block transmission over dispersive channels transmit filter optimization and realization and mmse dfe receiver performance optimal transmit filters for packet based data transmission on dispersive gaussian noise linear time invariant channels are derived by maximizing the mutual information subject to a fixed input power budget a quasi stationary approximation to the optimal nonstationary input covariance process is derived and shown to exhibit negligible mutual information loss from the optimal case for situations of most practical interest moreover this quasi stationary approximation results in efficiently computed lattice or pole zero implementations of the transmit filter by considering the popular finite impulse response minimum mean square error decision feedback equalizer fir mmse dfe as a receiver structure we show that transmitter optimization results in an appreciable improvement in the decision point signal to noise ratio finally we show that as the output blocklength becomes infinite the optimum finite dimensional nonstationary input covariance process converges to a stationary process whose power spectrum obeys the well known water pouring distribution

some results on kwong functions and related inequalities abstract we investigate some relations between kwong functions and operator monotone functions as an application we present an arithmetic geometric mean type inequality by showing that for two continuous functions f g on 0 such that h t f t g t is a kwong function and f t g t t any positive matrices a b and any matrix x it holds that f a x g b g a x f b a x x b for each unitarily invariant norm

a singular value inequality for heinz means we prove a matrix inequality for matrix monotone functions and apply it to prove a singular value inequality for heinz means recently conjectured by zhan

more results on singular value inequalities of matrices abstract the arithmetic geometric mean inequality for singular values due to bhatia and kittaneh says that 2 s j ab s j a a b b j 1 2 for any matrices a b we give a new equivalent form and some relevant generalizations of this inequality in particular we show that s j a 1 4 b 3 4 a 3 4 b 1 4 s j a b j 1 n for any n n positive semidefinite matrices a b which proves a special case of zhan s conjecture posed in 2000

mmse decision feedback equalizers and coding ii coding results for pt i see ibid vol 43 no 10 p 2582 1995 the minimum mean squared error decision feedback equalizer mmse dfe has properties that suggest that it is a canonical equalization structure in systems that combine equalization with coded modulation with a given symbol rate 1 t and transmit spectrum the output signal to noise ratio snr sub mmse dfe u of a mmse dfe with an unbiased decision rule is a single parameter that characterizes the channel for coding purposes indeed the transmit spectrum that maximizes snr sub mmse dfe u is the capacity achieving water pouring spectrum and the capacity c t in bits per two dimensions is given by c t log sub 2 1 snr sub mmse dfe u regardless of the channel characteristics the performance of a coded system with a mmse dfe equalization structure may be accurately estimated using the gain of the coding scheme at a given pr e this performance is shown to be approximately the same as that of a multicarrier system using the same transmit spectrum and similar coding such systems are known to be able to approach capacity arbitrarily closely the mmse dfe can perform significantly better than a zero forcing decision feedback equalizer particularly at moderate to low snr s and on severe isi channels simulation results indicate that performance of the mmse dfe is surprisingly insensitive to transmit spectral shaping as long as the transmit spectrum exceeds the capacity achieving band but that there is an optimal symbol rate that should approximately be used

positive semidefinite solutions of the operator equation j 1nan jxaj 1 b abstract let a be a positive definite operator and b be a self adjoint operator we discuss the existence of positive semidefinite solutions of the operator equation j 1 n a n j xa j 1 b via an order preserving operator inequality by using the solutions we also give concrete examples of positive semidefinite matrices

interpolating the arithmetic geometric mean inequality and its operator version two families of means called heinz means and heron means that interpolate between the geometric and the arithmetic mean are considered comparison inequalities between them are established operator versions of these inequalities are obtained failure of such extensions in some cases is illustrated by a simple example

finding global minimizer with one parameter filled function on unconstrained global optimization abstract in this paper a filled function with one parameter is proposed for solving unconstrained global optimization problems this filled function is modified from the previously reported one see paper y j yang y l shang a new filled function method for unconstrained global optimization applied mathematics and computations 173 1 2006 501 512 the properties of the filled function are discussed in this paper an algorithm for unconstrained global optimization is developed from this new filled function the computational results show that this algorithm is efficient and reliable

