On the /spl theta/-coverage and connectivity of large random networks

Wireless planar networks have been used to model wireless networks in a tradition that dates back to 1961 to the work of E. N. Gilbert. Indeed, the study of connected components in wireless networks was the motivation for his pioneering work that spawned the modern field of continuum percolation theory. Given that node locations in wireless networks are not known, random planar modeling can be used to provide preliminary assessments of important quantities such as range, number of neighbors, power consumption, and connectivity, and issues such as spatial reuse and capacity. In this paper, the problem of connectivity based on nearest neighbors is addressed. The exact threshold function for /spl theta/-coverage is found for wireless networks modeled as n points uniformly distributed in a unit square, with every node connecting to its /spl phi//sub n/ nearest neighbors. A network is called /spl theta/-covered if every node, except those near the boundary, can find one of its /spl phi//sub n/ nearest neighbors in any sector of angle /spl theta/. For all /spl theta//spl isin/(0,2/spl pi/), if /spl phi//sub n/=(1+/spl delta/)log/sub 2/spl pi//2/spl pi/-/spl theta//n, it is shown that the probability of /spl theta/-coverage goes to one as n goes to infinity, for any /spl delta/>0; on the other hand, if /spl phi//sub n/=(1-/spl delta/)log/sub 2/spl pi//2/spl pi/-/spl theta//n, the probability of /spl theta/-coverage goes to zero. This sharp characterization of /spl theta/-coverage is used to show, via further geometric arguments, that the network will be connected with probability approaching one if /spl phi//sub n/=(1+/spl delta/)log/sub 2/n. Connections between these results and the performance analysis of wireless networks, especially for routing and topology control algorithms, are discussed.     

Asymptotic distribution of the number of isolated nodes in wireless ad hoc networks with Bernoulli nodes

Nodes in wireless ad hoc networks may become inactive or unavailable due to, for example, internal breakdown or being in the sleeping state. The inactive nodes cannot take part in routing/relaying, and thus may affect the connectivity. A wireless ad hoc network containing inactive nodes is then said to be connected, if each inactive node is adjacent to at least one active node and all active nodes form a connected network. This paper is the first installment of our probabilistic study of the connectivity of wireless ad hoc networks containing inactive nodes. We assume that the wireless ad hoc network consists of n nodes which are distributed independently and uniformly in a unit-area disk, and are active (or available) independently with probability p for some constant 0

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The rank and kernel of extended 1-perfect /spl Zopf//sub 4/-linear and additive non-/spl Zopf//sub 4/-linear codes

A binary extended 1-perfect code of length n + 1 = 2/sup t/ is additive if it is a subgroup of /spl Zopf//sub 2//sup /spl alpha// /spl times/ /spl Zopf//sub 4//sup /spl beta//. The punctured code by deleting a /spl Zopf//sub 2/ coordinate (if there is one) gives a perfect additive code. 1-perfect additive codes were completely characterized and by using that characterization we compute the possible parameters /spl alpha/, /spl beta/, rank, and dimension of the kernel for extended 1-perfect additive codes. A very special case is that of extended 1-perfect /spl Zopf//sub 4/-linear codes.     

Time-frequency localisation of Shannon wavelet packets

It is well known that the 2/spl pi/ minimally supported frequency scaling function /spl phi//sup /spl alpha//(x) satisfying /spl phi//spl circ//sup /spl alpha//(/spl omega/)=/spl chi//sub (-/spl alpha/,2/spl pi/-/spl alpha/)/(/spl omega/), 0相似文献   

Optimal linear identifying codes

Identifying codes can be used to locate malfunctioning processors. We say that a code C of length n is a linear (1,/spl les/l)-identifying code if it is a subspace of F/sub 2//sup n/ and for all X,Y/spl sube/F/sub 2//sup n/ such that |X|, |Y|/spl les/l and X/spl ne/Y, we have /spl cup//sub x/spl isin/X/(B(x)/spl cap/C)/spl ne//spl cup/y/spl isin/Y(B(y)/spl cap/C). Strongly (1,/spl les/l)-identifying codes are a variant of identifying codes. We determine the cardinalities of optimal linear (1,/spl les/l)-identifying and strongly (1,/spl les/l)-identifying codes in Hamming spaces of any dimension for locating any at most l malfunctioning processors.     

