Error performance analysis for an optical heterodyne FSK communication system with a limiter-discriminator-integrator detector

The error probability for a coherent optical heterodyne FSK system with a limiter-discriminator-integrator (LDI) detector is analysed. The analysis includes laser quantum phase noise, the correlated receiver additive white Gaussian noise (AWGN), Gaussian narrow-band IF filtering and intersymbol interference (ISI) caused by it. It is shown that, for 1 dBEb/N ₀penalty at bit error rate (BER) 10^–9, (i) the normalized IF beat spectral linewidth T0.35% for frequency deviation ratioh=0.5(MSK), and T0.5% forh=0.7 (the receiver is insensitive to laser quantum phase noise ath=1.0, if ISI is not included.); (ii) if ISI is incorporated, T0.15% forh=0.5, T0.5% forh=0.7, both with 3dB bandwidth-bit period product (3.0>BT1.5), and T0.5% forh=1.0 with BT1.0 ifT0.35% when ISI exists,h=0.7 is optimum;h=1.0 otherwise.     

Exact calculation of one-dimensional irreducible integrals

One-dimensional irreducible integrals (_k) are computed in the form of Mayerf-function polynomials for a general interparticle potential. Obeisance to the exact specification of the irreducible integral definition produces regularities in the interaction of star graphs with the integration process. Tables of _k fork 5 and test solutions are presented.     

Current distributions in a two-dimensional random-resistor network

The current and logarithm-of-the-current distributionsn(i) andn(ln i) on bond diluted two-dimensional random-resistor networks at the percolation threshold are studied by a modified transfer matrix method. Thek th moment (–9k8) of n(ln i) i.e., ln i&^k, is found to scale with the linear sizeL as (InL)^(k). The exponents (k) are not inconsistent with the recent theoretical prediction (k)=k, with deviations which may be attributed to severe finitesize effects. For small currents, ln n(y)–, yielding information on the threshold below which the multifractality of (i) breaks down. Our numerical results for the moments of the currents are consistent with other available results.     

The lattice covering time problem for sites visitedk-times

The problem of the covering time for sites visitedk-times is defined as the mean time taken by a random walk to visit each site of a lattice at leastk times. We performed the investigation using Monte Carlo simulations over one dimensional lattices, ofN sites, with periodic boundary conditions. Two different regions are investigated:Nk1 andkN1. In the former region, we obtain a behaviour of the typet _k/t ₁=a –Bk ^–0.35+A(k)N ^–0.75, (a <1.6). In the latter region we obtain two possible behaviours:t _k k ^0.95 andt _k k(lnk)^–0.5. Two formulas which have a very close behaviour.     

On the critical exponent for random walk intersections

The exponent _d for the probability of nonintersection of two random walks starting at the same point is considered. It is proved that 1/2<₂3/4. Monte Carlo simulations are done to suggest ₂=0.61 and ₃0.29.     

The pivot algorithm: A highly efficient Monte Carlo method for the self-avoiding walk

The pivot algorithm is a dynamic Monte Carlo algorithm, first invented by Lal, which generates self-avoiding walks (SAWs) in a canonical (fixed-N) ensemble with free endpoints (hereN is the number of steps in the walk). We find that the pivot algorithm is extraordinarily efficient: one effectively independent sample can be produced in a computer time of orderN. This paper is a comprehensive study of the pivot algorithm, including: a heuristic and numerical analysis of the acceptance fraction and autocorrelation time; an exact analysis of the pivot algorithm for ordinary random walk; a discussion of data structures and computational complexity; a rigorous proof of ergodicity; and numerical results on self-avoiding walks in two and three dimensions. Our estimates for critical exponents are=0.7496±0.0007 ind=2 and= 0.592±0.003 ind=3 (95% confidence limits), based on SAWs of lengths 200N10000 and 200N 3000, respectively.     

Symmetries and correlation inequalities for classicaln-vector models

We describe a new class of single spin measures on then-dimensional sphereS _r ⁿ of radiusr (n 4) for which Lebowitz-type [J. Lebowitz,J. Stat. Phys. 16:463 (1977)] inequalities hold. This is achieved by an appropriate parametrization ofS _r ⁿ . The above class includes the uniform measures onxs Rⁿ ¦x¦ r for any 0 p r. The second topic of this paper is an abstract formulation of the first Griffiths inequality [R. B. Griffiths,J. Math. Phys. 8:478 (1967)] and the underlying symmetry property.     

Random surfaces with two-sided constraints: An application of the theory of dominant ground states

We consider some models of classical statistical mechanics which admit an investigation by means of the theory of dominant ground states. Our models are related to the Gibbs ensemble for the multidimensional SOS model with symmetric constraints _x m/2. The main result is that for ₀, where ₀ does not depend onm, the structure of thermodynamic phases in the model is determined by dominant ground states: for an evenm a Gibbs state is unique and for an oddm the number of space-periodic pure Gibbs states is two.     

