Inverse Chebyshev Research Articles

EXSHOF-II: Active filter design, from approximation function to graphic display of the circuit

This paper presents an expert-system-based package for active filter synthesis, called EXSHOF-II. It starts with approximation function and ends with graphic display of the complete filter circuit. The package is an improved and enhanced version of its mother EXSHOF ( EX pert system based S ynthesis of H igh O rder F ilter). EXSHOF-II gives quantitative and qualitative help at various stages of filter design. Besides the Butterworth, Chebyshev and Elliptic functions which were considered in EXSHOF, the inverse Chebyshev function is also included in EXSHOF-II. For choosing particular approximation EXSHOF-II gives the order of the transfer function. The group delay characteristics of all four approximation functions are plotted on the screen by EXSHOF-II. This aids the user to make a proper choice of an approximation function. Like its mother, EXSHOF-II also considers four high-order structures. The filtering requirement in EXSHOF-II can be directly input on the attenuation characteristics, making the system more user-friendly. The circuit implementation has been done using nine different biquadratic active RC building blocks. The delay requirement can be easily satisfied by cascaded all-pass second-order networks.

A modified low-pass filter with progressively diminishing ripples

A modified low-pass filter functions are proposed. A modified inverse Chebyshev function possesses progressively diminishing ripples in the stopband whereas the modified Chebyshev function exhibits ripples diminishing towardω = 0 in the passband. Both are realizable in the doubly-terminated ladder structures for the ordern even or odd, thus lending themselves amenable to high-quality activeRC or switched capacitor filters through the simulation techniques.

Iterative algorithms for computing the singular subspace of a matrix associated with its smallest singular values

Many problems in science require the computation of only one singular vector or, more generally, a singular subspace of a matrix. Instead of computing the complete singular-value decomposition, iterative methods can be used to improve the computational speed, in particular when a priori information about the solution is available. This paper deals with the computation of a basis of a singular subspace of a matrix associated with its smallest singular values. Three iterative methods—inverse, Chebyshev, and inverse Chebyshev iteration—are compared by analyzing their convergence properties. Based on the convergence rate and the operation counts per iteration step, it is shown in which problems a particular iterative algorithm is most efficient. If the gap between the singular values associated with the desired and undesired singular subspace is large, inverse iteration is clearly the best choice. If not, convergence can be accelerated by applying inverse Chebyshev iteration provided sufficiently tight bounds delineating the undesired singular-value spectrum are known. The smaller the gap, the larger the gain in speed. Unless the matrix is structured or sparse, this method is shown to be always more efficient than ordinary Chebyshev iteration.

Some prominent aspects of the inverse Chebyshev functions

It is shown that, given the magnitude specifications in the passband and stopband, the inverse Chebyshev functions require the identical order n as the Chebyshev counterparts. The phase and time-delay characteristics of the inverse Chebyshev functions as compared to Chebyshev functions are investigated. Unlike the Chebyshev or elliptic filters, the inverse Chebyshev functions possess the maximally flat magnitude at DC, and it decreases monotonically throughout the passband. This fact implies that the phase and time-delay characteristics of the inverse Chebyshev functions must be preferable to those of the Chebyshev functions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Improved version of the inverse Chebyshev filter

The inverse Chebyshev function is formed by a transformation of a rational equiripple function defined by two polynomial Chebyshev functions of the same order. Since the transformation moves the finite complex zeros of the original rational function to the imaginary axis, its main imperfections, such as low selectivity and relatively large Q factors of the zeros, are eliminated. Extremely low pole-Q factors of the initial function are retained, providing a low sensitivity and a good group delay.

Design of MURROMAF filters with equiripple stopband attenuation

A class of filters known as multiple real-pole equal-ripple rational (MRPER) filters has already been proposed. It provides the advantages of third-order block realization or the Cauer type of filter. A design technique is proposed here for a multiple real-root maximally flat (MURROMAF) filter with equiripple stopband. Hence, the filter designer is now provided with a third-order block realization of the inverse Chebyshev type of filter.

A new algorithm for second-order Inverse-Chebyshev response

It is the usual practice to obtain an Inverse-Chebyshev response by transforming Chebyshev response of a filter. The procedure required is to determine the Chebyshev poles and then the transformation to the inverse Chebyshev poles. This paper considers an alternate approach of directly obtaining Inverse-Chebyshev response using singularity locations, this seems to be a straight forward approach and gives more insight into the transfer function of the filter.

