a resolution of paz s conjecture in the presence of a nonderogatory matrix abstract let m n f be the algebra of n n matrices over the field f and let s be a generating set of m n f as an f algebra the length of a finite generating set s of m n f is the smallest number k such that words of length not greater than k generate m n f as a vector space it is a long standing conjecture of paz that the length of any generating set of m n f cannot exceed 2 n 2 we prove this conjecture under the assumption that the generating set s contains a nonderogatory matrix in addition we find linear bounds for the length of generating sets that include a matrix with some conditions on its jordan canonical form finally we investigate cases when the length 2 n 2 is achieved

criteria for embedding of spaces constructed by the method of means with arbitrary quasi concave functional parameters we consider a generalization x0 x1 p0 p1 of the method of means to arbitrary non degenerate functional parameter in this case non trivial embedding x0 x1 p0 p1 x0 x1 q0 q1 take place we find necessary and sufficient condition for such embedding if 1 q0 p0 and 1 q1 p1 or 1 p0 q0 and 1 p1 q1

characterizing the statistical properties of mutual information in mimo channels we consider gaussian multiple input multiple output mimo frequency selective spatially correlated fading channels assuming that the channel is unknown at the transmitter and perfectly known at the receiver for gaussian codebooks using results from multivariate statistics we derive an analytical expression for a tight lower bound on the ergodic capacity of such channels at any signal to noise ratio snr we show that our bound is tighter than previously reported analytical lower bounds and we proceed to analytically quantify the impact of spatial fading correlation on ergodic capacity based on a closed form approximation of the variance of mutual information in correlated flat fading mimo channels we provide insights into the multiplexing diversity tradeoff for gaussian code books furthermore for a given total number of antennas we consider the problem of finding the optimal ergodic capacity maximizing number of transmit and receive antennas and we reveal the snr dependent nature of the maximization strategy finally we present numerical results and comparisons between our capacity bounds and previously reported bounds

higher radix square rooting a general discussion on nonrestoring square root algorithms is presented showing bounds and constraints delimiting the space of feasible algorithms for all the choices of radix digit set and representation of the partial remainder two classes of algorithms are then derived from the general discussion and it is shown how it is possible to determine two parameters with a relevant impact on the implementation the number of radicand bits to be inspected in order to obtain a starting value and the number of partial remainder bits to be examined for digit selection the algorithms for the specific case of radix 4 digit set 2 1 0 1 2 and partial remainder represented in carry save form are derived in order to show that the algorithms introduced can lead to better results than those obtained with algorithms previously presented

lengths of quasi commutative pairs of matrices abstract in this paper we discuss some partial solutions of the length conjecture which describes the length of a generating system for matrix algebras we consider mainly the sets of two matrices which are quasi commuting it is shown that in this case the length function is linearly bounded we also analyze which particular natural numbers can be realized as the lengths of certain special generating sets and prove that for commuting or product nilpotent pairs all possible numbers are realizable however there are non realizable values between lower and upper bounds for the other quasi commuting pairs in conclusion we also present several related open problems

minimal and maximal methods of interpolation abstract we show that several interpolation functors including the widely used real and complex methods are minimal or maximal extensions from a single couple of banach spaces various consequences are drawn from this property

differential space time modulation space time coding and modulation exploit the presence of multiple transmit antennas to improve the performance on multipath radio channels thus far most work on space time coding has assumed that perfect channel estimates are available at the receiver in certain situations however it may be difficult or costly to estimate the channel accurately in which case it is natural to consider the design of modulation techniques that do not require channel estimates at the transmitter or receiver we propose a general approach to differential modulation for multiple transmit antennas based on group codes this approach ran be applied to any number of transmit and receive antennas and any signal constellation we also derive low complexity differential receivers error bounds and modulator design criteria which we use to construct optimal differential modulation schemes for two transmit antennas these schemes can be demodulated with or without channel estimates this permits the receiver to exploit channel estimates when they are available the performance degrades by approximately 3 db when estimates are not available

