Filters Design by Z Transformation and Pascal Matrix (original) (raw)

The frequency transformation by matrix operation and its application in IIR filters design

IEEE Signal Processing Letters, 2005

It is known that the transfer function () of the desired digital infinite-impulse response filter can be obtained from the normalized transfer function () of the analog low-pass filter. Recently it has been shown that Pascal matrix allows the appropriate transformation for design of the low-pass, high-pass, and bandpass digital filters. Unfortunately, this method is difficult to use in case of transforming () to the high-order bandpass filter. This letter proposes a similar method, especially suitable for design of the bandpass and bandstop filters without order limitation. The presented design algorithm is illustrated by a numerical example.

FREQUENCY TRANSFORMATIONS FOR DIGITAL FILTERS

Transformations to convert lowpass systems to highpass, bandpass and band-elimination systems in the case of pulse transfer functions of digital filters are given in this paper. It is believed that these results, for z plane transformations, are the first ones to be published in this field.

Design of Bandpass Digital Filters

A method for synthesizing bandpass digital filters is described.and illustrated by a particular example. This method rests on a complex plane mapping performed on the z-plane, and has several advantages over others proposed by various authors.

Accuracy and stability of discrete-time filters generated by higher-order s-to-z mapping functions

Ieee Transactions on Automatic Control, 1994

Abs&act-This note compares several diClerent approaches to s e l d n g an s-to-s mapping function for the purpose of converting a continuoustime transfer function F(s) to an "equivalent" discrete-time transfer function F g (z). The primary bases of comparison are stability and accuracy. Secondary bases of comparison include: 1) the amount of computation needed to process one input to produce one output, 2) the ease with which the coefficients of F g (z) can be determined, in particular, as the sampling period T is varied, 3) the ease of determining stability, in particular, the maximum value of T for stability, 4) the self-starting capability, and 5) compatibility with decomposition in the s-domain. I. INTRODUCTION Recent work [l], [2] presented a group of higher order s-toz transformations (mapping functions) for converting a continuoustime transfer function F (s) to a discrete-time equivalent FD(z) for control, digital simulation, numerical integration, or other purposes, offering far superior accuracy compared to the usual Tustin's rule or the Boxer-Thaler z-forms. One would use a higher order transformation in applications in which accuracy is necessary or desirable, for example, in a nonlinear robotic control system which depends critically on accurate numerical integration of signals in the feedback path. These new transformation, referred to collectively as the Schneider group, appear as the (1,2), (1,3), (1,4), (1,s) elements of the matrix in Table I. (It is essential that [ 11 be read for a thorough understanding of that which follows.) Hartley and Filicky developed a larger set of s-to-z transformation [3]-[5], presented (with corrections) as Table I. In this table, (U) is the input to the pth-order integrator (l /~)~ and u (~) is its mth derivative. Hartley refers to the members of one column as a family and proposes that a family be used to convert F (s) to FD (z). The collection of elements of the nth column will be referred to as the n-step family because they are all derived from the linear n-step method for the numerical solution of differential equations [6]. References [l], [2] documented the accuracy and stability obtainable with individual members of the Schneider group. The present note examines the accuracy and stability of the n-step families, and compares results for an individual family with results from the corresponding member of the Schneider group. 11. DISCUSSION To convert a continuous-time transfer function F (s) to discrete form by the use of element (1, 2) of Table I, one substitutes this element for each (l/s) in F (s). If F (s) is of order n, that is, if it

The modified Pascal polynomial approximation and filter design method

International Journal of Circuit Theory and Applications, 2012

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.

D03: IIR filters design procedures based on digital frequency transformations

IFAC Proceedings Volumes, 2004

It is known that the transfer function of the desired digital IIR filter can be obtained from the analog lowpass filter by use of the digitizing procedures. Many useful design formulas for continuous-time filters relate to lowpass filters. The other types of filters, like highpass, bandpass and bandstop are obtained by frequency transformation. In general, analog or digital frequency transformations can be used, however available digital signal processing books give the practical design formulas for analog frequency transformation only. This paper proposes new procedures that applies digital frequency transformations.

Design of digital filters from LC ladder networks

A design procedure is given which enables digital filters to be derived directly from ladder LC filters. The derivation is based on an element-by-element transfer of the ladder components to the digital domain. The manner in which such a transfer is achieved is dependent on linear voltage-current transformations, a special case of which is that relying on scattering parameters which has already been described by this author and by others elsewhere. A specific set of constraints are imposed on the transformations in this paper so as to satisfy the readability condition and also to ensure that the digital filters thus derived have transfer functions which are identical to the original ladder transfer function, generally modified by a constant multiplier, as viewed through the bilinear complex frequency transformation. Two special cases of the general approach presented in this paper show that the digital filter structures so derived have a low attenuation sensitivity to multiplier variation, which is consistent with the expectation for the low sensitivity properties of the doubly-terminated lossless ladder filters.

The generation of equivalent digital filter structures by a modified multiplier extraction approach

IEEE Transactions on Circuits and Systems, 1982

The multiplier extraction approach provides a matrix representation of digital filters which determines both their structure and transfer function. The properties of this representation are investigated, and a modified version is proposed for representing the structure of a class of digital filters which do not introduce product of multiplier terms in the transfer function. This method is used as the

Search of Optimal s-to-z Mapping Function for IIR Filter Designing without Frequency Pre-warping

IETE Journal of Research, 2019

Bilinear transformation is one of the most popular s-to-z mapping functions and widely used to design digital filters, disregarding its susceptibility to "frequency warping". Although "frequency pre-warping" is used to counter the warping effect, it cannot be applied uniformly in all the cases. Moreover, the pre-warping technique is unable to remove the warping-effect completely. It is a general convention that the higher-order s-to-z transform gives more linear mapping than the lower order counterparts. Surprisingly, the 1st-order bilinear transformation is still preferred over higherorder s-to-z mapping functions. An investigation has been performed to develop a generalized criterion for selecting optimal s-to-z mapping function so that IIR filters can be designed without the need of "frequency pre-warping". A couple of IIR filters (integer and fractional order) have been designed using several s-to-z mapping schemes, without applying the frequency pre-warping technique. MATLAB and Simulink responses have been given to support and validate the developed generalized criterion, for selecting optimal s-to-z mapping function.