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.     

Successive encoding of correlated sources

The encoding of a discrete memoryless multiple source{( X_{i}, Y_{i})}_{i=1}^{infty}for reconstruction of a sequence{Z_{i}}_{i=1}^{infty}}, withZ_{i} = F( X_{i}, Y_{i}); i = 1,2, cdotsis considered. We require that the encoding should be such that{X_{i}}_{i=1}^{infty}is encoded first without any consideration of{Y_{i}}_{i=1}^{infty}, while in a second part of the encoding, this latter sequence is encoded based on knowledge of the outcome of the first encoding. The resulting scheme is called successive encoding. We find general outer and inner bounds for the corresponding set of achievable rates along with a complete single letter characterization for the special caseH( X_{i}|Z_{i}, Y_{i}) = 0. Comparisons with the Slepian-Wolf problem and the Ahlswede-Korner-Wyner side information problem are carried out.     

On single-sample robust detection of known signals with additive unknown-mean amplitude-bounded random interference--II: The randomized decision rule solution (Corresp.)

A randomized decision rule is derived and proved to be the saddlepoint solution of the robust detection problem for known signals in independent unknown-mean amplitude-bounded noise. The saddlepoint solutionphi^{0}uses an equaUy likely mixed strategy to chose one ofNBayesian single-threshold decision rulesphi_{i}^{0}, i = 1,cdots , Nhaving been obtained previously by the author. These decision rules are also all optimal against the maximin (least-favorable) nonrandomized noise probability densityf_{0}, wheref_{0}is a picket fence function withNpickets on its domain. Thee pair(phi^{0}, f_{0})is shown to satisfy the saddlepoint condition for probability of error, i.e.,P_{e}(phi^{0} , f) leq P_{e}(phi^{0} , f_{0}) leq P_{e}(phi, f_{0})holds for allfandphi. The decision rulephi^{0}is also shown to be an eqoaliir rule, i.e.,P_{e}(phi^{0}, f ) = P_{e}(phi^{0},f_{0}), for allf, with4^{-1} leq P_{e}(phi^{0},f_{0})=2^{-1}(1-N^{-1})leq2^{-1} , N geq 2. Thus nature can force the communicator to use an {em optimal} randomized decision rule that generates a large probability of error and does not improve when less pernicious conditions prevail.     

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 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 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.     

On the quality factor of strip and line source antennas and its relationship to superdirectivity ratio

Exact formulas are derived for the quality factorQof strip and line source antennas. Contrary to popular opinion, none of them is equal to Taylor's superdirectivity ratiogammaor togamma - 1. But in the case ofE-plane strip sources (the complement of the type of strip source treated by Woodward and Lawson) the value ofQis precisely equal togamma_{alpha}^{-1/2}- 1, wheregamma_{alpha}^{beta}is a generalized superdirectivity ratio that reduces to Taylor'sgammawhen the edge exponentalphaand the pattern weighting exponentbetaare both zero. In the case ofH-plane strip sources the value ofQis approximately equal togamma_{alpha}^{1/2} - 1, and forH-plane line sources of vanishing widthait is approximately equal to[(2/pi) ln (2.516lambda/pi a)]gamma_{alpha}^{1}.     

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.     

The imperfectly conducting cylindrical transmitting antenna: Numerical results

The complex wave number, the distribution of current, the admittance, and the radiating efficiency of cylindrical antennas made of imperfect conductors are evaluated numerically from a previously derived theory[1]. The quantity2lambda r^{i}/zeta_{0}(wherer^{i}is the resistance per unit length,lambdais the free-space wavelength, andzeta_{0} = 377ohms) is used as the parameter in a range that extends from zero to 200. Extensive graphs and tables are given.     

On the distribution of de Bruijn sequences of given complexity

The distributiongamma (c, n)of de Bruijn sequences of ordernand linear complexitycis investigated. It is shown that forn geq 4, gamma (2^{n} - 1, n) equiv 0 pmod{8}, and fork geq 3, gamma (2^{2k} - 1,2k) equiv 0 pmod{l6}. It is also shown thatgamma (c, n) equiv 0 pmod{4}for allc, andn geq 3such thatcnis even.     

Sliding-block coding for weakly continuous channels

For a weakly continuous stationary channel with operational sliding-block capacityC_{s}, it is shown that every stationary, ergodic, aperiodic source with entropy less thanC_{s}is sliding-block transmissible over the channel. It had been shown earlier that a stationary and ergodic source is block transmissible over the channel if its entropy is less than the operational block capacityC_{b}. If, in addition, the channel is ergodic, it is shown thatC_{s}=C_{b}. An example of a nonergodic weakly continuous channel is given for which0.     

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.     

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.     

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.     

Stability criterion for Gunn diodes with injection-limiting cathodes

According to McCumber [1] a Gunn diode with an ohmic cathode (i.e., "differential cathode conductivity"sigma_{c} = delta) is stable in a constant-voltage circuit ifn_{0}L le (n_{0}L)_{crit} equiv 2.7 times 10^{11}cm^-2wheren_{0}Lis the doping-length product. We show that the same stability criterion applies to Gunn diodes with an injection-limiting cathode(sigma_{c} rightarrow 0), if(n_{0}L)_{crit}is allowed to be a function ofsigma_{c}L. The value of(n_{0}L)_{crit}increases by 30 percent if(sigma_{c}L)varies from infinity (ohmic cathode) to zero (injection-limiting cathode). If a cathode with negative differential conductivity is realizable, it may be possible to extend the(n_{0}L)region of stable operation of Gunn diodes drastically.     

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.     

Some cross correlation properties for distorted signals

For a nondecreasing distortion characteristicphi(cdot)and a given signalx(cdot), the "cross correlation" function defined byR_{phi} (tau) triangleq int_{-infty}^{infty} phi[x(t)]x(t - tau) dtis shown to satisfy the inequalityR_{phi}(tau) leq R_{phi}(0), for alltau, generalizing an earlier result of Richardson that requiredphi(cdot)to be continuous and strictly increasing. The methods of the paper also show that, under weak conditions, begin{equation} R_{phi,psi}(tau) triangleq int_{-infty}^{infty} phi[x(t)]psi[x(t - tau)] dt leq R_{phi,psi}(0) end{equation} whenpsiis strictly increasing andphiis nondecreasing. In the case of hounded signals (e.g., periodic functions), the appropriate cross correlation function is begin{equation} mathcal{R}_{phi,psi}(tau} triangleq lim_{T rightarrow infty} (2T)^{-l} int_{-T}^T phi[x(t)]psi[x(t - tau)] dt. end{equation} For this case it is shown thatmathcal{R}_{phi,psi} (tau) leq mathcal{R}_{phi,psi}(0)for any nondecreasing (or nonincreasing) distortion functionsphiandpsi. The result is then applied to generalize an inequality on correlation functions for periodic signals due to Prosser. Noise signals are treated and inequalities of a similar nature are obtained for ensemble-average cross correlation functions under suitable hypotheses on the statistical properties of the noise. Inequalities of this type are the basis of a well-known method of estimating the unknown time delay of an observed signal. The extension to nondecreasing discontinuous distortion functions allows the use of hard limiting or quantization to facilitate the cross correlation calculation.     

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.     

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.