limited feedback unitary precoding for spatial multiplexing systems multiple input multiple output mimo wireless systems use antenna arrays at both the transmitter and receiver to provide communication links with substantial diversity and capacity spatial multiplexing is a common space time modulation technique for mimo communication systems where independent information streams are sent over different transmit antennas unfortunately spatial multiplexing is sensitive to ill conditioning of the channel matrix precoding can improve the resilience of spatial multiplexing at the expense of full channel knowledge at the transmitter which is often not realistic this correspondence proposes a quantized precoding system where the optimal precoder is chosen from a finite codebook known to both receiver and transmitter the index of the optimal precoder is conveyed from the receiver to the transmitter over a low delay feedback link criteria are presented for selecting the optimal precoding matrix based on the error rate and mutual information for different receiver designs codebook design criteria are proposed for each selection criterion by minimizing a bound on the average distortion assuming a rayleigh fading matrix channel the design criteria are shown to be equivalent to packing subspaces in the grassmann manifold using the projection two norm and fubini study distances simulation results show that the proposed system outperforms antenna subset selection and performs close to optimal unitary precoding with a minimal amount of feedback

computing functions cos sup 1 and sin sup 1 using cordic an extension of the cordic coordinate rotation digital computer algorithm that makes it possible to compute the functions cos sup 1 sin sup 1 square root 1 t sup 2 sinh sup 1 cosh sup 1 and square root 1 t sup 2 is presented the algorithms are suitable for vlsi implementation and require only a slight modification of the original cordic algorithm

generalized linear precoder and decoder design for mimo channels using the weighted mmse criterion we address the problem of designing jointly optimum linear precoder and decoder for a mimo channel possibly with delay spread using a weighted minimum mean squared error mmse criterion subject to a transmit power constraint we show that the optimum linear precoder and decoder diagonalize the mimo channel into eigen subchannels for any set of error weights furthermore we derive the optimum linear precoder and decoder as functions of the error weights and consider specialized designs based on specific choices of error weights we show how to obtain 1 the maximum information rate design 2 qos based design we show how to achieve any set of relative snrs across the subchannels and 3 the unweighted mmse and equal error design for fixed rate systems

calderon couples of lipschitz spaces two real interpolation functors f0 f1 are said to form a c pair if f0 x f1 x is a caldero couple for every banach couple x we give a necessary condition for f0 and f1 to form a c pair in terms of the fundamental functions of these functors this condition turns out to be sufficient for the functors defined by weighted lp parameters as a corollary the complete description of all calderon couples of lipschitz spaces is given some new results are also obtained for couples of lorentz spaces

equal gain transmission in multiple input multiple output wireless systems multiple input multiple output mimo wireless systems are of interest due to their ability to provide substantial gains in capacity and quality the paper proposes equal gain transmission egt to provide diversity advantage in mimo systems experiencing rayleigh fading the applications of egt with selection diversity combining equal gain combining and maximum ratio combining are addressed it is proven that systems using egt with any of these combining schemes achieve full diversity order when transmitting over a memoryless flat fading rayleigh matrix channel with independent entries since in practice full channel knowledge at the transmitter is difficult to realize a quantized version of egt is proposed an algorithm to construct a beamforming vector codebook that guarantees full diversity order is presented monte carlo simulation comparisons with various beamforming and combining systems illustrate the performance as a function of quantization

iterative superlinear convergence svd beamforming algorithm and vlsi architecture for mimo ofdm systems in this paper we propose a singular value decomposition svd algorithm with superlinear convergence rate which is suitable for the beamforming mechanism in mimo ofdm channels with short coherent time or short training sequence the proposed superlinear convergence svd sl svd algorithm has the following features 1 superlinear convergence rate 2 the ability of being extended smaller numbers of transmit and receive antennas 3 being insensitive to dynamic range problems during the iterative process in hardware implementations and 4 low computational cost we verify the proposed design by using the vlsi implementation with cmos 90 nm technology the post layout result of the design has the feature of 0 48 core area and 18 mw power consumption our design can achieve 7 m channel matrices s and can be extended to deal with different transmit and receive antenna sets

