grassmannian beamforming for multiple input multiple output wireless systems transmit beamforming and receive combining are simple methods for exploiting the significant diversity that is available in multiple input multiple output mimo wireless systems unfortunately optimal performance requires either complete channel knowledge or knowledge of the optimal beamforming vector both are hard to realize in this article a quantized maximum signal to noise ratio snr beamforming technique is proposed where the receiver only sends the label of the best beamforming vector in a predetermined codebook to the transmitter by using the distribution of the optimal beamforming vector in independent and identically distributed rayleigh fading matrix channels the codebook design problem is solved and related to the problem of grassmannian line packing the proposed design criterion is flexible enough to allow for side constraints on the codebook vectors bounds on the codebook size are derived to guarantee full diversity order results on the density of grassmannian line packings are derived and used to develop bounds on the codebook size given a capacity or snr loss monte carlo simulations are presented that compare the probability of error for different quantization strategies

analysis of multiple antenna systems with finite rate feedback using high resolution quantization theory this paper considers the development of a general framework for the analysis of transmit beamforming methods in multiple antenna systems with finite rate feedback inspired by the results of classical high resolution quantization theory the problem of finite rate quantized communication system is formulated as a general fixed rate vector quantization problem with side information available at the encoder or the quantizer but unavailable at the decoder the framework of the quantization problem is sufficiently general to include quantization schemes with general non mean squared distortion functions and constrained source vectors asymptotic distortion analysis of the proposed general quantization problem is provided by extending the vector version of the bennett s integral specifically tight lower and upper bounds of the average asymptotic distortion are proposed sufficient conditions for the achievability of the distortion bounds are also provided and related to corresponding classical fixed rate quantization problems the proposed general methodology provides a powerful analytical tool to study a wide range of finite rate feedback systems to illustrate the utility of the framework we consider the analysis of a finite rate feedback multiple input single output miso beamforming system over independent and identically distributed i i d rayleigh flat fading channels numerical and simulation results are presented that further confirm the accuracy of the analytical results

new extremal problems for the riemannian recognition program via alexandrov geometry in its most general form the recognition problem in riemannian geometry asks for the identification of an unknown riemannian manifold via measurements of metric invariants on the manifold optimally one wants to recognize a manifold having made as few measurements as possible many results in riemannian geometry including pinching theorems can be viewed this way here we are only interested in measurements that assign real numbers to each complete riemannian manifold typical examples of such invariants are diameter volume curvature bounds etc when viewing one or several invariants i ii ii of this type as a map on a suitable class l of riemannian manifolds the following problems pose themselves

performance analysis of linear modulation schemes with generalized diversity combining on rayleigh fading channels with noisy channel estimates generalized diversity combining gdc also known as hybrid selection maximal ratio combining or generalized selection combining is a low complexity diversity combining technique by which a fixed subset of a large number of available diversity channels is chosen and then combined using the rules of maximal ratio combining in this paper we analyze the performance of gdc on time correlated rayleigh fading channels with noisy channel estimates we derive expressions for the probability of error for various linear modulation schemes with coherent detection and discuss the conditions under which the analysis can be extended to noncoherent and differentially coherent receiver structures throughout the paper using a fundamental approach to obtain the decision statistic at the combiner output a number of new expressions for the error probabilities are obtained in a rigorous way along with a presentation of their performance with channel estimation errors the final expressions have roughly the same complexity of evaluation as that for the channel with only additive gaussian noise our results correct various inaccuracies in the literature and show that coherent receivers based on imperfectly estimated channel knowledge incur a significant performance loss

unbalanced expanders and randomness extractors from parvaresh vardy codes we give an improved explicit construction of highly unbalanced bipartite expander graphs with expansion arbitrarily close to the degree which is polylogarithmic in the number of vertices both the degree and the number of right hand vertices are polynomially close to optimal whereas the previous constructions of ta shma et al 2007 required at least one of these to be quasipolynomial in the optimal our expanders have a short and self contained description and analysis based on the ideas underlying the recent list decodable error correcting codes of parvaresh and vardy 2005 our expanders can be interpreted as near optimal randomness condensers that reduce the task of extracting randomness from sources of arbitrary min entropy rate to extracting randomness from sources of min entropy rate arbitrarily close to 1 which is a much easier task using this connection we obtain a new self contained construction of randomness extractors that is optimal up to constant factors while being much simpler than the previous construction of lu et al 2003 and improving upon it when the error parameter is small e g 1 poly n

