Order selection statistical test for nonstationary AR models

A new statistical test for selecting the order of a nonstationary AR modelyk is presented based on the predictive least-squares principle. This test is of the same order as the accumulated cost function _n = _k=1 ⁿ ( _k ^* – _k )²;i.e., ^* wherey _k ^* is the predictive least-square estimate. It is constructed to show how many times the integrated AR processy _k is differenced in order to obtain a stationary AR process given that the exact order of the process is unknown.     

Construction of Time-Relaxed c-Broadcast Networks

C-broadcasting is an information dissemination task where a message, originated at any node in a network, is transmitted to all other nodes with the restriction that each node having the message can transmit it to almost c neighbors simultaneously. If the transmission time of the message is set to be one time unit, a minimal c-broadcast network (c-MBN) is a communication network in which c-broadcasting from any node can be accomplished in log _c+1 n time units, where n is the number of nodes and log _c+1 n is the fastest possible broadcast time. If networks are sparse, additional time units may be required to perform c-broadcasting. A time-relaxed c-broadcast network, denoted as (t,c)-RBN, is a network where c-broadcasting from any node can be completed in log _c+1 n+t time units. In this paper, a network compounding algorithm is proposed to construct large sparse (t,c)-RBNs by linking multiple copies of a time-relaxed network of small size using the structure of another time-relaxed network. Computational results are presented to show the effectiveness of the proposed algorithm.     

Shorts of block operators and infinite networks—A note on the shorted operator: II

Let A_ij i, j=1, 2,..., be operators on a Hilbert spaceX, such that the compound operatorA =A _{ij i, j=1} induces a bounded positive operator onl ₂(X). We show that S(A , theshorted operator (orgeneralized Schur complement), of A can be obtained as the limits of shorts of the operators A_n, where A_n is the truncated version ofA , thenA _n=A _{ij i, j=1} ⁿ . We use these results to study the short-circuit approximations to infinite networks.     

β-bit serial/parallel multipliers

A generalized -bit least-significant-digit (LSD) first, serial/parallel multiplier architecture is presented with 1n wheren is the operand size. The multiplier processes both the serial input operand and the double precision product -bits per clock cycle in an LSD first, synchronous fashion. The complete two's complement double precision product requires 2n/ clock cycles. This generalized architecture creates a continuum of multipliers between traditional bit-serial/parallel multipliers (=1) and fully-parallel multipliers (=n). -bit serial/parallel multipliers allow anoptimized integrated circuit arithmetic to be designed based on a particular application's area, power, throughput, latency, and numerical precision constraints.This project was pratically funded by the UCSD-NSF I/UCR Center on Ultra-High Speed Intergrated Circuits and Systems.     

On the robustness of stability for uncertain large-scale systems subjected to multiple time delays

The robustness problem of stability for large-scale uncertain systems with a class of multiple time delays is addressed in this paper. By applying the complex Lyapunov stability theorem, the matrix measure techniques, and norm inequalities, a new approach for solving a general case of the above problem is proposed. Several robust stability conditions, delay-dependent or delay-independent, are derived to guarantee the asymptotic stability and exponential stability of the uncertain large-scale time-delay systems. Moreover, these obtained results can also be applied to the stabilization design.Notation real number field - complex number field - x x=(x ₁,x ₂,...,x _n ) ^T R ⁿ - x ^T transpose of vectorx - x* complex conjugate transpose of vectorx - Re(·) real part of (·) - x norm of vectorx; x=(x*x)^1/2 - A ^T transpose of matrixA - A* complex conjugate transpose of matrixA - ¯(·) maximal absolute value of eigenvalue of matrixA - (A) matrix measure of matrixA; (A)=¯((A + A*)/2) - A induced norm of matrix A; A=[¯(A*A)]^1/2 - ¦a_ij ¦ absolute value of element a_ij - ¦ A¦ {¦a_ij¦} for matrix A={a_ij} - A >B a_ij > b_ij for alli andj where A={a_ij} andB={b_ij} - z complex number - ¯z complex conjugate ofz Supported by National Science Council, Taiwan, Republic of China, Grant NSC83-0404-E006-001.     