optimization of signal sets for partial response channels i numerical techniques given a linear time invariant discrete time channel the problem of constructing n input signals of finite length k that maximize minimum l sub 2 distance between pairs of outputs is considered two constraints on the input signals are considered a power constraint on each of the n inputs hard constraint and an average power constraint over the entire set of inputs soft constraint the hard constraint problem is equivalent to packing n points in an ellipsoid in min k n 1 dimensions to maximize the minimum euclidean distance between pairs of points gradient based numerical algorithms and a constructive technique based on dense lattices are used to find locally optimal solutions to the preceding signal design problems two numerical examples are shown for which the average spectrum of an optimized signal set resembles the water pouring spectrum that achieves shannon capacity assuming additive white gaussian noise

a new filled function method for unconstrained global optimization in this paper a new filled function which has better properties is proposed for identifying a global minimum point for a general class of nonlinear programming problems within a closed bounded domain an algorithm for unconstrained global optimization is developed from the new filled function theoretical and numerical properties of the proposed filled function are investigated the implementation of the algorithm on seven test problems is reported with satisfactory numerical results

convexity of the inverse and moore penrose inverse abstract convexity properties of the inverse of positive definite matrices and the moore penrose inverse of nonnegative definite matrices with respect to the partial ordering induced by nonnegative definiteness are studied for the positive definite case null space characterizations are derived and lead naturally to a concept of strong convexity of a matrix function extending the conventional concept of strict convexity the positive definite results are shown to allow for a unified analysis of problems in reproducing kernel hilbert space theory and inequalities involving matrix means the main results comprise a detailed study of the convexity properties of the moore penrose inverse providing extensions and generalizations of all the earlier work in this area

a class of filled functions for box constrained continuous global optimization in this paper we show that the unconstrained continuous global minimization problem cannot be solved by any algorithm so we consider the box constrained continuous global minimization problem we present a definition of filled function for the problem moreover a new class of filled functions are constructed which contains only one parameter a randomized algorithm is designed to solve the box constrained continuous global minimization problem based on the filled function numerical experiments are presented to show the practicability of the algorithm

improved young and heinz inequalities for matrices we give refinements of the classical young inequality for positive real numbers and we use these refinements to establish improved young and heinz inequalities for matrices

optimal transmitter eigen beamforming and space time block coding based on channel mean feedback optimal transmitter designs obeying the water filling principle are well documented they are widely applied when the propagation channel is deterministically known and regularly updated at the transmitter because channel state information is impossible to be known perfectly at the transmitter in practical wireless systems we design in this paper an optimal multiantenna transmitter based on the knowledge of mean values of the underlying channels our optimal transmitter design turns out to be an eigen beamformer with multiple beams pointing to orthogonal directions along the eigenvectors of the correlation matrix of the estimated channel at the transmitter and with proper power loading across beams the optimality pertains to minimizing an upper bound on the symbol error rate which leads to better performance than maximizing the expected signal to noise ratio snr at the receiver coupled with orthogonal space time block codes two directional eigen beamforming emerges as a more attractive choice than conventional one directional beamforming with uniformly improved performance without rate reduction and without essential increase in complexity with multiple receive antennas and reasonably good feedback quality the two directional eigen beamformer is also capable of achieving the best possible performance in a large range of transmit power to noise ratios without a rate penalty

on some inequalities for unitarily invariant norms and singular values abstract in this note we improve and generalize some existing inequalities for unitarily invariant norms and singular values and present some other inequalities for unitarily invariant norms

a filled function method applied to nonsmooth constrained global optimization the filled function method is an effective approach to find a global minimizer in this paper based on a new definition of the filled function for nonsmooth constrained programming problems a one parameter filled function is constructed to improve the efficiency of numerical computation then a corresponding algorithm is presented it is a global optimization method which modify the objective function as a filled function and which find a better local minimizer gradually by optimizing the filled function constructed on the minimizer previously found illustrative examples are provided to demonstrate the efficiency and reliability of the proposed filled function method