Order statistics from a logistic distribution and applications to survival and reliability analysis

Joint moments involving arbitrary powers of order statistics are the main concern. Consider order statistics u/sub 1/ /spl les/ u/sub 2/ /spl les/ /spl middot//spl middot//spl middot/ /spl les/ u/sub k/ coming from a simple random sample of size n from a real continuous population where u/sub 1/ = x/sub r(1):n/ is order-statistic #r/sub 1/, u/sub 2/ = x/sub r(1)+r(2):n/ is order statistic #(r/sub 1/ + r/sub 2/), et al., and u/sub k/ = x/sub r(1)+/spl middot//spl middot//spl middot/+r(k):n/ is order statistic #(r/sub 1/ +/spl middot//spl middot//spl middot/+ r/sub k/). Product moments are examined of the type E[u/sub 1//sup /spl alpha/(1)/ /spl middot/ u/sub 2//sup /spl alpha/(2)//sub /spl middot/ /spl middot//spl middot//spl middot//spl middot//u/sub k//sup /spl alpha/(k)/] where /spl alpha//sub 1/, ..., /spl alpha//sub k/ are arbitrary quantities that might be complex numbers, and E[/spl middot/] denotes the s-expected value. Some explicit evaluations are considered for a logistic population. Detailed evaluations of all integer moments of u/sub 1/ and recurrence relations, recurring only on the order of the moments, are given. Connections to survival functions in survival analysis, hazard functions in reliability situations, real type-1, type-2 /spl beta/ and Dirichlet distributions are also examined. Arbitrary product moments for the survival functions are evaluated. Very general results are obtained which can be used in many problems in various areas.     

A class of 1-generator quasi-cyclic codes

If R = F/sub q/[x/spl rceil/]/(x/sup m/ - 1), S = F/sub qn/[x]/(x/sup m/ - 1), we define the mapping a_(x) /spl rarr/ A(x) =/spl sigma//sub 0//sup n-1/a/sub i/(x)/spl alpha//sub i/ from R/sup n/ onto S, where (/spl alpha//sub 0/, /spl alpha//sub i/,..., /spl alpha//sub n-1/) is a basis for F/sub qn/ over F/sub q/. This carries the q-ray 1-generator quasicyclic (QC) code R a_(x) onto the code RA(x) in S whose parity-check polynomial (p.c.p.) is defined as the monic polynomial h(x) over F/sub q/ of least degree such that h(x)A(x) = 0. In the special case, where gcd(q, m) = 1 and where the prime factorizations of x/sub m/ 1 over F/sub q/ and F/sub qn/ are the same we show that there exists a one-to-one correspondence between the q-ary 1-generator quasis-cyclic codes with p.c.p. h(x) and the elements of the factor group J* /I* where J is the ideal in S with p.c.p. h(x) and I the corresponding quantity in R. We then describe an algorithm for generating the elements of J*/I*. Next, we show that if we choose a normal basis for F/sub qn/ over F/sub q/, then we can modify the aforementioned algorithm to eliminate a certain number of equivalent codes, thereby rending the algorithm more attractive from a computational point of view. Finally in Section IV, we show how to modify the above algorithm in order to generate all the binary self-dual 1-generator QC codes.     

Dynamic power allocation and routing for time-varying wireless networks

We consider dynamic routing and power allocation for a wireless network with time-varying channels. The network consists of power constrained nodes that transmit over wireless links with adaptive transmission rates. Packets randomly enter the system at each node and wait in output queues to be transmitted through the network to their destinations. We establish the capacity region of all rate matrices (/spl lambda//sub ij/) that the system can stably support-where /spl lambda//sub ij/ represents the rate of traffic originating at node i and destined for node j. A joint routing and power allocation policy is developed that stabilizes the system and provides bounded average delay guarantees whenever the input rates are within this capacity region. Such performance holds for general arrival and channel state processes, even if these processes are unknown to the network controller. We then apply this control algorithm to an ad hoc wireless network, where channel variations are due to user mobility. Centralized and decentralized implementations are compared, and the stability region of the decentralized algorithm is shown to contain that of the mobile relay strategy developed by Grossglauser and Tse (2002).     

