Exponentials and Logarithms Properties in an Extended Complex Number Field

It is well established the complex exponential and logarithm are multivalued functions, both failing to maintain most identities originally valid over the positive integers domain. Moreover the general case of complex logarithm, with a complex base, is hardly mentionned in mathematic litterature. We study the exponentiation and logarithm as binary operations where all operands are complex. In a redefined complex number system using an extension of the C field, hereafter named E, we prove both operations always produce single value results and maintain the validity of identities such as logu (w v) = logu (w) + logu (v) where u, v, w in E. There is a cost as some algebraic properties of the addition and subtraction will be diminished, though remaining valid to a certain extent. In order to handle formulas in a C and E dual number system, we introduce the notion of set precision and set truncation. We show complex numbers as defined in C are insufficiently precise to grasp all subtleties of some complex operations, as a result multivaluation, identity failures and, in specific cases, wrong results are obtained when computing exclusively in C. A geometric representation of the new complex number system is proposed, in which the complex plane appears as an orthogonal projection, and where the complex logarithm an exponentiation can be simply represented. Finally we attempt an algebraic formalization of E.

Exponentials and Logarithms Properties in an Extended Complex Number Field

It is well established the complex exponential and logarithm are multivalued functions, both failing to maintain most identities originally valid over the positive integers domain. Moreover the general case of complex logarithm, with a complex base, is hardly mentionned in mathematic litterature. We study the exponentiation and logarithm as binary operations where all operands are complex. In a redefined complex number system using an extension of the C field, hereafter named E, we proove both operations always produce single value results and maintain the validity of identities such as logu (w v) = logu (w) + logu (v) where u, v, w in E. There is a cost as some algebraic properties of the addition and subtraction will be diminished, though remaining valid to a certain extent. In order to handle formulas in a C and E dual number system, we introduce the notion of set precision and set truncation. We show complex numbers as defined in C are insufficiently precise to grasp all subtleties of some complex operations, as a result multivaluation, identity failures and, in specific cases, wrong results are obtained when computing exclusively in C. A geometric representation of the new complex number system is proposed, in which the complex plane appears as an orthogonal projection, and where the complex logarithm an exponentiation can be simply represented. Finally we attempt an algebraic formalization of E.

Geometrically Invariant NSCT–GPCET-Based Image Watermarking with Accurate Moment Calculation

Generic polar complex exponential transform (GPCET), as continuous orthogonal moment, has the advantages of computational simplicity, numerical stability, and resistance to geometric transforms, which make it suitable for watermarking. However, errors in kernel function discretization can degrade these advantages. To maximize the GPCET utilization in robust watermarking, this paper proposes a secondary grid-division (SGD)-based moment calculation method that divides each grid corresponding to one pixel into nonoverlapping subgrids and increases the number of sampling points. Using the accurate moment calculation method, a nonsubsampled contourlet transform (NSCT)–GPCET-based watermarking scheme with resistance to image processing and geometrical attacks is proposed. In this scheme, the accurate moment calculation can reduce the numerical error and geometrical error of the traditional methods, which is verified by an image reconstruction comparison. Additionally, NSCT and accurate GPCET are utilized to achieve watermark stability. Subsequent experiments test the proposed watermarking scheme for its invisibility and robustness, and verify that the robustness of the proposed scheme outperforms that of other schemes when its level of invisibility is significantly higher.

High-Accuracy Parameter Estimation for Damped Signal via Three Weighted DFT Samples

Recently, accurate parameter estimation of the damped complex exponential plays an increasingly important role in the field of precise measurement. However, the estimation variance of interpolation-based algorithms for the parameter estimation cannot be asymptotic to the Crámer-Rao lower bound (CRLB). This paper originally proposes a generalized, fast, and the accurate two-iteration estimator (TIE) based on the discrete Fourier transform (DFT). It can be operated by an arbitrary window (symmetric or asymmetric window). Theoretical estimation variances of the frequency and the damping factor for arbitrary windows are derived, respectively. Furthermore, extensive computer simulations are performed to compare the performance of the TIE with other state-of-the-art algorithms in the literature. The results support the theoretical findings and verify that high-accuracy parameter estimation can be ensured by the proposed algorithm. More importantly, the estimation variances returned by the TIE with the rectangle window exactly track the CRLB for a damped single tone.

Two Points Taylor’s Type Representations for Analytic Complex Functions with Integral Remainders

In this paper we establish some two point weighted Taylor’s expansions for analytic functions f : D ⊆ ℂ→ ℂ defined on a convex domain D. Some error bounds for these expansions are also provided. Examples for the complex logarithm and the complex exponential are also given.

A Polar Complex Exponential Transform-Based Zero-Watermarking for Multiple Medical Images with High Discrimination

Zero-watermarking is one of the solutions for image copyright protection without tampering with images, and thus it is suitable for medical images, which commonly do not allow any distortion. Moment-based zero-watermarking is robust against both image processing and geometric attacks, but the discrimination of watermarks is often ignored by researchers, resulting in the high possibility that host images and fake host images cannot be distinguished by verifier. To this end, this paper proposes a PCET- (polar complex exponential transform-) based zero-watermarking scheme based on the stability of the relationships between moment magnitudes of the same order and stability of the relationships between moment magnitudes of the same repetition, which can handle multiple medical images simultaneously. The scheme first calculates the PCET moment magnitudes for each image in an image group. Then, the magnitudes of the same order and the magnitudes of the same repetition are compared to obtain the content-related features. All the image features are added together to obtain the features for the image group. Finally, the scheme extracts a robust feature vector with the chaos system and takes the bitwise XOR of the robust feature and a scrambled watermark to generate a zero-watermark. The scheme produces robust features with both resistance to various attacks and low similarity among different images. In addition, the one-to-many mapping between magnitudes and robust feature bits reduces the number of moments involved, which not only reduces the computation time but also further improves the robustness. The experimental results show that the proposed scheme meets the performance requirements of zero-watermarking on the robustness, discrimination, and capacity, and it outperforms the state-of-the-art methods in terms of robustness, discrimination, and computational time under the same payloads.

Quality Classification of Lithium Battery in Microgrid Networks Based on Smooth Localized Complex Exponential Model

Accurate prediction of battery quality using early-cycle data is critical for battery, especially lithium battery in microgrid networks. To effectively predict the lifetime of lithium-ion batteries, a time series classification method is proposed that classifies batteries into high-lifetime and low-lifetime groups using features extracted from early-cycle charge-discharge data. The proposed method is based on a smooth localized complex exponential model that can extract battery features from time-frequency maps and self-adaptively select the time-frequency resolution to maximize the discrepancy of data from the two groups. A smooth localized complex exponential periodogram is then calculated to obtain the time-frequency decomposition of the whole time series data for further classification. The experimental results show that, by using battery features extracted from the first 128 charge-discharge processes, the proposed method can accurately classify batteries into high-lifetime and low-lifetime groups, with classification accuracy and specificity as high as 95.12% and 92.5%, respectively.