Design of selective lowpass sampled-data and digital filters exhibiting equiripple amplitude and phase error characteristics

A method is presented for the design of reciprocal reactant sampled-data and digital filters exhibiting equiripple passband and stopband amplitude characteristics and approximating a linear phase characteristic such that the phase error response is equiripple too. the passband amplitude characteristic approximates unity transmission and the stopband amplitude characteristic approximates zero. Odd- and even-degree cases are considered separately. the stopband amplitude response is controlled by an arbitrary number of finite transmission zeros together with two (one) zeros at infinity for even (odd) transfer functions. All the available degrees of freedom are used for controlling the filter performances. Thus the resulting solutions are selective and have good amplitude and phase characteristics. A comparison between our method and a recently published method shows that this approach has higher stopband loss and smaller phase error ripple with wider bandwidth.     

Odd-degree selective bandpass digital filters interpolating linear phase and constant group delay

A novel simultaneous amplitude and phase approximation method is presented for the construction of non-minimum phase odd-degree transfer functions of distributed (or digital) bandpass filters interpolating linear phase and constant group delay at a set of frequencies in the passband. the passband amplitude characteristic is equiripple interpolating exact transmission at the same set of frequencies. In addition, the stopband amplitude characteristic is maximally flat and the filter selectivity is achieved by introducing a number of transmission zeros distributed arbitrarily between zero and infinite frequencies. the resulting transfer functions are always stable for all the cases considered. Typical characteristics for filters of ninth to 15th degree are graphically presented. the transfer functions presented are also capable of realization as distributed microwave, switched-capacitor, digital or wave-digital filters.     

Robust simultaneous amplitude and phase approximations oriented to bandpass wave digital lattice filters

A straightforward design method is delivered for bandpass wave digital lattice filters satisfying arbitrary amplitude and linear phase specifications. The optimality and efficiency of this method are ensured by two sources. On the one hand, the approximation is carried out directly without the need to apply frequency transformation techniques. On the other, the amplitude and phase specifications are approximated simultaneously. The approximation process is relying on the pre-construction of one of the two branch all-pass functions to exhibit exact linear phase. The other all-pass function is determined such that it controls the amplitude and/or phase responses in the passband and the two stopbands. Accordingly, the approximation problem is reduced to constructing a strictly Hurwitz polynomial specified by its phase within the Nyquist range. The approximation problem is solved by applying interpolation techniques combined with the iterative Remez-exchange algorithm. The realization of the resulting filter is carried out according to explicit formulae. The delivered method is applied through two examples illustrating its efficiency and reliability. © 1998 John Wiley & Sons, Ltd.     

Simultaneous amplitude and phase approximation for fir filters

An FIR filter design method with simultaneous amplitude and phase approximation is presented. Two components are proposed in the design method: simultaneous amplitude and phase interpolation in the passband and amplitude interpolation in the stopband. These interpolations are implemented by the Remez exchange algorithm. The polynomials for the interpolations are generated by recurrence formulae and the recurrence coefficients are calculated by a recursive method. the properties of the filters designed with the simultaneous approximation are investigated and compared with those of exact linear and minimum phase filters.     

Selective linear phase filters possessing a pair of J-axis transmission zeros

A design technique is developed for prototype selective linear phase filters which have an equiripple passband amplitude response and a pair of j-axis transmission zeros in the stopband while maintaining a linear phase characteristic in the passband. Since the basic technique requires numerical optimization techniques, the element values for several cases have been tabulated together with detailed response characteristics. One important result is that for certain characteristics, one internal cross-coupling element may be designed to be zero. As a result, in the sixth degree case with only one extra cross-coupling element between elements 1 and 6 a significant improvement in both the amplitude and group delay performance can be achieved as compared to the Chebyshev case. Owing to its importance, this case is investigated in detail and a 2.6 GHz post decoupled combline filter has been constructed based upon this prototype.     

Selective,constant delay wave digital filters

A wave digital filter design method is discussed which produces good selectivity and flat delay characteristics. The filter is designed in the microwave domain with monotonic stopband response using Carlin and Wu's technique for minimum-phase linear phase commensurate line transducers, then realized in the digital domain with Fettweis relations. A comparison is made of frequency response sensitivity to multiplier coefficient truncation between this wave filter and some FIR's, all with approximately the same number of multipliers. The cascade realizations of the FIR's show considerable passband amplitude response deterioration due to coefficient truncation, whereas the direct realizations for the FIR's show significant deterioration of stopband amplitude response owing to coefficient truncation. Coefficient truncation for the wave digital filter caused relatively small deterioration of both passband and stopband amplitude responses as well as small distortion of the flat delay characteristic in the passband.     