some results on matrix monotone functions abstract let a and b be hermitian matrices we say that a b if a b is nonnegative definite a function 0 r is said to be matrix monotone m m if a b 0 implies that a b matrix convex and concave functions can be similarly defined in this paper several known results some in sharper forms are given alternative proofs these include the following a a positive function t is m m iff t t is m m b if t is m m then both t t and t t are matrix convex and are almost always strictly convex on the other hand t is always concave and almost always strictly concave the concavity of m m functions and an inequality of hansen are consequences of a more general inequality on m m functions some complementary inequalities are also established we then prove an extension of lyapunov s theorem let a and p be positive definite matrices and either or 1 be a m m function then the unique matrix solution x of the matrix equation ax xa a p p a is positive definite a similar result holds for a related equation most of the results are obtained by exploiting the integral representation of m m functions

the optimum combination of block codes and receivers for arbitrary channels a determination is made of the combinations of transmitted block codewords and matching receiver filters that maximize the signal to noise ratio at the detector with arbitrary channel impulse response and arbitrary noise spectrum this permits representing the channel as a set of parallel uncoupled subchannels with uncorrelated noise the representation is used to calculate the capacity of an arbitrary block coded time discrete channel it is shown that substantial coding gains compared to symbol by symbol processing can be achieved the block code used can be concatenated with trellis coded modulation so that the two coding gains are additive

operator inequalities related to the corach porta recht inequality abstract we prove some refinements of an inequality due to x zhan in an arbitrary complex hilbert space by using some results on the heinz inequality we present several related inequalities as well as new variants of the corach porta recht inequality we also characterize the class of operators satisfying sxs 1 s 1 xs kx k 2 x under certain conditions

on the definiteness of the solutions of certain matrix equations abstract it is shown that for each dimension n 2 there is a real number t n such that for all t 2 t n the unique n n solution x of the matrix equation a 2 x xa 2 taxa b must be positive semidefinite for any positive definite a and positive semidefinite b this generalizes the classical lyapunov result it is established that t n 8 4 for n 2 3 4 respectively and that t n 2 for all n upper bounds are given for n up to 6 related equations are also discussed

ber minimized ofdm systems with channel independent precoders we consider the minimization of uncoded bit error rate ber for the orthogonal frequency division multiplexing ofdm system with an orthogonal precoder we analyze the ber performance of precoded ofdm systems with zero forcing and minimum mean squared error mmse receivers in the case of mmse receivers we show that for quadrature phase shift keying qpsk there exists a class of optimal precoders that are channel independent examples of this class include the discrete fourier transform dft matrix and the hadamard matrix when the precoder is the dft matrix the resulting optimal transceiver becomes the single carrier system with cyclic prefix sc cp system we also show that the worst solution corresponds to the conventional ofdm system the conventional ofdm system has the largest ber in the case of zero forcing receivers the design of optimal transceiver depends on the signal to noise ratio snr for higher snr solutions of optimal precoders are the same as those of mmse receivers

novel matrix singular value inequalities and their applications to uncertain mimo channels novel matrix singular value inequalities are established for a sum product of three matrices their application to the uncertain compound multiple input multiple output mimo channel subject to normed additive uncertainty establishes the saddle point property for a wide range of performance metrics monotonic in the channel singular values including among others the mutual information mmse error exponent and pairwise error probability this in turn implies that the transmission on the eigenmodes of the nominal or worst case channel is also optimal for the whole set of channels under a general power constraint and hence achieves the compound channel capacity the worst case channel turns out to be antiparallel of the nominal one for all these performance metrics an application of these results to beamforming over compound mimo channels is discussed an optimal robust precoder for the uncertain mimo channel is obtained in a closed form under the sum mse criterion and the total power constraint the saddle point property is shown to hold and the optimal strategy is to diagonalize the nominal or worst case channel

a filled function method with one parameter for box constrained global optimization in this paper a new auxiliary function with one parameter on box constrained for escaping the current local minimizer of global optimization problem is proposed first a new definition of the filled function for box constrained minimization problem is given and under mild assumptions this new auxiliary function is really a filled function then a new solution algorithm is proposed according to the theoretical analysis and some numerical results demonstrate the efficiency of this method for box constrained global optimization

