both zero. In the case ofH-plane strip sources the value ofQis approximately equal togamma_{alpha}^{1/2} - 1, and forH-plane line sources of vanishing widthait is approximately equal to[(2/pi) ln (2.516lambda/pi a)]gamma_{alpha}^{1}.     

Maximum entropy and conditional probability

It is well-known that maximum entropy distributions, subject to appropriate moment constraints, arise in physics and mathematics. In an attempt to find a physical reason for the appearance of maximum entropy distributions, the following theorem is offered. The conditional distribution ofX_{l}given the empirical observation(1/n)sum^{n}_{i}=_{l}h(X_{i})=alpha, whereX_{1},X_{2}, cdotsare independent identically distributed random variables with common densitygconverges tof_{lambda}(x)=e^{lambda^{t}h(X)}g(x)(Suitably normalized), wherelambdais chosen to satisfyint f_{lambda}(x)h(x)dx= alpha. Thus the conditional distribution of a given random variableXis the (normalized) product of the maximum entropy distribution and the initial distribution. This distribution is the maximum entropy distribution whengis uniform. The proof of this and related results relies heavily on the work of Zabell and Lanford.     

On mean-square aliasing error in the cardinal series expansion of random processes (Corresp.)

An upper bound is derived for the mean-square error involved when a non-band-limited, wide-sense stationary random processx(t)(possessing an integrable power spectral density) is approximated by a cardinal series expansion of the formsum^{infty}_{-infty}x(n/2W)sinc2W(t-n/2W), a sampling expansion based on the choice of some nominal bandwidthW > 0. It is proved thatlim_{N rightarrow infty} E {|x(t) - x_{N}(t)|^{2}} leq frac{2}{pi}int_{| omega | > 2 pi W}S_{x}( omega) d omega,wherex_{N}(t) = sum_{-N}^{N}x(n/2W)sinc2W(t-n/2W), andS_{x}(omega)is the power spectral density forx(t). Further, the constant2/ piis shown to be the best possible one if a bound of this type (involving the power contained in the frequency region lying outside the arbitrarily chosen band) is to hold uniformly int. Possible reductions of the multiplicative constant as a function oftare also discussed, and a formula is given for the optimal value of this constant.     

Boresight-gain loss and gore-related sidelobes of an umbrella reflector

It is shown that energy in the gore-related sidelobes of an umbrella reflector comes from energy lost from the main beam. Both boresight gore-loss and sidelobe level correlate directly with cyclic aperture phase error caused by the geometry. The gore-sidelobe peak is located neartheta_{p}, wheresin theta_{p} = 1.2N_{G} (pi/Dlambda)whereN_{G}is the number of gores andDis the reflector diameter. An expression is also derived for the amplitude of the gore-related sidelobe.     

Bounds for truncation error in sampling expansions of band-limited signals

Forf(t)a real-valued signal band-limited to- pi r leq omega leq pi r (0 < r < 1)and represented by its Fourier integral, upper bounds are established for the magnitude of the truncation error whenf(t)is approximated at a generic timetby an appropriate selection ofN_{1} + N_{2} + 1terms from its Shannon sampling series expansion, the latter expansion being associated with the full band[-pi, pi]and thus involving samples offtaken at the integer points. Results are presented for two cases: 1) the Fourier transformF(omega)is such that|F(omega)|^{2}is integrable on[-pi, pi r](finite energy case), and 2)|F(omega)|is integrable on[-pi r, pi r]. In case 1) it is shown that the truncation error magnitude is bounded above byg(r, t) cdot sqrt{E} cdot left( frac{1}{N_{1}} + frac{1}{N_{2}} right)whereEdenotes the signal energy andgis independent ofN_{1}, N_{2}and the particular band-limited signal being approximated. Correspondingly, in case 2) the error is bounded above byh(r, t) cdot M cdot left( frac{1}{N_{1}} + frac{1}{N_{2}} right)whereMis the maximum signal amplitude andhis independent ofN_{1}, N_{2}and the signal. These estimates possess the same asymptotic behavior as those exhibited earlier by Yao and Thomas [2], but are derived here using only real variable methods in conjunction with the signal representation. In case 1), the estimate obtained represents a sharpening of the Yao-Thomas bound for values ofrdose to unity.     

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.     

