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.     

High-frequency scattering from a conducting cylinder with an inhomogeneous plasma sheath

The high-frequency backscattering characteristics of an infinite conducting cylinder enveloped in a radially inhomogeneous plasma sheath are studied in detail. When the permittivity of the sheath has a general power-law dependence,epsilon(rho) = epsilon(rho_{0}) (rho/rho_{0})^{p}, whererho_{0}is the outer radius of the sheath andpis any real numberneq -2, the wave function in the plasma is expressible in terms of Bessel functions of a fractional order. The slowly convergent series form of the backscattering coefficient is first recast into an integral by means of the Watson transformation, which is then asymptotically evaluated by the method of stationary phase. The mathematical result is conveniently interpreted in terms of geometrical optics by identifying the contributions due to the central ray and the trapped "interrupted" and "uninterrupted" rays. In contrast to the rather unpredictable and violent backscattering coefficient variations with frequency when a conducting cylinder is clad in a homogeneous plasma sleeve, the change in the coefficient for the same cylinder enveloped in a plasma sheath with a power-law radial inhomogeneity is much more smooth and, in most cases, the approximate locations of the maximums and minimums can be predicted. Numerical results showing the dependence of the backscattering coefficient on the type of power-law inhomogeneity, sheath thickness, and permittivity level are presented in graphical form.     

Modes in unstable optical resonators and lens waveguides

Optical resonators and/or lens waveguides are "unstable" when they have divergent focusing properties such that they fall in the unstable region of the Fox and Li mode chart. Although such resonators have large diffraction losses, their large mode volume and good transverse-mode discrimination may nonetheless make them useful for high-gain diffraction-coupled laser oscillators. A purely geometrical mode analysis (valid for Fresnel numberN = infty) shows that the geometrical eigenmodes of an unstable system are spherical waves diverging from unique virtual centers. As Burch has noted, the higher-order transverse modes in the geometrical limit have the formu_{n}(x) = x^{n}with eigenvaluesgamma_{n} = 1/M^{n+1/2}, whereMis the linear magnification of the spherical wave per period. The higher-order modes have nodes on-axis only, and there is substantial transverse-mode discrimination. More exact computer results for finiteNshow that the spherical-wave phase approximation remains very good even at very lowN, but the exact mode amplitudes become more complicated than the geometrical results. The exact mode loss versusNexhibits an interesting quasi-periodicity, withn = 0andn = 2mode degeneracy occurring at the loss peaks. Defining a new equivalent Fresnel number based on the actual spherical waves rather than plane waves shows that the loss peaks occur at integer values of Neqfor all values ofM.     

Fundamental mode size and bend sensitivity of graded and step-index single-mode fibers with zero-dispersion near 1.55 µm

Single-mode fibers, for near 1.55-μm operating system wavelength with triangular (alpha = 1), parabolic (alpha = 2), and step-index (alpha=infty) profiles were fabricated by the modified chemical vapor deposition (MCVD) technique. Optical transmission losses under zero-tension and with the basket weave under 10, 40, and 70 gm tensions were measured, respectively. Fundamental mode size was obtained as a function of wavelength by using the transverse offset technique. The triangular-profile fiber shows lower loss (≃0.25 dB/km at 1.55μm) under zero-tension and a larger spot size than the other fibers. However, the basket weave test showed the triangular-profile fiber incurred higher loss with tension than the other profiles.     

The covering radius of the (128,8) Reed-Muller code is 56 (Corresp.)

Letr_{i}be the covering radius of the(2^{i},i+ 1)Reed-Muller code. It is an open question whetherr_{2m+1}=2^{2_{m}}-2mholds for allm. It is known to be true form=0,1,2, and here it is shown to be also true form=3.     

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.     

Dual product constructions of Reed - Muller type codes (Corresp.)

Various linear and nonlinearR(r,m)codes having parameters(2^{m}, 2^{k}, 2^{m-r})withk=sum_{i=0}^{r}left(^{m}_{i}right)are constructed fromR(r,q)andR(r,p)codes,m=p+q. A dual construction forR(m-r,m)codes fromR(p-r,p)andR(q-r,q)codes is also presented,m=p+q. As a simple corollary we have that the number of nonequivalentR(r,m)codes is at least exponential in the length (forr>1). ForR(m-r,m)codes, the lower bound is doubly exponential in the length (forr>1).     