an on line square root algorithm in this correspondence a systematic derivation of an on line square root algorithm is presented the algorithm operates on variables represented in the normalized radix r floating point system in a digit by digit fashion with an on line delay of 1 the approach used in deriving the square root algorithm is described in detail the basic characteristics of hardware level implementation are also discussed

a remark on reducing iteration time when result digit is zero for radix 2 srt division and square root with redundant remainders in a previous paper by p montuschi and l ciminiera ibid vol 42 no 2 p239 246 feb 1993 an architecture for shared radix 2 division and square root has been presented whose main characteristic is the ability to avoid any addition subtraction when the digit 0 has been selected here we emphasize the characteristics of the digit selection mechanism used by montuschi and ciminiera by presenting a small modification of the digit selection hardware which has the benefit to further reduce the computation delay with respect to the time estimated in that work

gaussian multiaccess channels with isi capacity region and multiuser water filling the capacity region of a two user gaussian multiaccess channel with intersymbol interference isi in which the inputs pass through respective linear systems and are superimposed before being corrupted by an additive gaussian noise process is discussed a geometrical method for obtaining the optimal input power spectral densities and the capacity region is presented this method can be viewed as a nontrivial generalization of the single user water filling argument it is shown that as in the traditional memoryless multiaccess channel frequency division multiaccess fdma with optimally selected frequency bands for each user achieves the total capacity of the multiuser gaussian multiaccess channel with isi however the capacity region of the two user channel with memory is in general not a pentagon unless the channel transfer functions for both users are identical

coset codes i introduction and geometrical classification practically all known good constructive coding techniques for bandlimited channels including lattice codes and various trellis coded modulation schemes can be characterized as coset codes a coset code is defined by a lattice partition lambda lambda and by a binary encoder c that selects a sequence of cosets of the lattice lambda the fundamental coding gain of a coset code as well as other important parameters such as the error coefficient the decoding complexity and the constellation expansion factor are purely geometric parameters determined by c lambda lambda the known types of coset codes as well as a number of new classes that systematize and generalize known codes are classified and compared in terms of these parameters

spatio temporal coding for wireless communication multipath signal propagation has long been viewed as an impairment to reliable communication in wireless channels this paper shows that the presence of multipath greatly improves achievable data rate if the appropriate communication structure is employed a compact model is developed for the multiple input multiple output mimo dispersive spatially selective wireless communication channel the multivariate information capacity is analyzed for high signal to noise ratio snr conditions the mimo channel can exhibit a capacity slope in bits per decibel of power increase that is proportional to the minimum of the number multipath components the number of input antennas or the number of output antennas this desirable result is contrasted with the lower capacity slope of the well studied case with multiple antennas at only one side of the radio link a spatio temporal vector coding stvc communication structure is suggested as a means for achieving mimo channel capacity the complexity of stvc motivates a more practical reduced complexity discrete matrix multitone dmmt space frequency coding approach both of these structures are shown to be asymptotically optimum an adaptive lattice trellis coding technique is suggested as a method for coding across the space and frequency dimensions that exist in the dmmt channel experimental examples that support the theoretical results are presented

higher radix square root with prescaling a scheme for performing higher radix square root based on prescaling of the radicand is presented to reduce the complexity of the result digit selection the scheme requires several steps namely multiplication for prescaling the radicand square root multiplication for prescaling for the division and division online algorithms are used to reduce the overall time and pipelining to reuse the different modules an estimate of the execution time for a radix 256 unit for double precision square root and a comparison with other implementations indicate that the proposed approach is an alternative to consider when designing a square root unit

upper bounds for the length of non associative algebras abstract we introduce the notion of length for non associative finite dimensional unitary algebras and obtain a sharp upper bound for the lengths of algebras belonging to this class we also put forward a new method of characteristic sequences based on linear algebra technique which provides an efficient tool for computing the length function in non associative case then we apply the introduced method to obtain an upper bound for the length of an arbitrary locally complex algebra in the last case the length is bounded in terms of the fibonacci sequence we present concrete examples demonstrating the sharpness of our bounds

