periodic points of holomorphic maps via lefschetz numbers in this paper we study the set of periods of holomorphic maps on compact manifolds using the periodic lefschetz numbers introduced by dold and llibre which can be computed from the homology class of the map we show that these numbers contain information about the existence of periodic points of a given period and if we assume the map to be transversal then they give us the exact number of such periodic orbits we apply this result to the complex projective space of dimension n and to some special type of hopf surfaces partially characterizing their set of periods in the first case we also show that any holomorphic map of cp n of degree greater than one has infinitely many distinct periodic orbits hence generalizing a theorem of fornaess and sibony we then characterize the set of periods of a holomorphic map on the riemann sphere hence giving an alternative proof of baker s theorem

on the design of mimo block fading channels with feedback link capacity constraint in this paper we propose a combined adaptive power control and beamforming framework for optimizing multiple input multiple output mimo link capacity in the presence of feedback link capacity constraint the feedback channel is used to carry channel state information only it is assumed to be noiseless and causal with a feedback capacity constraint in terms of maximum number of feedback bits per fading block we show that the hybrid design could achieve the optimal mimo link capacity and we derive a computationally efficient algorithm to search for the optimal design under a specific average power constraint finally we shall illustrate that a minimum mean square error spatial processor with a successive interference canceller at the receiver could be used to realize the optimal capacity we found that feedback effectively enhances the forward channel capacity for all signal to noise ratio snr values when the number of transmit antennas n sub t is larger than the number of receive antennas n sub r the snr gain with feedback is contributed by focusing transmission power on active eigenchannel and temporal power waterfilling the former factor contributed at most 10log sub 10 n sub t n sub r db snr gain when n sub t n sub r while the latter factor s snr gain is significant only for low snr values

mimo systems with quantized covariance feedback a unified approach is developed for the study of multi input multi output mimo systems under fast fading with quantized covariance feedback in such systems the receiver computes using perfect channel state information csi the covariance matrices to be adopted by the transmitter corresponding to the current channel realization and feeds back a quantized version of this information to the transmitter using finite say nf bits per channel realization our analysis is based on a geometric framework we developed in a companion paper for manifolds of positive semi definite covariance matrices with various trace and rank constraints that analysis applies to mimo systems in a unified manner to optimal as well as various suboptimal precoding methods for obtaining input covariances these include for example the capacity achieving spatio temporal water filling strategy the incorporation of just spatial water filling and reduced rank beamforming with power allocation in both cases as well as mimo systems with antenna selection for a given system strategy including the one that achieves capacity the gap between the communication rates in the perfect and quantized covariance cases is shown to be o 2 nf n where n is the dimension of the particular pn manifold used for quantization this dimension depends on the precoding strategy adopted and strongly determines how quickly the communication rate with quantized covariance feedback approaches that with perfect csi at the transmitter csit our results show how the choice of covariance quantization manifold can be tailored to the available feedback rate in mimo system design

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

fixed point indices of iterated smooth maps in arbitrary dimension abstract let f be a smooth self map of r m when m is an arbitrary natural number we give a complete description of possible sequences of indices of iterations of f at an isolated fixed point answering in affirmative the chow mallet paret and yorke conjecture posed in s n chow j mallet parret j a yorke a periodic point index which is a bifurcation invariant in geometric dynamics rio de janeiro 1981 in lecture notes in math vol 1007 springer berlin 1983 pp 109 131

fixed point indices and periodic points of holomorphic mappings let n be the ball x 1 in the complex vector space mathbb c n let f delta n rightarrow mathbb c n be a holomorphic mapping and let m be a positive integer assume that the origin 0 0 dots 0 is an isolated fixed point of both f and the mth iteration f m of f then for each factor m of m the origin is again an isolated fixed point of f m and the fixed point index mu f m 0 of f m at the origin is well defined and so is the local dold s index invent math 74 3 419 435 1983 at the origin p m f 0 sum tau subset p m 1 tau mu f m tau 0 where p m is the set of all primes dividing m the sum extends over all subsets of p m is the cardinal number of and m tau m prod p in tau p 1 p m f 0 can be interpreted to be the number of periodic points of period m of f overlapped at the origin any holomorphic mapping f 1 delta n rightarrow mathbb c n sufficiently close to f has exactly p m f 0 distinct periodic points of period m near the origin provided that all the fixed points of f 1 m near the origin are simple note that f itself has no periodic point of period m near the origin if m 1 according to shub and sullivan s work topology 13 189 191 1974 a necessary condition so that p m f 0 0 is that the linear part of f at the origin has a periodic point of period m the goal of this paper is to prove that this condition is sufficient as well for holomorphic mappings

