A bound on lightweight sequences with application to definite decoding (Corresp.)

It is shown that the numberMof binary-valuedn-tuples having fractional weightdeltaor less,0 < delta leq frac{1}{3}, such that no twon-tuples agree in anyLconsecutive positions, is bounded by2^{2LH(delta)+1}. A set ofn-tuples is constructed to show that this bound is not likely to be improved upon by any significant factor. This bound is used to show that the ratiod_{DD}/n_{DD}of definite-decoding minimum distance to definite-decoding constraint length is lower bounded byH^{-l}[frac{1}{6} cdot (1 - R)/ (1+R)]asn_{DD}grows without bound.     

Hybrid low autocorrelation sequences (Corresp.)

Skew-symmetric sequences of(2n + 1)terms,a_0,a_1,cdots,a_{2n}, are described for which the "merit factor" begin{equation} F_h = frac{biggl[sum_{i=0}^{2n} mid a_i mid biggr] ^2}{ 2 sum_{k=1}^{2n} biggl[ sum_{i=0}^{2n-k} text{sign} (a_i) cdot a_{i+k} biggl] ^2} end{equation} is unusually high.     

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 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).     

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.     

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.     

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.     

4C6--Concentration and temperature dependences of spin-lattice relaxation times in ruby at helium temperatures: Relaxation in a zero magnetic field

The relaxation times T1for the grouud state levels in ruby were measured in the temperature range 4.2 to 1.6°K for the various concentrations of theCr^{3+}ionsffrom 0.05 to 0.7 percent. The dependenceT_{1}(f)of the formT_{1}^{-1}(f) = T_{1}^{-1}(0) + T_{1}^{-1}(I)f ^{n}withn simeq 2has been obtained for the different transitions. The measurements of relaxation times forpm frac{1}{2} leftrightarrow pm frac{3}{2}transition at zero magnetic field were especially aimed at establishing a form of dependenceT_{1}(f)because of the absence of the cross relaxation effects in this case. The normal temperature dependenceT_{1} propto T_{1}^{-1}has been obtained at all concentrations in comparison with anomalous dependences observed at high concentrations by some researchers.     

An upper bound to the capacity of the band-limited Gaussian autoregressive channel with noiseless feedback

Upper bounds to the capacity of band-limited Gaussianmth-order autoregressive channels with feedback and average energy constraintEare derived. These are the only known hounds on one- and two-way autoregressive channels of order greater than one. They are the tightest known for the first-order case. In this case letalpha_1be the regression coefficient,sigma^2the innovation variance,Nthe number of channel iterations per source symbol, ande = E/N; then the first-order capacityC^1is bounded by begin{equation} C^1 leq begin{cases} frac{1}{2} ln [frac{e}{sigma^2}(1+ mid alpha_1 mid ) ^ 2 +1], & frac{e}{sigma^2} leq frac{1}{1- alpha_1^2} \ frac{1}{2} ln [frac{e}{sigma^2} + frac{2mid alpha_1 mid}{sqrt{1-alpha_1^2}} sqrt{frac{e}{simga^2}} + frac{1}{1-alpha_1^2}], & text{elsewhere}.\ end{cases} end{equation} This is equal to capacity without feedback for very low and very highe/sigma^2and is less than twice this one-way capacity everywhere.     

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.     

Multiplication noise in uniform avalanche diodes

A general expression is derived from which the spectral density of the noise generated in a uniformly multiplying p-n junction can be calculated for any distribution of injected carriers. The analysis is limited to the white noise part of the noise spectrum only, and to diodes having large potential drops across the multiplying region of the depletion layer. It is shown for the special case in whichbeta = kalpha, wherekis a constant and α and β are the ionization coefficients of electrons and holes, respectively, that the noise spectral density is given by2eI_{in}M^{3}[1 + (frac{1 - k}{k})(frac{M - 1}{M})^{2}]where M is the current multiplication factor and Iinthe injected current, if the only carriers injected into the depletion layer are holes, and by2eI_{in}M^{3}[1 - (1 - k)(frac{M - 1}{M})^{2}]if the only injected carriers are electrons. An expression is also derived for the noise power which will be delivered to an external load for the limitM rightarrow infin.     

Sequences achieving the boundary of the entropy region for a two-source are virtually memoryless (Corresp.)