generalized geometric mean decomposition and dfe transceiver design part i design and complexity this paper considers a new 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 inner most triangular matrix has its diagonal elements equal to the geometric mean of the singular values of the target complex matrix the complexity defined in terms of the number of floating point operations of the new decomposition generalized geometric mean decomposition ggmd depends on its parameters but is always less than or equal to that of geometric mean decomposition gmd the optimal parameters which yield the minimal complexity are derived the paper also shows how to use ggmd to design an optimal decision feedback equalizer dfe transceiver for multiple input multiple output mimo channels without zero forcing constraint a novel iterative receiving detection algorithm for the specific receiver is also proposed for the application to cyclic prefix systems in which the svd of the equivalent channel matrix can be easily computed the proposed ggmd transceiver has k log2 k times complexity advantage over the gmd transceiver where is the number of data symbols per data block and is a power of 2 in a companion paper performance analyses of the proposed ggmd transceiver in terms of arithmetic mean square error mse symbol error rate ser and gaussian mutual information are performed and comparisons with well known transceivers are made the results show that the proposed transceiver reaches the same optimality that a gmd mmse transceiver can possibly achieve

the effect of assuming the identity as a generator on the length of the matrix algebra abstract let m n f be the algebra of n n matrices and let s be a generating set of m n f as an f algebra the length of a finite generating set s of m n f is the smallest number k such that words of length not greater than k generate m n f as a vector space traditionally the identity matrix is assumed to be automatically included in all generating sets s and counted as a word of length 0 in this paper we discuss how the problem changes if this assumption is removed

efficient adaptive svd algorithm for mimo applications many multiple input multiple output mimo applications require the computation of some or all of the nonzero singular values and the corresponding left and right singular vectors of a time varying channel response matrix an adaptive algorithm is derived to achieve this goal based on a first order perturbation which updates a full or partial singular value decomposition svd using input and noisy output vectors the updates can be computed recursively resulting in a highly efficient algorithm that has lower complexity than the earlier least mean square lms based algorithm and achieves better performance at low signal to noise ratio snr the performance is demonstrated using measured mimo channel data obtained in an urban microcellular environment

length realizability for pairs of quasi commuting matrices abstract for the pairs of quasi commuting matrices we completely characterize natural numbers that can be realized as the lengths of these pairs of generators

an upper bound for the length of a finite dimensional algebra abstract letfbe a field and letabe a finite dimensionalf algebra writed dimf a and letebe the largest degree of the minimal polynomial for anya a define the function f d e e 2d e 1 1 4 e 2 2 we prove that ifsis any finite generating set foraas anf algebra the words insof length less thanf d e spanaas anf vector space in the special case ofn by nmatrices this bound becomes f n 2 n n 2n 2 n 1 1 4 n 2 2 o n 3 2 this is a substantial improvement over previous bounds which have all beeno n2 we also prove that for particular setssof matrices the bound can be sharpened to one that is linear inn as an application of these results we reprove a theorem of small stafford and warfield about semiprime affinef algebras

interpolation based transmit beamforming for mimo ofdm with limited feedback transmit beamforming and receive combining are simple methods for exploiting spatial diversity in multiple input multiple output orthogonal frequency division multiplexing mimo ofdm system optimal beamforming requires channel state information in the form of the beamforming vectors for each ofdm subcarrier this paper proposes a limited feedback architecture that combines beamforming vector quantization and smart vector interpolation in the proposed system the receiver sends a fraction of information about the optimal beamforming vectors to the transmitter and the transmitter computes the beamforming vectors for all subcarriers through interpolation a new spherical interpolator is developed that exploits parameters for phase rotation to satisfy the phase invariance and unit norm properties of the transmitted beamforming vectors the beamforming vectors and phase parameters are quantized at the receiver and the quantized information is provided to the transmitter the proposed quantization system provides only a moderate increase in complexity versus over comparable approaches numerical simulations show that the proposed scheme performs better than existing diversity techniques with the same feedback data rate

commutative matrix subalgebras and length function it is proved that if the length of a commutative matrix subalgebra is maximal then this subalgebra is maximal under inclusion the examples are given showing that the converse does not hold to establish this result we prove several fundamental properties of the length function