Symmetry relations in multidimensional Fourier transform pairs

A relation between the types of symmetries that exist in signal and Fourier transform domain representations is derived for continuous as well as discrete domain signals. The symmetry is expressed by a set of parameters, and the relations derived in this paper will help to find the parameters of a symmetry in the signal or transform domain resulting from a given symmetry in the transform or signal domain respectively. A duality among the relations governing the conversion of the parameters of symmetry in the two domains is also brought to light. The application of the relations is illustrated by a number of two-dimensional examples.Notation R the set of real numbers - R ^m R × R × ... × R m-dimensional real vector space - continuous domain real vector - L {¦ – _i , i = 1,2,..., m} - m-dimensional frequency vector - W { – _i ,i=1,2,..., m} - m-dimensional normalized frequency vector - _P {¦ – _i , i=1,2,...,m} - g(ol) g (₁,₂,..., _m ) continuous domain signal - () ( _{1 2},..., _m )=G (j ₁,j ₂,..., j _m ) Fourier transform ofg (ol) - (A,b,,,) parameters ofT- symmetry - N the set of integers - N ^m N × N × ... × N m-dimensional integer vector spacem-dimensional lattice - h(n) h (n ₁,.,n _m ) discrete domain signal - H() Fourier transform ofh (n) - v ₁,v ₂,..., v_m m sample-direction and interval vectors - V (v ₁ v ₂ ...v _m ) sampling basis matrix - [x]^* complex conjugate ofx - detA determinant ofA - X {x¦ – x _i , i=1,2,..., m} - A ^–t [A ^–1] ^t ,t stands for transpose This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant A-7739 to M. N. S. Swamy and in part by Tennessee Technological University under its Faculty Research support program to P. K. Rajan.     

The prime and generalized nullspaces of right regular pencils

The classical notion of the -generalized nullspace, defined on a matrixA R ^n×n,where is an eigenvalue, is extended to the case of ordered pairs of matrices(F, G), F, G R ^m×nwhere the associated pencilsF – G is right regular. It is shown that for every C {} generalized eigenvalue of (F, G), an ascending nested sequence of spaces {P ⁱ ,i=1, 2,...} and a descending nested sequence of spaces {ie495-02 i=1, 2,...} are defined from the -Toeplitz matrices of (F, G); the first sequence has a maximal elementM ^* , the -generalized nullspace of (F, G), which is the element of the sequence corresponding to the index , the -index of annihilation of (F, G), whereas the second sequence has the first elementP ^* as its maximal element, the -prime space of (F, G). The geometric properties of the {M ⁱ ,i=1, 2,..., and {P ⁱ ,i=1, 2,...sets, as well as their interrelations are investigated and are shown to be intimately related to the existence of nested basis matrices of the nullspaces of the -Toeplitz matrices of (F, G). These nested basis matrices characterize completely the geometry ofM ^* and provide a systematic procedure for the selection of maximal length linearly independent vector chains characterizing the-Segre characteristic of (F, G).     

Estimation of Energy Aliasing Error for Nonbandlimited Signals

In this paper, we investigate the problem of approximating a given (not necessarily bandlimited) signal, x(t), by a (bandlimited) interpolation or sampling series of the form:

