Transceiver designs and analysis for LTI, LTV and broadcast channels - new matrix decompositions and majorization theory

Signal processing techniques play important roles in the design of digital communication systems. These include information manipulation, transmitter signal processing, channel estimation, channel equalization and receiver signal processing. By interacting with communication theory and system implementing technologies, signal processing specialists develop efficient schemes for various communication problems by wisely exploiting various mathematical tools such as analysis, probability theory, matrix theory, optimization theory, and many others. In recent years, researchers realized that multiple-input multiple-output (MIMO) channel models are applicable to a wide range of different physical communications channels. Using the elegant matrix-vector notations, many MIMO transceiver (including the precoder and equalizer) design problems can be solved by matrix and optimization theory. Furthermore, the researchers showed that the majorization theory and matrix decompositions, such as singular value decomposition (SVD), geometric mean decomposition (GMD) and generalized triangular decomposition (GTD), provide unified frameworks for solving many of the point-to-point MIMO transceiver design problems.

In this thesis, we consider the transceiver design problems for linear time invariant (LTI) flat MIMO channels, linear time-varying narrowband MIMO channels, flat MIMO broadcast channels, and doubly selective scalar channels. Additionally, the channel estimation problem is also considered. The main contributions of this dissertation are the development of new matrix decompositions, and the uses of the matrix decompositions and majorization theory toward the practical transmit-receive scheme designs for transceiver optimization problems. Elegant solutions are obtained, novel transceiver structures are developed, ingenious algorithms are proposed, and performance analyses are derived.

The first part of the thesis focuses on transceiver design with LTI flat MIMO channels. We propose a novel matrix decomposition which decomposes a complex matrix as a product of several sets of semi-unitary matrices and upper triangular matrices in an iterative manner. The complexity of the new decomposition, generalized geometric mean decomposition (GGMD), is always less than or equal to that of geometric mean decomposition (GMD). The optimal GGMD parameters which yield the minimal complexity are derived. Based on the channel state information (CSI) at both the transmitter (CSIT) and receiver (CSIR), GGMD is used to design a butterfly structured decision feedback equalizer (DFE) MIMO transceiver which achieves the minimum average mean square error (MSE) under the total transmit power constraint. A novel iterative receiving detection algorithm for the specific receiver is also proposed. For the application to cyclic prefix (CP) systems in which the SVD of the equivalent channel matrix can be easily computed, the proposed GGMD transceiver has K/log_2(K) times complexity advantage over the GMD transceiver, where K is the number of data symbols per data block and is a power of 2. The performance analysis shows that the GGMD DFE transceiver can convert a MIMO channel into a set of parallel subchannels with the same bias and signal to interference plus noise ratios (SINRs). Hence, the average bit rate error (BER) is automatically minimized without the need for bit allocation. Moreover, the proposed transceiver can achieve the channel capacity simply by applying independent scalar Gaussian codes of the same rate at subchannels.

In the second part of the thesis, we focus on MIMO transceiver design for slowly time-varying MIMO channels with zero-forcing or MMSE criterion. Even though the GGMD/GMD DFE transceivers work for slowly time-varying MIMO channels by exploiting the instantaneous CSI at both ends, their performance is by no means optimal since the temporal diversity of the time-varying channels is not exploited. Based on the GTD, we develop space-time GTD (ST-GTD) for the decomposition of linear time-varying flat MIMO channels. Under the assumption that CSIT, CSIR and channel prediction are available, by using the proposed ST-GTD, we develop space-time geometric mean decomposition (ST-GMD) DFE transceivers under the zero-forcing or MMSE criterion. Under perfect channel prediction, the new system minimizes both the average MSE at the detector in each space-time (ST) block (which consists of several coherence blocks), and the average per ST-block BER in the moderate high SNR region. Moreover, the ST-GMD DFE transceiver designed under an MMSE criterion maximizes Gaussian mutual information over the equivalent channel seen by each ST-block. In general, the newly proposed transceivers perform better than the GGMD-based systems since the super-imposed temporal precoder is able to exploit the temporal diversity of time-varying channels. For practical applications, a novel ST-GTD based system which does not require channel prediction but shares the same asymptotic BER performance with the ST-GMD DFE transceiver is also proposed.

