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.     

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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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.     

Detection of buried targets via active selection of labeled data: application to sensing subsurface UXO

When sensing subsurface targets, such as landmines and unexploded ordnance (UXO), the target signatures are typically a strong function of environmental and historical circumstances. Consequently, it is difficult to constitute a universal training set for design of detection or classification algorithms. In this paper, we develop an efficient procedure by which information-theoretic concepts are used to design the basis functions and training set, directly from the site-specific measured data. Specifically, assume that measured data (e.g., induction and/or magnetometer) are available from a given site, unlabeled in the sense that it is not known a priori whether a given signature is associated with a target or clutter. For N signatures, the data may be expressed as {x/sub i/,y/sub i/}/sub i=1,N/, where x/sub i/ is the measured data for buried object i, and y/sub i/ is the associated unknown binary label (target/nontarget). Let the N x/sub i/ define the set X. The algorithm works in four steps: 1) the Fisher information matrix is used to select a set of basis functions for the kernel-based algorithm, this step defining a set of n signatures B/sub n//spl sube/X that are most informative in characterizing the signature distribution of the site; 2) the Fisher information matrix is used again to define a small subset X/sub s//spl sube/X, composed of those x/sub i/ for which knowledge of the associated labels y/sub i/ would be most informative in defining the weights for the basis functions in B/sub n/; 3) the buried objects associated with the signatures in X/sub s/ are excavated, yielding the associated labels y/sub i/, represented by the set Y/sub s/; and 4) using B/sub n/,X/sub s/, and Y/sub s/, a kernel-based classifier is designed for use in classifying all remaining buried objects. This framework is discussed in detail, with example results presented for an actual buried-UXO site.     

A critical analysis on global convergence of Hopfield-type neural networks

This paper deals with the global convergence and stability of the Hopfield-type neural networks under the critical condition that M/sub 1/(/spl Gamma/)=L/sup -1/D/spl Gamma/-(/spl Gamma/W+W/sup T//spl Gamma/)/(2) is nonnegative for any diagonal matrix /spl Gamma/, where W is the weight matrix of the network, L=diag{L/sub 1/,L/sub 2/,...,L/sub N/} with L/sub i/ being the Lipschitz constant of g/sub i/ and G(u)=(g/sub 1/(u/sub 1/),g/sub 2/(u/sub 2/),...,g/sub N/(u/sub N/))/sup T/ is the activation mapping of the network. Many stability results have been obtained for the Hopfield-type neural networks in the noncritical case that M/sub 1/(/spl Gamma/) is positive definite for some positive definite diagonal matrix /spl Gamma/. However, very few results are available on the global convergence and stability of the networks in the critical case. In this paper, by exploring two intrinsic features of the activation mapping, two generic global convergence results are established in the critical case for the Hopfield-type neural networks, which extend most of the previously known globally asymptotic stability criteria to the critical case. The results obtained discriminate the critical dynamics of the networks, and can be applied directly to a group of Hopfield-type neural network models. An example has also been presented to demonstrate both theoretical importance and practical significance of the critical results obtained.     

Network information flow with correlated sources

Consider the following network communication setup, originating in a sensor networking application we refer to as the "sensor reachback" problem. We have a directed graph G=(V,E), where V={v/sub 0/v/sub 1/...v/sub n/} and E/spl sube/V/spl times/V. If (v/sub i/,v/sub j/)/spl isin/E, then node i can send messages to node j over a discrete memoryless channel (DMC) (X/sub ij/,p/sub ij/(y|x),Y/sub ij/), of capacity C/sub ij/. The channels are independent. Each node v/sub i/ gets to observe a source of information U/sub i/(i=0...M), with joint distribution p(U/sub 0/U/sub 1/...U/sub M/). Our goal is to solve an incast problem in G: nodes exchange messages with their neighbors, and after a finite number of communication rounds, one of the M+1 nodes (v/sub 0/ by convention) must have received enough information to reproduce the entire field of observations (U/sub 0/U/sub 1/...U/sub M/), with arbitrarily small probability of error. In this paper, we prove that such perfect reconstruction is possible if and only if H(U/sub s/ | U/sub S(c)/) < /spl Sigma//sub i/spl isin/S,j/spl isin/S(c)/ for all S/spl sube/{0...M},S/spl ne/O,0/spl isin/S(c). Our main finding is that in this setup, a general source/channel separation theorem holds, and that Shannon information behaves as a classical network flow, identical in nature to the flow of water in pipes. At first glance, it might seem surprising that separation holds in a fairly general network situation like the one we study. A closer look, however, reveals that the reason for this is that our model allows only for independent point-to-point channels between pairs of nodes, and not multiple-access and/or broadcast channels, for which separation is well known not to hold. This "information as flow" view provides an algorithmic interpretation for our results, among which perhaps the most important one is the optimality of implementing codes using a layered protocol stack.     

