the computation of the fundamental unit of totally complex quartic orders we describe an efficient algorithm for the computation of the regulator and a fundamental unit of an arbitrary totally complex quartic order we analyze its complexity and 4 we present tables with computational results for the orders z v 1 d 500

some periodic continued fractions with long periods let p d be the period length of the continued fraction for jd under the extended riemann hypothesis for 9 jd one would expect that p d o d1 2 log log d in order to test this it is necessary to find values of d for which p d is large this in turn requires that we be able to find solutions to large sets of simultaneous linear congruences the university of manitoba sieve unit umsu a machine similar to d h lehmer s dls 127 was used to find such values of d for example if d 46257585588439 then p d 25679652 some results are also obtained for the voronoi continued fraction for vd

rice s representation for cyclostationary processes the well known formulas relating power spectral densities of a bandpass process and its low pass representors in rice s representation are generalized to accommodate cyclostationary processes which can be more appropriate than stationary processes as models for modulated signals it is explained that improper phase randomization methods for stationarizing cyclostationary processes can yield misleading formulas for power spectral densities in rice s representation in addition the relationships among the spectral correlation functions for a cyclostationary bandpass process and its cyclostationary low pass representors are derived

a rapid method of evaluating the regulator and class number of a pure cubic field let 2 0 be the algebraic number field formed by adjoining 9 to the rationals 2 let r and h be respectively the regulator and class number of shanks has described a method of evaluating r for 2 vd where d is a positive integer his technique improved the speed of the usual continued fraction algorithm for finding r by allowing one to proceed almost directly from the nth to the mth step where m is approximately in in the continued fraction expansion of id this paper shows how shanks idea can be extended to the voronoi algorithm which is used to find r in cubic fields of negative discriminant it also discusses at length an algorithm for finding r and h for pure cubic fields 1 fd d an integer under a certain generalized riemann hypothesis the ideas developed here will provide a new method which will find r and h in 0 02 5 e operations when h is small this is an improvement over the 0 d h operations required by voronoi s algorithm to find r for example with d 200171999 it required only 5 minutes for an amdahl 470 vi computer to find that r 518594546 969083 and h 1 this same calculation would require about 8 days of computer time if it used only the standard voronoi algorithm a 0 and gcd w m2 m3 n2 n3 a 1 also if e m2n3 m3n2 then e a if u e we denote its conjugates by w and we also write the norm of as nia and define it as nico oco w from the simple fact that

a local version of karamardian s theorem on lower semicontinuous strictly quasiconvex functions abstract it is known that lower semicontinuous strictly quasiconvex functions are quasiconvex it is shown that semilocally strictly quasiconvex functions are semilocally quasiconvex on a locally star shaped domain if some condition of semilocal lower semicontinuity is satisfied these things are analyzed in some detail the functions involved have a connected quasiorder as their range as a motivation it is shown that a multiobjective function is a quasiconvex and strictly quasiconvex with respect to the lexminmax order if its components have the same properties there is a discussion of some applications to multiobjective decision particularly with lexminmax as the underlying decision criterion

voronoi s algorithm in purely cubic congruence function fields of unit rank 1 the first part of this paper classifies all purely cubic function fields over a finite field of characteristic not equal to 3 in the remainder we describe a method for computing the fundamental unit and regulator of a purely cubic congruence function field of unit rank 1 and characteristic at least 5 the technique is based on voronoi s algorithm for generating a chain of successive minima in a multiplicative cubic lattice which is used for calculating the fundamental unit and regulator of a purely cubic number field

cycloergodic properties of discrete parameter nonstationary stochastic processes it is shown that a large class of nonstationary discrete parameter stochastic processes possess novel ergodic properties which are referred to as em cycloergodic properties specifically it is shown that periodic components of time varying probabilistic parameters can be consistently estimated from time averages on one sample path the cycloergodic theory developed herein extends and generalizes existing ergodic theory for asymptotically mean stationary and n stationary cyclostationary processes and is presented in both wide sense and strict sense contexts