Convergence and loss bounds for Bayesian sequence prediction

The probability of observing x/sub t/ at time t, given past observations x/sub 1/...x/sub t-1/ can be computed if the true generating distribution /spl mu/ of the sequences x/sub 1/x/sub 2/x/sub 3/... is known. If /spl mu/ is unknown, but known to belong to a class /spl Mscr/ one can base one's prediction on the Bayes mix /spl xi/ defined as a weighted sum of distributions /spl nu/ /spl isin/ /spl Mscr/. Various convergence results of the mixture posterior /spl xi//sub t/ to the true posterior /spl mu//sub t/ are presented. In particular, a new (elementary) derivation of the convergence /spl xi//sub t///spl mu//sub t/ /spl rarr/ 1 is provided, which additionally gives the rate of convergence. A general sequence predictor is allowed to choose an action y/sub t/ based on x/sub 1/...x/sub t-1/ and receives loss /spl lscr//sub x(t)y(t)/ if x/sub t/ is the next symbol of the sequence. No assumptions are made on the structure of /spl lscr/ (apart from being bounded) and /spl Mscr/. The Bayes-optimal prediction scheme /spl Lambda//sub /spl xi// based on mixture /spl xi/ and the Bayes-optimal informed prediction scheme /spl Lambda//sub /spl mu// are defined and the total loss L/sub /spl xi// of /spl Lambda//sub /spl xi// is bounded in terms of the total loss L/sub /spl mu// of /spl Lambda//sub /spl mu//. It is shown that L/sub /spl xi// is bounded for bounded L/sub /spl mu// and L/sub /spl xi///L/sub /spl mu// /spl rarr/ 1 for L/sub /spl mu// /spl rarr/ /spl infin/. Convergence of the instantaneous losses is also proven.     

A Catalog of Low Power Loss Parameters and High Power Thresholds for Partially Magnetized Ferrites

The low power loss and high power threshold properties have been measured on a number of candidate ferrite phase-shifting materials. The low power loss is characterized by /spl mu//sub 0/', the imaginary part of the diagonal component of the permeability tensor for the completely demagnetized state. /spl mu/sub 0/' was measured from 3.0 to 16.8 GHz. The high power properties are characterized by the parallel pump threshold at a bias field correspontig to H/sub i/ /spl equiv/ 0 and to 4/spl pi/M /spl equiv/ 4/spl pi/M/sub s/. The threshold was measured between 3.0 and 16.8 GHz. For the purposes of computer calculation /spl mu//sub 0/' and h/sub crit/ were fit to an equation of the form A (/spl gamma/4/spl pi/M/sub s/ / /spl omega/)/sup N/. Translating /spl mu//sub 0/' and h/sub crit/ to /spl Delta/H/sub eff/ and /spl Delta/H/sub k/ gives the YIG plus Al as the lowest loss and lowest threshold materials followed by the Gd garnets and MgMn spinels. The Ni spinels are very Iossy.     

Computer-Aided Design of Three-Port Waveguide Junction Circulators

The complete performance of a lossless three-port H-plane waveguide junction loaded coaxially with various inhomogeneous ferrite cylanders has been evaluated over the waveguide bandwidth and compared with experiment. Qualitative agreement between the predicted and measured performance was generally good using only the first three modes, n=0/spl plusmn/1. It has been shown theoretically and verified experimentally that if the 4/spl pi/M/sub s/ of a homogeneous rod or the internal field is increased, the circulation frequency f/sub 0/ increases; conversely, if the pemittivity is increased, f/sub 0/ decreases. These conflicting effects are modified when the magnetization 4/spl pi/M/sub s/ and permittivity /spl epsiv/ are inhomogeneous. For example, if the 4/spl pi/M/sub s/(/spl gamma/) is small at the outer surface of the rod (with permittivity held constant), the effect on f/sub 0/ is very small; but if 4/spl pi/M/sub s/(spl gamma) approaches zero for /spl gamma/ small, then f/sub 0/ may decrease significantly. On the other hand, if /spl epsiv//sub/spl gamma//(/spl gamma/) approaches unity near the outer surface of the rod, f/sub 0/ may increase significantly; but if /spl epsiv//sub/spl gamma//(/spl gamma/) approaches unity near the center of the rod, f/sub 0/ is affected relatively little. The inhomogeneous structure has also shown that decreasing the ferrite volume may improve the performance, and high-power applications are suggested. With a conducting pin down the center of the ferrite, relative bandwidths of 40-50 percent are predicted.     