The third part of the thesis considers two quality of service (QoS) transceiver design problems for flat MIMO broadcast channels. The first one is the power minimization problem (min-power) with a total bitrate constraint and per-stream BER constraints. The second problem is the rate maximization problem (max-rate) with a total transmit power constraint and per-stream BER constraints. Exploiting a particular class of joint triangularization (JT), we are able to jointly optimize the bit allocation and the broadcast DFE transceiver for the min-power and max-rate problems. The resulting optimal designs are called the minimum power JT broadcast DFE transceiver (MPJT) and maximum rate JT broadcast DFE transceiver (MRJT), respectively. In addition to the optimal designs, two suboptimal designs based on QR decomposition are proposed. They are realizable for arbitrary number of users.

Finally, we investigate the design of a discrete Fourier transform (DFT) modulated filterbank transceiver (DFT-FBT) with LTV scalar channels. For both cases with known LTV channels and unknown wide sense stationary uncorrelated scattering (WSSUS) statistical channels, we show how to optimize the transmitting and receiving prototypes of a DFT-FBT such that the SINR at the receiver is maximized. Also, a novel pilot-aided subspace channel estimation algorithm is proposed for the orthogonal frequency division multiplexing (OFDM) systems with quasi-stationary multi-path Rayleigh fading channels. Using the concept of a difference co-array, the new technique can construct M^2 co-pilots from M physical pilot tones with alternating pilot placement. Subspace methods, such as MUSIC and ESPRIT, can be used to estimate the multipath delays and the number of identifiable paths is up to O(M^2), theoretically. With the delay information, a MMSE estimator for frequency response is derived. It is shown through simulations that the proposed method outperforms the conventional subspace channel estimator when the number of multipaths is greater than or equal to the number of physical pilots minus one.

Effects of particle size ratio on single particle motion in colloidal dispersions

The motion of a single Brownian particle of arbitrary size through a dilute colloidal dispersion of neutrally buoyant bath spheres of another characteristic size in a Newtonian solvent is examined in two contexts. First, the particle in question, the probe particle, is subject to a constant applied external force drawing it through the suspension as a simple model for active and nonlinear microrheology. The strength of the applied external force, normalized by the restoring forces of Brownian motion, is the Péclet number, Pe. This dimensionless quantity describes how strongly the probe is upsetting the equilibrium distribution of the bath particles. The mean motion and fluctuations in the probe position are related to interpreted quantities of an effective viscosity of the suspension. These interpreted quantities are calculated to first order in the volume fraction of bath particles and are intimately tied to the spatial distribution, or microstructure, of bath particles relative to the probe. For weak Pe, the disturbance to the equilibrium microstructure is dipolar in nature, with accumulation and depletion regions on the front and rear faces of the probe, respectively. With increasing applied force, the accumulation region compresses to form a thin boundary layer whose thickness scales with the inverse of Pe. The depletion region lengthens to form a trailing wake. The magnitude of the microstructural disturbance is found to grow with increasing bath particle size -- small bath particles in the solvent resemble a continuum with effective microviscosity given by Einstein's viscosity correction for a dilute dispersion of spheres. Large bath particles readily advect toward the minimum approach distance possible between the probe and bath particle, and the probe and bath particle pair rotating as a doublet is the primary mechanism by which the probe particle is able to move past; this is a process that slows the motion of the probe by a factor of the size ratio. The intrinsic microviscosity is found to force thin at low Péclet number due to decreasing contributions from Brownian motion, and force thicken at high Péclet number due to the increasing influence of the configuration-averaged reduction in the probe's hydrodynamic self mobility. Nonmonotonicity at finite sizes is evident in the limiting high-Pe intrinsic microviscosity plateau as a function of bath-to-probe particle size ratio. The intrinsic microviscosity is found to grow with the size ratio for very small probes even at large-but-finite Péclet numbers. However, even a small repulsive interparticle potential, that excludes lubrication interactions, can reduce this intrinsic microviscosity back to an order one quantity. The results of this active microrheology study are compared to previous theoretical studies of falling-ball and towed-ball rheometry and sedimentation and diffusion in polydisperse suspensions, and the singular limit of full hydrodynamic interactions is noted.