digital implementations of spectral correlation analyzers the issues involved in the design of computationally efficient algorithms for spectral correlation estimation and the resulting impact on high speed digital realization are addressed the spectral correlation analyzer is characterized as a periodically time varying quadratic system with a kernel possessing certain gross properties the mean and variance of the output are expressed in terms of the kernel and the spectral correlation function of the input three realizations are analyzed in detail one is based on the frequency smoothing method of cross spectral analysis the others are variants of the time smoothing method for each of these realizations an exact expression for the quadratic system kernel is given the digital implementation is developed and a detailed complexity analysis is presented high speed pipeline realizations of the algorithms are analyzed and related special issues are discussed examples involving the calculation of the spectral correlation function in near real time for broadband communications signals are discussed

generalizations of convex and related functions abstract the concept of semilocally convexity was recently introduced by ewing 2 this paper defines semilocally quasiconvex semi locally pseudoconvex and other related functions and investigates some of their properties

random modulation a review a unified treatment of various aspects of random modulation is presented with emphasis on the spectral properties of fm signals the study includes reexamination of rice s representation woodward s theorem fsk and processes with normal markov and cyclostationary frequencies

programming with semilocally convex functions abstract a theorem of the alternatives is derived for semilocally convex functions defined on locally starshaped sets this result is applied to constrained minimization problems to obtain optimality conditions and duality theorems

on principal ideal testing in totally complex quartic fields and the determination of certain cyclotomic constants let y be any totally complex quartic field two algorithms are described for determining whether or not any given ideal in y is principal one of these algorithms is very efficient in practice but its complexity is difficult to analyze the other algorithm is computationally more elaborate but in this case a complexity analysis can be provided these ideas are applied to the problem of determining the cyclotomic numbers of order 5 for a prime p 1 mod 5 given any quadratic or quintic nonresidue of p it is shown that these cyclotomic numbers can be efficiently computed in 0 log p 3 binary operations

common pitfalls in the application of stationary process theory to time sampled and modulated signals the common practice of applying the theory of stationary stochastic processes to a cyclostationary process by introducing random phase s into the probabilistic model in order to stationarize the process can lead to erroneous results such as incorrect formulas for power spectral density this is illustrated by showing that commonly used formulas for signals that have undergone frequency conversion or time sampling can be incorrect the source of error is shown to be inappropriate phase randomization procedures the correct procedure is described and corrected formulas are given the problem is further illustrated by showing that commonly used resolution and reliability mean and variance formulas for spectrum analyzers must be corrected for cyclostationary signals it is explained that all corrections to formulas reflect the effects of spectral correlation these effects are inappropriately averaged out by inappropriate phase randomization procedures it is further explained that these inappropriate procedures destroy the important property of ergodicity of the probabilistic model

the role of spectral correlation in design and performance analysis of synchronizers the popular class of synchronizers that consist of a quadratic nonlinearity followed by a phase lock loop is investigated and it is shown that the optimum design of the quadratic transformation is characterized in terms of a spectral correlation function for the signal to be synchronized to it is also shown that the snr performance of this quadratic transformation and the mean square phase jitter of the phaselock loop are both characterized in terms of spectral correlation functions the conditions under which the optimum quadratic transformations for symbol synchronization of bpsk qpsk sqpsk and msk and for carrier synchronization of bpsk reduce to the well known matched filter squarer are identified in addition the well known zeromean square phase jitter condition is generalized from pam to all synchronizable signals and is characterized in terms of the spectral correlation function the low snr maximum likelihood synchronizer for all quadratically synchronizable signals is characterized in terms of a multiplicity of maximum snr quadratic spectral line generators a closed form implementation in terms of a matched filter squarer and symbol rate synchronized averager is obtained for bpsk and qpsk signals

the performance of time dependent adaptive filters for interference rejection time dependent adaptive filters tdafs that allow for the cyclostationary nature of communication signals by periodically changing the filter and adaptation parameters are examined the tdaf has an advantage over the conventional time independent adaptive filter in achieving better performance i e reduced mean square error mse for signals with periodic statistics the basic theory of the tdaf is presented the tdaf is shown to be more effective than the time independent adaptive filter for interference rejection this is verified by theoretical analysis and computer simulation of specific cases of extracting a signal in noise or interference the criteria for judging the performance of the tdaf for interference rejection are mse bit error rate measurements and constellation diagrams