lattice filter parameterization and modeling of nonstationary processes a general theory of constant parameter modular lattice models for discrete time nonstationary second order processes is presented a complete parametrization of such processes in terms of schur and congruence coefficients is derived by developing a natural connection between the displacement structure of a covariance matrix and schur s test for positive definiteness of matrices schur coeffieients provide a simple solution to problems of covariance extension and rational spectral approximation for nonstationary covariances and they coincide with the well known reflection or parcor coefficients when the covariance is stationary the congruence coefficients provide the time varying gains of a tapped delay line realization of the whitening filter for the process a constant parameter realization of the same filter is derived by combining a lattice filter structure with a tapped delay line both with time invariant gains this configuration also provides a recursive relation for the congruence coefficients namely a generalized levinson szego recursion the tapped delay line part of the realization can be eliminated by introducing the concept of admissibility admissibility also reduces the parametrization of a nonstationary process to schur coefficients alone in analogy to stationary processes which are completely characterized by their parcor coefficients

spectral compensation for multicarrier communication spectral compensation is an information processing technique applied in the transmitter to improve the spectral efficiency of multicarrier modulation under a given power spectral density constraint a set of carefully chosen tones the so called information tones carries data the remaining tones referred to as compensation tones are modulated with a linear combination of the data such that the spectral characteristic of the transmit signal and thus the throughput are improved this paper investigates strategies to find the set of compensation tones and presents the optimal solution for the linear combination of the data given the tone set split the optimality criterion is maximization of throughput for a time dispersive gaussian channel known to the transmitter furthermore a suboptimal design is proposed which has low runtime complexity and achieves near optimal performance

global descent method for constrained continuous global optimization abstract in this paper we consider the problem of constrained global optimization of a continuous multivariable function we propose a global descent function technique which requires an easily adjustable single parameter the characteristic property of the proposed function is that each of its local minimizers verifying constraints is a better local minimizer of the objective function or at less an approximated local minimizer with a given tolerance several other properties of the new function are investigated in order to establish a corresponding optimization algorithm we have performed numerical experiments on a set of standard test problems using this algorithm the results illustrate the efficiency of our approach

mean matrices and infinite divisibility abstract we consider matrices m with entries mij m i j where 1 n are positive numbers and m is a binary mean dominated by the geometric mean and matrices w with entries wij 1 m i j where m is a binary mean that dominates the geometric mean we show that these matrices are infinitely divisible for several much studied classes of means

symmetric function means of positive operators abstract many authors have studied means involving ratios of the elementary symmetric functions this work involves generalizing these means to an operator theoretic setting and extending many of the classical scalar inequalities to operators it is shown that these operator means are related to electrical networks

zhan s inequality on a g mean inequalities abstract in this paper we show some generalizations of zhan s inequality by virtue of hadamard product as follows let a b x be n n matrices such that a b are positive definite and f g be two continuous functions on 0 k max f g a b where a is the spectrum of a if f x g x is kwong i e the matrix f g i f g j i j is positive semidefinite for any positive numbers 1 n then for any 0 i a 1 2 f a x g b g a x f b b 1 2 k 2 1 2 2 r 0 a x b 4 1 r 0 t 2 a 2 x t a x b x b 2 holds for 2 t 2 2 where r 0 min 1 2 1 r 1 1 r i i a 1 2 f a x g b g a x f b b 1 2 2 k t 2 a 2 x t a x b x b 2 holds for any 2 t 2 where is any unitarily invariant norm

mmse decision feedback equalizers and coding i equalization results the minimum mean squared error decision feedback equalizer mmse dfe has properties that suggest that it is a canonical equalization structure for systems that combine equalization with coded modulation the structure and performance of the mmse dfe are succinctly derived using linear estimation theoretic principles in this first part of this two part paper the front end of the mmse dfe called the mean square whitened matched filter ms wmf is preferable in some ways to a matched filter or a whitened matched filter as a canonical receiver front end in a coded system the feedback filter of the mmse dfe may be implemented in the transmitter using precoding the mmse dfe can perform significantly better than a zero forcing decision feedback equalizer particularly at moderate to low snr s and on severe isi channels the mmse dfe is biased the optimum unbiased mmse dfe is the mmse dfe with the bias removed removing bias improves error probability but reduces the snr to snr sub mmse dfe u snr sub mmse dfe 1 it is shown that this snr relationship is a particular case of a very general result and that snr sub mmse dfe u gives a more realistic estimate of snr the results are extended to partial response equalization and to equalization with correlated inputs in an appendix

