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.     

A method for robust stabilization

LetP(y,M) be a family of polynomials, depending on a controller parameter vector y in ⁿ⁺¹, defined as follows: a family of interval matrices, and y in ⁿ⁺¹, set Given an initial stabilizing controller y⁰, this paper provides a simple method to robustify y⁰, i.e., to obtain a controller parameter vector y¹ which is more robust than y⁰. Furthermore we define the stability factort for a given robustly stable controller y, which serves as a measure of the robustness of y.     

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).     

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.     

On the convergence rate ofs-numbers of compact Hankel operators

It is well known that the sequence ofs-numbers {s_n},n=0, 1,..., of a compact operator, and particularly a compact Hankel operator =[h _j+k –1], converges monotonically to zero. Since the (n + 1)sts-number s_n measures the error ofL (¦z¦=1) approximation, modulo an additive H function, by nth degree proper rational functions whose poles are restricted to ¦z¦ < 1, it is very important to study how fast {s _n } converges to zero. It is not difficult to see that ifh _n =O(n ^–), for some > 1, thens _n =O(n ^–). In this paper we construct, for any given sequence _n 0, a compact Hankel operator such thats _{n n} for alln.The research of C. K. Chui was supported by the SDIO/IST managed by the U.S. Army under Contract No. DAAL-03-87-K-0025, and by the NSF under Grant No. DMS-8901345. X. Li's research was supported by the SDIO/IST managed by the U.S. Army under Contract No. DAAL-03-87-K-0025. The research of J. D. Ward was supported by the NSF under Grant No. DMS-8901345.     

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.     

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.     

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.     

Theory of PLL fractional-N frequency synthesizers

This paper presents an overview of the evolution of frequency synthesizers based on phase-locked loops (PLLs). The main limitations of the digital PLLs are described, and the consequent necessity of using fractional-N techniques is justified. The origin of the typical spurious noise lines on the sidelobes of the synthesized frequency is explained. It is shown how to eliminate these spurious noise lines by using digital modulators to control the frequency division value. Finally, the implications of using digital modulators together with fractional-N PLLs on the output phase noise are analysed.     

CBC MACs for Arbitrary-Length Messages: The Three-Key Constructions

We suggest some simple variants of the CBC MAC that enable the efficient authentication of arbitrary-length messages. Our constructions use three keys, K1, K2, K3, to avoid unnecessary padding and MAC any message M {0,1}* using max{1, |M|/n} applications of the underlying n-bit block cipher. Our favorite construction, XCBC, works like this: if |M| is a positive multiple of n then XOR the n-bit key K2 with the last block of M and compute the CBC MAC keyed with K1; otherwise, extend Ms length to the next multiple of n by appending minimal 10 padding ( 0), XOR the n-bit key K3 with the last block of the padded message, and compute the CBC MAC keyed with K1. We prove the security of this and other constructions, giving concrete bounds on an adversarys inability to forge in terms of his inability to distinguish the block cipher from a random permutation. Our analysis exploits new ideas which simplify proofs compared with prior work.     

Geometric properties of nonlinear networks containing capacitor-only cutsets and/or inductor-only loops. Part I: Conservation laws

This paper is the first in a two part sequence which studies nonlinear networks, containing capacitor-only cutsets and/or inductor-only loops from the geometric coordinate-free point of view of differentiable manifolds. Given such a nonlinear networkN, with °₀ equal to the sum of the number of independent capacitor-only cutsets and the number of independent inductor-only loops, we establish the following: (i) circuit theoretic sufficient conditions to guarantee that the set ⁰, of equilibrium points is a ₀-dimensional submanifold of the state space ofN; (ii) circuit theoretic sufficient conditions for the condition thatN has ₀ independent conservation laws and hence that through each point of the state space ofN, there passes a codimension ₀ invariant submanifold ^* of the network dynamics; (iii) circuit theoretic sufficient conditions to guarantee that the manifolds ^* and ⁰ intersect transversely.This work was supported by the Natural Sciences and Engineering Research Council of Canada, under Grant Number A7113, and by scholarships from the Natural Sciences and Engineering Research Council of Canada and the Ontario Provincial Government.     

