a new upper bound on the minimum distance of turbo codes in this paper a new upper bound on the minimum distance of turbo codes is derived the new bound is obtained by construction of an undirected graph which reflects the characteristics of the constituent codes and the interleaver the resulting expression shows that the minimum distance of a turbo code grows approximately with the base 3 logarithm of the information word length the new bound is easy to compute applies to rate k sub 0 n sub 0 constituent encoders and often improves over existing results

analysis and design of tuned turbo codes it has been widely observed that there exists a fundamental tradeoff between the minimum hamming distance properties and the iterative decoding convergence behavior of turbo like codes while capacity achieving code ensembles typically are asymptotically bad in the sense that their minimum distance does not grow linearly with block length and they therefore exhibit an error floor at moderate to high signal to noise ratios asymptotically good codes usually converge further away from channel capacity in this paper we introduce the concept of tuned turbo codes a family of asymptotically good hybrid concatenated code ensembles where asymptotic minimum distance growth rates convergence thresholds and code rates can be tradedoff using two tuning parameters and by decreasing the asymptotic minimum distance growth rate is reduced in exchange for improved iterative decoding convergence behavior while increasing raises the asymptotic minimum distance growth rate at the expense of worse convergence behavior and thus the code performance can be tuned to fit the desired application by decreasing a similar tuning behavior can be achieved for higher rate code ensembles

analysis and regularization of the td efie low frequency breakdown low frequency breakdown phenomenon plaguing marching on in time solutions of time domain electric field integral equations is analyzed and cured a theoretical analysis of the breakdown identifies deficiencies in the treatments proposed to date a scheme for eliminating low frequency breakdown phenomena that leverages hierarchical regularization is presented it applies to arbitrary meshes and gives rise to linear systems that are better conditioned than those obtained with standard loop star tree bases the scheme s analytical properties and numerical results demonstrate its efficacy in accelerating the iterative solution of the time domain electric field integral equations

electromagnetic scattering from 3 d curved dielectric bodies using time domain integral equations a boundary integral equation bie approach is developed to calculate transient scattering from dielectric bodies the treatment is directly in terms of the e and h fields rather than magnetic and electric currents it employs curvilinear quadratic modeling which allows accurate representation of arbitrarily shaped curved bodies the treatment is isoparametric with the same quadratic representation of the spatial field variation and with the temporal variation modeled by similar quadratic elements integration employs high order gaussian quadrature with special treatment of the singular and hypersingular integrals that arise the treatment is implicit requiring the solution of a sparse matrix equation at each timestep this adds only trivially to the cost at each timestep and by freeing the timestep from the constraint that it be smaller than the smallest nodal spacing can greatly reduce the number of timesteps that must be employed additionally it produces stable results without resort to the averaging processes proposed elsewhere example calculations of scattering from a sphere a cube and an almond are presented and compared with earlier published transient results and with results from a frequency domain treatment good agreement and improved accuracy is found

joint permutor analysis and design for multiple turbo codes in this paper we study the problem of joint permutor analysis and design for j dimensional multiple turbo codes with j constituent encoders j 2 the concept of summary distance is extended to multiple permutors of size n and used as the design metric using the sphere packing concept we prove that the minimum length 2 summary distance spread dmin 2 is asymptoticly upper bounded by o n j 1 j we also show that the asymptotic minimum length 2l summary distance dmin 2l for the class of random permutors is lower bounded by o nj 2j epsi where epsi 0 can be arbitrarily small then using the technique of expurgating bad symbols we show that the spread of random permutors can achieve the optimum growth rate i e o nj 1 j and that the asymptotic growth rate of dmin 2l can also be improved the minimum length 2 and length 4 summary distances are studied for an important practical class of permutors linear permutors we prove that there exist j dimensional multiple linear permutors with optimal spread dmin 2 o nj 1j finally we present several joint permutor construction algorithms applicable to multiple turbo codes of short and medium lengths

a distance spectrum interpretation of turbo codes the performance of turbo codes is addressed by examining the code s distance spectrum the error floor that occurs at moderate signal to noise ratios is shown to be a consequence of the relatively low free distance of the code it is also shown that the error floor can be lowered by increasing the size of the interleaver without changing the free distance of the code alternatively the free distance of the code may be increased by using primitive feedback polynomials the excellent performance of turbo codes at low signal to noise ratios is explained in terms of the distance spectrum the interleaver in the turbo encoder is shown to reduce the number of low weight codewords through a process called spectral thinning this thinned distance spectrum results in the free distance asymptote being the dominant performance parameter for low and moderate signal to noise ratios