For a joint distribution{rm dist}(X,Y), the functionT(t)=min { H(Y|U): I(U wedge Y|X)=O, H(X|U)geq t}is an important characteristic. It equals the asymptotic minimum of(1/n)H(Y^{n})for random pairs of sequences(X^{n}, Y^{n}), wherefrac{1}{n} sum ^{n}_{i=1}{rm dist} X_{i} sim {rm dist} X, {rm dist} Y^{n}|X^{n} = ({rm dist} Y|X)^{n}, frac{1}{n}H(X^{n})geq t.We show that if, for(X^{n}, Y^{n})as given, the rate pair[(1/n)H(X^{n}),(1/n)H(Y^{n})]approaches the nonlinear part of the curve(t,T(t)), then the sequenceX^{n}is virtually memoryless. Using this, we determine some extremal sections of the rate region of entropy characterization problems and find a nontrivial invariant for weak asymptotic isomorphy of discrete memoryless correlated sources.     

A simple analysis of the stable field profile in the supercritical TEA

An analytical investigation supported by numerical calculations has been performed of the stable field profile in a supercritical diffusion-stabilized n-GaAs transferred electron amplifier (TEA) with ohmic contacts. In the numerical analysis, the field profile is determined by solving the steady-state continuity and Poisson equations. The diffusion-induced short-circuit stability is checked by performing time-domain computer simulations under constant voltage conditions. The analytical analysis based on simplifying assumptions gives the following results in good agreement with the numerical results. 1) A minimum doping level required for stability exists, which is inversely proportional to the field-independent diffusion coefficient assumed in the simple analysis. 2) The dc current is bias independent and below the threshold value, and the current drop ratio increases slowly and almost linearly with the doping level. 3) The domain width normalized to the diode lengthLvaries almost linearly with(V_{B}/V_{T}-1)^{frac{1}{2}}/(n_{0}L)^{frac{1}{2}}where VBis the bias voltage VTis the threshold voltage, and no is the doping level. 4) The peak domain field varies almost linearly with (V_{B}/V_{T}-1)^{frac{1}{2}} (n_{0}L)^{frac{1}{2}}. Those results contribute to the understanding of the highn_{0}L-product switch and the stability of the supercritical TEA.     

An improved error bound for equal-strength diversity channels (Corresp.)

An upper bound on the minimum probability of error for an equal-strength diversity channel is simply derived that improves a previously known bound by the factor[4(1 - p)]^(-1),0 leq p leq frac{1}{2}.     

Temperature dependence of MOSFET characteristics in weak inversion

A knowledge of the MOSFET operating in weak inversion is important for circuits with low leakage specifications. This paper discusses the effect of temperature on the MOSFET in weak inversion. The reciprocal slopenof the log IDSversus VGSrelationship between source-drain current IDSand gate bias VGSmay be given byfrac{1}{(n - 1 - gamma)^{2}} = frac{2Cmin{ox}max{2}}{qepsilon_{s}N_{B}} [frac{3}{4} frac{E_{g^{0}}{q} - (frac{3}{2}alpha + frac{k}{q})T]withalpha equiv (k/q)(38.2 - ln N_{B} + (3/2) ln T)and γ ≡C_{ss}/C_{ox}, where Coxis the oxide capacitance per unit area, Cssthe surface states capacitance per unit area,qthe electronic charge, εsthe permittivity of silicon, NBthe bulk doping concentration,kthe Boltzmann's constant,Tthe absolute temperature, andE_{g0}the extrapolated value of the energy gap of lightly doped silicon atT = 0K. This theoretical formula was in good agreement with experimental results in a temperature range of interest.     

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}.     

An upper bound for codes for the noisy two-access binary adder channel (Corresp.)

Using earlier methods a combinatorial upper bound is derived for|C|. cdot |D|, where(C,D)is adelta-decodable code pair for the noisy two-access binary adder channel. Asymptotically, this bound reduces toR_{1}=R_{2} leq frac{3}{2} + elog_{2} e - (frac{1}{2} + e) log_{2} (1 + 2e)= frac{1}{2} - e + H(frac{1}{2} - e) - frac{1}{2}H(2e),wheree = lfloor (delta - 1)/2 rfloor /n, n rightarrow inftyandR_{1}resp.R_{2}is the rate of the codeCresp.D.     

An estimate of the variation of a band-limited process

Upper and lower bounds are established for the mean-square variation of a stationary processX(t)whose power spectrum is bounded byomega_{c}, in terms of its average powerP_{0}and the average powerP_{1}of its derivative. It is shown thatleft( frac{2}{pi} right)^{2} P_{1} tau^{2} leq E {|X(t+tau )-X(t)|^{2}} leq P_{1} tau^{2} leq omega_{c}^{2}P_{0}tau^{2}where the upper bounds are valid for anytauand the lower bound fortau < pi / omega_{c}. These estimates are applied to the mean-square variation of the envelope of a quasi-monochromatic process.