An algorithm for stability determination of two-dimensional delta-operator formulated discrete-time systems

The recent interest in delta-operator (or, -operator) formulated discrete-time systems (or, -systems) is due mainly to (a) their superior finite wordlength characteristics as compared to their more conventional shift-operator (or,q-operator) counterparts (or,q-systems), and (b) the possibility of a more unified treatment of both continuous- and discrete-time systems. With such advantages, design, analysis, and implementation of two-dimensional (2-D) discrete-time systems using the -operator is indeed warranted. Towards this end, the work in this paper addresses the development of an easily implementabledirect algorithm for stability checking of 2-D -system transfer function models.Indirect methods that utilize transformation techniques are not pursued since they can be numerically unreliable. In developing such an algorithm, a tabular form for stability checking of -system characteristic polynomials with complex-valued coefficients and certain quantities that may be regarded as their corresponding Schur-Cohn minors are also proposed.     

A note on error bounds for function approximation using nonlinear networks

For many problems in classification, compensation, adaptivity, identification, and signal processing, results concerning the representation and approximation of nonlinear functions can be of particular interest to engineers. Here we consider a large class of functionsf that map ⁿ into the set of real or complex numbers, and we give bounds on the number of parameters of a certain approximation network so thatf can be approximated to within a prescribed degree of accuracy using an appropriate configuration of the network. We also describe related work in the neural networks literature.     

The discrete Einstein relation

In this paper the connection between massM, resistance and commute time for random walks on graphs is further explored, and the relation=2M · is proved. An extension of the result is made to multigraphs, which are an extension of the graph concept where a black box is treated like an edge.     

Quality Determination for Gate Delay Fault Tests Considering Three-State Elements

Most industrial digital circuits contain three-state elements besides pure logic gates. This paper presents a gate delay fault simulator for combinational circuits that can handle three-state elements like bus drivers, transmission gates and pulled busses. The well known delay faults--slow-to-rise and slow-to-fall--are considered as well as delayed transitions from isolating signal state high impedance to binary states 0 and 1 and vice versa. The presented parallel delay fault simulator distinguishes between non-robust, robust and hazard free tests and determines the quality of a test. Experimental results for ISCAS85/89 benchmark circuits are presented as well as results for industrial circuits containing three-state elements.     

Performance Monitoring for In-life Capacity Management

The key to providing a meaningful management view of the performance of complex systems lies in having a co-ordinated process for data collection, analysis and presentation. TeleMarketing Services, such as the Freefone service, provide a good example. Managers of diverse, remote subsystems receive summarised performance and customer service information, mostly in the form of simple graphs, in exchange for providing reliable data. The aim is to bring together a structured view of all the components, including a red-amber-green status table, for use by senior customer-facing staff. Generally, the analyses focus on capacity, forecast and measurement, covering a flexible range of parameters. Managers then have an early view of service degradation, potential capacity exhaustion, workload imbalance and resource wastage, enabling timely remedial action, fault clearance and system enhancements.     

Parametric toeplitz systems

This paper investigates the properties of the two-variable polynomialu (, z) built on the first column of the adjoint matrix ofI -C, whereC is a given Hermitian Toeplitz matrix. In particular, the stability properties ofu (,z) are discussed and are shown to depend essentially on the location of X with respect to the eigenvalues ofC. The eigenvectors ofC, which have recently found some applications in signal processing and estimation theory, are obtained from the polynomialu(,z) when tends to the eigenvalues ofC. This allows one to derive several results concerning the eigenpolynomials, including those for the case of multiple eigenvalues.     

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.     

Efficient linear-phase FIR, maximally flat error approximations for the amplitude response |1/ω r |,r,r=1, 2, 3,..., and a versatile realization

In this paper, designs of linear-phase finite impulse response (FIR) filters approximating the ideal frequency responseH (,r) |1/ ^r r=1, 2, 3,..., by using maximally flat error criteria have been proposed. Exact weights required for the realizations have been derived and are shown to be independent of the filter orderN. The suggested designs are particularly suitable for operation in the midband frequency range of 0.10–0.90 radians.     

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.