Second, the probe particle in question is no longer subject to a constant applied external force. Rather, the particle is considered to be a catalytically-active motor, consuming the bath reactant particles on its reactive face while passively colliding with reactant particles on its inert face. By creating an asymmetric distribution of reactant about its surface, the motor is able to diffusiophoretically propel itself with some mean velocity. The effects of finite size of the solute are examined on the leading order diffusive microstructure of reactant about the motor. Brownian and interparticle contributions to the motor velocity are computed for several interparticle interaction potential lengths and finite reactant-to-motor particle size ratios, with the dimensionless motor velocity increasing with decreasing motor size. A discussion on Brownian rotation frames the context in which these results could be applicable, and future directions are proposed which properly incorporate reactant advection at high motor velocities.

Topics in randomized numerical linear algebra

This thesis studies three classes of randomized numerical linear algebra algorithms, namely: (i) randomized matrix sparsification algorithms, (ii) low-rank approximation algorithms that use randomized unitary transformations, and (iii) low-rank approximation algorithms for positive-semidefinite (PSD) matrices.

Randomized matrix sparsification algorithms set randomly chosen entries of the input matrix to zero. When the approximant is substituted for the original matrix in computations, its sparsity allows one to employ faster sparsity-exploiting algorithms. This thesis contributes bounds on the approximation error of nonuniform randomized sparsification schemes, measured in the spectral norm and two NP-hard norms that are of interest in computational graph theory and subset selection applications.

Low-rank approximations based on randomized unitary transformations have several desirable properties: they have low communication costs, are amenable to parallel implementation, and exploit the existence of fast transform algorithms. This thesis investigates the tradeoff between the accuracy and cost of generating such approximations. State-of-the-art spectral and Frobenius-norm error bounds are provided.

The last class of algorithms considered are SPSD "sketching" algorithms. Such sketches can be computed faster than approximations based on projecting onto mixtures of the columns of the matrix. The performance of several such sketching schemes is empirically evaluated using a suite of canonical matrices drawn from machine learning and data analysis applications, and a framework is developed for establishing theoretical error bounds.

In addition to studying these algorithms, this thesis extends the Matrix Laplace Transform framework to derive Chernoff and Bernstein inequalities that apply to all the eigenvalues of certain classes of random matrices. These inequalities are used to investigate the behavior of the singular values of a matrix under random sampling, and to derive convergence rates for each individual eigenvalue of a sample covariance matrix.

Social saliency : visual psychophysics and single-neuron recordings in humans

My thesis studies how people pay attention to other people and the environment. How does the brain figure out what is important and what are the neural mechanisms underlying attention? What is special about salient social cues compared to salient non-social cues? In Chapter I, I review social cues that attract attention, with an emphasis on the neurobiology of these social cues. I also review neurological and psychiatric links: the relationship between saliency, the amygdala and autism. The first empirical chapter then begins by noting that people constantly move in the environment. In Chapter II, I study the spatial cues that attract attention during locomotion using a cued speeded discrimination task. I found that when the motion was expansive, attention was attracted towards the singular point of the optic flow (the focus of expansion, FOE) in a sustained fashion. The more ecologically valid the motion features became (e.g., temporal expansion of each object, spatial depth structure implied by distribution of the size of the objects), the stronger the attentional effects. However, compared to inanimate objects and cues, people preferentially attend to animals and faces, a process in which the amygdala is thought to play an important role. To directly compare social cues and non-social cues in the same experiment and investigate the neural structures processing social cues, in Chapter III, I employ a change detection task and test four rare patients with bilateral amygdala lesions. All four amygdala patients showed a normal pattern of reliably faster and more accurate detection of animate stimuli, suggesting that advantageous processing of social cues can be preserved even without the amygdala, a key structure of the “social brain”. People not only attend to faces, but also pay attention to others’ facial emotions and analyze faces in great detail. Humans have a dedicated system for processing faces and the amygdala has long been associated with a key role in recognizing facial emotions. In Chapter IV, I study the neural mechanisms of emotion perception and find that single neurons in the human amygdala are selective for subjective judgment of others’ emotions. Lastly, people typically pay special attention to faces and people, but people with autism spectrum disorders (ASD) might not. To further study social attention and explore possible deficits of social attention in autism, in Chapter V, I employ a visual search task and show that people with ASD have reduced attention, especially social attention, to target-congruent objects in the search array. This deficit cannot be explained by low-level visual properties of the stimuli and is independent of the amygdala, but it is dependent on task demands. Overall, through visual psychophysics with concurrent eye-tracking, my thesis found and analyzed socially salient cues and compared social vs. non-social cues and healthy vs. clinical populations. Neural mechanisms underlying social saliency were elucidated through electrophysiology and lesion studies. I finally propose further research questions based on the findings in my thesis and introduce my follow-up studies and preliminary results beyond the scope of this thesis in the very last section, Future Directions.

