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.     

Spectral Analysis of Digital Messages Through Finite-Memory Transformations

The spectral analysis is performed for the stochastic sequence{b_{n}} = {h(a_{n}, a_{n-1},cdots, a_{n-M+1})}, where{a_{n}}is an independent identically distributed digital sequence andhis an arbitrary complex function. The computation of the spectrum is based on a general matrix method for the spectral analysis of memoryless functions of Markov chains. The approach employs a recursive algorithm, whose computational time and storage requirements are sufficiently small for the algorithm to be implemented on a modern general purpose computer, even for moderately high values of both the size of the source alphabet and the parameterM. The method is also extended to the case in which the input sequence{a_{n}}is a stationary finite Markov chain.     

Measurement of fourth-order aberration in a lens-like medium

Deformation of the spot of an off-axis Gaussian beam after passing through a lens-like glass fiber (SELFOC®) with the fourth-order aberration is investigated by using first-order perturbation theory and numerical calculation. A method for determining the coefficient of the fourth-order aberration is proposed. By this method, it is found that a SELFOC sample (0.5 mm in diameter and 150 mm in length) has a coefficienthof about 1.1, if the dielectric constant in the meridional plane of the sample is expressed asepsilon(x) = epsilon(0) (1 - g^{2}x^{2} + hg^{4}x^{4}), wherexis the distance from the axis of the medium andgis a positive constant.     

The fast decoding of Reed-Solomon codes using Fermat transforms (Corresp.)

It is shown thatsqrt[8]{2}is an element of order2^{n+4}inGF(F_{n}), whereF_{n}=2^{2^{n}}+1is a Fermat prime forn=3,4. Hence it can be used to define a fast Fourier transform (FFT) of as many as2^{n+4}symbols inGF(F_{n}). Sincesqrt[8]{2}is a root of unity of order2^{n+4}inGF(F_{n}), this transform requires fewer muitiplications than the conventional FFT algorithm. Moreover, as Justesen points out [1], such an FFT can be used to decode certain Reed-Solomon codes. An example of such a transform decoder for the casen=2, wheresqrt{2}is inGF(F_{2})=GF(17), is given.     

On the accuracy of physical optics

The first-order correction to the physical optics (PO) solution for the axial radiation field from parabolic reflector antennas is found. The correction is of orderk^{-1}and is in quadrature with the PO solution. By means of the correction term the accuracy of the PO surface integral is round to be better than0.22(lambda/D)^{2}dB on axis, whereDis the diameter of the reflector. Thus, the PO solution for the directivity is extremely accurate for commun reflector sizes of several wavelengthslambda.     

Comment on 'The capacity of the photon counting channel' by Pierce, J.R., et al.

The capacity of the photon counting channel with an average-power constraint equalsP_{s}/(kT ln2)bit/s at all noise temperaturesT, whereP_{s}is the average signal power andkis Boltzmann's constant. The channel can carry arbitrary orthogonal signals and is not restricted to the poise-position modulation (PPM) signals adduced in the above paper.^{1}     

Asymptotic rate-distortion functions for coding precedence relations (Corresp.)

An improved analysis for an information system is described in which the input and output alphabets consist of (equiprobable) ordered lists of m items, with two distortion criteria: 1) the fraction of item-pairs found out of order and 2) the fraction of items found in wrong positions. For the former it is shown that, asm rightarrow infty, the rate-distortion functionR_{1} (D)is equivalent tomG(D)with, for smallD,G(D)=log (2/D)-2+ O(D). For the latter,R_{2} (D) = m log m(1 - D) + m(l - D)[log(1 - D) - 1] + o(m).     

Writing on dirty paper (Corresp.)

A channel with outputY = X + S + Zis examined, The stateS sim N(0, QI)and the noiseZ sim N(0, NI)are multivariate Gaussian random variables (Iis the identity matrix.). The inputX in R^{n}satisfies the power constraint(l/n) sum_{i=1}^{n}X_{i}^{2} leq P. IfSis unknown to both transmitter and receiver then the capacity isfrac{1}{2} ln (1 + P/( N + Q))nats per channel use. However, if the stateSis known to the encoder, the capacity is shown to beC^{ast} =frac{1}{2} ln (1 + P/N), independent ofQ. This is also the capacity of a standard Gaussian channel with signal-to-noise power ratioP/N. Therefore, the stateSdoes not affect the capacity of the channel, even thoughSis unknown to the receiver. It is shown that the optimal transmitter adapts its signal to the stateSrather than attempting to cancel it.     

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.     

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.     

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.     

Optimal source codes for geometrically distributed integer alphabets (Corresp.)

LetP(i)= (1 - theta)theta^ibe a probability assignment on the set of nonnegative integers wherethetais an arbitrary real number,0 < theta < 1. We show that an optimal binary source code for this probability assignment is constructed as follows. Letlbe the integer satisfyingtheta^l + theta^{l+1} leq 1 < theta^l + theta^{l-1}and represent each nonnegative integeriasi = lj + rwhenj = lfloor i/l rfloor, the integer part ofi/l, andr = [i] mod l. Encodejby a unary code (i.e.,jzeros followed by a single one), and encoderby a Huffman code, using codewords of lengthlfloor log_2 l rfloor, forr < 2^{lfloor log l+1 rfloor} - l, and lengthlfloor log_2 l rfloor + 1otherwise. An optimal code for the nonnegative integers is the concatenation of those two codes.     

