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.     

Asymptotic properties on codeword lengths of an optimal FV code for general sources

This correspondence is concerned with asymptotic properties on the codeword length of a fixed-to-variable length code (FV code) for a general source {X/sup n/}/sub n=1//sup /spl infin// with a finite or countably infinite alphabet. Suppose that for each n /spl ges/ 1 X/sup n/ is encoded to a binary codeword /spl phi//sub n/(X/sup n/) of length l(/spl phi//sub n/(X/sup n/)). Letting /spl epsiv//sub n/ denote the decoding error probability, we consider the following two criteria on FV codes: i) /spl epsiv//sub n/ = 0 for all n /spl ges/ 1 and ii) lim sup/sub n/spl rarr//spl infin///spl epsiv//sub n/ /spl les/ /spl epsiv/ for an arbitrarily given /spl epsiv/ /spl isin/ [0,1). Under criterion i), we show that, if X/sup n/ is encoded by an arbitrary prefix-free FV code asymptotically achieving the entropy, 1/nl(/spl phi//sub n/(X/sup n/)) - 1/nlog/sub 2/ 1/PX/sup n/(X/sup n/) /spl rarr/ 0 in probability as n /spl rarr/ /spl infin/ under a certain condition, where P/sub X//sup n/ denotes the probability distribution of X/sup n/. Under criterion ii), we first determine the minimum rate achieved by FV codes. Next, we show that 1/nl(/spl phi//sub n/(X/sup n/)) of an arbitrary FV code achieving the minimum rate in a certain sense has a property similar to the lossless case.     

Physical-layer secrecy in AWGN via a class of chaotic DS/SS systems: analysis and design

We study a class of pseudo-chaotic spread spectrum systems for secure communication over additive white Gaussian noise (AWGN) channels, whereby a symbol stream is linearly modulated on a spreading sequence generated by iterating an initial condition through a suitably chosen chaotic map. We compare the uncoded probability of error (Pr(/spl epsiv/)) attainable by intended receivers that know the initial condition to the associated Pr(/spl epsiv/) of unintended receivers that know the modulation scheme but not the initial condition. The sensitive dependence of chaotic sequences on initial conditions, together with the presence of channel noise, can be exploited to provide substantially lower Pr(/spl epsiv/) to intended than to unintended receivers. We develop computationally efficient methods for obtaining tight bounds on the best P r(/spl epsiv/) performance of intended and unintended receivers. In the process, we identify chaotic map attributes that affect the relative Pr(/spl epsiv/) advantages provided to intended receivers and develop methods for designing maps that achieve a target gap between the intended and unintended receiver Pr(/spl epsiv/).     

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 filtering problem for stationary random /spl Zopf//sup 2/-fields

It is shown that whenever a stationary random field (Z/sub n,m/)/sub n,m/spl isin/z/ is given by a Borel function f:/spl Ropf//sup z/ /spl times/ /spl Ropf//sup z/ /spl rarr/ /spl Ropf/ of two stationary processes (X/sub n/)/sub n/spl isin/z/ and (Y/sub m/)/sub m/spl isin/z/ i.e., then (Z/sub n, m/) = (f((X/sub n+k/)/sub k/spl epsi/z/, (Y/sub m + /spl lscr// )/sub /spl lscr/ /spl epsi/z/)) under a mild first coordinate univalence assumption on f, the process (X/sub n/)/sub n/spl isin/z/ is measurable with respect to (Z/sub n,m/)/sub n,m/spl epsi/z/ whenever the process (Y/sub m/)/sub m/spl isin/z/ is ergodic. The notion of universal filtering property of an ergodic stationary process is introduced, and then using ergodic theory methods it is shown that an ergodic stationary process has this property if and only if the centralizer of the dynamical system canonically associated with the process does not contain a nontrivial compact subgroup.     

