Improved method for calculation of Fourier coefficients of a function given at a set of arbitrary points

A recently published method for the calculation of Fourier coefficients of a function given at a discrete set of arbitrary points is improved. The function is approximated by a sum of Legendre polynomials, rather than by Cheby?shev polynomials. The Fourier coefficients are then readily expressed in terms of spherical Bessel functions.     

Approximation errors in the computation of Cheby?shev expansion coefficients

Exact and approximate formulas for computing the Cheby?shev expansion coefficients of a rational function are analysed. A simple expression is then given for evaluating the approximation errors of a classic numerical quadrature formula.     

Generalised Cheby?shev lowpass filters

An expression is presented from which the pole positions of any shifted Cheby?shev lowpass filter can be evaluated. The standard Cheby?shev filters are shown to be special cases.     

Identification of double-valued nonlinearities from their harmonic response

A method is presented for determining an equation for a double-valued nonlinearity from a measurement of its response to a sinusoidal input. It is shown that the nonlinearity can be represented as a series in Cheby?shev polynomials of the first and second kind with argument dependent on the input magnitude. The coefficients of the Cheby?shev-polynomial terms in the series are the measured in-phase and quadrature output-harmonic magnitudes.     

Calculation of transient response from frequency-response data using Cheby?shev polynomials

The method presented in this letter converts frequency-response data into an analytical expression for the impulse (step response) as a linear combination of polynomials whose coefficients are given. The constants of this linear combination are, instead, the coefficients of the Cheby?shev approximation of the real (or imaginary) part of the frequency response.     

Transfer functions obtained by a transform derived from the Darlington transform

A method is given for determining a family of transfer functions by transforming the poles of a known transfer function. By means of a transformation derived from the Darlington one, the attenuation characteristics and the group transit-time characteristics of the system thus obtained are expressed by means of Cheby?shev polynomials.     

Transfer functions with quasiCheby?shev-type characteristics obtained by a Darlington-derived transformation

The letter establishes the classes of transfer functions corresponding to amplitude characteristics of the Cheby?shev type as well as to transit-time characteristics of the quasiCheby?shev type, starting from the transform given in a previous letter.     

Optimum distributed-parameter Butterworth-filter type as a limiting case of the optimum Cheby?shev approximation

It is shown that the optimum distributed-parameter Butter-worth approximation function for a cascade of m LC elements and nunit elements is a limiting case of the Cheby?shev type when the value of the passband ripple approaches zero.     

Family of equiripple lowpass digital filters

The letter deals with the special case of general-parameter filters that yield Cheby?shev characteristics. The solution gives a family of Cheby?shev responses with monotonic stopband and equiripple passband characteristics. These characteristics belong to combinations of the sin (?T/2) and tan (?T/2) formulations and it is shown that under the condition of maximum sharpness at cutoff the tan (?T/2) formulation is unique.     

Broadbanding the quarter-wave transformer using two parallel stubs

The quarter-wave transformer is broadbanded by including two parallel, open and short, eighth-wavelength stubs. A bandpass transfer function results and Butterworth or Cheby?shev designs can be achieved. Tables for practical use are presented.     

Generalisation of m.f.m.b.o. and transitional Butterworth-Cheby?shev filters

The maximally flat magnitude beyond the origin is generalised to give filters having a controlled passband characteristic. The Cheby?shev polynomials are then used as the variable for these generalised approximants to yield the transitional Butterworth-Cheby?shev filters. Optimisation of these filters is also discussed.     

Generalised patterns of nonuniformity for solvable algebraic and transcendental transmission lines

The aim of this letter is to show that, on transforming the independent variable x of the telegrapher's equation for nonuniform lines to another independent variable w=F(x) which is some function of the distance variable x, and then constraining the transformed equation, so obtained, to represent either Su's trigonometric or his hyperbolic line, a generalised pattern of nonuniformity, expressible either in terms of any arbitrary w=F(x), or in terms of another arbitrary f(x) [where f(x)=Z(x)/Z0, Z0 being an impedance constant, defines the distribution of nonuniformity for the series impedance Z(x) per unit length] may be obtained. In addition, the generalised expressions for the distributions of nonuniformity for a prototype of Cheby?shev line are derived, and it is shown that the Cheby?shev line is a subclass of Hellstrom's generalised proportional line.     

Cheby?shev time-domain deconvolution for transversal filter equalisers

Design equations are presented for computing the N tap weights of a transversal filter equaliser. The associated time-domain deconvolution is performed subject to the Cheby?shev (minimax) error criterion. Individual or pointwise deconvolution errors are controlled and minimised uniformly. This method offers an alternative to least square error or discrete Fourier transform designs wherein only the cumulative de-convolution error is minimised. Computationally, the Ascent algorithm is used which requires inversion of two (N + 1) × (N + 1) matrices followed by a finite number of elementary algebraic exchange operations. The equations needed to implement this algorithm are presented in a form suitable for digital computation.     

Coaxially fed hollow cylindrical monopole in a rectangular waveguide

A formally exact solution that can easily be put in a form suitable for accurate numerical calculation is presented for a hollow coaxially fed cylindrical monopole in a rectangular waveguide. The solution utilises a well known Fourier-transform technique in conjunction with a Cheby?shev polynomial representation for the unknown function. The result is an almost diagonalised infinite matrix?a form eminently suitable for the purpose of numerical inversion.     

Search procedure for efficient superdirective array and image enhancement functions

A technique is presented for generation of efficient superdirective array functions, using a composite polynomial based on the Cheby?shev polynomials. The generated radiation pattern has a narrow main beam with high radiation efficiency, indicative of an array function which is tolerant to elemental amplitude and phase errors.     

Cheby?shev linear phase approximation using iterative technique

A computer program for determining the exact equiripple approximation of a linear phase characteristic is described. The program is based on a recently introduced method for the approximation of a linear phase characteristic in which only linear equations are involved. A comparison with the Cheby?shev type of approximation of a constant-delay characteristic is also given.     

C.C.D. Cheby?shev filter for radar m.t.i. applications

A 3-pole Cheby?shev analogue filter is described in which charge-coupled devices are used as the delay elements. The use of this type of filter in radar moving-target-indicator applications is illustrated.     

Gaussian-filter approximation

A new approximation to the Gaussian-filter response is presented that uses a power-series-economisation technique with Cheby?shev polynomials of the first kind. For high-order filters, the new method appears to yield a closer approximation to the Gaussian response than is given by the Taylor or Laguerre approximations.     

Cheby?shev quarter-wave stepped balun transformer

The design of a balun transformer from coaxial to open-wire line is presented, in which conventional quarter-wave-matching theory is used to produce a compact device with optimal Cheby?shev performance over its passband. Included in the design are the evaluation and corrections for the junction capacitances caused by stepping the outer coaxial conductor.     

New approach to the approximation of RC-active-filter transmission functions with low Q factor

A new polynomial class is considered for the design of the transmission functions of lowpass RC active filters. These transmission functions have higher degree but lower Q factors than those computed with the classical Cheby?shev method.