General mode analysis of a gyrotron dispersion equation

A linear dispersion relation for the gyrotron operating at generalTEmin{ln}max{S}modes has been derived within the Maxwell-Vlasov system under the tenuous beam assumption. Unlike previous analyses, the dispersion equation accurately predicts the linear gain of the gyrotron on the entire range of wave frequency near the electron cyclotron instability. By careful choice of variables for the beam distribution both the instability driving and the stabilizing terms are obtained accurately. Not only is the dispersion equation valid for arbitrary beam distribution but it describes the negative mass instability as well. The explicit expression for the growth rate of the cold beam and its dependence on the wavenumber and the wall radius are examined. The optimization conditions on the wall radius, the beam center location (r0), the radial mode number (n), and the azimuthal mode number (l) are also found. It is found that for a given harmonic numbers, the negative mass in-Stability withTEmin{S1}max{S}(i.e.,l = s, n = 1) for a "rotating" beam whose guiding center coincides with the waveguide axis (i.e.,r_0 = 0), yields the highest linear gain, typically twice of that forl = 0annular beam.     

RCS of a coated circular waveguide terminated by a perfect conductor

The radar cross section (RCS) of a circular waveguide terminated by a perfect electric conductor is calculated by the geometrical theory of diffraction (GTD) for the rim diffraction and by a physical optics approximation for the interior irradiation. The interior irradiation is generally more than 10 dB higher than the rim diffraction fora/lambda geq 1(ais the waveguide radius,lambdais the free-space wavelength). At low frequencies (a/lambda sim 1), the interior irradiation can be significantly reduced over a broad range of incident angle if the interior waveguide wall is coated with a thin layer (1 percent of the radius) of lossy magnetic material. Our theoretical prediction is confirmed by measurements. At higher frequencies (a/lambda sim 3), a thin layer of coating is effective for the case of near axial incidence, provided that a good transition of theTE_{11}mode near the waveguide opening to theHE_{11}mode inside the waveguide is made. A thicker layer of coating is required for the RCS reduction over wider incident angle.     

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.     

Backscattering from a Gaussian-distributed perfectly conducting rough surface

An analytical approach to the problem of scattering by composite random surfaces is presented. The surface is assumed to be Gaussian so that the surface height can be split (in the mean-square sense) into large (zeta_{l}) and small (zeta_{s}) scale components relative to the electromagnetic wavelength. A first-order perturbation approach developed by Burrows is used wherein the scattering solution for the large-scale structure is perturbed by the small-scale diffraction effects. The scattering from the large-scale structure (the zeroth-order perturbation solution) is treated via geometrical optics since4k_{0}^{2}bar{zeta_{l}^{2}} gg 1. The first-order perturbation result comprises a convolution in wavenumber space of the height spectrum, the shadowing function, a polarization dependent factor, the joint density function for the large-scale slopes, and a truncation function which restricts the convolution to the domain corresponding to the small-scale height spectrum. The only "free" parameter is the surface wavenumber separating the large and small height contributions. For a given surface height spectrum, this wavenumber can be determined by a combination of mathematical and physical arguments.     

An approximation to the weight distribution of binary primitive BCH codes with designed distances 9 and 11 (Corresp.)

Recently Kasami {em et al.} presented a linear programming approach to the weight distribution of binary linear codes [2]. Their approach to compute upper and lower bounds on the weight distribution of binary primitive BCH codes of length2^{m} - 1withm geq 8and designed distance2t + 1with4 leq t leq 5is improved. From these results, the relative deviation of the number of codewords of weightjleq 2^{m-1}from the binomial distribution2^{-mt} left( stackrel{2^{m}-1}{j} right)is shown to be less than 1 percent for the following cases: (1)t = 4, j geq 2t + 1andm geq 16; (2)t = 4, j geq 2t + 3and10 leq m leq 15; (3)t=4, j geq 2t+5and8 leq m leq 9; (4)t=5,j geq 2t+ 1andm geq 20; (5)t=5, j geq 2t+ 3and12 leq m leq 19; (6)t=5, j geq 2t+ 5and10 leq m leq 11; (7)t=5, j geq 2t + 7andm=9; (8)t= 5, j geq 2t+ 9andm = 8.     

Diffraction by a circular aperture in a unidirectionally conducting screen

The diffraction of a normally incident plane electromagnetic wave with wave numberkby a circular aperture of radiusain a unidirectionally conducting plane screen of zero thickness and infinite extent is considered. In the limit of largeka, the ratio of the transmission cross section to the geometrical optics valuepi a^{2}, is found up to the order(ka)^{-3/2}.     