some convex and monotone matrix functions we prove some matrix monotonicity and matrix convexity properties for functions derived from operator connections as a consequence we prove that a certain product of matrix monotone functions is a matrix monotone function similar results are proved for matrix convex functions these results include some known results for matrix monotone and matrix convex functions

a one parameter filled function method applied to nonsmooth constrained global optimization in this paper we propose a filled function method for solving nonsmooth constrained global optimization problems based on a new definition of the filled function a more practical one parameter filled function is constructed which overcomes some drawbacks of the previous filled functions then a corresponding algorithm is presented it attains a local minimizer by implementing a local search procedure and finds a better local minimizer gradually by optimizing the filled function constructed on the minimizer previously found by repeating these steps a global minimizer is obtained numerical experiments are presented to show the practicability of the proposed filled function method in the end extension conceivable applications are given in order to evaluate the merits of this method

mmse decision feedback equalizers finite length results this paper extends a number of results on the infinite length minimum mean square error decision feedback equalizer mmse dfe reported by cioffi dudevoir eyuboglu and forney see ieee trans commun 1995 to the finite length case cholesky factorization and displacement structure theory are demonstrated to be two powerful analytical tools for analyzing the finite length mmse dfe our objective throughout the paper is to establish finite length analogs of the well known infinite length mmse dfe results similarities and differences between the two cases are examined and delineated finally convergence of our derived finite length results to their well established infinite length counterparts is shown

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

time varying versus time invariant finite length mmse dfe on stationary dispersive channels a time varying structure and a time invariant structure for the finite length minimum mean square error decision feedback equalizer mmse dfe are presented and their performances are compared on a stationary channel impaired by intersymbol interference isi and additive gaussian noise the time varying structure has an innovations error sequence but incurs a throughput loss on isi channels because of its block processing nature conditions under which the time invariant structure exhibits near optimal performance are described both structures converge to the canonical mmse dfe of cioffi et al see ibid vol 43 p 2582 94 1995 and vol 43 p 2595 2604 1995 as their filters lengths become infinite

a new filled function method for unconstrained global optimization in this paper a new definition of the filled function is given it is different from the primary definition which was given by ge in paper r p ge a filled function method for finding a global minimzer of a function of several variables math program 46 1990 191 204 based on the definition a new filled function is proposed and it has better properties an algorithm for unconstrained global optimization is developed from the new filled function the implementation of the algorithm on several test problems is reported with satisfactory numerical results

optimal transmitter eigen beamforming and space time block coding based on channel correlations optimal transmitter designs obeying the water filling principle are well documented and widely applied when the propagation channel is deterministically known and regularly updated at the transmitter because channel state information csi may be costly or impossible to acquire in rapidly varying wireless environments we develop in this paper statistical water filling approaches for stationary random fading channels these approaches require only knowledge of the channel correlations that do not necessitate frequent updates and can be easily acquired applied to a multiple transmit antenna paradigm our optimal transmitter design turns out to be an eigen beamformer with multiple beams pointing to orthogonal directions along the eigenvectors of the channel s correlation matrix and with proper power loading across the beams the optimality pertains to minimizing a tight bound on the symbol error rate the resulting loaded eigen beamforming outperforms not only the equal power allocation across all antennas but also the conventional beamformer that transmits the available power along the strongest direction coupled with orthogonal space time block codes two dimensional 2 d eigen beamforming emerges as a more attractive choice than conventional one dimensional 1 d beamforming with uniformly better performance without rate reduction and without complexity increase

a new filled function algorithm for constrained global optimization problems a new filled function with one parameter is proposed for solving constrained global optimization problems without the coercive condition in which the filled function contains neither exponential term nor fractional term and is easy to be calculated a corresponding filled function algorithm is established based on analysis of the properties of the filled function at last we perform numerical experiments on some typical test problems using the algorithm and the detailed numerical results show that the algorithm is effective