Multiple scattering in random media I. Restricted two-body additive approximation

We present a microscopic theory of the problem of finding the properties of a particle interacting with potentials located at random sites. The sites are governed by a general probability distribution. The starting point is the multiple scattering equations for the amplitude k ₁|T |k ₂ in terms of the individual scattering amplitudes k ₁|T |k ₂. We work with quantitiesA defined by k ₁|T |k ₂=k ₁|T |k ₂exp[i(k ₁–k ₂)R ]. The theory is based on a splitting of the fundamental equation forA into equations for the mean A and the fluctuationsAA . Neglect of the fluctuations yields the quasicrystalline approximation. We rearrange the equation forAA to isolate the collective part of the fluctuations. We then make the simplest microscopic truncation which is thatAA is a restricted two-body additive function of the site positions. With the contribution of the collective fluctuations, this yields results forA that are accurate to ordert ⁴.Work supported in part by the National Science Foundation under Contract No. NSF DMRWork supported in part by the National Science Foundation under Contract No. NSF DMR     

Analytic study of power spectra of the tent maps near band-splitting transitions

Successive band-splitting transitions occur in the one-dimensional map x_i+1=g(x_i),i=0, 1, 2,... withg(x)=x, (0 x 1/2) –x +, (1/2 <x 1) as the parameter is changed from 2 to 1. The transition point fromN (=2ⁿ) bands to 2Nbands is given by=(2)^1/N (n=0, 1,2,...). The time-correlation function _i=x_ix₀/(x₀)²,x_i x_i–x_i is studied in terms of the eigenvalues and eigenfunctions of the Frobenius-Perron operator of the map. It is shown that, near the transition point=2, _i–[(10–42)/17] _i,0-[(102-8)/51]_i,1 + [(7 + 42)/17](–1)ⁱe^–yi, where2(–2) is the damping constant and vanishes at=2, representing the critical slowing-down. This critical phenomenon is in strong contrast to the topologically invariant quantities, such as the Lyapunov exponent, which do not exhibit any anomaly at=2. The asymptotic expression for _i has been obtained by deriving an analytic form of _i for a sequence of which accumulates to 2 from the above. Near the transition point=(2)^1/N, the damping constant of _i fori N is given by _N=2(^N-2)/N. Numerical calculation is also carried out for arbitrary a and is shown to be consistent with the analytic results.     

Correlation inequalities for the truncated two-point function of an Ising ferromagnet

We establish the following new correlation inequalities for the truncated twopoint function of an Ising ferromagnet in a positive external field: _j ; _l ^T _j ; _k ^T _k ; _l ^T , and _j ; _l ^T _{k K j} ; _k ^T _{k l} , whereK is any set of sites which separatesj froml. The inequalities are also valid for the pure phases with zero magnetic field at all temperatures. Above the critical temperature they reduce to known inequalities of Griffiths and Simon, respectively.NSERC Postgraduate Fellow, 1978–1981. Research supported in part by NSF Grant No. PHY-78-25390-A02.     

More inequalities for critical exponents

A variety of rigorous inequalities for critical exponents is proved. Most notable is the low-temperature Josephson inequalitydv +2 2–. Others are 1 1 +v, 1 1 , 1,d 1 + 1/ (for d),dv, ₃ + (for d), ₄ , and _{2m 2m+2} (form 2). The hypotheses vary; all inequalities are true for the spin-1/2 Ising model with nearest-neighbor ferromagnetic pair interactions.NSF Predoctoral Fellow (1976–1979). Research supported in part by NSF Grant PHY 78-23952.     

Spatial structure in diffusion-limited two-particle reactions

We analyze the limiting behavior of the densities _A(t) and _B(t), and the random spatial structure(r) = ( _A(t)., _B(t)), for the diffusion-controlled chemical reaction A+Binert. For equal initial densities _B(0) = _b(0) there is a change in behavior fromd 4, where _A(t) = _B(t) C/t^d/4, tod 4, where _A(t) = _b(t) C/t ast ; the termC depends on the initial densities and changes withd. There is a corresponding change in the spatial structure. Ind < 4, the particle types separate with only one type present locally, and , after suitable rescaling, tends to a random Gaussian process. Ind >4, both particle types are, after large times, present locally in concentrations not depending on type or location. Ind=4, both particle types are present locally, but with random concentrations, and the process tends to a limit.     

Random walks on lattices. IV. Continuous-time walks and influence of absorbing boundaries

The general study of random walks on a lattice is developed further with emphasis on continuous-time walks with an asymmetric bias. Continuous time walks are characterized by random pauses between jumps, with a common pausing time distribution(t). An analytic solution in the form of an inverse Laplace transform for P(l, t), the probability of a walker being atl at timet if it started atl _o att=0, is obtained in the presence of completely absorbing boundaries. Numerical results for P(l, t) are presented for characteristically different (t), including one which leads to a non-Gaussian behavior for P(l, t) even for larget. Asymptotic results are obtained for the number of surviving walkers and the mean l showing the effect of the absorption at the boundary.This study was partially supported by ARPA and monitored by ONR(N00014-17-0308).     