Design of single mesh flashlamp driving circuits with resistive losses

The normalized nonlinear equation(dI/dtau) pm [alpha + beta|I|^{1/2}] + intmin{0}max{tau}I dbar{tau}= 1describing current evolution in a single mesh flashlamp driving circuit has been solved using numerical methods for the case of a critically damped pulse (alpha simeq 0.84) and beta = r/Z_{0} neq 0. Normalized current, power, and energy waveforms for values of β in the range0 leq beta leq 0.5are obtained. The energy efficiency ηE, which is numerically equal to the flashlamp dissipated energy di-Vided by the energy stored in the driving circuit, is also shown as a function of β. Increased numerical accuracy has improved the previously used value of α for critical damping and leads to a slightly different form for the curve representing the locus of values (α β) to obtain critically damped pulses in the presence of restrictive losses.     

On the Schalkwijk-Kailath coding scheme with a peak energy constraint

In a recent series of papers, [2]-[4] Schalkwijk and Kailath have proposed a block coding scheme for transmission over the additive white Gaussian noise channel with one-sided spectral densityN_{0}using a noiseless delayless feedback link. The signals have bandwidthW (W leq infty)and average powerbar{P}. They show how to communicate at ratesR < C = W log (1 + bar{P}/N_{0}W), the channel capacity, with error probabilityP_{e} = exp {-e^{2(C-R)T+o(T)}}(whereTis the coding delay), a "double exponential" decay. In their scheme the signal energy (in aT-second transmission) is a random variable with only its expectation constrained to bebar{P}T. In this paper we consider the effect of imposing a peak energy constraint on the transmitter such that whenever the Schalkwijk-Kailath scheme requires energy exceeding abar{P}T(wherea > 1is a fixed parameter) transmission stops and an error is declared. We show that the error probability is degraded to a "single exponential" formP_{e} = e^{-E(a)T+o(T)}and find the exponentE(a). In the caseW = infty , E(a) = (a - 1)^{2}/4a C. For finiteW, E(a)is given by a more complicated expression.     

Weight enumerator for second-order Reed-Muller codes

In this paper, we establish the following result. Theorem:A_i, the number of codewords of weightiin the second-order binary Reed-Muller code of length2^mis given byA_i = 0unlessi = 2^{m-1}or2^{m-1} pm 2^{m-l-j}, for somej, 0 leq j leq [m/2], A_0 = A_{2^m} = 1, and begin{equation} begin{split} A_{2^{m-1} pm 2^{m-1-j}} = 2^{j(j+1)} &{frac{(2^m - 1) (2^{m-1} - 1 )}{4-1} } \ .&{frac{(2^{m-2} - 1)(2^{m-3} -1)}{4^2 - 1} } cdots \ .&{frac{(2^{m-2j+2} -1)(2^{m-2j+1} -1)}{4^j -1} } , \ & 1 leq j leq [m/2] \ end{split} end{equation} begin{equation} A_{2^{m-1}} = 2 { 2^{m(m+1)/2} - sum_{j=0}^{[m/2]} A_{2^{m-1} - 2^{m-1-j}} }. end{equation}     

On the covering radius of binary codes (Corresp.)

Upper bounds on the covering radius of binary codes are studied. In particular it is shown that the covering radiusr_{m}of the first-order Reed-Muller code of lenglh2^{m}satisfies2^{m-l}-2^{lceil m/2 rceil -1} r_{m} leq 2^{m-1}-2^{m/2-1}.     

All binary 3-error-correcting BCH codes of length2^m-ihave covering radius 5 (Corresp.)

Van der Horst and Berger have conjectured that the covering radius of the binary 3-error-correcting Bose-Chaudhuri-Hocquenghem (BCH) code of length2^{m} - l, m geq 4is 5. Their conjecture was proved earlier whenm equiv 0, 1, or 3 (mod 4). Their conjecture is proved whenm equiv 2(mod 4).     

On the survival of sequence information in filters (Corresp.)

Given a binary data streamA = {a_i}_{i=o}^inftyand a filterFwhose output at timenisf_n = sum_{i=0}^{n} a_i beta^{n-i}for some complexbeta neq 0, there are at most2^{n +1)distinct values off_n. These values are the sums of the subsets of{1,beta,beta^2,cdots,beta^n}. It is shown that all2^{n+1}sums are distinct unlessbetais a unit in the ring of algebraic integers that satisfies a polynomial equation with coefficients restricted to +1, -1, and 0. Thus the size of the state space{f_n}is2^{n+1}ifbetais transcendental, ifbeta neq pm 1is rational, and ifbetais irrational algebraic but not a unit of the type mentioned. For the exceptional values ofbeta, it appears that the size of the state space{f_n}grows only as a polynomial innifmidbetamid = 1, but as an exponentialalpha^nwith1 < alpha < 2ifmidbetamid neq 1.