A minimum cost heterogeneous sensor network with a lifetime constraint

We consider a heterogeneous sensor network in which nodes are to be deployed over a unit area for the purpose of surveillance. An aircraft visits the area periodically and gathers data about the activity in the area from the sensor nodes. There are two types of nodes that are distributed over the area using two-dimensional homogeneous Poisson point processes; type 0 nodes with intensity (average number per unit area) /spl lambda//sub 0/ and battery energy E/sub 0/; and type 1 nodes with intensity /spl lambda//sub 1/ and battery energy E/sub 1/. Type 0 nodes do the sensing while type 1 nodes act as the cluster heads besides doing the sensing. Nodes use multihopping to communicate with their closest cluster heads. We determine them optimum node intensities (/spl lambda//sub 0/, /spl lambda//sub 1/) and node energies (E/sub 0/, E/sub 1/) that guarantee a lifetime of at least T units, while ensuring connectivity and coverage of the surveillance area with a high probability. We minimize the overall cost of the network under these constraints. Lifetime is defined as the number of successful data gathering trips (or cycles) that are possible until connectivity and/or coverage are lost. Conditions for a sharp cutoff are also taken into account, i.e., we ensure that almost all the nodes run out of energy at about the same time so that there is very little energy waste due to residual energy. We compare the results for random deployment with those of a grid deployment in which nodes are placed deterministically along grid points. We observe that in both cases /spl lambda//sub 1/ scales approximately as /spl radic/(/spl lambda//sub 0/). Our results can be directly extended to take into account unreliable nodes.     

On The General Relation Between /spl alpha/ and Q (Correspondence)

The relation between the quality factor Q and the attenuation constant /spl alpha/ of a transmission line has been known as follows: /spl alpha/ = /spl beta/ / /2Q where /spl beta/ is the phase constant. Recently from the following relation of propagation constant at resonance /spl Gamma/(/spl omega//sub 0/) + /spl part//spl Gamma/ / /spl part//spl omega/ /spl Delta//spl omega//spl cong/ i/spl beta/(/spl omega//sub 0/), where /spl Gamma/(/spl omega//sub 0/) = /spl alpha/(/spl omega//sub 0/) + i/spl beta/(/spl omega//sub 0/). Yeh derived a general relation between Q and /spl alpha/, namely, /spl alpha/ = /spl upsi//sub p/ / /spl upsi//sub g/ /spl beta/ / /2Q where /spl upsi//sub p/, and /spl upsi//sub g/ are the phase velocity and group velocity of the wave respectively. This general relation can be derived very simply from the generally accepted definition of /spl alpha/ and Q.     

Semi-empirical model of the ensemble-averaged differential Mueller matrix for microwave backscattering from bare soil surfaces

A semi-empirical model of the ensemble-averaged differential Mueller matrix for microwave backscattering from bare soil surfaces is presented. Based on existing scattering models and data sets measured by polarimetric scatterometers and the JPL AirSAR, the parameters of the co-polarized phase-difference probability density function, namely the degree of correlation /spl alpha/ and the co-polarized phase-difference /spl sigmav/, in addition to the backscattering coefficients /spl sigma//sub /spl nu//spl nu///sup 0/,/spl sigma//sub hh//sup 0/ and /spl sigma//sub /spl nu/h//sup 0/, are modeled empirically in terms of the volumetric soil moisture content m/sub /spl nu// and the surface roughness parameters ks and kl, where k=2/spl pi/f/c, s is the rms height and l is the correlation length. Consequently, the ensemble-averaged differential Mueller matrix (or the differential Stokes scattering operator) is specified completely by /spl sigma//sub /spl nu//spl nu///sup 0/,/spl sigma//sub hh//sup 0/,/spl sigma//sub /spl nu/h//sup 0/,/spl alpha/, and /spl zeta/.     