quantization bounds on grassmann manifolds and applications to mimo communications the grassmann manifold gn p l is the set of all p dimensional planes through the origin in the n dimensional euclidean space ln where l is either r or c this paper considers the quantization problem in which a source in gn p l is quantized through a code in gn q l with p and q not necessarily the same the analysis is based on the volume of a metric ball in gn p l with center in gn q l and our chief result is a closed form expression for the volume of a metric ball of radius at most one this volume formula holds for arbitrary n p q and l while previous results pertained only to some special cases based on this volume formula several bounds are derived for the rate distortion tradeoff assuming that the quantization rate is sufficiently high the lower and upper bounds on the distortion rate function are asymptotically identical and therefore precisely quantify the asymptotic rate distortion tradeoff we also show that random codes are asymptotically optimal in the sense that they achieve the minimum possible distortion in probability as n and the code rate approach infinity linearly finally as an application of the derived results to communication theory we quantify the effect of beamforming matrix selection in multiple antenna communication systems with finite rate channel state feedback

systematic design of unitary space time constellations we propose a systematic method for creating constellations of unitary space time signals for multiple antenna communication links unitary space time signals which are orthonormal in time across the antennas have been shown to be well tailored to a rayleigh fading channel where neither the transmitter nor the receiver knows the fading coefficients the signals can achieve low probability of error by exploiting multiple antenna diversity because the fading coefficients are not known the criterion for creating and evaluating the constellation is nonstandard and differs markedly from the familiar maximum euclidean distance norm our construction begins with the first signal in the constellation an oblong complex valued matrix whose columns are orthonormal and systematically produces the remaining signals by successively rotating this signal in a high dimensional complex space this construction easily produces large constellations of high dimensional signals we demonstrate its efficacy through examples involving one two and three transmitter antennas

fixed point indices of iterated maps on determine toutes les relations universelles qui existent entre les indices de point fixe i f n n 1 2 des iterees f n d une application f v y ou y est un enr vcy est un sous ensemble ouvert

information capacity of a random signature multiple input multiple output channel a closed form expression for information capacity of the random signature multiple input multiple output channel is given a direct calculation of the capacity is provided additional to the proof involving the capacity of minimum mean square error systems provided by verdu and shamai see ieee trans inform theory vol 45 p 622 40 1999

the effect of finite rate feedback on cdma signature optimization and mimo beamforming vector selection we analyze the effect of finite rate feedback on code division multiple access cdma signature optimization and multiple input multiple output mimo beamforming vector selection in cdma signature optimization for a particular user the receiver selects a signature vector from a codebook to best avoid interference from other users and then feeds the corresponding index back to the specified user for mimo beamforming vector selection the receiver chooses a beamforming vector from a given codebook to maximize the instantaneous information rate and feeds back the corresponding index to the transmitter these two problems are dual both can be modeled as selecting a unit norm vector from a finite size codebook to ldquomatchrdquo a randomly generated gaussian matrix assuming that the feedback link is rate limited our main result is an exact asymptotic performance formula where the length of the signature beamforming vector the dimensions of interference channel matrix and the feedback rate approach infinity with constant ratios the proof rests on the large deviations of the underlying random matrix ensemble further we show that random codebooks generated from the isotropic distribution are asymptotically optimal not only on average but also in probability

periodic points for area preserving birational maps of surfaces it is a basic problem to count the number of periodic points of a surface mapping since the growth rate of this number as the period tends to infinity is an important dynamical invariant however this problem becomes difficult when the map admits curves of periodic points in this situation we give a precise estimate of the number of isolated periodic points for an area preserving birational map of a projective complex surface

periods for transversal maps via lefschetz numbers for periodic points let f m m be a c1 map on a c differentiable manifold the map f is called transversal if for all m e n the graph of fm intersects transversally the diagonal of m x m at each point x x such that x is a fixed point of fm we study the set of periods of f by using the lefschetz numbers for periodic points we focus our study on transversal maps defined on compact manifolds such that their rational homology is ho z q hi q e q and hk t 0 for k 0 1