Chains of Non-regular de Branges Spaces

We consider canonical systems with singular left endpoints, and discuss the concept of a scalar spectral measure and the corresponding generalized Fourier transform associated with a canonical system with a singular left endpoint. We use the framework of de Branges’ theory of Hilbert spaces of entire functions to study the correspondence between chains of non-regular de Branges spaces, canonical systems with singular left endpoints, and spectral measures.

We find sufficient integrability conditions on a Hamiltonian H which ensure the existence of a chain of de Branges functions in the first generalized Pólya class with Hamiltonian H. This result generalizes de Branges’ Theorem 41, which showed the sufficiency of stronger integrability conditions on H for the existence of a chain in the Pólya class. We show the conditions that de Branges came up with are also necessary. In the case of Krein’s strings, namely when the Hamiltonian is diagonal, we show our proposed conditions are also necessary.

We also investigate the asymptotic conditions on chains of de Branges functions as t approaches its left endpoint. We show there is a one-to-one correspondence between chains of de Branges functions satisfying certain asymptotic conditions and chains in the Pólya class. In the case of Krein’s strings, we also establish the one-to-one correspondence between chains satisfying certain asymptotic conditions and chains in the generalized Pólya class.

Theory of the positive column of a gas discharge

An attempt is made to provide a theoretical explanation of the effect of the positive column on the voltage-current characteristic of a glow or an arc discharge. Such theories have been developed before, and all are based on balancing the production and loss of charged particles and accounting for the energy supplied to the plasma by the applied electric field. Differences among the theories arise from the approximations and omissions made in selecting processes that affect the particle and energy balances. This work is primarily concerned with the deviation from the ambipolar description of the positive column caused by space charge, electron-ion volume recombination, and temperature inhomogeneities.

The presentation is divided into three parts, the first of which involved the derivation of the final macroscopic equations from kinetic theory. The final equations are obtained by taking the first three moments of the Boltzmann equation for each of the three species in the plasma. Although the method used and the equations obtained are not novel, the derivation is carried out in detail in order to appraise the validity of numerous approximations and to justify the use of data from other sources. The equations are applied to a molecular hydrogen discharge contained between parallel walls. The applied electric field is parallel to the walls, and the dependent variables—electron and ion flux to the walls, electron and ion densities, transverse electric field, and gas temperature—vary only in the direction perpendicular to the walls. The mathematical description is given by a sixth-order nonlinear two-point boundary value problem which contains the applied field as a parameter. The amount of neutral gas and its temperature at the walls are held fixed, and the relation between the applied field and the electron density at the center of the discharge is obtained in the process of solving the problem. This relation corresponds to that between current and voltage and is used to interpret the effect of space charge, recombination, and temperature inhomogeneities on the voltage-current characteristic of the discharge.