on the general ber expression of one and two dimensional amplitude modulations quadrature amplitude modulation qam is an attractive technique to achieve high rate transmission without increasing the bandwidth a great deal of attention has been devoted to the study of bit error rate ber performance of qam and approximate expressions for the bit error probability of qam have been developed in many places in the literature however the exact and general ber expression of qam with an arbitrary constellation size has not been derived yet we provide an exact and general closed form expression of the ber for one dimensional and two dimensional amplitude modulations i e pam and qam under an additive white gaussian noise awgn channel when gray code bit mapping is employed the provided ber expressions offer a convenient way to evaluate the performance of pam and qam systems for various cases of practical interest moreover simple approximations can be found from our expressions which are the same as the well known approximations if only the dominant terms are considered

the shrinkage exponent of de morgan formulas is 2 we prove that if we hit a de morgan formula of size l with a random restriction from rp then the expected remaining size is at most o p 2 log frac 1 p 3 2 l p sqrt l as a corollary we obtain an omega n 3 o 1 formula size lower bound for an explicit function in p this is the strongest known lower bound for any explicit function in np

index parity of closed geodesics and rigidity of hopf fibrations the diameter rigidity theorem of gromoll and grove 1987 states that a riemannian manifold with sectional curvature 1 and diameter 2 is either homeomorphic to a sphere locally isometric to a rank one symmetric space or it has the cohomology ring of the cayley plane ca the reason that they were only able to recognize the cohomology ring of ca is due to an exceptional case in another theorem gromoll and grove 1988 a riemannian submersion m b b with connected fibers that is defined on the euclidean sphere m is metrically congruent to a hopf fibration unless possibly m b 15 8 we will rule out the exceptional cases in both theorems our argument relies on a rather unusal application of morse theory for that purpose we give a general criterion which allows to decide whether the morse index of a closed geodesic is even or odd

performance analysis of transmit beamforming for miso systems with imperfect feedback in this paper we analyze the performance of transmit beamforming on multiple antenna rayleigh fading channels with imperfect channel feedback we characterize the feedback imperfections in terms of noisy channel estimation feedback delay and finite rate channel quantization we develop a general framework that is valid for an arbitrary two dimensional linear modulation to capture the aforementioned imperfections and derive the symbol and bit error probability expressions for both m psk and m ary rectangular qam constellations with gray code mapping we show that the proposed analytical formulation is valid for a frequency domain duplexing system with without finite rate channel quantization and a time domain duplexing system we validate the accuracy of the analysis through simulations and assess the relative effects of channel estimation inaccuracy feedback delay and finite rate quantization on the symbol and bit error performances for various constellations

computation of the exact bit error rate of coherent m ary psk with gray code bit mapping the problem of calculating the average bit error probability bep of coherent m ary phase shift keying psk over a gaussian channel has been studied previously in the literature a solution to the problem for systems using a binary reflected gray code brgc to map bits to symbols was first presented by p j lee see ibid vol com 34 p 488 91 1986 we show that the results obtained by lee are incorrect for m spl ges 16 we show that the reason for this is an invalid assumption that the bit error rate ber is independent of the transmitted symbols an assumption which has also propagated to textbooks we give a new expression for the ber of m psk systems using the brgc and compare this with lee s results

on the shrinkage exponent for read once formulae we prove that the size of any read once de morgan formula reduces on average by a factor of at least p o 1 when all but a fraction p of the input variables are randomly assigned to 0 1 here llog2 5 1 3 27 this resolves in the affirmative a conjecture of paterson and zwick the bound is shown to be tight up to a polylogarithmic factor for all p n 1

on the optimality of bit detection of certain digital modulations in this paper we study the optimality of bit detection for coherent m ary phase shift keying psk and m ary quadrature amplitude modulation qam and noncoherent m ary frequency shift keying fsk signal sets for m psk and m qam signal constellations we employ gray mapping consider 8 psk and 16 qam signal sets as representative of the general results and derive the log likelihood ratio llr for each bit forming the symbol using the llrs we derive the average bit error probability bep for the individual bits and show that the decision regions and the corresponding average bep for the case of m psk coincide with those obtained with the optimal symbol based detector whereas in general this is not the case for both m qam and m fsk

transmit beamforming in multiple antenna systems with finite rate feedback a vq based approach this paper investigates quantization methods for feeding back the channel information through a low rate feedback channel in the context of multiple input single output miso systems we propose a new quantizer design criterion for capacity maximization and develop the corresponding iterative vector quantization vq design algorithm the criterion is based on maximizing the mean squared weighted inner product mswip between the optimum and the quantized beamforming vector the performance of systems with quantized beamforming is analyzed for the independent fading case this requires finding the density of the squared inner product between the optimum and the quantized beamforming vector which is obtained by considering a simple approximation of the quantization cell the approximate density function is used to lower bound the capacity loss due to quantization the outage probability and the bit error probability the resulting expressions provide insight into the dependence of the performance of transmit beamforming miso systems on the number of transmit antennas and feedback rate computer simulations support the analytical results and indicate that the lower bounds are quite tight

