Some Prominent Characteristics of the Modified Elliptic Low-Pass Function

A modified elliptic low-pass filter function is proposed. The modified elliptic function possesses progressively diminishing ripples in both passband and stopband, and improves the frequency and the time-domain characteristics as compared with the classical elliptic function. Also it is realizable in the doubly-terminated ladder structures for the order n even or odd, thus lending themselves amenable to high-quality active RC or switched capacitor filters through simulation techniques.

This article provides an overview of elliptic filter synthesis for RF and microwave frequencies. The characteristics of an elliptic filter function are discussed and compared to other classical filter function responses. An overview of the mathematics of the lowpass elliptic filter function is discussed. Methods for transforming the low‐pass elliptic function to highpass, bandpass, and band‐reject functions are provide. The article concludes with a discussion of the realizations for RF and microwave elliptic filters and discusses issues related to RF and microwave elliptic filter synthesis.

This article provides an overview of elliptic filter synthesis for RF and microwave frequencies. The characteristics of an elliptic filter function are discussed and compared to other classical filter function responses. An overview of the mathematics of the lowpass elliptic filter function is discussed. Methods for transforming the low‐pass elliptic function to highpass, bandpass, and band‐reject functions are provide. The article concludes with a discussion of the realizations for RF and microwave elliptic filters and discusses issues related to RF and microwave elliptic filter synthesis.

A projection‐recurrence method is applied to active and passive ladder networks with a tridiagonal interaction matrix to obtain a factorization of the response functions. The factors are partial chain parameters, determined by third order recurrences. For a ladder with N nodes, the number of arithmetic operations to calculate response functions ranges from 4N to 6N. A diagrammatic representation is developed for passive ladders which makes it possible to obtain the partial chain parameters by inspection, and a compact formula for the node‐voltage response is presented for active doubly‐terminated ladders.

This paper takes into consideration a number of different ladder networks showing links with the so-called Golden Ratio φ = 1.6180339⋯ , which is embedded into the Fibonacci sequence. Discreteness and periodicity in electronic ladder networks seem to appear as the main features able to include Fibonacci numbers. For instance, in the telecommunication environment, transmission lines can be studied with a given approximation by simplified discrete ladder networks made by a sequence of passive identical cells, such as R-R; R-C; R-L; C-R; L-R; L-C; C-L; directly and electrically coupled each other. In this context, the so called Golden Ratio frequently appears in particular conditions, without a specific reason, at least to our best knowledge, and its presence, so far, still does represent a sort of mystery. © 2014 IEEE.

Wilker- and Huygens-type inequalities for Jacobian elliptic functions and classical theta functions are established. For the limiting values of the modulus parameter of the elliptic functions obtained results simplify to known ones which have been established earlier for circular and hyperbolic functions. The main results in this paper are derived with the aid of two inequalities proven in Neuman [Inequalities for weighted sums of powers and their applications. Math Inequal Appl. 2012;15(4):995–1005].

Switched-capacitor immittance converters (SCIC's) which are the switched-capacitor (SC) counterparts of GIC's are proposed. In this paper the definition and analysis of the SCIC's are given. Their fundamental properties are derived from the definition. Three methods for LC ladder simulation using the concept of SCIC's are presented. In the proposed bottom-plate stray-insensitive SCIC filter structure, only one operational amplifier is needed per node (excluding the grounded node) in the passive ladder network. This saving in the number of active elements is an advantage of this method especially in case of bandpass filter simulation.

The problem of representing an integer as a sum of squares of integers is one of the oldest and most significant in mathematics. It goes back at least 2000 years to Diophantus, and continues more recently with the works of Fermat, Euler, Lagrange, Jacobi, Glaisher, Ramanujan, Hardy, Mordell, Andrews, and others. Jacobi's elliptic function approach dates from his epic Fundamenta Nova of 1829. Here, the author employs his combinatorial/elliptic function methods to derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's (1829) 4 and 8 squares identities to 4n2 or 4n(n+1) squares, respectively, without using cusp forms such as those of Glaisher or Ramanujan for 16 and 24 squares. These results depend upon new expansions for powers of various products of classical theta functions. This is the first time that infinite families of non-trivial exact explicit formulas for sums of squares have been found. The author derives his formulas by utilizing combinatorics to combine a variety of methods and observations from the theory of Jacobi elliptic functions, continued fractions, Hankel or Turanian determinants, Lie algebras, Schur functions, and multiple basic hypergeometric series related to the classical groups. His results (in Theorem 5.19) generalize to separate infinite families each of the 21 of Jacobi's explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions in sections 40-42 of the Fundamental Nova. The author also uses a special case of his methods to give a derivation proof of the two Kac and Wakimoto (1994) conjectured identities concerning representations of a positive integer by sums of 4n2 or 4n(n+1) triangular numbers, respectively. These conjectures arose in the study of Lie algebras and have also recently been proved by Zagier using modular forms. George Andrews says in a preface of this book, 'This impressive work will undoubtedly spur others both in elliptic functions and in modular forms to build on these wonderful discoveries.' Audience: This research monograph on sums of squares is distinguished by its diversity of methods and extensive bibliography. It contains both detailed proofs and numerous explicit examples of the theory. This readable work will appeal to both students and researchers in number theory, combinatorics, special functions, classical analysis, approximation theory, and mathematical physics.