The complete solution of the equations is impractical both numerically and analytically, and in Part II the gas temperature is assumed uniform so as to focus on the combined effects of space charge and recombination. The terms representing these effects are treated as perturbations to equations that would otherwise describe the ambipolar situation. However, the term representing space charge is not negligible in a thin boundary layer or sheath near the walls, and consequently the perturbation problem is singular. Separate solutions must be obtained in the sheath and in the main region of the discharge, and the relation between the electron density and the applied field is not determined until these solutions are matched.

In Part III the electron and ion densities are assumed equal, and the complicated space-charge calculation is thereby replaced by the ambipolar description. Recombination and temperature inhomogeneities are both important at high values of the electron density. However, the formulation of the problem permits a comparison of the relative effects, and temperature inhomogeneities are shown to be important at lower values of the electron density than recombination. The equations are solved by a direct numerical integration and by treating the term representing temperature inhomogeneities as a perturbation.

The conclusions reached in the study are primarily concerned with the association of the relation between electron density and axial field with the voltage-current characteristic. It is known that the effect of space charge can account for the subnormal glow discharge and that the normal glow corresponds to a close approach to an ambipolar situation. The effect of temperature inhomogeneities helps explain the decreasing characteristic of the arc, and the effect of recombination is not expected to appear except at very high electron densities.

I. Current commutators in quantum electrodynamics. II. Electromagnetic mass differences of the pseudoscalar meson and baryon octets

This thesis is in two parts. In the first section, the operator structure of the singular terms in the equal-time commutator of space and time components of the electromagnetic current is investigated in perturbation theory by establishing a connection with Feynman diagrams. It is made very plausible that the singular term is a c number. Some remarks are made about the same problem in the electrodynamics of a spinless particle.

In the second part, an SU(3) symmetric multi-channel calculation of the electromagnetic mass differences in the pseudoscalar meson and baryon octets is carried out with an attempt to include some of the physics of the crossed (pair annihilation) channel along the lines of the recent work by Ball and Zachariasen. The importance of the tensor meson Regge trajectories is emphasized. The agreement with experiment is poor for the isospin one mass differences, but excellent for those with isospin two.

On the set of eigenvalues of a class of equimodular matrices

The structure of the set ϐ(A) of all eigenvalues of all complex matrices (elementwise) equimodular with a given n x n non-negative matrix A is studied. The problem was suggested by O. Taussky and some aspects have been studied by R. S. Varga and B.W. Levinger.

If every matrix equimodular with A is non-singular, then A is called regular. A new proof of the P. Camion-A.J. Hoffman characterization of regular matrices is given.

The set ϐ(A) consists of m ≤ n closed annuli centered at the origin. Each gap, ɤ, in this set can be associated with a class of regular matrices with a (unique) permutation, π(ɤ). The association depends on both the combinatorial structure of A and the size of the a_ii. Let A be associated with the set of r permutations, π₁, π₂,…, π_r, where each gap in ϐ(A) is associated with one of the π_k. Then r ≤ n, even when the complement of ϐ(A) has n+1 components. Further, if π(ɤ) is the identity, the real boundary points of ɤ are eigenvalues of real matrices equimodular with A. In particular, if A is essentially diagonally dominant, every real boundary point of ϐ(A) is an eigenvalues of a real matrix equimodular with A.

Several conjectures based on these results are made which if verified would constitute an extension of the Perron-Frobenius Theorem, and an algebraic method is introduced which unites the study of regular matrices with that of ϐ(A).

I. The effect of droplet solidification upon two-phase flow in a rocket nozzle. II. A kinetic theory investigation of some condensation-evaporation phenomena by a moment method

Part I:

The perturbation technique developed by Rannie and Marble is used to study the effect of droplet solidification upon two-phase flow in a rocket nozzle. It is shown that under certain conditions an equilibrium flow exists, where the gas and particle phases have the same velocity and temperature at each section of the nozzle. The flow is divided into three regions: the first region, where the particles are all in the form of liquid droplets; a second region, over which the droplets solidify at constant freezing temperature; and a third region, where the particles are all solid. By a perturbation about the equilibrium flow, a solution is obtained for small particle slip velocities using the Stokes drag law and the corresponding approximation for heat transfer between the particle and gas phases. Singular perturbation procedure is required to handle the problem at points where solidification first starts and where it is complete. The effects of solidification are noticeable.