The inequalities of quantum information theory

Let /spl rho/ denote the density matrix of a quantum state having n parts 1, ..., n. For I/spl sube/N={1, ..., n}, let /spl rho//sub I/=Tr/sub N/spl bsol/I/(/spl rho/) denote the density matrix of the state comprising those parts i such that i/spl isin/I, and let S(/spl rho//sub I/) denote the von Neumann (1927) entropy of the state /spl rho//sub I/. The collection of /spl nu/=2/sup n/ numbers {S(/spl rho//sub I/)}/sub I/spl sube/N/ may be regarded as a point, called the allocation of entropy for /spl rho/, in the vector space R/sup /spl nu//. Let A/sub n/ denote the set of points in R/sup /spl nu// that are allocations of entropy for n-part quantum states. We show that A~/sub n/~ (the topological closure of A/sub n/) is a closed convex cone in R/sup /spl nu//. This implies that the approximate achievability of a point as an allocation of entropy is determined by the linear inequalities that it satisfies. Lieb and Ruskai (1973) have established a number of inequalities for multipartite quantum states (strong subadditivity and weak monotonicity). We give a finite set of instances of these inequalities that is complete (in the sense that any valid linear inequality for allocations of entropy can be deduced from them by taking positive linear combinations) and independent (in the sense that none of them can be deduced from the others by taking positive linear combinations). Let B/sub n/ denote the polyhedral cone in R/sup /spl nu// determined by these inequalities. We show that A~/sub n/~=B/sub n/ for n/spl les/3. The status of this equality is open for n/spl ges/4. We also consider a symmetric version of this situation, in which S(/spl rho//sub I/) depends on I only through the number i=/spl ne/I of indexes in I and can thus be denoted S(/spl rho//sub i/). In this case, we give for each n a finite complete and independent set of inequalities governing the symmetric allocations of entropy {S(/spl rho//sub i/)}/sub 0/spl les/i/spl les/n/ in R/sup n+1/.     

Compression mappings on primitive sequences over Z/(p/sup e/)

Let Z/(p/sup e/) be the integer residue ring with odd prime p/spl ges/5 and integer e/spl ges/2. For a sequence a_ over Z/(p/sup e/), there is a unique p-adic expansion a_=a_/sub 0/+a_/spl middot/p+...+a_/sub e-1//spl middot/p/sup e-1/, where each a_/sub i/ is a sequence over {0,1,...,p-1}, and can be regarded as a sequence over the finite field GF(p) naturally. Let f(x) be a primitive polynomial over Z/(p/sup e/), and G'(f(x),p/sup e/) the set of all primitive sequences generated by f(x) over Z/(p/sup e/). Set /spl phi//sub e-1/ (x/sub 0/,...,x/sub e-1/) = x/sub e-1//sup k/ + /spl eta//sub e-2,1/(x/sub 0/, x/sub 1/,...,x/sub e-2/) /spl psi//sub e-1/(x/sub 0/,...,x/sub e-1/) = x/sub e-1//sup k/ + /spl eta//sub e-2,2/(x/sub 0/,x/sub 1/,...,x/sub e-2/) where /spl eta//sub e-2,1/ and /spl eta//sub e-2,2/ are arbitrary functions of e-1 variables over GF(p) and 2/spl les/k/spl les/p-1. Then the compression mapping /spl phi//sub e-1/:{G'(f(x),p/sup e/) /spl rarr/ GF(p)/sup /spl infin// a_ /spl rarr/ /spl phi//sub e-1/(a_/sub 0/,...,a_/sub e-1/) is injective, that is, a_ = b_ if and only if /spl phi//sub e-1/(a_/sub 0/,...,a_/sub e-1/) = /spl phi//sub e-1/(b_/sub 0/,...,b_/sub e-1/) for a_,b_ /spl isin/ G'(f(x),p/sup e/). Furthermore, if f(x) is a strongly primitive polynomial over Z/(p/sup e/), then /spl phi//sub e-1/(a_/sub 0/,...,a_/sub e-1/) = /spl psi//sub e-1/(b_/sub 0/,...,b_/sub e-1/) if and only if a_ = b_ and /spl phi//sub e-1/(x/sub 0/,...,x/sub e-1/) = /spl psi//sub e-1/(x/sub 0/,...,x/sub e-1/) for a_,b_ /spl isin/ G'(f(x),p/sup e/).     