Transcendental numbers form a fascinating subject: so little is known about the nature of analytic constants that more research is needed in this area. Even when one is interested only in numbers like $\pi$ and $e^\pi$ which are related with the classical exponential function, it turns out that elliptic functions are required (so far -- this should not last forever!) to prove transcendence results and get a better understanding of the situation. First we briefly recall some of the basic transcendence results related with the exponential function. Next, we survey the main properties of elliptic functions that are involved in transcendence theory. We survey transcendence theory of values of elliptic functions, linear independence and algebraic independence. This splitting is somewhat artificial but convenient. Moreover, we restrict ourselves to elliptic functions, even when many results are only special cases of statements valid for abelian functions. A number of related topics are not considered here (e.g. heights, $p$-adic theory, theta functions, diophantine geometry on elliptic curves...).

Presents closed-form solutions in explicit form for the direct derivation of discrete-time elliptic transfer functions having denormalized lowpass (LP), highpass (HP), bandpass (BP), or bandstop (BS) loss-frequency characteristics. The proposed method is based on the derivation of a suitable discrete-time z-domain normalized lowpass elliptic prototype reference transfer function and on the transformation of that transfer function into the desired discrete-time z-domain denormalized LP, HP, BP, or BS elliptic transfer function using a recently advanced z-to z frequency transformation technique. The required discrete-time z-domain normalized lowpass prototype reference transfer function is derived by establishing the fact that if it is related to a continuous-time s-domain normalized lowpass elliptic prototype reference transfer function through the conventional s-to-z bilinear frequency transformation. The parameters of the former as expressed in terms of Jacobian elliptic functions, are related to those of the latter through the ascending Landen transformation. >

Elliptic functions and Riemann surfaces played an important role in nineteenth-century mathematics. At the present time there is a great revival of interest in these topics not only for their own sake but also because of their applications to so many areas of mathematical research from group theory and number theory to topology and differential equations. In this book the authors give elementary accounts of many aspects of classical complex function theory including Möbius transformations, elliptic functions, Riemann surfaces, Fuchsian groups and modular functions. A distinctive feature of their presentation is the way in which they have incorporated into the text many interesting topics from other branches of mathematics. This book is based on lectures given to advanced undergraduates and is well-suited as a textbook for a second course in complex function theory. Professionals will also find it valuable as a straightforward introduction to a subject which is finding widespread application throughout mathematics.

The Fourier expansion of the hypermonogenic generalized trigonometric and elliptic functions

This paper presents a solution to the problem of accommodating the important new subject of switched capacitor networks in already crowded electrical engineering curricula. This is achieved by combining in a one-semester course three different, but related, subjects. This is made possible 1) by placing a strong emphasis on the underlying principles that are common to active, digital, and switched capacitor filters, and 2) by concentrating on the realization of these three types of filters by simulation of passive ladder networks.

By using the results on explicit formulas for element values in 1-variable ladder networks dependent on one or two auxiliary parameters, and by using appropriate frequency transformations, certain generalisations may be made to 2-variable networks. For the solution to the approximation problem, which results in an elliptic function or Cheby̅shev response, nonreciprocal 2-variable networks result, but for the matched inverse-Cheby̅shev-response case, a simple 2-variable ladder network is obtained. This case is treated in detail, and typical responses for networks containing lumped and commensurate distributed elements are presented, with a specific example on a waveguide bandstop filter.

This paper investigates the design of low-voltage low-power switched-capacitor (SC) filters for high-frequency applications by using the clock-booster approach. In particular, our proposed SC filter architecture uses single-ended double-sampling integrator cells based on low-voltage operational transconductance amplifiers which take advantage of dynamic biasing and the clock-booster technique to drive the switch transistors. To validate its high-frequency capability, two low-pass elliptic SC filters respectively with a corner frequency of 6 and 8-MHz, were designed in a 0.35-/spl mu/m CMOS process. Both are suitable for telecom applications and can operate with a power supply as low as 1.5 V, while dissipating 11 mW. Measurements showed that for an output amplitude of 1 V/sub pp/, their total harmonic distortions were maintained well below -40 dB in their bandwidths. Comparisons with other SC filter implementations in the literature, which highlight the quality of our implementation are also provided.