how to generate cryptographically strong sequences of pseudo random bits we give a set of conditions that allow one to generate 50 50 unpredictable bits based on those conditions we present a general algorithmic scheme for constructing polynomial time deterministic algorithms that stretch a short secret random input into a long sequence of unpredictable pseudo random bits we give an implementation of our scheme and exhibit a pseudo random bit generator for which any efficient strategy for predicting the next output bit with better than 50 50 chance is easily transformable to an equally efficient algorithm for solving the discrete logarithm problem in particular if the discrete logarithm problem cannot be solved in probabilistic polynomial time no probabilistic polynomial time algorithm can guess the next output bit better than by flipping a coin if head guess 0 if tail guess 1

pseudorandomness from shrinkage one powerful theme in complexity theory and pseudorandomness in the past few decades has been the use of lower bounds to give pseudorandom generators prgs however the general results using this hardness vs randomness paradigm suffer from a quantitative loss in parameters and hence do not give nontrivial implications for models where we don t know super polynomial lower bounds but do know lower bounds of a fixed polynomial we show that when such lower bounds are proved using random restrictions we can construct prgs which are essentially best possible without in turn improving the lower bounds more specifically say that a circuit family has shrinkage exponent if a random restriction leaving a p fraction of variables unset shrinks the size of any circuit in the family by a factor of p o 1 our prg uses a seed of length s1 1 o 1 to fool circuits in the family of size s by using this generic construction we get prgs with polynomially small error for the following classes of circuits of size s and with the following seed lengths 1 for de morgan formulas seed length s1 3 o 1 2 for formulas over an arbitrary basis seed length s1 2 o 1 3 for read once de morgan formulas seed length s 234 4 for branching programs of size s seed length s1 2 o 1 the previous best prgs known for these classes used seeds of length bigger than n 2 to output n bits and worked only for size seo n 8

analysis of rake reception with multiple symbol weight estimation for antipodal signaling consider a rake receiver for coherent binary antipodal signaling with 1 a delayed received signal configuration 2 weight estimation by matched filtering using the reference signal along with the decisions of the previous m symbol intervals and 3 predetection maximal ratio combining mrc the weight estimation errors here are not independent of the additive noise and do not fit into the gaussian weighting error model for mrc here we analyze the error performance of the receiver by obtaining the conditional symbol error probability conditioned on past decisions from the characteristic function of the decision variable and getting the unconditional error probability uep for a block of m consecutive symbols using a markov model of the decision process the channel is rayleigh fading with independent and identically distributed branch gains results show that the error performance of the gaussian distributed weighting error model is a bound for that of multiple symbol weight estimation by matched filtering and the steady state uep decreases with increase of m but the amount of decrease reduces as m increases

comments on exact error rate analysis of diversity 16 qam with channel estimation error recently cao and beaulieu presented an insightful analysis on the bit error performance of 16 quadrature amplitude modulation qam scheme on rayleigh fading channels with maximal ratio combining diversity and channel estimation errors unfortunately although exact the results presented in that paper are based on a complicated parabolic cylinder function and a two dimensional numerical integration has to be performed to evaluate the average probability of bit error in this letter we present a surprisingly simple expression for the average probability of error in a well known closed form we point out that the error rates for any n ary square rectangular qam constellation with gray mapping can also be expressed in this form furthermore we show that the presented results can also be modified in a simple fashion for a generalized selection combining scheme in the presence of channel estimation errors

capacity and power allocation for fading mimo channels with channel estimation error in this correspondence we investigate the effect of channel estimation error on the capacity of multiple input multiple output mimo fading channels we study lower and upper bounds of mutual information under channel estimation error and show that the two bounds are tight for gaussian inputs assuming gaussian inputs we also derive tight lower bounds of ergodic and outage capacities and optimal transmitter power allocation strategies that achieve the bounds under perfect feedback for the ergodic capacity the optimal strategy is a modified waterfilling over the spatial antenna and temporal fading domains this strategy is close to optimum under small feedback delays but when the delay is large equal powers should be allocated across spatial dimensions for the outage capacity the optimal scheme is a spatial waterfilling and temporal truncated channel inversion numerical results show that some capacity gain is obtained by spatial power allocation temporal power adaptation on the other hand gives negligible gain in terms of ergodic capacity but greatly enhances outage performance