some new classes of complex symmetric operators we say that an operator t e b h is complex symmetric if there exists a conjugate linear isometric involution c so that t ct c we prove that binormal operators operators that are algebraic of degree two including all idempotents and large classes of rank one perturbations of normal operators are complex symmetric from an abstract viewpoint these results explain why the compressed shift and volterra integration operator are complex symmetric finally we attempt to describe all complex symmetric partial isometries obtaining the sharpest possible statement given only the data dim ker t dim ker t

reliable channel regions for good binary codes transmitted over parallel channels we study the average error probability performance of binary linear code ensembles when each codeword is divided into j subcodewords with each being transmitted over one of j parallel channels this model is widely accepted for a number of important practical channels and signaling schemes including block fading channels incremental redundancy retransmission schemes and multicarrier communication techniques for frequency selective channels our focus is on ensembles of good codes whose performance in a single channel model is characterized by a threshold behavior e g turbo and low density parity check ldpc codes for a given good code ensemble we investigate reliable channel regions which ensure reliable communications over parallel channels under maximum likelihood ml decoding to construct reliable regions we study a modifed 1961 gallager bound for parallel channels by allowing codeword bits to be randomly assigned to each component channel the average parallel channel gallager bound is simplified to be a function of code weight enumerators and channel assignment rates special cases of this bound average union bhattacharyya ub shulman feder sf simplified sphere ss and modified shulman feder msf parallel channel bounds allow for describing reliable channel regions using simple functions of channel and code spectrum parameters parameters describing the channel are the average parallel channel bhattacharyya noise parameter the average channel mutual information and parallel gaussian channel signal to noise ratios snrs code parameters include the union bhattacharyya noise threshold and the weight spectrum distance to the random binary code ensemble reliable channel regions of repeat accumulate ra codes for parallel binary erasure channels becs and of turbo codes for parallel additive white gaussian noise awgn channels are numerically computed and compared with simulation results based on iterative decoding in addition an examp

on quasi cyclic interleavers for parallel turbo codes in this correspondence we present an interleaving scheme that yields quasi cyclic turbo codes we prove that randomly chosen members of this family yield with probability almost 1 turbo codes with asymptotically optimum minimum distance i e growing as a logarithm of the interleaver size these interleavers are also very practical in terms of memory requirements and their decoding error probabilities for small block lengths compare favorably with previous interleaving schemes

an upper bound on the minimum distance of serially concatenated convolutional codes this correspondence presents an upper bound on the minimum distance of serially concatenated codes with interleaver where the inner code is a systematic recursive convolutional encoder and the outer code is any convolutional encoder the resulting expression shows that the minimum distance of the concatenated code cannot grow more than o k1 1 df o where k is the information word length and df o is the free distance of the outer code the obtained upper bound is shown to agree with and in some cases improve over previously known results

optimal puncturing ratios and energy allocation for multiple parallel concatenated codes we propose a systematic design framework for optimal low complexity punctured multiple parallel concatenated codes mpccs based on minimizing the convergence threshold using extrinsic information transfer exit charts as the convergence threshold is related to the area between the two exit curves the corresponding optimization problem is equivalent to a curve fitting problem the exit curves are determined by the respective exit functions of the constituents which can be conveniently shaped through the use of random puncturing and unequal energy allocations across parallel coding streams the design task is therefore to find the optimal combination of constituents puncturing ratios and energy allocation for matching the exit curves a search over all rate one convolutional codes of memory length four or less is performed identifying 98 classes of codes with unique exit functions out of a total of 310 codes low complexity mpccs with up to four constituents are found where the convergence thresholds are observed to be within 0 1 db or less of the fundamental minimum signal to noise ratio snr corresponding to the binary phase shift keying bpsk capacity for code rates 1 3 les r les 7 8 further allowing for unequal energy allocation the convergence thresholds for lower code rates are similarly improved