Electromagnetic reflection from an extended turbulent medium: Cumulative forward-scatter single-backscatter approximation

The backscatter cross sectionQfor high-frequency irradiated turbulent dielectric media, many mean free pathsL_{1}wide, is computed. The lengthL_{1}is the distance into the medium over which the mean electric field decreases in amplitude by a factore^{-1}. Previous calculations have always been restricted toL ll L_{1}. It is found thatQincreases from the Born approximationQ = Q_{1}for medium widthL ll L_{1}toQ = 2Q_{1}forL gg L_{1}, and the theory is valid as long asL ll (kL_{0})^{5/3} L_{1}, a significant improvement over the Born approximation, when the macroscaleL_{0}is much larger than the wavelength2_{pi}k^{-1}. The improvement is due to incorporation of the dominant effects of cumulative forward scattering in the local electric field in the medium. A rigorous and a heuristic derivation are given. The transitional behavior is discussed and a simple physical interpretation is given.     

The wide-band matching area for a small antenna

A special frequency scale is proposed for use in describing the fundamental limitation on wide-band matching of a small antenna by means of a fixed reactive network. This scale isu = - 1/omega^{3}, normalized to unit frequency (omega_{1} = 1)-It has unit width for the high-pass band above this frequency. In terms of the familiar matching exponentmu = ln 1/|rho|, the matching area over this scale is limited to3 pi p_{1}, whereP_{1}is the (small) radiation power factor atomega_{1}. The corresponding limit on matching efficiency is6 pi p_{1}. A double-tuned matching network can approach2/pithis area over any bandwidth on this scale.     

A method for computing addition tables inGF(p^n)(Corresp.)

Conway showed that a table of Zech's logarithms is useful to perform addition in GF(p^{n})when the elements are represented as powers of a primitive element. The Zech's logarithmZ(x)ofxis defined by the equationalpha^{z(x)}=alpha^{x} + 1, wherealphais a primitive element, zero is written asalpha^{ast}, andx=ast,O,1, cdots ,p^{n}-2. A simple algorithm for making a table of Zech's logarithms is presented.     

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.     

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.     

Two methods to deconvolve: L1-method using simplex algorithm and L2-method using least squares and parameter

Ifr(t)is the linear scattering response of an object to an excitation waveforme(t), thenr(t) = (e ast h) (t). One would like to deconvolve and solve forh(t), the impulse response. It is well-known that this is often an ill-conditioned problem. Two methods are discussed. The first method replaces the discretized matrix formE cdot H = Rby the following problem. Minimize|h_{1}|+ ldots + |h_{n}|subject toR - lambda leq E cdot H leq R + lambdawherelambdais a column vector chosen sufficiently small to yield acceptable residuals, yet large enough to make the problem well-conditioned. This problem is converted to a linear programming problem so that the simplex algorithm can be used. The second method is to minimizeparallel E cdot H - R parallel^{2} +lambda parallel H parallel^{2}where againlambdais chosen small enough to yield acceptable residuals and large enough to make the problem well-conditioned. The method will be demonstrated with a Hilbert matrix inversion problem, and also by the deconvolution of the impulse response of a simple target from measured data.     

Antenna radiation characteristics with partially coherent illumination

The radiation characteristics of a partially coherent illuminated antenna are found for both a circular aperture and a linear antenna. The antenna illumination function is assumed to be Gaussian, and the phase structure function in the plane of the antenna is assumed to be of the formcr^{nu}, withnu = 1, 5/3, and 2. The beam broadening and reduction in gain in decibels are found to be of the formKb^{alpha}, whereKandalphaare functions of the antenna illumination and the "law" of the phase structure function, andbis a function of the "level" of the phase structure function.     

An upper bound on the aperiodic autocorrelation function for a maximal-length sequence (Corresp.)

The magnitude of the out-of-phase aperiodic autocorrelation function for a maximal-length linear feedback shift register sequence of periodNis at most1 + (2/ pi)(N + 1)^{1/2} ln(4N/ pi). Previously, the best upper bound was(N + 1)^{1/2} ln (eN).     

On certain projective geometry codes (Corresp.)

LetVbe an(n, k, d)binary projective geometry code withn = (q^{m}-1)/(q - 1), q = 2^{s}, andd geq [(q^{m-r}-1)/(q - 1)] + 1. This code isr-step majority-logic decodable. With reference to the GF(q^{m}) = {0, 1, alpha , alpha^{2} , cdots , alpha^{n(q-1)-1} }, the generator polynomialg(X), ofV, hasalpha^{nu}as a root if and only ifnuhas the formnu = i(q - 1)andmax_{0 leq l < s} W_{q}(2^{l} nu) leq (m - r - 1)(q - 1), whereW_{q}(x)indicates the weight of the radix-qrepresentation of the numberx. LetSbe the set of nonzero numbersnu, such thatalpha^{nu}is a root ofg(X). LetC_{1}, C_{2}, cdots, C_{nu}be the cyclotomic cosets such thatSis the union of these cosets. It is clear that the process of findingg(X)becomes simpler if we can find a representative from eachC_{i}, since we can then refer to a table, of irreducible factors, as given by, say, Peterson and Weldon. In this correspondence it was determined that the coset representatives for the cases ofm-r = 2, withs = 2, 3, andm-r=3, withs=2.     