spread spectrum despreading without the code a new method for despreading direct sequence spreadspectrum signals without use of the spreading code is evaluated by analysis and simulation the new method exploits the cyclic autocorrelation in place of the conventional autocorrelation which was used in a prior method both methods require that the period of the code be equal to a multiple of the data symbol interval broad band noise and narrowband interference rejection capabilities of both methods are compared it is shown that the new method unlike the prior method can provide substantial immunity to strong narrow band interference

spectral correlation of modulated signals part i analog modulation the importance of the concept of cyclostationarity in design and analysis of signal detectors synchronizers and extractors in communication systems is briefly discussed and the central role of spectral correlation in the characterization of random processes that are cyclostationary in the wide sense is explained a spectral correlation function that is a generalization of the power spectral density function is described and a corresponding generalization of the wiener khinchine relation and several other fundamental spectral correlation relations also are described explicit formulas for the spectral correlation function for various types of analog modulated signals are derived this includes pulse and carrier amplitude modulation quadrature amplitude carrier modulation and phase and frequency carrier modulation to illustrate the differing spectral correlation characteristics of different modulation types the magnitudes of the spectral correlation functions are graphed or described in graphical terms as the heights of surfaces above a bifrequency plane

on the computation of unit groups and class groups of totally complex quartic fields in this paper we describe the computation of a system of fundamental units and of the class group for each totally real quartic field y of discriminant less than 106 generating equations integral bases and the galois groups for all those fields were recently given by buchmann and ford

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

frequency domain interpretation of the cramer rao bound for carrier and clock synchronization the cramer rao bound is a fundamental lower bound on the variance of unbiased estimates applied to carrier phase and clock phase synchronization for digital communications it is shown that the corresponding bounds can be derived from the power density spectrum of the transmitted signal irrespective of the particular modulation scheme this allows the localization of the available information about the parameters in the frequency domain if the incoming signal is only partly exploited the resulting performance loss is now easily attainable

second order blind separation of first and second order cyclostationary sources application to am fsk cpfsk and deterministic sources most of the second order so and higher order ho blind source separation bss methods developed this last decade aim at blindly separating statistically independent sources that are assumed zero mean stationary and ergodic nevertheless in many situations of practical interest such as in radiocommunications contexts the sources are nonstationary and very often cyclostationary digital modulations the behavior of the current so and fourth order fo cumulant based bss methods in the presence of cyclostationary sources has been analyzed recently in a previous paper by ferre spl acute ol and chevalier assuming zero mean sources however some cyclostationary sources used in practical situations are not zero mean but have a first order fio cyclostationarity property which is in particular the case for some amplitude modulated am signals and for some nonlinearly modulated digital sources such as frequency shift keying fsk or some continuous phase frequency shift keying cpfsk sources for such sources the results presented in the previous paper by ferre spl acute ol and chevalier no longer hold and the purpose of this paper is to analyze the behavior and to propose adaptations of the current so bss methods for sources that are both fio and so cyclostationary and cyclo ergodic an extension for deterministic sources is also proposed in the paper

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

improved methods for calculating vectors of short length in a lattice including a complexity analysis the standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of z in a suitable box however it suffices to consider those x e z which lie in a suitable ellipsoid having a much smaller volume than the box we show in this paper that searching through that ellipsoid is in many cases much more efficient if combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions several randomly constructed numerical examples illustrate the superiority of our new method over the known ones

the distortion rate function of cyclostationary gaussian processes a general expression for the quadratic distortion rate function drf of cyclostationary gaussian processes in terms of their spectral properties is derived this expression can be seen as the result of orthogonalization over the different components in the polyphase decomposition of the process we use this expression to derive in a closed form the drf of several cyclostationary processes arising in practice we first consider the drf of a combined sampling and source coding problem it is known that the optimal coding strategy for this problem involves source coding applied to a signal with the same structure as one resulting from pulse amplitude modulation pam since a pam modulated signal is cyclostationary our drf expression can be used to solve for the minimal distortion in the combined sampling and source coding problem we also analyze in more detail the drf of a source with the same structure as a pam modulated signal and show that it is obtained by reverse waterfilling over an expression that depends on the energy of the pulse and the baseband process modulated to obtain the pam signal this result is then used to explore the effect of the symbol rate in pam on the drf of its output in addition we also study the drf of sources with an amplitude modulation structure and show that the drf of a narrow band gaussian stationary process modulated by either a deterministic or a random phase sine wave equals the drf of the baseband process