On the inequivalence of generalized Preparata codes

Ifmis odd andsigma /in Aut GF(2^{m})is such thatx rightarrow x^{sigma^{2}-1}is1-1, there is a[2^{m+1}-1,2^{m+l}-2m-2]nonlinear binary codeP(sigma)having minimum distance 5. All the codesP(sigma)have the same distance and weight enumerators as the usual Preparata codes (which rise asP(sigma)whenx^{sigma}=x^{2}). It is shown thatP(sigma)andP(tau)are equivalent if and only iftau=sigma^{pm 1}, andAut P(sigma)is determined.     

On efficient majority logic decodable codes

A particular shortening technique is applied to majority logic decodable codes of length2^{t}. The shortening technique yields new efficient codes of lengthsn = 2^{p}, wherepis a prime, e.g., a (128,70) code withd_{maj} = 16. For moderately long code lengths (e.g.,n = 2^{11} or 2^{13}), a 20-25 percent increase in efficiency can be achieved over the best previously known majority logic decodable codes. The new technique also yields some efficient codes for lengthsn = 2^{m}wheremis a composite number, for example, a (512,316) code withd_{maj} = 32code which has 42 more information bits than the previously most efficient majority logic decodable code.     

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.     

A proof of the data compression theorem of Slepian and Wolf for ergodic sources (Corresp.)

If{(X_i, Y_i)}_{i=1}^{infty}is a sequence of independent identically distributed discrete random pairs with(X_i, Y_i) ~ p(x,y), Slepian and Wolf have shown that theXprocess and theYprocess can be separately described to a common receiver at ratesR_XandR_Yhits per symbol ifR_X + R_Y > H(X,Y), R_X > H(XmidY), R_Y > H(YmidX). A simpler proof of this result will be given. As a consequence it is established that the Slepian-Wolf theorem is true without change for arbitrary ergodic processes{(X_i,Y_i)}_{i=1}^{infty}and countably infinite alphabets. The extension to an arbitrary number of processes is immediate.     

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.     

The distribution of gains in uniformly multiplying avalanche photodiodes: Theory

Expressions are derived for the probabilityP_{n,m}that a pulse initiated bynelectrons (or holes) in a uniformly multiplying semiconductor diode will result in a total number of electrons (or holes)m, to give a gainm/n, and for the probabilityQ_{n,m}that the gain will bem/nor greater. It is shown that the distributions are far from Gaussian. The gain distributionP_{1,m}for a single photoelectron, for example, is shown to have a maximum value form = 1for any value of the average gainM=m/n. The derivations are valid for any electric field distribution and assume only that the hole ionization coefficientbeta(E) can be approximated by the relationbeta(E) =kalpha(E), wherealpha(E)is the electron ionization coefficient andkis a constant. A method of determining an effective value ofk, for cases wherebeta=kalphais not a good approximation, is presented. The results can be used to calculate the average gain and the mean square deviation from the average, giving results in agreement with previously published relations [1], [2]. The implications of this theory on the use of avalanche diodes for low-level photodetection are discussed. It is shown that in the near infrared, cooled avalanche photodiodes can compare favorably with the best available photomultiplier when used either in a photon-counting mode, or for the reliable detection of low-level laser pulses.     

Cutoff Rate for PPM Noisy Photon Counting Channel with Soft Decision

A new method for bounding the cutoff rateRis presented. As an application, the case of ideal laser input signal with thermal noise on the photon counting PPM channel with soft decision decoding is given. The main result is that the cutoff rate efficiency (defined aslim_{srightarrow + infty} R/s, wheresis the average signal photon number) is given by (1- gamma)/(1 + gamma)(wheregamma = n_{0}/(1 + n_{0}), n_{0}being the average thermal noise photons). This value is much smaller than the capacity efficiency,log (1/gamma), given by Pierce et al. in [1].     

Capacity theorems for the relay channel

A relay channel consists of an inputx_{l}, a relay outputy_{1}, a channel outputy, and a relay senderx_{2}(whose transmission is allowed to depend on the past symbolsy_{1}. The dependence of the received symbols upon the inputs is given byp(y,y_{1}|x_{1},x_{2}). The channel is assumed to be memoryless. In this paper the following capacity theorems are proved. 1)Ifyis a degraded form ofy_{1}, thenC : = : max !_{p(x_{1},x_{2})} min ,{I(X_{1},X_{2};Y), I(X_{1}; Y_{1}|X_{2})}. 2)Ify_{1}is a degraded form ofy, thenC : = : max !_{p(x_{1})} max_{x_{2}} I(X_{1};Y|x_{2}). 3)Ifp(y,y_{1}|x_{1},x_{2})is an arbitrary relay channel with feedback from(y,y_{1})to bothx_{1} and x_{2}, thenC: = : max_{p(x_{1},x_{2})} min ,{I(X_{1},X_{2};Y),I ,(X_{1};Y,Y_{1}|X_{2})}. 4)For a general relay channel,C : leq : max_{p(x_{1},x_{2})} min ,{I ,(X_{1}, X_{2};Y),I(X_{1};Y,Y_{1}|X_{2}). Superposition block Markov encoding is used to show achievability ofC, and converses are established. The capacities of the Gaussian relay channel and certain discrete relay channels are evaluated. Finally, an achievable lower bound to the capacity of the general relay channel is established.