Influence of the law of electrical conductivity on thermal switching and breakdown

Influences of the law of electrical conductivity σ on thermal switching and breakdown events were investigated. The conductivities selected weresigma_{1}propto exp(aT), orsigma_{1}propto exp(-B/T), and also voltageVdependentpropto exp (bV), whenTis temperature and a, B, and b are constants. Simple models, thin film, and cylindrical specimens were considered. The influence of σ was examined by comparing steady-state V-I characteristics, the values of maximum voltages Vm, times for temperature runaway when voltages larger than Vmwere applied, and instabilities in the negative differential resistance ranges. Some significant differences were found as σ1rises much faster with temperature than σ2. Thus current filaments were very narrow whensigma_{1}propto exp(aT), but orders of magnitude wider whensigma_{1}propto exp(-B/T). For a given current, much larger temperature rises occur in the σ2than in the σ1case of conductivity. Transition to a branch line V-I characteristic and filament formation can occur in the σ2case at considerably higher temperatures than in the σ1case.     

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.     

Equidistant binary arithmetic codes (Corresp.)

LetC(B)denote the binary cyclicANcode with generatorA, whereAB=2^{n} - 1. It is known thatC(B)is equidistant ifBis a prime powerp^{k}, where either2or-2is primitive moduloBprovidedpequiv 1 pmod{3}{rm if} k > 1. It is conjectured that these are the onlyBsuch thatC(B)is equidistant. We have verified this forB < 100 000. Several results are established that further limit the possibilities for counterexamples to the conjecture.     

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].     

Mutual information of the white Gaussian channel with and without feedback

The following model for the white Gaussian channel with or without feedback is considered: begin{equation} Y(t) = int_o ^{t} phi (s, Y_o ^{s} ,m) ds + W(t) end{equation} wheremdenotes the message,Y(t)denotes the channel output at timet,Y_o ^ {t}denotes the sample pathY(theta), 0 leq theta leq t. W(t)is the Brownian motion representing noise, andphi(s, y_o ^ {s} ,m)is the channel input (modulator output). It is shown that, under some general assumptions, the amount of mutual informationI(Y_o ^{T} ,m)between the messagemand the output pathY_o ^ {T}is directly related to the mean-square causal filtering error of estimatingphi (t, Y_o ^{t} ,m)from the received dataY_o ^{T} , 0 leq t leq T. It follows, as a corollary to the result forI(Y_o ^ {T} ,m), that feedback can not increase the capacity of the nonband-limited additive white Gaussian noise channel.     

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.     

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.     

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 stochastic process associated with the EAR(l) model

A stochastic bivariate process{(T_{n}, X_{n}): n=0,1,2, cdots }is considered. TheT_{n}are the occurrence times of a random event generated by a Poisson stochastic point process. EachX_{n}is the amplitude associated with thenth event at random timeT_{n}and is constructed fromX_{n}-1and the interarrival timeT_{n}-T_{n-1}, according to the first-order autoregressive exponential time series model (EAR(l)). Moments and joint distributions for the bivariate process are obtained, as well as the distribution of some extreme values related to the bivariate process.     

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 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.     

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.     

The influence of post-emitter processing on the current gain of bipolar transistors

The effects of post-emitter processing on the current gain of n-p-n and p-n-p transistors used in linear integrated circuits, is the subject of this paper. It is seen that during the emitter diffusion deep level impurities segregate into the bulk and space-charge regions of the emitter. The aluminum alloy process at 500°C provides a powerful scavenging action that getters the impurities. The post-alloy current gain increases by a factor of ten. This gettering is effective in phosphorus diffused emitters if the dielectric is either a CVD oxide, a plasma deposited oxide, or phosphorus glass under Si3N4. With thermally oxidized emitters there is no improvement in current gain. Boron diffused emitters, however, respond to the aluminum alloy process regardless of the dielectric. This study also identifies two processes that contribute to bandgap narrowing (DeltaE_{g}) these are the emitter diffusion and the post-emitter thermal oxidation. A value ofDeltaE_{g}= 0.04 eV measured after the emitter diffusion was found to increase to 0.09 eV after thermal oxidation at 975°C. This additional component ofDeltaE_{g}is a result of the anomalous accumulation of phosphorus at the silicon surface during oxidation.     

Analyses of mode-hopping phenomena in an AlGaAs laser

Intensity fluctuations of the longitudinal modes of a 0.8 μm AlGaAs laser were precisely measured during the occurrence of hopping between two modes. It was found from this result that mode hopping follows the stochastics of a Poisson process. The frequency of mode hopping was measured asf_{c} = [exp [-95(I/I_{th} - 1)]] times 10^{7}(Hz). whereI/I_{th}is the injection current normalized to its threshold value. Results of analog computer simulations showed that spontaneous emission worked as a triggering force for mode hopping. Results of the analysis based on the Fokker-Planck equation were compared to the experimental results, from which the root-mean-square value of the fluctuating electric field of spontaneous emission was estimated as2.3 times 10^{2)(V/m)leqlanglesim{E}_{N} leq 3.2 times 10^{2}(V/m). It is concluded that an effective reduction of mode hopping is achieved if the laser is operated at a higher bias or if the coupling constant between the two modes is increased.     

Equipartition laws of thermal noise

Classically, the thermal noise in electricalRCcircuits andLCRseries circuits is governed by the equipartition lawfrac{1}{2}overline{CV^{2}} = frac{1}{2}kT, whereV(t)is the noise voltage developed acrossC. When quantum effects are taken into account, the equipartition law no longer holds forRCcircuits, although an equipartition law can be deemed for the measured mean square noise voltage under certain conditions. InLCRseries circuits the equipartition lawfrac{1}{2}overline{CV^{2}} = frac{1}{2}kT, changes intofrac{1}{2}overline{CV^{2}} = frac{1}{2}bar{E}(f_{0})for high-Qtuned circuits, wherebar{E}(f_{0})is the average energy of a harmonic oscillator tuned at the tuning frequency of the tuned circuit.     

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.     

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.     

Linear feedback rate bounds for regressive channels (Corresp.)

This article presents new tighter upper bounds on the rate of Gaussian autoregressive channels with linear feedback. The separation between the upper and lower bounds is small. We havefrac{1}{2} ln left( 1 + rho left( 1+ sum_{k=1}^{m} alpha_{k} x^{- k} right)^{2} right) leq C_{L} leq frac{1}{2} ln left( 1+ rho left( 1+ sum_{k = 1}^{m} alpha_{k} / sqrt{1 + rho} right)^{2} right), mbox{all rho}, whererho = P/N_{0}W, alpha_{l}, cdots, alpha_{m}are regression coefficients,Pis power,Wis bandwidth,N_{0}is the one-sided innovation spectrum, andxis a root of the polynomial(X^{2} - 1)x^{2m} - rho left( x^{m} + sum^{m}_{k=1} alpha_{k} x^{m - k} right)^{2} = 0.It is conjectured that the lower bound is the feedback capacity.     

Complete decoding of triple-error-correcting binary BCH codes

An extensive study of binary triple-error-correcting codes of primitive lengthn = 2^{m} - 1is reported that results in a complete decoding algorithm whenever the maximum coset weightW_{max}is five. In this regard it is shown thatW_{max} = 5when four dividesm, and strong support is provided for the validity of the conjecture thatW_{max} = 5for allm. The coset weight distribution is determined exactly in some cases and bounded in others.     

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.