the asymptotic capacity of multiple antenna rayleigh fading channels we consider the asymptotic behavior of the capacity of multiple antenna rayleigh fading channels in the limit as the transmit and receive arrays become large we show that the capacity converges in distribution to a gaussian random variable and give closed form formulas for its mean and variance these results enable us to derive the first asymptotic formula for outage rates as well as a sharper estimate of the error in previously reported asymptotic formulas for ergodic capacity although these formulas are asymptotic we show by simulation that they are often quite accurate even for relatively small arrays

outage mutual information of space time mimo channels we derive analytical expressions for the probability density function pdf of the random mutual information between transmitted and received vector signals of a random space time independent and identically distributed i i d multiple input multiple output mimo channel assuming that the transmitted signals from the multiple antennas are gaussian i i d we show that this pdf can be well approximated by a gaussian distribution and such a gaussian approximation is based on expressions for the given pdfs mean and variance that we derive we prove that at high signal to noise ratio snr every factor of 2 increase in snr leads to an increase in outage rate in the amount of min m n bits where m and n denote the number of transmit and receive antennas respectively a simple expression for the moment generating function mgf of the mutual information pdf is also provided based on which we establish normality of the pdf when both m and n are large and the snr is large

spectral efficiency of cdma with random spreading the cdma channel with randomly and independently chosen spreading sequences accurately models the situation where pseudonoise sequences span many symbol periods furthermore its analysis provides a comparison baseline for cdma channels with deterministic signature waveforms spanning one symbol period we analyze the spectral efficiency total capacity per chip as a function of the number of users spreading gain and signal to noise ratio and we quantify the loss in efficiency relative to an optimally chosen set of signature sequences and relative to multiaccess with no spreading white gaussian background noise and equal power synchronous users are assumed the following receivers are analyzed a optimal joint processing b single user matched filtering c decorrelation and d mmse linear processing

on the information rate of mimo systems with finite rate channel state feedback using beamforming and power on off strategy it is well known that multiple input multiple output mimo systems have high spectral efficiency especially when channel state information at the transmitter csit is available in many practical systems it is reasonable to assume that the csit is obtained by a limited i e finite rate feedback and is therefore imperfect we consider the design problem of how to use the limited feedback resource to maximize the achievable information rate in particular we develop a low complexity power on off strategy with beamforming or grassmann precoding and analytically characterize its performance given the eigenvalue decomposition of the covariance matrix of the transmitted signal refer to the eigenvectors as beams and to the corresponding eigenvalues as the beam s power a power on off strategy means that a beam is either turned on with a constant power or turned off we will first assume that the beams match the channel perfectly and show that the ratio between the optimal number of beams turned on and the number of antennas converges to a constant when the numbers of transmit and receive antennas approach infinity proportionally this motivates our power on off strategy where the number of beams turned on is independent of channel realizations but is a function of the signal to noise ratio snr when the feedback rate is finite beamforming cannot be perfect and we characterize the effect of imperfect beamforming by quantization bounds on the grassmann manifold by combining the results for power on off and beamforming a good approximation to the achievable information rate is derived simulations show that the proposed strategy is near optimal and the performance approximation is accurate for all experimented snrs

bounds on packings of spheres in the grassmann manifold we derive the gilbert varshamov and hamming bounds for packings of spheres codes in the grassmann manifolds over r and c asymptotic expressions are obtained for the geodesic metric and projection frobenius chordal metric on the manifold

limited feedback unitary precoding for orthogonal space time block codes orthogonal space time block codes ostbcs are a class of easily decoded space time codes that achieve full diversity order in rayleigh fading channels ostbcs exist only for certain numbers of transmit antennas and do not provide array gain like diversity techniques that exploit transmit channel information when channel state information is available at the transmitter though precoding the space time codeword can be used to support different numbers of transmit antennas and to improve array gain unfortunately transmitters in many wireless systems have no knowledge about current channel conditions this motivates limited feedback precoding methods such as channel quantization or antenna subset selection this paper investigates a limited feedback approach that uses a codebook of precoding matrices known a priori to both the transmitter and receiver the receiver chooses a matrix from the codebook based on current channel conditions and conveys the optimal codebook matrix to the transmitter over an error free zero delay feedback channel a criterion for choosing the optimal precoding matrix in the codebook is proposed that relates directly to minimizing the probability of symbol error of the precoded system low average distortion codebooks are derived based on the optimal codeword selection criterion the resulting design is found to relate to the famous applied mathematics problem of subspace packing in the grassmann manifold codebooks designed by this method are proven to provide full diversity order in rayleigh fading channels monte carlo simulations show that limited feedback precoding performs better than antenna subset selection