Part II:

When a liquid surface, in contact with only its pure vapor, is not in the thermodynamic equilibrium with it, a net condensation or evaporation of fluid occurs. This phenomenon is studied from a kinetic theory viewpoint by means of moment method developed by Lees. The evaporation-condensation rate is calculated for a spherical droplet and for a liquid sheet, when the temperatures and pressures are not too far removed from their equilibrium values. The solutions are valid for the whole range of Knudsen numbers from the free molecule to the continuum limit. In the continuum limit, the mass flux rate is proportional to the pressure difference alone.

Part I. Approximate Hartree-Fock wavefunctions one-electron properties and electronic structure of the water molecule. Part II. Perturbation-variational calculation of the nuclear spin-spin isotope coupling constant in HD

Part I

Several approximate Hartree-Fock SCF wavefunctions for the ground electronic state of the water molecule have been obtained using an increasing number of multicenter s, p, and d Slater-type atomic orbitals as basis sets. The predicted charge distribution has been extensively tested at each stage by calculating the electric dipole moment, molecular quadrupole moment, diamagnetic shielding, Hellmann-Feynman forces, and electric field gradients at both the hydrogen and the oxygen nuclei. It was found that a carefully optimized minimal basis set suffices to describe the electronic charge distribution adequately except in the vicinity of the oxygen nucleus. Our calculations indicate, for example, that the correct prediction of the field gradient at this nucleus requires a more flexible linear combination of p-orbitals centered on this nucleus than that in the minimal basis set. Theoretical values for the molecular octopole moment components are also reported.

Part II

The perturbation-variational theory of R. M. Pitzer for nuclear spin-spin coupling constants is applied to the HD molecule. The zero-order molecular orbital is described in terms of a single 1s Slater-type basis function centered on each nucleus. The first-order molecular orbital is expressed in terms of these two functions plus one singular basis function each of the types e^-r/r and e^-r ln r centered on one of the nuclei. The new kinds of molecular integrals were evaluated to high accuracy using numerical and analytical means. The value of the HD spin-spin coupling constant calculated with this near-minimal set of basis functions is J_HD = +96.6 cps. This represents an improvement over the previous calculated value of +120 cps obtained without using the logarithmic basis function but is still considerably off in magnitude compared with the experimental measurement of J_HD = +43 0 ± 0.5 cps.

On concentrated loads and Green's functions in elastostatics

This investigation is concerned with the notion of concentrated loads in classical elastostatics and related issues. Following a limit treatment of problems involving concentrated internal and surface loads, the orders of the ensuing displacements and stress singularities, as well as the stress resultants of the latter, are determined. These conclusions are taken as a basis for an alternative direct formulation of concentrated-load problems, the completeness of which is established through an appropriate uniqueness theorem. In addition, the present work supplies a reciprocal theorem and an integral representation-theorem applicable to singular problems of the type under consideration. Finally, in the course of the analysis presented here, the theory of Green's functions in elastostatics is extended.

A kinetic theory description for external spherical flows with arbitrary Knudsen number by a moment method

The Maxwell integral equations of transfer are applied to a series of problems involving flows of arbitrary density gases about spheres. As suggested by Lees a two sided Maxwellian-like weighting function containing a number of free parameters is utilized and a sufficient number of partial differential moment equations is used to determine these parameters. Maxwell's inverse fifth-power force law is used to simplify the evaluation of the collision integrals appearing in the moment equations. All flow quantities are then determined by integration of the weighting function which results from the solution of the differential moment system. Three problems are treated: the heat-flux from a slightly heated sphere at rest in an infinite gas; the velocity field and drag of a slowly moving sphere in an unbounded space; the velocity field and drag torque on a slowly rotating sphere. Solutions to the third problem are found to both first and second-order in surface Mach number with the secondary centrifugal fan motion being of particular interest. Singular aspects of the moment method are encountered in the last two problems and an asymptotic study of these difficulties leads to a formal criterion for a "well posed" moment system. The previously unanswered question of just how many moments must be used in a specific problem is now clarified to a great extent.