Novel Ion-Lens Configurations: A Further Step to 2-nm Ion-Beam Lithography

Composite axially symmetric immersion ion lenses are considered that consist of an electrostatic and a magnetic lens. For the first time, their performance is evaluated over the entire range of operating conditions: from the case of a zero magnetic field to the case of a zero ion energy on the target. Operating conditions are characterized in terms of = W _t/W ₀, where W ₀ is the energy of an ion at the boundary of the region in which the trajectories are parallel to the axis and W _t is that on the target. For the first time, simple analytical approximations are derived for C _c/r, C _s/r, f/r, and NI, where C _c is the chromatic-aberration coefficient, C _s is the third-order spherical-aberration coefficient, f is the focal distance, NI is the magnetomotive force of the coil, and r is the outer radius of the coil. The behavior of the four quantities is explored as a function of . The following conclusions are drawn: (i) The aberrations are maximum for a zero magnetic field. (ii) The aberration coefficients decrease monotonically with increasing NIand decreasing , the lens changing from an accelerating to a decelerating one. (iii) If , then C _s/r – ^1/4, C _c/r – ^1/6, f/r– ^1/3, and NI – ^–1/2. (iv) The lenses are suitable for resistless heavy-ion projection lithography and can provide 20 × 10¹¹ pixels of area 2 × 2 nm² for an exposed area of 3 × 3 mm². (v) Used in heavy-ion microprobe systems, the lenses could enable resistless lithography over much larger areas than existing equipment.     

A 70-MHz Continuous-Time CMOS Band-Pass ΣΔ Modulator for GSM Receivers

A 70-MHz continuous-time CMOS band-pass modulator for GSM receivers is presented. Impulse-invariant-transformation is used to transform a discrete-time loop-filter transfer function into continuous-time. The continuous-time loop-filter is implemented using a transconductor-capacitor (G _m -C) filter. A latched-type comparator and a true-single-phase-clock (TSPC) D flip-flop are used as the quantizer of the modulator. Implemented in a MOSIS HP 0.5-m CMOS technology, the chip area is 857 m × 420 m, and the total power consumption is 39 mW. At a supply voltage of 2.5 V, the maximum SNDR is measured to be 42 dB, which corresponds to a resolution of 7 bits.     

Gaussian radial basis functions and inner product spaces

An approximation result is given concerning Gaussian radial basis functions in a general inner product space. Applications are described concerning the classification of the elements of disjoint sets of signals, and also the approximation of continuous real functions defined on all of ⁿ using radial basis function (RBF) networks. More specifically, it is shown that an important large class of classification problems involving signals can be solved using a structure consisting of only a generalized RBF network followed by a quantizer. It is also shown that Gaussian radial basis functions defined on ⁿ can uniformly approximate arbitrarily well over all of ⁿ any continuous real functionalf on ⁿ that meets the condition that |f(x)|0 as x.     

Parallel sums and norm convergence

LetA andB be the impedance operators of two resistive Hilbert ports. Then theparallel sum ofA andB, denotedA: B, is defined to be the joint impedance of the parallel interconnection ofA andB. We survey some results about the norm convergence of lim₀(A+I): (B+I). By means of several new examples, we investigate the relation between such norm convergence and range inclusion conditions.     

Efficient signature generation by smart cards

We present a new public-key signature scheme and a corresponding authentication scheme that are based on discrete logarithms in a subgroup of units in _p where p is a sufficiently large prime, e.g., p 2⁵¹². A key idea is to use for the base of the discrete logarithm an integer in _p such that the order of is a sufficiently large prime q, e.g., q 2¹⁴⁰. In this way we improve the ElGamal signature scheme in the speed of the procedures for the generation and the verification of signatures and also in the bit length of signatures. We present an efficient algorithm that preprocesses the exponentiation of a random residue modulo p.European patent application 89103290.6 from February 24, 1989. U.S. patent number 4995082 of February 19, 1991.     

Digital Modulator with Bandpass Delta-Sigma Modulator

A digital quadrature modulator with a bandpass -modulator is presented that interpolates orthogonal input carriers by 16 and performs a digital quadrature modulation at carrier frequencies f_s/4, –f_s/4 (f_s is the sampling frequency). After quadrature modulation, the signal is converted into an analogue IF signal using a bandpass modulator and a 1-bit D/A converter. The die area of the chip is 5.2 mm² (0.13 m CMOS technology). The total power consumption is 139 mW at 1.5 V with a clock frequency of 700 MHz (D/A converter full-scale output current 11.5 mA).     

