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Many practical adaptive feedforward systems are overparametrized and for this reason will not satisy persistent excitation (PE) conditions. For these systems, a weaker PE condition is poposed under which it is shown that the tracking error (the error between the desired and estimated outputs) converges exponentially.
This paper presents the best obtainable random coding and expurgated upper bounds on the probabilities of undetectable error, of t-order failure (advance to depth t into an incorrect subset), and of likelihood rise in the incorrect subset, applicable to sequential decoding when the metric bias G is arbitrary. Upper bounds on the Pareto exponent are also presented. The G-values optimizing each of the parameters of interest are determined, and are shown to lie in intervals that in general have nonzero widths. The G-optimal expurgated bound on undetectable error is shown to agree with that for maximum likelihood decoding of convolutional codes, and that on failure agrees with the block code expurgated bound. Included are curves evaluating the bounds for interesting choices of G and SNR for a binary-input quantized-output Gaussian additive noise channel.
An upper bound on the average probability of error for maximum-likelihood decoding of the ensemble of random L-branch binary trellis codes of rate R = 1/n with distinction between memory length and tail length is given. It is shown that the bound is independent of the length L of the information sequence if the memory length exceeds the tail length by a specified amount that depends on L. Sequential decoding simulations using the stack algorithm were conducted to test the dependence of the undetected error probability on tail length and memory length, and the results corroborated the theory.
Paper extends theory of decoder-error probability for linear maximum-distance separable (MDS) codes. General class of error-correcting codes includes Reed-Solomon codes, important in communications with distant spacecraft, military communications, and compact-disk recording industry. Advancing beyond previous theoretical developments that placed upper bounds on decoder-error probabilities, author derives an exact formula for probability PE(u) that decoder will make error when u code symbols in error.
In this article we apply transfer function bounding techniques to obtain upper bounds on the bit-error rate for maximum likelihood decoding of turbo codes constructed with random permutations. These techniques are applied to two turbo codes with constraint length 3 and later extended to other codes. The performance predicted by these bounds is compared with simulation results. The bounds are useful in estimating the 'error floor' that is difficult to measure by simulation, and they provide insight on how to lower this floor. More refined bounds are needed for accurate performance measures at lower signal-to-noise ratios.
A procedure for calculating a kxn rank k matrix B for data compression using the Bhattacharyya bound on the probability of error and an iterative construction using Householder transformation was developed. Two sets of remotely sensed agricultural data are used to demonstrate the application of the procedure. The results of the applications gave some indication of the extent to which the Bhattacharyya bound on the probability of error is affected by such transformations for multivariate normal populations.
The multipath communication channel for space communications is modeled as a multiplicative channel. This paper discusses the effects of multiplicative channel processes on the symbol error rate for quadrature modulation (QM) digital modulation schemes. An expression for the upper bound on the probability of error is derived and numerically evaluated. The results are compared with those obtained for additive channels.
Pseudo-data with simulated experimental errors can be generated to train an ensemble of Artificial Neural Networks (ANN) implemented on a regression to extract Transverse Momentum-dependent Distributions (TMDs). A preliminary analysis is presented on the reliability in extraction of the Sivers function imposed in the pseudo-data given the bounds on the experimental errors, data sparsity, and complexity of phase-space.
We develop a remarkably tight upper bound on the performance of a parameterized family of bounded angle maximum-likelihood (BA-ML) incomplete decoders. The new bound for this class of incomplete decoders is calculated from the code's weight enumerator, and is an extension of Poltyrev-type bounds developed for complete ML decoders. This bound can also be applied to bound the average performance of random code ensembles in terms of an ensemble average weight enumerator. We also formulate conditions defining a parameterized family of optimal incomplete decoders, defined to minimize both the total codeword error probability and the undetected error probability for any fixed capability of the decoder to detect errors. We illustrate the gap between optimal and BA-ML incomplete decoding via simulation of a small code.