stable electric field tdie solvers via quasi exact evaluation of mot matrix elements prior theoretical studies and experience confirm that the stability of marching on in time mot solvers pertinent to the analysis of scattering from free standing three dimensional perfect electrically conducting surfaces hinges on the accurate evaluation of mot matrix elements resulting from a galerkin discretization of the underlying time domain integral equation tdie unfortunately the accurate evaluation of the four dimensional spatial integrals involved in the expressions for these matrix elements is prohibitively expensive when performed by computational means here a method that permits the quasi exact evaluation of mot matrix elements is presented specifically the proposed method permits the analytical evaluation of three out of the four spatial integrations leaving only one integral to be evaluated numerically since the latter has finite range and a piecewise smooth integrand it can be evaluated to very high accuracy using standard quadrature rules as a result the proposed method permits the fast evaluation of mot matrix elements with arbitrary user specified accuracy extensive numerical experiments show that an mot solver for the electric field tdie that uses the proposed quasi exact method is stable for a very wide range of time step sizes and yields solutions that decay exponentially after the excitation vanishes

analysis of transient electromagnetic scattering phenomena using a two level plane wave time domain algorithm a fast algorithm is presented for solving electric magnetic and combined field time domain integral equations pertinent to the analysis of surface scattering phenomena the proposed two level plane wave time domain pwtd algorithm permits a numerically rigorous reconstruction of transient near fields from their far field expansion and augments classical marching on in time mot based solvers the computational cost of analyzing surface scattering phenomena using pwtd enhanced mot schemes scales as o n sub i n sub s sup 2 3 log n sub s as opposed to spl oscr n sub t n sub s sup 2 for classical mot methods where n sub t and n sub s are the numbers of temporal and spatial basis functions discretizing the scatterer current numerical results that demonstrate the efficacy of the proposed solver in analyzing transient scattering from electrically large structures and that confirm the above complexity estimate are presented

fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm the computational complexity of classical marching on in time mot methods for solving time domain integral equations tdies pertinent to the analysis of transient scattering phenomena involving perfectly conducting targets grows as o n sub t n sub s sup 2 n sub t and n sub s denote the number of temporal and spatial degrees of freedom dof of the electric current on the target this scaling law impedes the application of these schemes to the analysis of large scale scattering phenomena the recently developed plane wave time domain pwtd algorithm permits the rapid evaluation of transient wave fields generated by temporally bandlimited sources and hence the acceleration of marching on in time based tdie solvers previously we described a two level pwtd enhanced tdie solver for analyzing electromagnetic scattering from perfectly conducting targets the computational complexity of this algorithm scales as o n sub t n sub s sup 1 5 logn sub s here a multilevel pwtd scheme for rapidly evaluating electric fields due to temporally bandlimited electric current sources is described in addition a multilevel pwtd enhanced tdie solver for analyzing electromagnetic scattering from perfectly conducting scatterers using o n sub t n sub s log sup 2 n sub s cpu resources is outlined last the accuracy and cpu memory efficiency of this solver are demonstrated by analyzing transient scattering from electrically large bodies

time domain adaptive integral method for surface integral equations an efficient marching on in time mot scheme is presented for solving electric magnetic and combined field integral equations pertinent to the analysis of transient electromagnetic scattering from perfectly conducting surfaces residing in an unbounded homogenous medium the proposed scheme is the extension of the frequency domain adaptive integral pre corrected fast fourier transform fft method to the time domain fields on the scatterer that are produced by space time sources residing on its surface are computed 1 by locally projecting for each time step all sources onto a uniform auxiliary grid that encases the scatterer 2 by computing everywhere on this grid the transient fields produced by the resulting auxiliary sources via global multilevel blocked space time ffts 3 by locally interpolating these fields back onto the scatterer surface as this procedure is inaccurate when source and observer points reside close to each other and 4 near fields are computed classically albeit pre corrected for errors introduced through the use of global ffts the proposed scheme has a computational complexity and memory requirement of o n sub t n sub s log sup 2 n sub s and o n sub s sup 3 2 when applied to quasiplanar structures and of o n sub t n sub s sup 3 2 log sup 2 n sub s and o n sub s sup 2 when used to analyze scattering from general surfaces here n sub s and n sub t denote the number of spatial and temporal degrees of freedom of the surface current density these computational cost and memory requirements are contrasted to those of classical mot solvers which scale as o n sub t n sub s sup 2 and o n sub s sup 2 respectively a parallel implementation of the scheme on a distributed memory computer cluster that uses the message passing interface is described simulation results demonstrate the accuracy efficiency and the parallel performance of the implementation