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/.     

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.     

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.     

On-line routing and wavelength assignment for dynamic traffic in WDM ring and torus networks

We develop on-line routing and wavelength assignment (RWA) algorithms for WDM bidirectional ring and torus networks with N nodes. The algorithms dynamically support all k-allowable traffic matrices, where k denotes an arbitrary integer vector [k/sub 1/, k/sub 2/,... k/sub N/], and node i, 1 /spl les/ i /spl les/ N, can transmit at most k/sub i/ wavelengths and receive at most k/sub i/ wavelengths. Both algorithms support the changing traffic in a rearrangeably nonblocking fashion. Our first algorithm, for a bidirectional ring, uses [(/spl Sigma//sub i=1//sup N/ k/sub i/)/3] wavelengths in each fiber and requires at most three lightpath rearrangements per new session request regardless of the number of nodes N and the amount of traffic k. When all the k/sub i/'s are equal to k, the algorithm uses [kN/3] wavelengths, which is known to be the minimum for any off-line rearrangeably nonblocking algorithm. Our second algorithm, for a torus topology, is an extension of a known off-line algorithm for the special case with all the k/sub i/'s equal to k. For an R /spl times/ C torus network with R /spl ges/ C nodes, our on-line algorithm uses [kR/2] wavelengths in each fiber, which is the same as in the off-line algorithm, and is at most two times a lower bound obtained by assuming full wavelength conversion at all nodes. In addition, the on-line algorithm requires at most C - 1 lightpath rearrangements per new session request regardless of the amount of traffic k. Finally, each RWA update requires solving a bipartite matching problem whose time complexity is only O (R), which is much smaller than the time complexity O(kCR/sup 2/) of the bipartite matching problem for an off-line algorithm.     

More results on the weight enumerator of product codes

We consider the product code C/sub p/ of q-ary linear codes with minimum distances d/sub c/ and d/sub r/. The words in C/sub p/ of weight less than d/sub r/d/sub c/+max(d/sub r//spl lceil/(d/sub c//g)/spl rceil/,d/sub c//spl lceil/(d/sub r//q)/spl rceil/) are characterized, and their number is expressed in the number of low-weight words of the constituent codes. For binary product codes, we give an upper bound on the number of words in C/sub p/ of weightless than min(d/sub r/(d/sub c/+/spl lceil/(d/sub c//2)/spl rceil/+1)), d/sub c/(d/sub r/+/spl lceil/(d/sub r//2)/spl rceil/+1) that is met with equality if C/sub c/ and C/sub r/ are (extended) perfect codes.     

Entropy and the law of small numbers

Two new information-theoretic methods are introduced for establishing Poisson approximation inequalities. First, using only elementary information-theoretic techniques it is shown that, when S/sub n/=/spl Sigma//sub i=1//sup n/X/sub i/ is the sum of the (possibly dependent) binary random variables X/sub 1/,X/sub 2/,...,X/sub n/, with E(X/sub i/)=p/sub i/ and E(S/sub n/)=/spl lambda/, then D(P(S/sub n/)/spl par/Po(/spl lambda/)) /spl les//spl Sigma//sub i=1//sup n/p/sub i//sup 2/+[/spl Sigma//sub i=1//sup n/H(X/sub i/)-H(X/sub 1/,X/sub 2/,...,X/sub n/)] where D(P(S/sub n/)/spl par/Po(/spl lambda/)) is the relative entropy between the distribution of S/sub n/ and the Poisson (/spl lambda/) distribution. The first term in this bound measures the individual smallness of the X/sub i/ and the second term measures their dependence. A general method is outlined for obtaining corresponding bounds when approximating the distribution of a sum of general discrete random variables by an infinitely divisible distribution. Second, in the particular case when the X/sub i/ are independent, the following sharper bound is established: D(P(S/sub n/)/spl par/Po(/spl lambda/))/spl les/1//spl lambda/ /spl Sigma//sub i=1//sup n/ ((p/sub i//sup 3/)/(1-p/sub i/)) and it is also generalized to the case when the X/sub i/ are general integer-valued random variables. Its proof is based on the derivation of a subadditivity property for a new discrete version of the Fisher information, and uses a recent logarithmic Sobolev inequality for the Poisson distribution.     