Some theorems in classical elastodynamic

This investigation is concerned with various fundamental aspects of the linearized dynamical theory for mechanically homogeneous and isotropic elastic solids. First, the uniqueness and reciprocal theorems of dynamic elasticity are extended to unbounded domains with the aid of a generalized energy identity and a lemma on the prolonged quiescence of the far field, which are established for this purpose. Next, the basic singular solutions of elastodynamics are studied and used to generate systematically Love's integral identity for the displacement field, as well as an associated identity for the field of stress. These results, in conjunction with suitably defined Green's functions, are applied to the construction of integral representations for the solution of the first and second boundary-initial value problem. Finally, a uniqueness theorem for dynamic concentrated-load problems is obtained.

On the distribution of the signs of the conjugates of the cyclotomic units in the maximal real subfield of the qth cyclotomic field, q A prime

Let F = Ǫ(ζ + ζ ^–1) be the maximal real subfield of the cyclotomic field Ǫ(ζ) where ζ is a primitive qth root of unity and q is an odd rational prime. The numbers u₁=-1, u_k=(ζ^k-ζ^-k)/(ζ-ζ^-1), k=2,…,p, p=(q-1)/2, are units in F and are called the cyclotomic units. In this thesis the sign distribution of the conjugates in F of the cyclotomic units is studied.

Let G(F/Ǫ) denote the Galoi's group of F over Ǫ, and let V denote the units in F. For each σϵ G(F/Ǫ) and μϵV define a mapping sgn_σ: V→GF(2) by sgn_σ(μ) = 1 iff σ(μ) ˂ 0 and sgn_σ(μ) = 0 iff σ(μ) ˃ 0. Let {σ₁, ... , σ_p} be a fixed ordering of G(F/Ǫ). The matrix M_q=(sgn_σj(v_i) ) , i, j = 1, ... , p is called the matrix of cyclotomic signatures. The rank of this matrix determines the sign distribution of the conjugates of the cyclotomic units. The matrix of cyclotomic signatures is associated with an ideal in the ring GF(2) [x] / (x^p+ 1) in such a way that the rank of the matrix equals the GF(2)-dimension of the ideal. It is shown that if p = (q-1)/ 2 is a prime and if 2 is a primitive root mod p, then M_q is non-singular. Also let p be arbitrary, let ℓ be a primitive root mod q and let L = {i | 0 ≤ i ≤ p-1, the least positive residue of defined by ℓⁱ mod q is greater than p}. Let H_q(x) ϵ GF(2)[x] be defined by H_q(x) = g. c. d. ((Σ xⁱ/I ϵ L) (x+1) + 1, x^p + 1). It is shown that the rank of M_q equals the difference p - degree H_q(x).

Further results are obtained by using the reciprocity theorem of class field theory. The reciprocity maps for a certain abelian extension of F and for the infinite primes in F are associated with the signs of conjugates. The product formula for the reciprocity maps is used to associate the signs of conjugates with the reciprocity maps at the primes which lie above (2). The case when (2) is a prime in F is studied in detail. Let T denote the group of totally positive units in F. Let U be the group generated by the cyclotomic units. Assume that (2) is a prime in F and that p is odd. Let F₍₂₎ denote the completion of F at (2) and let V₍₂₎ denote the units in F₍₂₎. The following statements are shown to be equivalent. 1) The matrix of cyclotomic signatures is non-singular. 2) U∩T = U². 3) U∩F²₍₂₎ = U². 4) V₍₂₎/ V₍₂₎² = ˂v₁ V₍₂₎²˃ ʘ…ʘ˂v_p V₍₂₎²˃ ʘ ˂3V₍₂₎²˃.

The rank of M_q was computed for 5≤q≤929 and the results appear in tables. On the basis of these results and additional calculations the following conjecture is made: If q and p = (q -1)/ 2 are both primes, then M_q is non-singular.