reflexivity properties of t 0 a construction is given of a reflexive operator t acting on a separable hilbert space h with the property that the direct sum t 0 fails to be reflexive this construction is then used to provide solutions to several other problems which have been studied concerning the direct sum splitting of operator algebras scott brown s technique the theory of bitriangular operators and parareflexivity

a logarithmic upper bound on the minimum distance of turbo codes we derive new upper bounds on the minimum distance which turbo codes can maximally attain with the optimum interleaver of a given length the new bounds grow approximately logarithmically with the interleaver length and they are tighter than all previously derived bounds for medium length and long interleavers an extensive discussion highlights the impacts of the new bounds in the context of interleaver design and provides some new design guidelines

on the minimum distance of turbo codes with quadratic permutation polynomial interleavers an interleaver is a critical component for the channel coding performance of turbo codes algebraic constructions are of particular interest because they admit analytical designs and simple practical hardware implementations also the recently proposed quadratic permutation polynomial qpp based interleavers by sun and takeshita have provided excellent performance for short to medium block lengths and have been selected for the 3gpp lte standard in this paper we derive some upper bounds on the best achievable minimum distance dmin of qpp based conventional binary turbo codes with tailbiting termination or dual termination when the interleaver length n is sufficiently large that are tight for larger block sizes in particular we show that the minimum distance is at most 2 2v 1 9 independent of the interleaver length when there exists an inverse polynomial of degree two where v is the degree of the primitive feedback and monic feedforward polynomials however allowing the qpp to have no inverse polynomials of degree two may give strictly larger minimum distances and lower multiplicities in particular we provide several qpps with no quadratic inverse for some of the 3gpp lte interleaver lengths giving a dmin with the 3gpp lte constituent encoders which is strictly larger than 50 for instance we have found a qpp for n 6016 which gives an estimated dmin of 57 furthermore we provide the exact minimum distances and the corresponding multiplicities for all 3gpp lte turbo codes with dual termination which shows that the best minimum distance is 51 finally we compute the best achievable minimum distances with qpp interleavers for all 3gpp lte interleaver lengths n 4096 and compare these minimum distances with the ones we get when using the 3gpp lte polynomials

combinatorial analysis of the minimum distance of turbo codes in this paper new upper bounds on the maximum attainable minimum hamming distance of turbo codes with arbitrary including the best interleavers are established using a combinatorial approach these upper bounds depend on the interleaver length the code rate and the scramblers employed in the encoder examples of the new bounds for particular turbo codes are given and discussed the new bounds are tighter than all existing ones and prove that the minimum hamming distance of turbo codes cannot asymptotically grow at a rate more than the third root of the codeword length

higher order interpolatory vector bases for computational electromagnetics low order vector basis functions compatible with the nedelec 1980 representations are widely used for electromagnetic field problems higher order functions are receiving wider application but their development is hampered by the complex procedures used to generate them and lack of a consistent notation for both elements and bases in this paper fully interpolatory higher order vector basis functions of the nedelec type are defined in a unified and consistent manner for the most common element shapes it is shown that these functions can be obtained as the product of zeroth order nedelec representations and interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties of the vector functions are discussed and expressions for the vector functions of arbitrary polynomial order are presented sample numerical results confirm the faster convergence of the higher order functions

time domain single source integral equations for analyzing scattering from homogeneous penetrable objects single source time domain electric and magnetic field integral equations for analyzing scattering from homogeneous penetrable objects are presented their temporal discretization is effected by using shifted piecewise polynomial temporal basis functions and a collocation testing procedure thus allowing for a marching on in time mot solution scheme unlike dual source formulations single source equations involve space time domain operator products for which spatial discretization techniques developed for standalone operators do not apply here the spatial discretization of the single source time domain integral equations is achieved by using the high order divergence conforming basis functions developed by graglia et al alongside the high order divergence and quasi curl conforming dqcc basis functions of valdes et al the combination of these two sets allows for a well conditioned mapping from div to curl conforming function spaces that fully respects the space mapping properties of the space time operators involved numerical results corroborate the fact that the proposed procedure guarantees accuracy and stability of the mot scheme

the minimum distance of turbo like codes worst case upper bounds are derived on the minimum distance of parallel concatenated turbo codes serially concatenated convolutional codes repeat accumulate codes repeat convolute codes and generalizations of these codes obtained by allowing nonlinear and large memory constituent codes it is shown that parallel concatenated turbo codes and repeat convolute codes with sub linear memory are asymptotically bad it is also shown that depth two serially concatenated codes with constant memory outer codes and sublinear memory inner codes are asymptotically bad most of these upper bounds hold even when the convolutional encoders are replaced by general finite state automata encoders in contrast it is proven that depth three serially concatenated codes obtained by concatenating a repetition code with two accumulator codes through random permutations can be asymptotically good