On the design of feedback controllers to achieve prescribed initial value constraints

The problem of designing a stabilizing compensator for a control system to achieve prescribed initial value constraints ⁽ⁱ⁾(0⁺)=y_i is considered. Indeed, modulo certain technical conditions, such a compensator exists if and only if y_i=0;i= 0,1,...,rp +rt –2; whererp is the relative degree of the plant andrt is the relative degree of the system input. This theorem is derived and a complete parameterization of the set of compensators that achieve the prescribed design constraints is formulated.This research was supported in part by NSF Grant No. 921106.     

Factorizations in the elementary Abelian p-group and their cryptographic significance

Let G be a finite group and let A _i 1 i s, be subsets of G where ¦A _i ¦ 2, 1 i s and s 2. We say that (A₁, A₂,..., A₃) is a factorization of G if and only if for each g G there is exactly one way to express g = a ₁ a ₁ a ₂··· a ₃, where a _j A _i , 1 i s.The problem of finding factorizations of this type was first introduced by Hajos [3] in 1941. Since then a number of papers have appeared on the subject. More recently, Magliveras [6] has applied factorization of permutation groups to cryptography to obtain a private-key cryptosystem. Factorizations in the elementary abelian p-group were exploited (but not explicitly stated in these terms) by Webb [13] to produce a public-key cryptosystem conceptually similar to cryptosystems based on the knapsack problem.Using the result that certain types of factorizations in the elementary abelian p-group are necessarily transversal (a term introduced by Magliveras), this paper shows that the public-key system proposed by Webb is insecure.     

On the distribution of characteristics in bijective mappings

Differential cryptanalysis is a method of attacking iterated mappings based on differences known as characteristics. The probability of a given characteristic is derived from the XOR tables associated with the iterated mapping. If is a mapping : Z ₂ ^m , then for each , X, Y Z ₂ ^m the XOR table for gives the number of input pairs of difference X=X+X for which gp(X)+(X)=Y.The complexity of a differential attack depends upon two properties of the XOR tables: the density of zero entries in the table, and the size of the largest entry in the table. In this paper we present the first results on the expected values of these properties for a general class of mappings . We prove that if : Z ₂ ^m Z ₂ ^m is a bijective mapping, then the expected size of the largest entry in the XOR table for is bounded by 2m, while the fraction of the XOR table that is zero approaches e ^–1/2=0.60653. We are then able to demonstrate that there are easily constructed classes of iterated mappings for which the probability of a differential-like attack succeeding is very small.The author is presently employed by the Distributed System Technology Center, Brisbane, Australia.     

Input-output systems whose transmission operators furnish nearly best approximations to a given map

Let {S(A):A A}, whereA is a subset of an infinite-dimensional normed linear spaceL, be a class of general nonlinear input-output systems that are governed by operator equations relating the input, state, and output, all of which are in extended spaces. IfQ is a given operator from a specified set ¯D _i, of inputs into the space of outputs ¯H ₀, the problem we consider is to find, for a given >0, a parameterA A such that the transmission operatorR(A ) ofS(A ) furnishes a nearly best (or -best) approximation toQ from allR(A),A A. Here the distance betweenQ andR(A) is defined as the supremum of distances betweenQz andR(A)z taken over allz ¯D _i. In Theorems 2 through 5 we show that ifS(A) is normal (Definition 2),A satisfies some mild requirement andL contains a fundamental sequence, then establishingA A reduces to minimizing a certain continuous functional on a compact subset ofR ⁿ, and thus can be carried out by conventional methods. The applications of results are illustrated by the example of a model-matching problem for a nonlinear system, and of optimal tracking.     

Higher-order terms in asymptotic expansion for information loss in quantized random processes

Uniform quantization of random vectors onto -grids ⁿ is considered. Higherorder terms in asymptotic expansions for the entropy of the -quantized random vector and for the loss of the mutual information between two random vectors under such quantization as 0+are obtained. The coefficients in these asymptotics are explicitly calculated for Gaussian distributed vectors. Taken for initial segments of stationary Gaussian sequences, these factors have limit average values per unit of time. For such sequences governed by state-space equations, computation of these average values is reduced to solutions of algebraic matrix Riccati and Lyapunov equations.Work supported by the Australian Research Council grant A 4970 2246.