Decoding by linear programming

This paper considers a natural error correcting problem with real valued input/output. We wish to recover an input vector f/spl isin/R/sup n/ from corrupted measurements y=Af+e. Here, A is an m by n (coding) matrix and e is an arbitrary and unknown vector of errors. Is it possible to recover f exactly from the data y? We prove that under suitable conditions on the coding matrix A, the input f is the unique solution to the /spl lscr//sub 1/-minimization problem (/spl par/x/spl par//sub /spl lscr/1/:=/spl Sigma//sub i/|x/sub i/|) min(g/spl isin/R/sup n/) /spl par/y - Ag/spl par//sub /spl lscr/1/ provided that the support of the vector of errors is not too large, /spl par/e/spl par//sub /spl lscr/0/:=|{i:e/sub i/ /spl ne/ 0}|/spl les//spl rho//spl middot/m for some /spl rho/>0. In short, f can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program). In addition, numerical experiments suggest that this recovery procedure works unreasonably well; f is recovered exactly even in situations where a significant fraction of the output is corrupted. This work is related to the problem of finding sparse solutions to vastly underdetermined systems of linear equations. There are also significant connections with the problem of recovering signals from highly incomplete measurements. In fact, the results introduced in this paper improve on our earlier work. Finally, underlying the success of /spl lscr//sub 1/ is a crucial property we call the uniform uncertainty principle that we shall describe in detail.     

List decoding from erasures: bounds and code constructions

We consider the problem of list decoding from erasures. We establish lower and upper bounds on the rate of a (binary linear) code that can be list decoded with list size L when up to a fraction p of its symbols are adversarially erased. Such bounds already exist in the literature, albeit under the label of generalized Hamming weights, and we make their connection to list decoding from erasures explicit. Our bounds show that in the limit of large L, the rate of such a code approaches the "capacity" (1 - p) of the erasure channel. Such nicely list decodable codes are then used as inner codes in a suitable concatenation scheme to give a uniformly constructive family of asymptotically good binary linear codes of rate /spl Omega/(/spl epsiv//sup 2//log(1//spl epsiv/)) that can be efficiently list-decoded using lists of size O(1//spl epsiv/) when an adversarially chosen (1 - /spl epsiv/) fraction of symbols are erased, for arbitrary /spl epsiv/ > 0. This improves previous results in this vein, which achieved a rate of /spl Omega/(/spl epsiv//sup 3/log(1//spl epsiv/)).     

Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information

This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f/spl isin/C/sup N/ and a randomly chosen set of frequencies /spl Omega/. Is it possible to reconstruct f from the partial knowledge of its Fourier coefficients on the set /spl Omega/? A typical result of this paper is as follows. Suppose that f is a superposition of |T| spikes f(t)=/spl sigma//sub /spl tau//spl isin/T/f(/spl tau/)/spl delta/(t-/spl tau/) obeying |T|/spl les/C/sub M//spl middot/(log N)/sup -1/ /spl middot/ |/spl Omega/| for some constant C/sub M/>0. We do not know the locations of the spikes nor their amplitudes. Then with probability at least 1-O(N/sup -M/), f can be reconstructed exactly as the solution to the /spl lscr//sub 1/ minimization problem. In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for C/sub M/ which depend on the desired probability of success. Our result may be interpreted as a novel kind of nonlinear sampling theorem. In effect, it says that any signal made out of |T| spikes may be recovered by convex programming from almost every set of frequencies of size O(|T|/spl middot/logN). Moreover, this is nearly optimal in the sense that any method succeeding with probability 1-O(N/sup -M/) would in general require a number of frequency samples at least proportional to |T|/spl middot/logN. The methodology extends to a variety of other situations and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one- or two-dimensional) object from incomplete frequency samples - provided that the number of jumps (discontinuities) obeys the condition above - by minimizing other convex functionals such as the total variation of f.     

Loss characteristics of high-/spl epsiv//sub r/ microstrip lines fabricated by an etchable thick-film on ceramic MCM technology