A variable structure control technique combined with an operator spline for bilinear systems is implemented to a fast moving deformable structure for purposes of achieving robust tracking performance. The control torque applied to the structure is based on estimates of the structure parameters and on upper bounds of the model errors. For a rapid rotation of the deformable structure, the elastic response can be modeled by oscillators driven by angular acceleration, where stiffness and damping coefficients are also angular velocity and acceleration dependent. By transforming this 'slew-driven' elastic dynamics into a bilinear form, an operator spline can be constructed, that gives a low order estimate of the induced disturbance. An upper bound between the estimated deformation and the unknown exact deformation is also generated, which can be used where required in the sliding control correction.
A time-averaged error indicator with POD- h Greedy is developed to drive parametric model order reduction (pMOR) for 2D unsteady natural convection in a high-aspect ratio slot parameterized with the Prandtl number, Rayleigh number, and slot angle with respect to the gravity. The error indicator is extended to accommodate the energy equation and Leray regularization. Despite being two-dimensional and laminar, the target flow regime presents several challenges: 1) there is a bifurcation in the angle parameter space; 2) the solution can be multivalued, even at steady state; and 3) the solution exhibits spatio-temporal chaos at several points in the parameter space. The authors explore several reduced-order models (ROMs) and demonstrate that Leray-regularized Galerkin ROMs provide a robust solution approach for this class of flows. They further demonstrate that error-indicated pMOR can efficiently predict several QOIs, such as mean flow, mean Nusselt number and mean turbulent kinetic energy, even in the presence of a bifurcation. Finally, they show that spatio-temporal chaos can lead to lack of reproducibility in both the full-order model and the reduced-order model and that the variance in the full-order model provides a lower bound on the pMOR error in these cases.
Preparing the ground state of a given Hamiltonian and estimating its ground energy are important but computationally hard tasks. However, given some additional information, these problems can be solved efficiently on a quantum computer. We assume that an initial state with non-trivial overlap with the ground state can be efficiently prepared, and the spectral gap between the ground energy and the first excited energy is bounded from below. With these assumptions we design an algorithm that prepares the ground state when an upper bound of the ground energy is known, whose runtime has a logarithmic dependence on the inverse error. When such an upper bound is not known, we propose a hybrid quantum-classical algorithm to estimate the ground energy, where the dependence of the number of queries to the initial state on the desired precision is exponentially improved compared to the current state-of-the-art algorithm proposed in [Ge et al. 2019]. These two algorithms can then be combined to prepare a ground state without knowing an upper bound of the ground energy. We also prove that our algorithms reach the complexity lower bounds by applying it to the unstructured search problem and the quantum approximate counting problem.
The first paper presents a new minimum-weight trellis-based soft-decision iterative decoding algorithm for binary linear block codes. The second paper derives an upper bound on the probability of block error for multilevel concatenated codes (MLCC). The bound evaluates difference in performance for different decompositions of some codes. The third paper investigates the bit error probability code for maximum likelihood decoding of binary linear codes. The fourth and final paper included in this report is concerns itself with the construction of multilevel concatenated block modulation codes using a multilevel concatenation scheme for the frequency non-selective Rayleigh fading channel.
The zero error capacity of a discrete memoryless channel is defined as the largest rate at which information can be transmitted over the channel with zero error probability. One channel with five inputs and outputs whose zero capacity remained unsolved until very recently is considered. An extremely powerful and general technique phased in terms of graph theory, for studying combinatorial packing problems is presented. In particular, Delsarte's linear programming bound for cliques in association schemes appears as a special case of the Lovasz bound.