signal interception a unifying theoretical framework for feature detection the unifying framework of the spectral correlation theory of cyclostationary signals is used to present a broad treatment of weak random signal detection for interception purposes the relationships among a variety of previously proposed ad hoc detectors optimum detectors and newly proposed detectors are established the spectral correlation plane approach to the interception problem is put forth as especially promising for detection classification and estimation in particularly difficult environments involving unknown and changing noise levels and interference activity a fundamental drawback of the popular radiometric methods in such environments is explained

duality in multiobjective nonlinear programming involving semilocally convex and related functions necessary optimality conditions are established for a multiobjective nonlinear programming problem involving semilocally convex and related functions in terms of their right differentials wolfe type and mond weir type duals are formulated and duality results are proved under the assumptions of semilocal convexity semilocal quasiconvexity and semilocal pseudoconvexity

modulation by means of linear periodic filtering linear periodically time varying filters are considered as modulators and demodulators for the transmission of analog information over a stationary gaussian channel a complete analysis is carried out based on a certain general decomposition of such a filter into a number of branches each containing a linear time invariant filter and a multiplication with a complex sinusoid receiver optimization is carried out under the assumption that the above decomposition is finite finally a joint optimization of transmitter and receiver is derived under the further restriction that the message signal is strictly band limited

a separability theorem for 2m conjugate symmetric signals impinging on an m element sensor array we show under reasonable assumptions that an m sensor linear conjugate linear lcl processor can separate up to 2m conjugate symmetric signals including the cases in which up to m of the signals each share the same direction of arrival as each of m other signals numerical evaluations illustrate ways in which the performance of the m sensor lcl processor is superior to that of a conventional 2m sensor linear processor when 2m conjugate symmetric signals are received

on the infrastructure of the principal ideal class of an algebraic number field of unit rank one let r be the regulator and let d be the absolute value of the discriminant of an order 0 of an algebraic number field of unit rank 1 it is shown how the infrastructure idea of shanks can be used to decrease the number of binary operations needed to compute r from the best known 0 rd for most continued fraction methods to 0 rll2de these ideas can also be applied to significantly decrease the number of operations needed to determine whether or not any fractional ideal of 0 is principal

structural characterization of locally optimum detectors in terms of locally optimum estimators and correlators explicit formulas for locally snr rightarrow 0 optimum mmse signal estimators smoother filter and predictor for discrete time observations of a random signal in additive random noise are derived and used to characterize the locally optimum likelihood ratio signal detector for on off signaling the characterizations are canonical distribution free detector structures involving estimator correlators these structural characterizations provide new interpretations of known detectors for various special cases if the one step signal predictor is recursive and the noise is white possibly non gaussian or nonstationary there is a canonical structure that admits recursive computation the primary motivation for these structural characterizations is to render the estimator correlator design philosophy applicable for the purpose of simplifying implementations and enhancing adaptability unlike the known esfimator correlator structural characterizations for continuous time globally optimum detectors the new characterizations apply for non gaussian as well as gaussian noise and the estimators are explicit rather than implicit

performance analysis of the cyclic music method of direction estimation for cyclostationary signals the cyclic music algorithm exploits the cyclostationarity exhibited by most man made communication signals to perform signal selective estimation of directions of arrival at an array of sensors in the paper the bias and mean squared error of these direction estimates are derived under the assumption that the signals are complex possibly non gaussian and cyclostationary the analytical results are shown to be excellent predictors of empirical results obtained from computer simulation of the cyclic music algorithm and are also compared with the cramer rao bound which in certain cases can be orders of magnitude less than the cramer rao bound for the directions of arrival of stationary signals