Design of digital filters with maximally flat passband and equiripple stopband magnitude

This paper introduces a general class of digital filters with maximally flat passband magnitude, equiripple stopband magnitude, and different order numerator and denominator. the classical Chebyschev type II (inverse Chebyschev) filters having equal numerator and denominator orders are considered as special members of this filter class. the filter types to be considered are lowpass, highpass and bandpass. the proposed filter class consists both of filters having all the zeros on the unit circle and of filters having some zeros off the unit circle to contribute to the passband shaping. An efficient iterative algorithm for the design of filters of the given type is presented. It is based on shaping the passband and stopband responses alternately until the difference between successive solutions is within given tolerance limits. Several comparisons show that in wideband applications filters with numerator order higher than denominator order present considerable advantages over equivalent Chebyschev type II designs, in terms of multiplication rate and/or frequency selectivity. In narrowband applications, in turn, filters with higher denominator order are shown to be the most effective.     

Selective digital filters with quadratic phase

A method for design of a new class of digital infinite impulse response filters realized as parallel connection of two all‐pass filters is presented in this paper. A new approach to approximation of quadratic phase of all‐pass filter at all frequencies is given. Chosen parallel structure offers opportunity for realization of filters with arbitrary shape phase. The presented algorithm is based on all‐pass filter phase approximation. Phases of both all‐pass filters approximate ideal quadratic phase in minimax sense at all frequencies. Such filters can be applied for chirp signal compression or expansion. Magnitude characteristic of described filters is very selective and elliptic‐like. Obtained filters are compared with elliptic filter and group delay corrector in cascade. For the same specifications, much better results are achieved by the proposed filters. Parallel connection of all‐pass filters introduces lower signal delay, and for a given maximal phase, approximation error demands less complex network. Examples to illustrate the proposed method are given. Copyright © 2016 John Wiley & Sons, Ltd.     

On optimal filters with maximum number of constraints on amplitude and phase characteristics

For optimal filters specified amplitude and phase (or group delay) characteristics, it is required that all the free parameters of the transfer function be used for the approximation. To achieve this requirement, the number of constraints on the amlitude and phase characteristics is defined first on general interpolation bases. Constant or arbitrarily prescribed lowpass or bandpass group delay and amplitude characteristics are approximated in the maximally flat, ripple or mixed sense for lumped filters, whereas highpass and band rejection characteristics are also considered for distributed or sampled data filters. The relationship between the number of free parameters and the number of amplitude and phase constraints for a given degree is derived for non-reciprocal as well as for reciprocal lossy and reciprocal reactant cases. Conditions are derived for the distribution of free parameters between the passband and stopband amplitude and phase characteristics for transfer functions with Hurwitzian denominator and also for monotonic amplitude characteristics in the transition band. Some published separate and simultaneous approximation methods are evaluated and compared on the above bases. It is pointed out that some of these methods do not satisfy the above requirements, although the optimal solution would exhibit higher performances.     

Microwave filter design from a systems perspective

The intention of this article is merely to provide the reader with some basic concepts that will enable them to better make intelligent decisions when specifying filters. We restrict the discussion to the most common type of filter, the bandpass filter. Bandpass filters are intended to minimize the signal attenuation in one band of frequencies (the passband), while achieving a specified attenuation in another band (the stopband).     

Inverse general Chebyshev bandpass filters

In this paper, the synthesis of inverse general Chebyshev bandpass filters is discussed. Unlike general Chebyshev filters that flexibly place transmission zeros to mainly control the performance of the filters in the stopband and keep constant ripple in the passband, inverse general Chebyshev filters are featured by flexible placement of reflection zeros in the passband and keep constant ripple in the stopband. Two approaches to directly derive filtering polynomials of inverse general Chebyshev bandpass filters in the bandpass domain are discussed in detail. By applying the zero‐extraction method, these filtering polynomials could be realized through appropriate networks. Finally, two examples are presented for demonstration.     

On the simultaneous amplitude and phase approximations of wave digital lattice filters

The paper presents new solution for the problem of simultaneously approximating the amplitude and phase functions of wave digital lattice filters. The approximation is relying on translating the amplitude and phase specifications into corresponding specifications for the difference and sum phase functions of the two branch polynomials. As a consequence, the phase specifications for each of the two branch polynomials is determined. Accordingly, these two polynomials are generated such that the amplitude and phase functions are approximated alternatively. This means that while one of these functions is approximated, the other is fixed. By iterating this alternative process, the two functions converge to their optimal response. Copyright © 2003 John Wiley & Sons, Ltd.     