The use of computational fluid dynamics in scramjet engine component development is widespread in the existing literature. Unfortunately, the quantification of model-form uncertainties is rarely addressed with anything other than sensitivity studies, requiring that the computational results be intimately tied to and calibrated against existing test data. This practice must be replaced with a formal uncertainty quantification process for computational fluid dynamics to play an expanded role in the system design, development, and flight certification process. Due to ground test facility limitations, this expanded role is believed to be a requirement by some in the test and evaluation community if scramjet engines are to be given serious consideration as a viable propulsion device. An effort has been initiated at the NASA Langley Research Center to validate several turbulence closure models used for Reynolds-averaged simulations of scramjet isolator flows. The turbulence models considered were the Menter BSL, Menter SST, Wilcox 1998, Wilcox 2006, and the Gatski-Speziale explicit algebraic Reynolds stress models. The simulations were carried out using the VULCAN computational fluid dynamics package developed at the NASA Langley Research Center. A procedure to quantify the numerical errors was developed to account for discretization errors in the validation process. This procedure utilized the grid convergence index defined by Roache as a bounding estimate for the numerical error. The validation data was collected from a mechanically back-pressured constant area (1 2 inch) isolator model with an isolator entrance Mach number of 2.5. As expected, the model-form uncertainty was substantial for the shock-dominated, massively separated flowfield within the isolator as evidenced by a 6 duct height variation in shock train length depending on the turbulence model employed. Generally speaking, the turbulence models that did not include an explicit stress limiter more closely matched the measured surface pressures. This observation is somewhat surprising, given that stress-limiting models have generally been developed to better predict shock-separated flows. All of the models considered also failed to properly predict the shape and extent of the separated flow region caused by the shock boundary layer interactions. However, the best performing models were able to predict the isolator shock train length (an important metric for isolator operability margin) to within 1 isolator duct height.
The decoder error probability for Reed-Solomon codes (more generally, linear maximum distance separable codes) is examined. McEliece and Swanson offered an upper bound on P sub E (u), the decoder error probability given that u symbol errors occurs. This upper bound is slightly greater than Q, the probability that a completely random error pattern will cause decoder error. By using a combinatoric technique, the principle of inclusion and exclusion, an exact formula for P sub E (u) is derived. The P sub e (u) for the (255, 223) Reed-Solomon Code used by NASA, and for the (31,15) Reed-Solomon code (JTIDS code), are calculated using the exact formula, and the P sub E (u)'s are observed to approach the Q's of the codes rapidly as u gets larger. An upper bound for the expression is derived, and is shown to decrease nearly exponentially as u increases. This proves analytically that P sub E (u) indeed approaches Q as u becomes large, and some laws of large numbers come into play.
The decoder error probability for Reed-Solomon codes (more generally, linear maximum distance separable codes) is examined. McEliece and Swanson offered an upper bound on P sub E (u), the decoder error probability given that u symbol errors occur. This upper bound is slightly greater than Q, the probability that a completely random error pattern will cause decoder error. By using a combinatoric technique, the principle of inclusion and exclusion, an exact formula for P sub E (u) is derived. The P sub E (u) for the (255,223) Reed-Solomon Code used by NASA, and for the (31,15) Reed-Solomon code (JTIDS code), are calculated using the exact formula, and the P sub E (u)'s are observed to approach the Q's of the codes rapidly as u gets larger. An upper bound for the expression is derived, and is shown to decrease nearly exponentially as u increases. This proves analytically that P sub E (u) indeed approaches Q as u becomes large, and some laws of large numbers come into play.
Abstract The regional seismic travel time (RSTT) model and software were developed to improve travel-time prediction accuracy by accounting for three-dimensional crust and upper mantle structure. Travel-time uncertainty estimates are used in the process of associating seismic phases to events and to accurately calculate location uncertainty bounds (i.e. event location error ellipses). We improve on the current distance-dependent uncertainty parameterization for RSTT using a random effects model to estimate slowness (inverse velocity) uncertainty as a mean squared error for each model parameter. The random effects model separates the error between observed slowness and model predicted slowness into bias and random components. The path-specific travel-time uncertainty is calculated by integrating these mean squared errors along a seismic-phase ray path. We demonstrate that event location error ellipses computed for a 90% coverage ellipse metric (used by the Comprehensive Nuclear-Test-Ban Treaty Organization International Data Centre (IDC)), and using the path-specific travel-time uncertainty approach, are more representative (median 82.5% ellipse percentage) of true location error than error ellipses computed using distance-dependent travel-time uncertainties (median 70.1%). We also demonstrate measurable improvement in location uncertainties using the RSTT method compared to the current station correction approach used at the IDC (median 74.3% coverage ellipse).