designing a good low rate sparse graph code this paper deals with the design of low rate sparse graph codes having a linear minimum distance d min in the blocklength n its main contributions are a a necessary condition on a general family of sparse graph codes with linear d min b a justification of having degree 1 bits in the low rate code structure c a new efficient ensemble of low rate sparse graph codes with bits of degree 1 designed so that the necessary condition a is satisfied

coding theorems for turbo code ensembles this paper is devoted to a shannon theoretic study of turbo codes we prove that ensembles of parallel and serial turbo codes are good in the following sense for a turbo code ensemble defined by a fixed set of component codes subject only to mild necessary restrictions there exists a positive number spl gamma sub 0 such that for any binary input memoryless channel whose bhattacharyya noise parameter is less than spl gamma sub 0 the average maximum likelihood ml decoder block error probability approaches zero at least as fast as n sup spl beta where spl beta is the interleaver gain exponent defined by benedetto et al in 1996

an analysis of the block error probability performance of iterative decoding asymptotic iterative decoding performance is analyzed for several classes of iteratively decodable codes when the block length of the codes n and the number of iterations i go to infinity three classes of codes are considered these are gallager s regular low density parity check ldpc codes tanner s generalized ldpc gldpc codes and the turbo codes due to berrou et al it is proved that there exist codes in these classes and iterative decoding algorithms for these codes for which not only the bit error probability p sub b but also the block frame error probability p sub b goes to zero as n and i go to infinity

upper bound on the minimum distance of turbo codes an upper bound on the minimum distance of turbo codes is derived which depends only on the interleaver length and the component scramblers employed the derivation of this bound considers exclusively turbo encoder input words of weight 2 the bound does not only hold for a particular interleaver but for all possible interleavers including the best it is shown that in contrast to general linear binary codes the minimum distance of turbo codes cannot grow stronger than the square root of the block length this implies that turbo codes are asymptotically bad a rigorous proof for the bound is provided which is based on a geometric approach

reflexivite d une extension d un operateur normal par un operateur nilpotent let t be an extension of a one to one normal operator a by a nilpotent operator n in this paper we calculate the defect of reflexivity of t we give a necessary and sufficient condition to insure the reflexivity of such extensions in particular it is shown that t is reflexive when n is reflexive

the performance of serial turbo codes does not concentrate minimum distances and maximum likelihood error probabilities of serial turbo codes with uniform interleaver are analyzed it is shown that for a fraction of interleavers approaching one as the block length grows large the minimum distance of serial turbo codes grows as a positive power of their block length while their error probability decreases exponentially fast in some positive power of their block length on sufficiently good memoryless channels such a typical code behavior contrasts the performance of the average serial turbo code whose error probability is dominated by an asymptotically negligible fraction of poorly performing interleavers and decays only as a negative power of the block length the analysis proposed in this paper relies on precise bounds of the minimum distance of the typical serial turbo code whose scaling law is shown to depend both on the free distance of its outer constituent encoder which determines the exponent of its sub linear growth in the block length and on the effective free distance of its inner constituent encoder the latter is defined as the smallest weight of codewords obtained when the input word of the inner encoder has weight two and appears as a linear scaling factor for the minimum distance of the typical serial turbo code hence despite the lack of concentration of the maximum likelihood error probability around its expected value the main design parameters suggested by the average code analysis turn out to characterize also the performance of the typical serial turbo code by showing for the first time that the typical serial turbo code s minimum distance scales linearly in the effective free distance of the inner constituent encoder the presented results generalize and improve upon the probabilistic bounds of kahale and urbanke as well as the deterministic upper bound of bazzi mahdian and spielman where only the dependence on the outer encoder s free distance was proved

variance of the turbo code performance bound over the interleavers we evaluate the variance of the union performance bound for a rate 1 3 turbo code over all possible interleavers of length n under the assumption of a maximum likelihood ml decoder theoretical and simulation results for turbo codes with two memory component codes indicate that the coefficient of variation of the bound increases with the signal to noise ratio and decreases with the interleaver length theoretical analysis for large interleaver lengths shows that the coefficient of variation asymptotically approaches a constant value the results also demonstrate that the majority of the interleavers have performance bounds very close to the average value of the bound this phenomenon is more palpable for larger interleaver lengths