Programmable switched-capacitor low-pass ladder filters

Switched-capacitor filters with digitally programmable capacitor arrays (DPCAs) offer a solution to the diverse needs to the market for audio-frequency high-order low-pass filters. Cascading switched-capacitor biquads which contain binary or geometrically weighted DPCAs are not a viable way to produce precision high-order responses, because of the errors introduced by capacitance quantization. To overcome this problem, a ladder-based structure is proposed. It uses DPCAs tailored to provide exactly the capacitances required to realize a restricted but useful set of responses. A simple procedure for synthesizing such filters is presented and illustrated with a detailed example. A CMOS prototype chip which provides eight seventh-order responses was fabricated. A 3-bit word programs its response: elliptic (with four different selectivities), Chebyshev, inverse Chebyshev, Butterworth, or modified Bessel (modified to have stopband notches). Experimental results are included which show that with this design philosophy, performance is not sacrificed for programmability.

Explicit formulas for lattice wave digital filters

Explicit formulas are derived for designing lattice wave digital filters of the most common filter types, for Butterworth, Chebyshev, inverse Chebyshev, and Cauer parameter (elliptic) filter responses. Using these formulas a direct top down design method is obtained and most of the practical design problems can be solved without special knowledge of filter synthesis methods. Since the formulas are simple enough also in the case of elliptic filters, the design process is sufficiently simple to serve as basis in the first part (filter design from specs to algorithm) of silicon compilers or applied to high level programmable digital signal processors.

Design of Digital Filter Suitable for Microprocessor Implementation

A technique of designing a digital filter suitable for microprocessor implementation is presented in this paper. A transfer function is approximated from a given specification, whose coefficients are expressed as simple combinations of two's power. Then, the signal can be processed with high speed by using only addition and shift operators instead of multipliers.Two sets of pole and zero locations are first obtained by means of ordinary design methods, which correspond to upper and lower bounds of a given specification. Then, the desired transfer function is derived, whose poles and zeros locate near the center of these two sets obtained earlier. Some spectral transformations are introduced to shift the pole and zero locations to the suitable places.The design examples for Butterworth, Chebyshev, inverse Chebyshev, and elliptic approximations are included, and their processing times are compared with those using multipliers by using 8-bit and 16-bit microprocessors.

An active circuit for elliptic-type filters

An active filter circuit is presented that realizes second-order stages of band-reject fiters of all types, and low-pass, high-pass, and bandpass filters of elliptic and inverse Chebyshev types. The circuit is comparable in complexity (having either one more or one less resistor depending on the mode of operation of one op amp) and in sensitivities to the popular biquad circuits that realize all-pole low-pass prototype filters.

Transitional rational filters

Transitional rational XY filters (TRXY) with rational transfer functions are considered having characteristics that vary smoothly from those of filter X to those of filter Y as certain parameters are varied continuously. The X and Y filters are chosen as various combinations of the ultraspherical rational filter, the generalized Bessel rational filter and the inverse Chebyshev filter.

Explicit design formulae for digital tan filters with low-pass, high-pass, band-pass, and band- stop characteristics

Explicit expressions of transfer functions for digital tan filters with low-pass, high-pass, band-pass, and band-stop characteristics approximating given design specifications are advanced. The higher order transfer function satisfying the design specification is expressed as the product of first-order and second-order filter sections of identical forms but with different coefficient values. All coefficients of these low-order filter sections are expressed in explicit forms related simply to the specification values. Thus, the result developed can be easily applied for the cascade or time-sharing realization. The low-order filter section developed here can be implemented with the fewest multipliers. The design formulae for Butterworth, Chebyshev, inverse Chebyshev and elliptic approximations are included. The transfer functions for all these filters are expressed in the same form of low-order filter section with different coefficient values.

On maximally flat low-pass filters with ripple in the stopband

Maximally flat (MF) low-pass filters of order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> with exactly <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> , or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n - 1</tex> finite zeros are investigated and are compared with the same order Butterworth filter, i.e., the MF filter with no finite zeros. It is seen that the MF filter with finite imaginary zeros exhibits sharper cutoff at the edge of the passband and ripple in the stopband. It is shown that the MF filter that maximizes the magnitude of the slope at cutoff is characterized by equal ripple in the stopband. This filter is shown to be the Inverse Chebyshev. Expressions for cutoff slope and for transition bandwidth are derived and are compared with those of the Butterworth filter. The step responses of the Inverse Chebyshev and of the Butterworth are also compared.

Attenuation Characteristics and Realization of Odd-Order Generalized Inverse Chebyshev Filters

Tables of transition bandwidth ratio, zero transmission and minimum attenuation frequencies in the stopband, and element values for two kinds of realizations of odd-order generalized inverse Chebyshev filters with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> up to 11 and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> up to 3, all using minimum stopband loss as a parameter, are given.

Maximally Flat Amplitude Low-Pass Filter with an Arbitrary Number of Pairs of Real Frequency Transmission Zeros

The synthesis of a general class of filter with maximally flat passband and Chebyshev stopband attenuation characteristic is presented. The filter has an nth order voltage transfer function with m pairs of transmission zeros in the stopband, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n \geq 2 m + 1</tex> . Butterworth and inverse Chebyshev filters are special cases of this general class with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m = 0</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n = 2m + 1</tex> , respectively.