On interesting walks in a graph

Notions of interesting walks and of their equivalence are introduced. A general formula for the number _l, of equivalence classes of interesting walks of lengthl in a given graphG is derived and applied forl 5 so as to express _l in terms of the adjacency matrix ofG.     

The projection approach to the Fokker-Planck equation. I. Colored Gaussian noise

It is shown that the Fokker-Planck operator can be derived via a projection-perturbation approach, using the repartition of a more detailed operator into a perturbation ₁ and an unperturbed part ₀. The standard Fokker-Planck structure is recovered at the second order in ₁, whereas the perturbation terms of higher order are shown to provoke the breakdown of this structure. To get rid of these higher order terms, a key approximation, local linearization (LL), is made. In general, to evaluate at the second order in ₁ the exact expression of the diffusion coefficient which simulates the influence of a Gaussian noise with a finite correlation time, a resummation up to infinite order in must be carried out, leading to what other authors call the best Fokker-Planck approximation (BFPA). It is shown that, due to the role of terms of higher order in ₁, the BFPA leads to predictions on the equilibrium distributions that are reliable only up to the first order in t. The LL, on the contrary, in addition to making the influence of terms of higher order in ₁ vanish, results in a simple analytical expression for the term of second order that is formally coincident with the complete resummation over all the orders in t provided by the Fox theory. The corresponding diffusion coefficient in turn is shown to lead in the limiting case to exact results for the steady-state distributions. Therefore, over the whole range 0 the LL turns out to be an approximation much more accurate than the global linearization proposed by other authors for the same purpose of making the terms of higher order in ₁ vanish. In the short- region the LL leads to results virtually coincident with those of the BFPA. In the large- region the LL is a more accurate approximation than the BFPA itself. These theoretical arguments are supported by the results of both analog and digital simulation.     

Singular Perturbations of Self-Adjoint Operators

Singular finite rank perturbations of an unbounded self-adjoint operator A ₀ in a Hilbert space ₀ are defined formally as A ₍₎=A ₀+GG ^*, where G is an injective linear mapping from = ^d to the scale space _-k(A₀)k , kN, of generalized elements associated with the self-adjoint operator A ₀, and where is a self-adjoint operator in . The cases k=1 and k=2 have been studied extensively in the literature with applications to problems involving point interactions or zero range potentials. The scalar case with k=2n>1 has been considered recently by various authors from a mathematical point of view. In this paper, singular finite rank perturbations A ₍₎ in the general setting ran G – _k (A ₀), kN, are studied by means of a recent operator model induced by a class of matrix polynomials. As an application, singular perturbations of the Dirac operator are considered.     

Single random walker on disordered lattices

Random walks on square lattice percolating clusters were followed for up to 2×105 steps. The mean number of distinct sites visited (S _N > gives a spectral dimension ofd _s = 1.30±0.03 consistent with superuniversality (d _s =4J3) but closer to the alternatived _s = 182/139, based on the low dimensionality correction. Simulations are also given for walkers on anenergetically disordered lattice, with a jump probability that depends on the local energy mismatch and the temperature. An apparent fractal behavior is observed for a low enough reduced temperature. Above this temperature, the walker exhibits a crossover from fractal-to-Euclidean behavior. Walks on two- and three-dimensional lattices are similar, except that those in three dimensions are more efficient.Supported by NSF Grant No. DMR 8303919 and Nato Grant No. SA 5205 RG 295J82.     

A characterization of Robertson-Walker spaces by null sectional curvature

ForN a null vector andA a vector perpendicular toN, define the null sectional curvature, with respect to TV, of the planeN A ask _N(N A) = R(N,A)A,N<A,A.Then Robertson-Walker metrics can be locally characterized as those for whichk _n at each point is a constant for all the null plans at that point (in each null direction,N must be appropriately chosen). A global characterization of Robertson-Walker spaces is achieved by adding completeness and causality hypotheses.     

The asymmetric contact process on a finite set

The contact process onZ has one phase transition; let _c be the critical value at which the transition occurs. Let _N be the extinction time of the contact process on {0,...,N}. Durrett and Liu (1988), Durrett and Schonmann (1988), and Durrett, Schonmann, and Tanaka (1989) have respectively proved that the subcritical, supercritical, and critical phases can be characterized using a large finite system (instead ofZ) in the following way. There are constants ₁() and ₂() such that if < _c , lim _{N N} /logN = 1/₁(); if > _c , lim _N log _N /N = ₂(); if = _c , lim _{N N} /N= and lim _{N N} /N ⁴=0 in probability. In this paper we consider the asymmetric contact process onZ when it has two distinct critical values _c1< _c2. The arguments of Durrett and Liu and of Durrett and Schonmann hold for < _c1 and > _c2. We show that for [ _c1< _c2), lim _{N N} /N=-1/, (where _i is an edge speed) and for = _c2, lim _N log _N /logN=2 in probability.