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.     

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.     

Short convolutional codes with maximal free distance for rates 1/2, 1/3, and 1/4 (Corresp.)

This paper gives a tabulation of binary convolutional codes with maximum free distance for ratesfrac{1}{2}, frac{1}{3}, andfrac{1}{4}for all constraint lengths (measured in information digits)nuup to and includingnu = 14. These codes should be of practical interest in connection with Viterbi decoders.     

The imperfectly conducting cylindrical transmitting antenna

The properties of a cylindrical antenna with a continuous ohmic resistance along its length are of interest in the design of certain types of directive broadband antennas and in the determination of the efficiency of dipole antennas. Conventionally, the contribution by ohmic resistance to the distribution of current and the impedance is contained in a particular integral that is either ignored or treated as a higher-order correction to formulas derived for perfectly conducting antennas. An alternative and more useful form has been developed in which the integral equation for the current is rearranged to permit the introduction of a complex wave numberk. An approximate solution of this equation is then obtained in terms of the three trigonometric functions,sin k(h-|z|),cos kz-cos kh, andcos frac{1}{2}k_{0}z - cos frac{1}{2}k_{0}h, wherek_{0}is the free-space wave number. Expressions are derived for the coefficients of these functions and fork. Explicit formulas are given for the distribution of current and the admittance.     

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).     

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.     

A theorem on the entropy of certain binary sequences and applications--I

In this, the first part of a two-part paper, we establish a theorem concerning the entropy of a certain sequence of binary random variables. In the sequel we will apply this result to the solution of three problems in multi-user communication, two of which have been open for some time. Specifically we show the following. LetXandYbe binary randomn-vectors, which are the input and output, respectively, of a binary symmetric channel with "crossover" probabilityp_0. LetH{X}andH{ Y}be the entropies ofXandY, respectively. Then begin{equation} begin{split} frac{1}{n} H{X} geq h(alpha_0), qquad 0 leq alpha_0 &leq 1, Rightarrow \ qquad qquad &qquad frac{1}{n}H{Y} geq h(alpha_0(1 - p_0) + (1 - alpha_0)p_0) end{split} end{equation} whereh(lambda) = -lambda log lambda - (1 - lambda) log(l - lambda), 0 leq lambda leq 1.     

A First-Order Markov Model for Understanding Delta Modulation Noise Spectra

A first-order Markov process is used to model the sequence of quantization noise samples in delta modulation. An autocorrelation parameterCin the Markov model controls the shape of the noise spectrum, and asCdecreases from 1 to 0 and then to -1, the spectrum changes from a low-pass to a flat, and then to a high-pass characteristic. One can also use the Markov model to predict the so-called out-of-band noise rejection that is obtained when delta modulation is performed with an oversampled input, and the resulting quantization noise is lowpass filtered to the input band. The noise rejectionGis a function ofCas well as an oversampling factorFand an interesting asymptotic result is thatG=frac{1-C}{1+C} dot FifF gg frac{1+C}{1-C} dot frac{pi}{2}. Delta modulation literature has noted the importance of the specialGvalues,Fand2F. These correspond to autocorrelation values of 0 and -1/3.     

A fundamental inequality between the probabilities of binary subgroups and cosets

The probability of a set of binaryn-tuples is defined to be the sum of the probabilities of the individualn-tuples when each digit is chosen independently with the same probabilitypof being a "one." It is shown that, under such a definition, the ratio between the probability of a subgroup of order2^{k}and any of its proper cosets is always greater than or equal to a functionF_{k}(p), whereF_{k}(p) geq 1forp leq frac{1}{2}with equality when and only whenp = frac{1}{2}. It is further shown thatF_{k}(p)is the greatest lower bound on this ratio, since a subgroup and proper coset of order2^{k}can always be found such that the ratio between their probabilities is exactlyF_{k}(p). It is then demonstrated that for a linear code on a binary symmetric channel the "tall-zero" syndrome is more probable than any other syndrome. This result is applied to the problem of error propagation in convolutional codes.     

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.     

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.     

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.     

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.     

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.     

An equivalent circuit for microwave noise at the potential minimum

It has been suggested that an inductive element should be added to Watkins RC equivalent circuit for the microwave noise current fluctuations at the potential minimum of a diode, and that this added element gives closer agreement with more recent work which shows a pronounced peak in noise current near the plasma resonance,a = frac{1}{2}. This paper presents an approximate analytical argument for the inclusion of an inductive element, based on the density-function method of analysis, which predicts the proper value for the inductance and yields a curve of reduction factor vs frequency in agreement with more detailed theories. The value of the inductance is such as to resonate with the capacitance of the potential minimum ata = frac{1}{2}. A discussion of the linearity of the potential minimum and of the idealized potential-minimum model is also included.     

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.     

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.     

Concentration-dependent diffusion of boron and phosphorus in silicon

A calculation is made of the tail of the impurity concentration profile resulting from concentration-dependent diffusion from a constant surface concentration into a semi-infinite medium. The calculation predicts that if the concentration dependence at low impurity concentrations is negligible, the low concentration portion of the doping profile should still take the familiar form,C = C'_{s} erfc (x/2 D_{i^{frac{1}{2}}}t^{frac{1}{2}}). Diis the commonly known diffusion coefficient at low impurity concentrations, whileC'_{s}is the "apparent" surface concentration.C'_{s}depends on the actual surface concentration and also depends on how the diffusion coefficient varies with impurity concentration at high concentrations. It is a constant for a given diffusion system but could be orders of magnitude higher than the actual surface concentration. Empirical data have been obtained for boron and phosphorus diffusions in silicon and found to be in good agreement with this prediction.     

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.