Etchable thick-film multi-chip-module (MCM) technology has led to the possibility of fabricating microwave integrated circuits (MICs) with performance similar to MICs produced using more expensive conventional thin-film MCM-D techniques. However, little data is available on the loss characteristics of the technology at microwave frequencies. This paper describes an experimental investigation into the loss properties of high-definition etchable thick-film MCM microstrip lines formed on a variety of high dielectric constant (high-/spl epsiv//sub r/) ceramic substrates. Substrates investigated comprise 96% alumina (/spl epsiv//sub r/=9.5), (Zr,Sn)TiO/sub 4/(/spl epsiv//sub r/=36.6) and BaO-PbO-Nd/sub 2/O/sub 3/-TiO/sub 2/ (/spl epsiv//sub r/=90.9). Microstrip loss properties are determined by fabricating a series of loosely coupled half-wave resonators on each substrate, with a range of characteristic impedance values. Measurements to 6 GHz are compared to those for similar lines fabricated using conventional thin-film MCM-D technology. The results demonstrate that etchable thick-film MCM technology provides many of the advantages of thin-film MCM-D technology, such as low-loss and high-definition conductors, and is suitable for the cost-effective fabrication of miniaturised high-performance microstrip MICs in high volume.     

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.     

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.     

A Simple Technique for the Accurate Determination of the Microwave Dielectric Constant for Microwave Integrated Circuit Substrates (Correspondence)

A method of determining the microwave dielectric constant of microwave integrated circuit substrates is described. The technique is especially suitable to substrates being prepared for MICs since they are, in general, regular, rectangular, and, therefore, simple resonators. The dielectric constant using this technique has been determined in the 2- to 12-GHz range for GaAs (/spl epsiv/R = 12.46), sapphire (/spl epsiv/R =9.37), polyguide (/spl epsiv/ =2.33), and Alsimag 772 (/spl epsiv/R = 10.08).     

Prime-length real-valued polynomial residue division algorithms

One class of efficient algorithms for computing a discrete Fourier transform (DFT) is based on a recursive polynomial factorization of the polynomial 1-z/sup -N/. The Bruun algorithm is a typical example of such algorithms. Previously, the Bruun algorithm, which is applicable only when system lengths are powers of two in its original form, is generalized and modified to be applicable to the case when the length is other than a power of two. This generalized algorithm consists of transforms T/sub d,f/ with prime d and real f in the range 0/spl les/f<0.5. T/sub d,0/ computes residues X(z)mod(1-z/sup -2/) and X(z)mod(1-2 cos(/spl pi/k/d)z/sup -1/+z/sup -2/), k=1, 2, ..., d-1, and T/sub d,f/ (f /spl ne/0) computes residues X(z)mod(1-2cos(2/spl pi/(f+k)/d)z/sup -1/+z/sup -2/), k=0, 1, ..., d-1 for a given real signal X(z) of length 2d. The purpose of this paper is to find efficient algorithms for T/sub d,f/. First, polynomial factorization algorithms are derived for T/sub d,0/ and T/sub d,1/4/. When f is neither 0 nor 1/4, it is not feasible to derive a polynomial factorization algorithm. Two different implementations of T/sub d,f/ for such f are derived. One implementation realizes T/sub d,f/ via a d-point DFT, for which a variety of fast algorithms exist. The other implementation realizes T/sub d,f/ via T/sub d, 1/4/, for which the polynomial factorization algorithm exists. Comparisons show that for d/spl ges/5, these implementations achieve better performance than computing each output of T/sub d,f/ separately.     

Synthesis of an incident aperiodic tapered beam

This paper presents a technique for generating numerically an incident tapered beam of a prescribed polarization and a prescribed incidence angle by summing a finite number of propagating uniform plane waves, each constituent plane wave having its own magnitude, phase, incidence angle and polarization. Our three ad-hoc schemes for choosing the magnitude, the incidence angle and the polarization of a constituent plane wave are different from those by Braunisch et al. Our choice of the propagating plane waves corresponds to a sampling of the physical angular (/spl theta/, /spl phi/) space. This contrasts with the two-dimensional Fourier transform technique by Braunisch et al. and its Cartesian sampling of the (k/sub x/, k/sub y/) spectral plane where k/sub x/=(k/sub o/sin/spl theta/cos/spl phi/) and k/sub y/=(k/sub o/sin/spl theta/sin/spl phi/). Our technique is based on the summation of only propagating plane waves, whereas the technique by Braunisch et al. is based on the summation of both propagating and evanescent plane waves.     

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/).     