Spectral analysis of variable-length coded digital signals

Spectral analysis is performed for a digital message produced by a variable-length encoder driven by a stationary memoryless source. In particular, closed form expressions are derived for both the continuous and the discrete part of the spectral density. The continuous part turns out to be a rational function ofz=exp (j2pi fT),whereTis the symbol period, whereas the discrete part exhibits in general spectral lines at multiple integers of(lambda_{0}T)^{-1}, wherelambda_{0}is the greatest common divisor of the codeword lengths. As an application of the theory explicit formulas are derived for the spectra of BnZS and HDBncodes.     

Uniform linear prediction of bandlimited processes from past samples (Corresp.)

Forx(t)either a deterministic or stochastic signal band-limited to the normalized frequency intervalmidomegamid leq pi, explicit coefficients{ a_{kn} }are exhibited that have the property that begin{equation} lim_{n rightarrow infty} parallel x(t) - sum_{1}^n a_{kn} x(t - kT) parallel = 0 end{equation} in an appropriate norm and for any constant intersample spacingTsatisfying0 < T < fac{1}{2}; that is,x(t)may be approximated arbitrarily well by a linear combination of past samples taken at any constant rate that exceeds twice the associated Nyquist rate. Moreover, the approximation ofx(t)is uniform in the sense that the coefficients{ a_{kn} }do not depend on the detailed structure ofx(t)but are absolute constants for any choice ofT. The coefficients that are obtained provide a sharpening of a previous result by Wainstein and Zubakov where a rate in excess of three times the Nyquist rate was required.     

An algorithm for maximizing expected log investment return

Let the random (stock market) vectorX geq 0be drawn according to a known distribution functionF(x), x in R^{m}. A log-optimal portfoliob^{ast}is any portfoliobachieving maximal expectedlogreturnW^{ast}=sup_{b} E ln b^{t}X, where the supremum is over the simplexb geq 0, sum_{i=1}^{m} b_{i} = 1. An algorithm is presented for findingb^{ast}. The algorithm consists of replacing the portfoliobby the expected portfoliob^{'}, b_{i}^{'} = E(b_{i}X_{i}/b^{t}X), corresponding to the expected proportion of holdings in each stock after one market period. The improvement inW(b)after each iteration is lower-bounded by the Kullback-Leibler information numberD(b^{'}|b)between the current and updated portfolios. Thus the algorithm monotonically improves the returnW. An upper bound onW^{ast}is given in terms of the current portfolio and the gradient, and the convergence of the algorithm is established.     

Asymptotic properties of delay-time-weighted probability of error (Corresp.)

Asymptotic properties of expected distortion are studied for the delay-time-weighted probability of error distortion measured_n(x,tilde{x}) = n^{-1} sum_{t=0}^{n-1} f(t + n)[l - delta(x_t,tilde{x}_t)],, wherex = (x_0,x_1,cdots,x_{n-1})andtilde{x} = (tilde{x}_0,tilde{x}_1,cdots,tilde{x}_{n-1})are source and reproducing vectors, respectively, anddelta (cdot, cdot)is the Kronecker delta. With reasonable block coding and transmission constraintsx_tis reproduced astilde{x}_twith a delay oft + ntime units. It is shown that if the channel capacity is greater than the source entropyC > H(X), then there exists a sequence of block lengthncodes such thatE[d_n(X,tilde{X})] rigjhtarrow 0asn rightarrow inftyeven iff(t) rightarrow inftyat an exponential rate. However, iff(t)grows at too fast an exponential rate, thenE[d_n(X,tilde{X})] rightarrow inftyasn rightarrow infty. Also, ifC < H(X)andf(t) rightarrow inftythenE[d_n(X,tilde{X})] rightarrow inftyasn rightarrow inftyno matter how slowlyf(t)grows.     

A Technique for Improving the Efficiency of M-ary Signaling

The coding gain ofM-ary communication systems using orthogonal alphabets have been shown to improve by using concatenated orthogonal codes, each of lengthm = sqrt{M}. In the receiver, a two-stage MF/correlator and ML detection are utilized to provide the final decision through only2(m- 1)comparisons instead of the(M- 1)comparisons normally required. A coding gain of 8 dB forP_{e} = 10^{-5}is obtained withm = 2^{l0}only. Higher coding gain is obtained with higher order concatenation.