On /spl Zopf//sub 4/-linear Preparata-like and Kerdock-like codes

We say that a binary code of length n is additive if it is isomorphic to a subgroup of /spl Zopf//sub 2//sup /spl alpha// /spl times/ /spl Zopf//sub 4//sup /spl beta//, where the quaternary coordinates are transformed to binary by means of the usual Gray map and hence /spl alpha/ + 2/spl beta/ = n. In this paper, we prove that any additive extended Preparata (1968) -like code always verifies /spl alpha/ = 0, i.e., it is always a /spl Zopf//sub 4/-linear code. Moreover, we compute the rank and the dimension of the kernel of such Preparata-like codes and also the rank and the kernel of the /spl Zopf//sub 4/-dual of these codes, i.e., the /spl Zopf//sub 4/-linear Kerdock-like codes.     

Mode Propagation in a Nonuniform Cylindrical Medium (Correspondence)

A simplified form of the coupling coefficient C(/spl beta//sub p/, /spl beta//sub q/) resulting from a coupled mode theory analysis of wave propagation in a nonuniform medium is derived. It is found for most situations of interest that C(/spl beta//sub p/, /spl beta//sub q/) is proportional to 1/(/spl beta//sub p/-/spl beta//sub q/) and the power transfer between two modes is proportional to 1/(/spl beta//sub p/ - /spl beta//sub q/)/sup 4/. /spl beta//sub p/ and /spl beta//sub q/ are the two different modal propagation constants. For a dielectric rod C(/spl beta//sub p/, /spl beta//sub q/) is a simple line integral around the rod boundary. Approximate forms are presented for optical waveguides.     

Achievable key rates for universal simulation of random data with respect to a set of statistical tests

We consider the problem of universal simulation of an unknown source from a certain parametric family of discrete memoryless sources, given a training vector X from that source and given a limited budget of purely random key bits. The goal is to generate a sequence of random vectors {Y/sub i/}, all of the same dimension and the same probability law as the given training vector X, such that a certain, prescribed set of M statistical tests will be satisfied. In particular, for each statistical test, it is required that for a certain event, /spl epsiv//sub /spl lscr//, 1 /spl les/ /spl lscr/ /spl les/ M, the relative frequency /sup 1///sub N/ /spl Sigma//sub i=1//sup N/ 1/sub /spl epsiv//spl lscr//(Y/sub i/) (1/sub /spl epsiv//(/spl middot/) being the indicator function of an event /spl epsiv/), would converge, as N /spl rarr/ /spl infin/, to a random variable (depending on X), that is typically as close as possible to the expectation of 1/sub /spl epsiv//spl lscr/,/ (X) with respect to the true unknown source, namely, to the probability of the event /spl epsiv//sub /spl lscr//. We characterize the minimum key rate needed for this purpose and demonstrate how this minimum can be approached in principle.     

Efficient routing and wavelength assignment for reconfigurable WDM ring networks with wavelength converters

We consider the problem of wavelength assignment in reconfigurable WDM networks with wavelength converters. We show that for N-node P-port bidirectional rings, a minimum number of /spl lceil/PN/4/spl rceil/ wavelengths are required to support all possible connected virtual topologies in a rearrangeably nonblocking fashion, and provide an algorithm that meets this bound using no more than /spl lceil/PN/2/spl rceil/ wavelength converters. This improves over the tight lower bound of /spl lceil/PN/3/spl rceil/ wavelengths required for such rings given in if no wavelength conversion is available. We extend this to the general P-port case where each node i may have a different number of ports P/sub i/, and show that no more than /spl lceil//spl sigma//sub i/P/sub i//4/spl rceil/+1 wavelengths are required. We then provide a second algorithm that uses more wavelengths yet requires significantly fewer converters. We also develop a method that allows the wavelength converters to be arbitrarily located at any node in the ring. This gives significant flexibility in the design of the networks. For example, all /spl lceil/PN/2/spl rceil/ converters can be collocated at a single hub node, or distributed evenly among the N nodes with min{/spl lceil/P/2/spl rceil/+1,P} converters at each node.