Performance of reduced size substrate lens antennas for Millimeter-wave communications

This paper presents the theoretical performance (input impedance, -10 dB return-loss bandwidth, radiation patterns and surface efficiencies) of reduced size substrate lenses fed by aperture-coupled microstrip patch antennas. The diameter of the extended hemispherical homogeneous dielectric (/spl epsiv//sub r,lens/) lenses varies between one and five wavelengths in free-space, in order to obtain radiating structures whose directivity is comprised between 10 and 25 dB. A lot of configurations of lenses are investigated using the finite-difference time-domain methods technique and compared in the 47-50 GHz band as a function of their diameter, extension length and dielectric constant. In particular, the analysis of internal reflections-in time and frequency domains-shows that the latter have potentially a strong influence on the input impedance of small lens antennas, even for low values of /spl epsiv//sub r,lens/(2.2), whereas the usual limit (beyond which anti-reflection coatings are required) is /spl epsiv//sub r,lens/=4. We also demonstrate that the diffraction limit of reduced size lenses is reached for extension lengths varying between 50% and 175% of the extension of synthesized ellipses, depending on the lens material and diameter. Finally, we show that superdirective structures with surface efficiencies reaching 250% can be obtained with small lens diameters, justifying the interest in reduced size lens antennas.     

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.     

Computing the error linear complexity spectrum of a binary sequence of period 2/sup n/

Binary sequences with high linear complexity are of interest in cryptography. The linear complexity should remain high even when a small number of changes are made to the sequence. The error linear complexity spectrum of a sequence reveals how the linear complexity of the sequence varies as an increasing number of the bits of the sequence are changed. We present an algorithm which computes the error linear complexity for binary sequences of period /spl lscr/=2/sup n/ using O(/spl lscr/(log/spl lscr/)/sup 2/) bit operations. The algorithm generalizes both the Games-Chan (1983) and Stamp-Martin (1993) algorithms, which compute the linear complexity and the k-error linear complexity of a binary sequence of period /spl lscr/=2/sup n/, respectively. We also discuss an application of an extension of our algorithm to decoding a class of linear subcodes of Reed-Muller codes.     

Capacity and lattice strategies for canceling known interference

We consider the generalized dirty-paper channel Y=X+S+N,E{X/sup 2/}/spl les/P/sub X/, where N is not necessarily Gaussian, and the interference S is known causally or noncausally to the transmitter. We derive worst case capacity formulas and strategies for "strong" or arbitrarily varying interference. In the causal side information (SI) case, we develop a capacity formula based on minimum noise entropy strategies. We then show that strategies associated with entropy-constrained quantizers provide lower and upper bounds on the capacity. At high signal-to-noise ratio (SNR) conditions, i.e., if N is weak relative to the power constraint P/sub X/, these bounds coincide, the optimum strategies take the form of scalar lattice quantizers, and the capacity loss due to not having S at the receiver is shown to be exactly the "shaping gain" 1/2log(2/spl pi/e/12)/spl ap/ 0.254 bit. We extend the schemes to obtain achievable rates at any SNR and to noncausal SI, by incorporating minimum mean-squared error (MMSE) scaling, and by using k-dimensional lattices. For Gaussian N, the capacity loss of this scheme is upper-bounded by 1/2log2/spl pi/eG(/spl Lambda/), where G(/spl Lambda/) is the normalized second moment of the lattice. With a proper choice of lattice, the loss goes to zero as the dimension k goes to infinity, in agreement with the results of Costa. These results provide an information-theoretic framework for the study of common communication problems such as precoding for intersymbol interference (ISI) channels and broadcast channels.     

Lognormal selection with applications to lifetime data

We consider selection problems on k lognormal (/spl mu//sub i/,/spl sigma//sub i//sup 2/) populations when the /spl sigma//sub i//sup 2/ are unequal, under both known & unknown cases. Selection based on linear functions of /spl mu/ & /spl sigma//sup 2/ are considered. For the unequal & unknown /spl sigma//sub i//sup 2/ case, a two-stage Rinott-type asymptotic procedure is proposed. In consideration to reliability & survival analysis applications, an asymptotic procedure is also proposed for the parametric selection based on the /spl alpha/-quantiles. Simulation studies to evaluate the procedures & to compare the latter procedure to the nonparametric procedure based on the /spl alpha/-quantiles are presented. The procedures are demonstrated using examples from reliability & quality control.