vector coding for partial response channels a linear technique for combining equalization and coset codes on partial response channels with additive white gaussian noise is developed the technique vector coding uses a set of transmit filters or vectors to partition the channel into an independent set of parallel intersymbol interference isi free channels for any given finite or infinite block length the optimal transmit vectors for such channel partitioning are shown to be the eigenvectors of the channel covariance matrix for the specified block length and the gains of the individual channels are the eigenvalues an optimal bit allocation and energy distribution are derived for the set of parallel channels under an accurate extension of the continuous approximation for power in optimal multidimensional signal sets for constellations with unequal signal spacing in different dimensions examples are presented that demonstrate performance advantages with respect to zero forcing decision feedback methods that use the same coset code on the same partial response channel only resampling the channel at an optimal rate and assuming no errors in the feedback path will bring the performance of the decision feedback methods up to the level of the vector coded system

discrete multiple tone modulation with coset coding for the spectrally shaped channel a discrete approach to multiple tone modulation is developed for digital communication channels with arbitrary intersymbol interference isi and additive gaussian noise multiple tone modulation is achieved through the concatenation of a finite block length modulator based on discrete fourier transform dft code vectors and high gain coset or trellis codes symbol blocks from an inverse dft idft are cyclically extended to generate isi free channel output symbols that decompose the channel into a group of orthogonal and independent parallel subchannels asymptotic performance of this system is derived and examples of asymptotic and finite block length coding gain performance for several channels are evaluated at different values of bits per sample this discrete multiple tone technique is linear in both the modulation and the demodulation and is free from the effects of error propagation that often afflict systems employing bandwidth optimized decision feedback plus coset codes

multistage iterative decoding with complexity reduction for concatenated space time codes space time encoders exploiting concatenated coding structures are efficient in attaining the high rates available to large dimensional multiple transmitter multiple receiver wireless systems under fading conditions while also providing maximal diversity benefits we present a multistage iterative decoding structure that takes full advantage of the concatenated nature of the transmission path treating the modulator and channel stages as an additional encoder in serial concatenation this iterative decoder architecture allows an encoder employing decoupled coding and modulation to reach the performance of coded modulation systems it also admits reduced complexity decoding with a computational load that is nonexponential in the number of antennas or the transmission bit rate and makes practical decoding for large transmitter arrays possible the performance curves for these methods follow the shape of the fano bound with only a modest power penalty

square rooting algorithms for high speed digital circuits two binary algorithms for the square rooting of a number or of a sum of two numbers are presented they are based on the classical nonrestoring method the main difference lies in the replacement of subtractions and additions by a parallel reduction f three summands which may be positive and negative to two summands to eliminate the carry propagation two of three summands form the successive partial remainder their most significant bit triples sometimes together with a sign bit of the earlier partial remainder are used to determine digits 1 0 1 of a redundant square root notation these digits are transformed during the square rooting process into the conventional notation square root bits which are next used in further square rooting steps to form the third reduced summands

burnside s theorem irreducible pairs of transformations abstract by burnside s theorem if the linear transformations a and b acting on a finite dimensional complex vector space h have no common nontrivial invariant subspaces the words in a and b span b h call the minimum spanning length of the pair a b the smallest positive integer l with the property that words in a and b of length at most l span b h let msl a b denote the minimum spanning length if dim h 2 msl a b 2 and if dim h 3 msl a b 3 or 4 if dim h 4 msl a b n 2 3 if dim h n 2 then i msl a b 2 n 2 if a b ab ba is linearly dependent ii if b is unicellular then msl a b 2 n 2 where the inequality is sharp and it can happen that msl a b n

joint transmitter receiver optimization for multi input multi output systems with decision feedback the joint transmit receive optimization problem for multiuser communication systems with decision feedback is investigated it is shown that minimization of the geometric mean squared error defined as the determinant of the error covariance matrix leads to a tractable transmitter optimization problem for general multi input multi output decision feedback systems several computational results are included that highlight system performance for a variety of useful transmission scenarios