cauchy schwarz norm inequalities for weak integrals of operator valued functions for a finite measures on and weakly measurable families at t and bt t of hilbert space operators we have the non commutative cauchy schwarz inequalities in schatten p ideals axbd p a aa d q 1ad 2qx b b bd r 1b d 2rp for all x p h and for all p q r 1 such that 1q 1r 2p if both at t and bt t consists of commuting normal operators then axbd a ad x b bd for all unitarily invariant norms and all x h if additionally a ad i and b bd i then i a ad xi b bd h and i a ad xi b bd x axbd applications include young s and arithmetic geometric logarithmic means inequalities for operators and the mean value theorem for operator monotone functions

application of fast subspace decomposition to signal processing and communication problems the authors previously described a class of fast subspace decomposition fsd algorithms though these algorithms can be applied to solve a variety of signal processing and communication problems with significant computational reduction they focus their discussion on two typical applications i e sensor array processing and time series analysis in many cases replacing the usual eigenvalue decomposition evd or singular value decomposition svd by the fsd is quite straightforward however the fsd approach can exploit more structure of some special problems to further simplify the implementation they first discuss the implementation details of the fsd such as how to choose an optimal starting vector how to handle correlated noise and how to exploit additional matrix structure for further computational reduction then they describe an fsd approach targeted at data matrices rectangular n spl times m n spl ges m which requires only o nmd flops where d denotes the signal subspace dimension versus a regular o nm sup 2 m sup 3 svd the computational reduction is substantial in typical scenarios i e d spl lt m spl les n in the spectrum estimation problems the data matrix has additional structure such as toeplitz or hankel they finally show how the fsd can exploit such structure for further computational reduction

on singular value inequalities for matrix means abstract for positive semidefinite n n matrices a and b the singular value inequality 2 t s j a r b 2 r a 2 r b r 2 s j a 2 t a b b 2 is shown to hold for r 1 2 1 3 2 and all 2 t 2

filterbank transceivers optimizing information rate in block transmissions over dispersive channels optimal finite impulse response fir transmit and receive filterbanks are derived for block based data transmissions over frequency selective additive gaussian noise agn channels by maximizing mutual information subject to a fixed transmit power constraint both fir and pole zero channels are considered the inherent flexibility of the proposed transceivers is exploited to derive as special cases zero forcing zf and minimum mean square error receive filterbanks the transmit filterbank converts transmission over a frequency selective fading channel affected by additive colored noise into a set of independent flat fading subchannels with uncorrelated noise samples two loading algorithms are also developed to distribute transmit power and number of bits across the usable subchannels while adhering to an upper bound on the bit error rate ber reduction of the signal to noise ratio snr margin required to satisfy the prescribed ber is achieved by coding each subchannel s bit stream the potential of the proposed transceivers is illustrated and compared to discrete multitone dmt with simulated examples

throughput maximizing fir transmit filters for linear dispersive channels optimum finite impulse response transmit filters for symbol by symbol transmission on linear dispersive additive gaussian noise channels are derived by maximizing the channel throughput subject to a fixed average input energy constraint this maximized throughput is compared to that achievable with water pour and flat transmit filters the effect of transmit filter optimization on the receiver performance is investigated by considering the popular minimum mean square error decision feedback equalizer receiver structure

on the sensitivity of a suboptimal precoding scheme for frequency selective block based channels with respect to channel inaccuracies water filling is the precoding scheme that achieves capacity in gaussian parallel or block based channels a suboptimal precoding scheme which has been observed to perform quite close to water filling in some cases of great interest is when all active eigenvectors of the input data covariance matrix receive the same power both techniques require perfect channel knowledge at the transmitter in this work we consider the sensitivity of the suboptimal scheme when a channel estimate is used at the transmitter as if it were the true channel using tools from matrix perturbation theory we derive closed form expressions and bounds relating the channel estimation error covariance matrix with the mean mutual information decrease we thus uncover the factors that determine the behavior of the suboptimal precoding scheme under channel uncertainties simulations are in agreement with our theoretical results

notes on matrix arithmetic geometric mean inequalities abstract for positive semi definite n n matrices the inequality 4 ab a b 2 isshown to hold for every unitarily invariant norm the connection of this with some other matrix arithmetic geometric mean inequalities and trace inequalities is discussed