Search

In this article, two new techniques of approximation of the reliability of a two-terminal network are developed based on the constructive theory of functions and related methods. Two methods of generating an approximation cubic spline are used: Lagrange-type interpolation procedures and Bernstein approximation operator. A possibility of minimizing the total error of approximation, based on keeping some properties invariant, is described in case of a large class of pairs of dual two-terminal networks. Simulations are included, showing that the error of approximation is negligible in case of some special initial data. [ABSTRACT FROM AUTHOR]

Efficient hardware implementations routinely approximate mathematical functions with look-up tables, while keeping the error of the approximation under control. For a certain class of commonly occurring 1D functions, namely monotonically increasing or decreasing functions, we found that it is possible to approximate such functions by repeated application of a very low resolution 1D look-up table. There are many advantages to cascading multiple identical LUTs, including the promise of a very simple hardware design and the use of standard linear interpolation. Further, the complexity associated with unequal bin sizes can be avoided. We show that for realistic applications, including gamma correction, high dynamic range encoding and decoding curves, as well as tone mapping and inverse tone mapping applications, multiple cascaded look-up tables can reduce the approximation error by more than 50% compared to a single look-up table with the same total memory footprint. [ABSTRACT FROM AUTHOR]

Downsampling and the computation of the bandlimited interpolation of discrete-time signals are two important concepts in signal processing. In this paper we analyze the downsampling operation regarding its impact on the existence and computability of the bounded bandlimited interpolation. We assume that the discrete-time signal is obtained by downsampling the samples of a bounded bandlimited signal that vanishes at infinity, and we study two problems. First, we investigate the existence of the bounded bandlimited interpolation for such discrete-time signals from a signal theoretic perspective and show that there exist signals for which the bounded bandlimited interpolation does not exist. Second, we analyze the algorithmic generation of the bounded bandlimited interpolation, using the concept of Turing computability. Turing computability models what is theoretically implementable on a digital computer. Interestingly, it turns out that even if the bounded bandlimited interpolation exists analytically, it is not always computable, which implies that there exists no algorithm on a digital computer that can always compute it. Computability is important in order that the approximation error be controlled. If a signal is not computable, we cannot ascertain whether the computed signal is sufficiently close to the true signal, i.e., we cannot verify every approximation accuracy. [ABSTRACT FROM AUTHOR]

Time delay or the time-of-flight is a most frequently used parameter in many ultrasonic applications. Delay estimation is based on the sampled signal, so the resolution is limited by the sampling grid. Higher accuracy is available if the signal-to-noise ratio is high, then the subsample estimate is desired. Techniques used for subsample interpolation suffer from bias error. Time-of-flight estimation that is free from bias errors is required. The proposed subsample estimation works in the frequency domain; it is based on the cross-correlation peak temporal position. The phase of the cross-correlation frequency response becomes linear thanks to multiplication by the complex conjugate, and its inclination angle is proportional to the delay. Then, subsample interpolation becomes free from the bias error. Twelve algorithmic implementations of this technique have been proposed in this paper. All algorithmic implementations have been analyzed for bias and random errors using simulation and MATLAB codes are given as supplementary material. Comparison with best-performing interpolation techniques (spline approximation, cosine interpolation, carrier phase) is given for both bias and random errors. It was demonstrated that frequency domain interpolation has no bias errors, and noise performance is better or comparable to other subsample estimation techniques. Weighted regression using L2 norm minimization has the best performance: total errors (bias and random) are within 3% of theoretical lower bound. [ABSTRACT FROM AUTHOR]

The modified improved interpolating moving least squares (MIIMLS) method for meshless approaches has been developed. In a meshless method, the approximation accuracy of the derivative is directly proportional to the simulation accuracy. For improving the approximation accuracy of a derivative, we propose MIIMLS using Chebyshev nodes. In the approximation of a function, the results of MIIMLS reach the precision limit. In the approximation of a derivative, the MIIMLS method has high approximation accuracy between the nodes as compared to the conventional methods. [ABSTRACT FROM AUTHOR]

Many applications of finite impulse response (FIR) digital filters impose strict format constraints on the filter coefficients. Such requirements increase the complexity of determining optimal designs for the problem at hand. We introduce a fast and efficient method, based on the computation of good nodes for polynomial interpolation and Euclidean lattice basis reduction. Experiments show that it returns quasi-optimal finite wordlength FIR filters; compared to previous approaches it also scales remarkably well (length 125 filters are treated in $<$ 9 s). It also proves useful for accelerating the determination of optimal finite wordlength FIR filters. [ABSTRACT FROM AUTHOR]

Orthogonal projection offers the optimal solution for many sampling-reconstruction problems in terms of the least square error. In the standard interpolation setting where the sampling is assumed to be ideal, however, the projection is impossible unless the interpolation kernel is related to the sinc function and the input is bandlimited. In this paper, we propose a notion of shifting kernel toward the orthogonal projection. For a given interpolation kernel, we formulate optimization problems whose solutions lead to shifted interpolations that, while still being interpolatory, are closest to the orthogonal projection in the sense of the minimax regret. The quality of interpolation is evaluated in terms of the average approximation error over input shift. For the standard linear interpolation, we obtain several values of optimal shift, dependent on a priori information on input signals. For evaluation, we apply the new shifted linear interpolations to a Gaussian signal, an ECG signal, a speech signal, a two-dimensional signal, and three natural images. Significant improvements are observed over the standard and the 0.21-shifted linear interpolation proposed early. [ABSTRACT FROM PUBLISHER]