electromagnetic scattering from rough surface using single integral equation and adaptive integral method an efficient algorithm for electromagnetic wave scattering from rough dielectric surfaces is developed the algorithm is based on the single magnetic field integral equation smfie and the surface is discretized using rao wilton glisson rwg triangular basis functions the new feature of the algorithm is the application of the adaptive integral method aim with smfie for speeding up the calculation the developed new method utilizes the flexibility of rwg functions to model arbitrary rough surface and the speed of fft enabling accurate simulations over large surfaces

performance of parallel concatenated coding schemes in this paper ensembles of parallel concatenated codes are studied and rigorous results on their asymptotic performance under the assumption of maximum likelihood ml decoding are presented in particular it is proven that in any parallel concatenation scheme with k branches where all k encoders are recursive and the bhattacharyya parameter of the channel is sufficiently small the bit error rate ber and the word error rate go to 0 exactly like n 1 k and n 2 k respectively different types of ensembles by changing the subgroup of permutations used to interconnect the various encoders are considered

a new cycle based joint permutor design for multiple turbo codes in this letter a permutor design parameter called the weight of a cycle through j 1 permutation matrices for multiple turbo codes is introduced analogously to the single permutor case of conventional turbo codes we show the connection of the new parameter to the minimum distance of special classes of multiple turbo codes in addition simulation results are presented for multiple turbo codes based on the new design

permutation polynomial interleavers an algebraic geometric perspective an interleaver is a critical component for the channel coding performance of turbo codes algebraic constructions are important because they admit analytical designs and simple practical hardware implementation the spread factor of an interleaver is a common measure for turbo coding applications maximum spread interleavers are interleavers whose spread factors achieve the upper bound an infinite sequence of quadratic pps over integer rings that generate maximum spread interleavers is presented new properties of pp interleavers are investigated from an algebraic geometric perspective resulting in a new non linearity metric for interleavers a new interleaver metric that is a function of both the nonlinearity metric and the spread factor is proposed it is numerically demonstrated that the spread factor has a diminishing importance with the block length a table of good interleavers for a variety of interleaver lengths according to the new metric is listed extensive computer simulation results with impressive frame error rates confirm the efficacy of the new metric further when tail biting constituent codes are used the resulting turbo codes are quasi cyclic

time domain calderon identities and their application to the integral equation analysis of scattering by pec objects part ii stability novel time domain integral equations for analyzing scattering from perfect electrically conducting objects are presented they are free from dc and resonant instabilities plaguing standard electric field integral equation the new equations are obtained using operator manipulations originating from the calderon identities theoretical motivations leading to the construction of the new equations are explored and numerical results confirming their theoretically predicted behavior are presented

rapid carrier acquisition from baud rate samples maximum likelihood estimation of phase and frequency offset involves the maximization of a nonlinear likelihood function we develop an estimator for carrier acquisition by linearizing the likelihood function and show that its performance is close to the cramer rao bound crb the estimator is then extended to blind frequency acquisition and combined with testing to eliminate unlikely hypotheses with proper modeling of the likelihood resulting from the blind search simulations show that the algorithm quickly rejects all but the correct frequency hypothesis

unveiling turbo codes some results on parallel concatenated coding schemes a parallel concatenated coding scheme consists of two simple constituent systematic encoders linked by an interleaver the input bits to the first encoder are scrambled by the interleaver before entering the second encoder the codeword of the parallel concatenated code consists of the input bits to the first encoder followed by the parity check bits of both encoders this construction can be generalized to any number of constituent codes parallel concatenated schemes employing two convolutional codes as constituent codes in connection with an iterative decoding algorithm of complexity comparable to that of the constituent codes have been previously shown to yield remarkable coding gains close to theoretical limits they have been named and are known as turbo codes we propose a method to evaluate an upper bound to the bit error probability of a parallel concatenated coding scheme averaged over all interleavers of a given length the analytical bounding technique is then used to shed some light on some crucial questions which have been floating around in the communications community since the proposal of turbo codes

reflexive binormal operators abstract this paper initiates a study of the problem of which binormal operators on a hilbert space are reflexive for a certain important class of binormal operators a simple measure theoretic characterization is given for reflexivity