A SIMPLE DESIGN OF LINEAR-PHASE,LOW-PASS,RECURSIVE DIGITAL FILTERS

A simple technique for designing low-pass recursive digital filters with nearly linear phase characteristic is presented. The linearity of phase is obtained by placing the poles in the neighbourhood of the passband at equal intervals on a circle of radius r, inside the unit circle on the z-plane. The zeros are placed in the stopband on another circle of radius 1/r. The phase characteristics of these filters are shown to be linear with a maximum periodic error of r^m radian, where m is the sum of the number of poles and zeros of the filter. This error can be made small by suitably choosing m and r. Stop-band transmission characteristics of these filters are also examined and an upper bound obtained for the asymptotic db-loss.     

On the problem of filters with arbitrary amplitude and phase constraints

A simplified method—based on the concept of reflection filters—is presented for designing linear phase filters with arbitrary amplitude to phase constraints. This method starts with transfer functions with 2: 1 amplitude to phase constraints, and through varying the recurrence relation of the polynomials involved, transfer functions with more selective amplitude are obtained.     

Effect of operational amplifier imperfections on single amplifier filters

A general method is given for the analysis of single amplifier RC-active filters using nonideal operational amplifiers (OA). It is shown that the output resistance of the OA has a significant effect in the passband and stopband in addition to the known deviations due to the finite gain—bandwidth product. The nonideal transfer functions in the factorized form, giving clearer concept of the effects of the nonideal OA, are obtained using some realistic approximations based on the typical specifications of modern general purpose OA. Expressions thus obtained closely follow the exact transfer functions, as demonstrated in the illustrative examples given, and their validity is confirmed by the experimental results.     

Using least-mean-square-approximation technique in filter design

The applications of the least-mean-square-approximation technique to filter synthesis are discussed, and explicit expressions for the characteristic function of all-pole lowpass filters are derived using a weighted least mean-square error norm. The weight function depends on one variable parameter which controls the shape of the magnitude response both in the passband and in the stopband. It is shown that most of the filter functions in common use are special cases of this approximation procedure, including Legendre monotonic passband filters as a limiting, degenerate case. Also, a useful generalization of Legendre filter functions is proposed.     

The modified Pascal polynomial approximation and filter design method

Polynomial approximations are extensively used in analog and IIR digital filter design. In this paper, a comprehensive filter design and an optimization procedure are presented explicitly using a filter‐appropriate modified Pascal polynomial. The so‐designed all‐pole Pascal filters exhibit non‐equiripple passband and monotonic transition and stopband responses. The order of the new Pascal filters is calculated from the order inequality which, although it cannot be analytically solved, leads to a nomograph that has been created and is presented here. Inevitably, the mathematical complexity introduced by the nature of the Pascal polynomials makes the analytical expression of the poles of the transfer function unfeasible and for that reason poles are given by means of appropriate tables. The design method is demonstrated in several detailed examples and Pascal filters are compared with their all‐pole counterparts, Butterworth and Chebyshev, over which they reveal certain advantages and disadvantages. Copyright © 2010 John Wiley & Sons, Ltd.     

A family of modular fully differential witched capacitor filters for simulating ladder filters

The switched capacitor filters proposed in this paper are based on the simulation of LC ladder filters through the state variable equations. They include all-pole and finite transmission zero, lowpass filters, bandpass and highpass filters. the introduced family of filters has a fully differential structure; consequently it has the advantages of improved power supply and common mode rejection ratios and extended dynamic range. Since the proposed family of filters simulates LC ladder filters, it inherits their low passband sensitivities. One of the main advantages of this family of filters is that it has a modular structure composed of key and auxiliary circuits. This modular structure results in easier design and implementation procedures in VLSI fabrication. the double-sampling concept is applied to the introduced modular blocks, which is essentially needed in high-frequency applications. A comparison with similar published work available in the literature is presented and illustrative examples are given.     

Microwave filters with nonperiodic transmission characteristic

A microwave filter with nonuniform-coupled transmission lines is suggested. In contrast to known transmission-line microwave filters the transmission characteristic does not change periodically with frequency. The synthesis procedure shown reduces the design problem to the known synthesis of directional couplers with nonuniform coupling factor. A table of calculated coupling factor values of the microwave filter with an equi-ripple passband of 0.05 dB up to 1 dB and an equi-ripple stopband of 60 dB is given. A practical design example shows good agreement with theoretical results.     

The use of Padé approximants in the derivation of distributed low-pass filters with simultaneous flat amplitude and delay characteristics

A unified approach—based on Padé Approximants—is presented for the design of simultaneous equal constraints non-minimum phase distributed filters. The results obtained indicate that this method is more general than previous methods, and allow considerable freedom for the control of both amplitude and delay. The procedure outlined can apply for any transfer function with arbitrary characteristics.