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

degrees of freedom in adaptive modulation a unified view we examine adaptive modulation schemes for flat fading channels where the data rate transmit power and instantaneous ber are varied to maximize spectral efficiency subject to an average power and ber constraint both continuous rate and discrete rate adaptation are considered as well as average and instantaneous ber constraints we find the general form of power ber and data rate adaptation that maximizes spectral efficiency for a large class of modulation techniques and fading distributions the optimal adaptation of these parameters is to increase the power and data rate and decrease the ber as the channel quality improves surprisingly little spectral efficiency is lost when the power or rate is constrained to be constant hence the spectral efficiency of adaptive modulation is relatively insensitive to which degrees of freedom are adapted

ber criterion and codebook construction for finite rate precoded spatial multiplexing with linear receivers precoded spatial multiplexing systems with rate limited feedback have been studied recently based on various precoder selection criteria instead of those based on indirect performance indicators we in this paper propose a new criterion directly based on the exact bit error rate ber that is applicable to systems with linear receivers and rectangular square quadrature amplitude modulation constellations the ber criterion outperforms any other alternative in terms of optimizing the ber performance for an uncoded system with linear receivers we then develop a precoder codebook construction method based on the generalized lloyd algorithm from the vector quantization literature this construction is not directly based on the ber criterion hence it is suboptimal in the ber sense however relative to those currently available our newfound codebooks improve considerably various minimum distances between any pair of codewords of the codebook finally we analyze the ber optimal precoder in the asymptotic case with infinite rate feedback that amounts to perfect channel knowledge at the transmitter the infinite rate optimal precoder based on the ber criterion is drastically different from the counterparts with other criteria and it leads to a benchmark performance for finite rate precoded spatial multiplexing systems we observe from numerical results that the ber performance of finite rate feedback with suboptimal codebooks approaches quickly the benchmark performance of infinite rate feedback this suggests that i the number of feedback bits in practical systems need not be large and ii the room for performance improvement via further codebook optimization shrinks quickly as the codebook size increases

communication on the grassmann manifold a geometric approach to the noncoherent multiple antenna channel we study the capacity of multiple antenna fading channels we focus on the scenario where the fading coefficients vary quickly thus an accurate estimation of the coefficients is generally not available to either the transmitter or the receiver we use a noncoherent block fading model proposed by marzetta and hochwald see ibid vol 45 p 139 57 1999 the model does not assume any channel side information at the receiver or at the transmitter but assumes that the coefficients remain constant for a coherence interval of length t symbol periods we compute the asymptotic capacity of this channel at high signal to noise ratio snr in terms of the coherence time t the number of transmit antennas m and the number of receive antennas n while the capacity gain of the coherent multiple antenna channel is min m n bits per second per hertz for every 3 db increase in snr the corresponding gain for the noncoherent channel turns out to be m 1 m t bits per second per hertz where m min m n t 2 the capacity expression has a geometric interpretation as sphere packing in the grassmann manifold

on beamforming with finite rate feedback in multiple antenna systems we study a multiple antenna system where the transmitter is equipped with quantized information about instantaneous channel realizations assuming that the transmitter uses the quantized information for beamforming we derive a universal lower bound on the outage probability for any finite set of beamformers the universal lower bound provides a concise characterization of the gain with each additional bit of feedback information regarding the channel using the bound it is shown that finite information systems approach the perfect information case as t 1 2 sup b t 1 where b is the number of feedback bits and t is the number of transmit antennas the geometrical bounding technique used in the proof of the lower bound also leads to a design criterion for good beamformers whose outage performance approaches the lower bound the design criterion minimizes the maximum inner product between any two beamforming vectors in the beamformer codebook and is equivalent to the problem of designing unitary space time codes under certain conditions finally we show that good beamformers are good packings of two dimensional subspaces in a 2t dimensional real grassmannian manifold with chordal distance as the metric