Cyclic codes over GR(4/sup m/) which are also cyclic over /spl Zopf//sub 4/

Let GR(4/sup m/) be the Galois ring of characteristic 4 and cardinality 4/sup m/, and /spl alpha/_={/spl alpha//sub 0/,/spl alpha//sub 1/,...,/spl alpha//sub m-1/} be a basis of GR(4/sup m/) over /spl Zopf//sub 4/ when we regard GR(4/sup m/) as a free /spl Zopf//sub 4/-module of rank m. Define the map d/sub /spl alpha/_/ from GR(4/sup m/)[z]/(z/sup n/-1) into /spl Zopf//sub 4/[z]/(z/sup mn/-1) by d/spl alpha/_(a(z))=/spl Sigma//sub i=0//sup m-1//spl Sigma//sub j=0//sup n-1/a/sub ij/z/sup mj+i/ where a(z)=/spl Sigma//sub j=0//sup n-1/a/sub j/z/sup j/ and a/sub j/=/spl Sigma//sub i=0//sup m-1/a/sub ij//spl alpha//sub i/, a/sub ij//spl isin//spl Zopf//sub 4/. Then, for any linear code C of length n over GR(4/sup m/), its image d/sub /spl alpha/_/(C) is a /spl Zopf//sub 4/-linear code of length mn. In this article, for n and m being odd integers, it is determined all pairs (/spl alpha/_,C) such that d/sub /spl alpha/_/(C) is /spl Zopf//sub 4/-cyclic, where /spl alpha/_ is a basis of GR(4/sup m/) over /spl Zopf//sub 4/, and C is a cyclic code of length n over GR(4/sup m/).     

A fast algorithm for computing multidimensional DCT on certain small sizes

This paper presents a new algorithm for the fast computation of multidimensional (m-D) discrete cosine transform (DCT) with size N/sub 1//spl times/N/sub 2//spl times//spl middot//spl middot//spl middot//spl times/N/sub m/, where N/sub i/ is a power of 2 and N/sub i//spl les/256, by using the tensor product decomposition of the transform matrix. It is shown that the m-D DCT or inverse discrete cosine transform (IDCT) on these small sizes can be computed using only one-dimensional (1-D) DCTs and additions and shifts. If all the dimensional sizes are the same, the total number of multiplications required for the algorithm is only 1/m times of that required for the conventional row-column method. We also introduce approaches for computing scaled DCTs in which the number of multiplications is considerably reduced.     

The empirical distribution of rate-constrained source codes

Let X = (X/sub 1/,...) be a stationary ergodic finite-alphabet source, X/sup n/ denote its first n symbols, and Y/sup n/ be the codeword assigned to X/sup n/ by a lossy source code. The empirical kth-order joint distribution Q/spl circ//sup k/[X/sup n/,Y/sup n//spl rceil/(x/sup k/,y/sup k/) is defined as the frequency of appearances of pairs of k-strings (x/sup k/,y/sup k/) along the pair (X/sup n/,Y/sup n/). Our main interest is in the sample behavior of this (random) distribution. Letting I(Q/sup k/) denote the mutual information I(X/sup k/;Y/sup k/) when (X/sup k/,Y/sup k/)/spl sim/Q/sup k/ we show that for any (sequence of) lossy source code(s) of rate /spl les/R lim sup/sub n/spl rarr//spl infin//(1/k)I(Q/spl circ//sup k/[X/sup n/,Y/sup n//spl rfloor/) /spl les/R+(1/k)H (X/sub 1//sup k/)-H~(X) a.s. where H~(X) denotes the entropy rate of X. This is shown to imply, for a large class of sources including all independent and identically distributed (i.i.d.). sources and all sources satisfying the Shannon lower bound with equality, that for any sequence of codes which is good in the sense of asymptotically attaining a point on the rate distortion curve Q/spl circ//sup k/[X/sup n/,Y/sup n//spl rfloor//spl rArr//sup d/P(X/sup k/,Y~/sup k/) a.s. whenever P(/sub X//sup k//sub ,Y//sup k/) is the unique distribution attaining the minimum in the definition of the kth-order rate distortion function. Consequences of these results include a new proof of Kieffer's sample converse to lossy source coding, as well as performance bounds for compression-based denoisers.     

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.     

Sum capacity of MIMO Gaussian broadcast channels with channel energy constraints

The motivation of this paper is to find the class of channels that provides the best sum capacity of a MIMO Gaussian broadcast channel with both transmit power and channel energy constraints when N/sub t//spl ges/KN/sub r/, where N/sub t/, N/sub r/, and K denote the number of transmit antennas, the number of receive antennas, and the number of users in the system, respectively. The best sum capacity is achieved when the user channels are mutually orthogonal to each other. For each individual user, equal energy is distributed to all non-zero spatial eigenmodes. Further, we optimize the number of non-zero eigenmodes for all users and the optimal power distribution among users. Although, we only study the case of N/sub t//spl ges/KN/sub r/ in this paper, we conjecture that similar results still hold for N/sub t/相似文献   

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/.