The linear canonical transform (LCT), which generalizes many classical transforms, has been shown to be a powerful tool for signal processing and optics. Sampling theory of the LCT for bandlimited signals has blossomed in recent years. However, in practice signals are never perfectly bandlimited, and in many cases measurement devices are nonideal. The objective of this paper is to develop a sampling theorem for the LCT from general measurements, which can provide a suitable and realistic model of sampling and approximation for real-world applications. We first describe a general class of approximation spaces for the LCT and provide a full characterization of their basis functions. Then, we propose a generalized sampling theorem for arbitrary measurement and approximation spaces associated with the LCT. Several properties of the proposed sampling theorem are also discussed. Furthermore, the approximation error is estimated. Finally, numerical results and several applications of the derived results are presented. [ABSTRACT FROM PUBLISHER]

State-of-the-art piecewise logarithmic converters employ nonuniform segments to reduce the variance of relative errors, particularly for small inputs. However, it is difficult to have an optimized algorithm to accurately choose the nonuniform segments for a required approximation error. In this brief, we present a relative error equal distribution (REED) algorithm that performs the nonuniform piecewise linear interpolation of logarithm. It is able to precisely set the nonuniform piecewise points so that the relative error of each segment is within an upper bound, and finally produce a flat relative error distribution. It can be applied to any number of segments, even if the number is not the power of 2. Experimental results show that our REED algorithm achieves over 70% reduction of average relative errors for the benchmark graphics application, compared to both the state-of-the-art uniform and nonuniform methods. Synthesis results show that our hardware resources are comparable to that of the state-of-the-art nonuniform methods. [ABSTRACT FROM AUTHOR]

The hardware computation of the logarithm function is required in several applications, ranging from signal and image processing to telecommunication systems. This brief shows that most of previous proposed logarithmic converters, based on piecewise linear approximations, suffer from large errors when dealing with fixed-point input values with many fractional bits, a situation often encountered in practical applications. Thus, this brief proposes a novel logarithmic converter, using nonuniform segmentation and piecewise linear approximation. A rigorous technique that allows computing the optimal segmentation and the coefficients values for a prescribed precision is described in this brief. For fixed-point input values, the proposed approach allows obtaining a sensibly lower error, for the same number of nonuniform segments, compared with previously published results. Implementation details and synthesis results in a 65-nm CMOS technology are also presented. [ABSTRACT FROM AUTHOR]

We present a method for real-time computation of clothoid coordinates that guarantees bounded approximation error over a wide range of clothoid parameters provided that the clothoid’s orientation change and length are bounded. It is shown that coordinates of clothoid with any parameters can be computed from those of a single clothoid (with fixed parameters), using appropriate geometrical transformations. A comprehensive analysis is given on how to determine a required set of clothoids and, based on this, how to sample a clothoid in a lookup table in order to achieve required approximation precision. The algorithm is computationally very efficient and therefore suitable for real-time path planning, as well as for other applications that benefit from fast clothoid computation. [ABSTRACT FROM PUBLISHER]

We present a new class of continuously defined parametric snakes using a special kind of exponential splines as basis functions. We have enforced our bases to have the shortest possible support subject to some design constraints to maximize efficiency. While the resulting snakes are versatile enough to provide a good approximation of any closed curve in the plane, their most important feature is the fact that they admit ellipses within their span. Thus, they can perfectly generate circular and elliptical shapes. These features are appropriate to delineate cross sections of cylindrical-like conduits and to outline bloblike objects. We address the implementation details and illustrate the capabilities of our snake with synthetic and real data. [ABSTRACT FROM AUTHOR]

The coefficient decimation technique for reconfigurable FIR filters was recently proposed as a filter structure with low computational complexity. In this brief, we propose to design these filters using linear programming taking all configuration modes into account, instead of only considering the initial reconfiguration mode as in previous works. Minimax solutions with significantly lower approximation errors compared to the straightforward design method in earlier works are obtained. In addition, some new insights that are useful when designing coefficient decimation filters are provided. [ABSTRACT FROM PUBLISHER]

A novel numerical method, which uses linear edge elements in eddy current region and nodal finite elements in non eddy current regions, is presented for solving 3D eddy current problems. The mathematical models are divided into two parts according to their properties. The theory of linear edge elements is developed. The test results show the validity of the algorithm. [ABSTRACT FROM PUBLISHER]

A novel technique for designing piecewise-polynomial interpolators for hardware implementation of elementary functions is investigated in this paper. In the proposed approach, the interval where the function is approximated is subdivided in equal length segments and two adjacent segments are grouped in a segment pair. Suitable constraints are then imposed between the coefficients of the two interpolating polynomials in each segment pair. This allows reducing the total number of stored coefficients. It is found that the increase in the approximation error due to constraints between polynomial coefficients can easily be overcome by increasing the fractional bits of the coefficients. Overall, compared with standard unconstrained piecewise-polynomial approximation having the same accuracy, the proposed method results in a considerable advantage in terms of the size of the lookup table needed to store polynomial coefficients. The calculus of the coefficients of constrained polynomials and the optimization of coefficients bit width is also investigated in this paper. Results for several elementary functions and target precision ranging from 12 to 42 bits are presented. The paper also presents VLSI implementation results, targeting a 90 nm CMOS technology, and using both direct and Horner architectures for constrained degree-1, degree-2, and degree-3 approximations. [ABSTRACT FROM AUTHOR]