Mirror Image Polynomial Research Articles

NEW Z-DOMAIN CONTINUED FRACTION EXPANSIONS BASED ON AN INFINITE NUMBER OF MIRROR-IMAGE AND ANTI-MIRROR-IMAGE POLYNOMIAL DECOMPOSITIONS

A magnitude response preserving modification of the denominator polynomial of a causal and stable digital transfer function leads to an infinite number of decompositions into a mirror-image polynomial (MIP) and an anti-mirror-image polynomial (AMIP). Properties and identifications of the MIP and AMIP are given. The identifications of Schussler and Davis, and the line spectral frequency formulation are special cases of the general MIP and AMIP decompositions introduced in this paper. Two types of Discrete Reactance Functions (DRF) are constructed. From these DRFs, five new continued fraction expansions (CFE) are developed, and some properties are obtained.

A new canonical expansion of z-transfer function for reduced-order modeling of discrete-time systems

On the basis of the unique decomposition of a polynomial into a mirror image polynomial (MIP) and an antimirror image polynomial (AMIP) and the expansion of A.M. Davis' discrete reactance function (ibid., vol.CAS-29, no.10, p.658-62, 1982) into a continued fraction which proceeds in terms of z/(z-1) and 1/(z-1) alternately, a new canonical expansion of the z-transfer function is presented. Although it has the same structure as the Routh canonical expansion of the s-transfer function, the new canonical expansion is suitable for deriving reduced-order models of discrete-time systems by direct truncation. Using this canonical expansion, frequency- and time-domain reduced-order modeling procedures are derived. The necessary and sufficient conditions imposed on the continued-fraction expansion of Davis' discrete reactance function for reduced-order models to be stable are also derived. It is shown that the reduced model has the partial Pade approximation property. >

Formation and Modification of Two-Dimensional Discrete Reactance Functions and its Continued Fraction Expansions

The procedure involves the decomposition of the denominator polynomial of 2-D discrete system into mirror image polynomial and an antimirror image polynomial. 2-D discrete reactance function is formed with these polynomials. This reactance function and its modified versions are expanded in continued fraction in Z1—Z2 domain in a number of ways. These expansions establish the stability of 2-D discrete systems along with probing the possibility of their physical realization.

Algorithms for Z-domain continued fraction expansion by Davis

The authors cast the algorithm for computing the Z-domain continued fraction expansion by A.M. Davis (1982) in the form of a three-term recurrence relation. Such a recurrence relation processes a mirror image polynomial and an anti-mirror image polynomial alternately. The algorithms derived are at least twice as efficient in computational complexity as the algorithms recently proposed by R. Parthasarthy and S.N. Iyer (1987). >

A frequency–domain design of two–dimensional recursive digital filters

AbstractThis paper presents a technique for the design of two‐dimensional recursive digital filters in the frequency domain. The filter consists of the denominator being structurally stable and the numerator being a mirror‐image polynomial. To find the denominator, a stable two‐dimensional all‐pole filter is designed first so as to approximate the constant group delay specified in each direction. This is done by using the Davidon‐Fletcher and Powell algorithm. The numerator is then designed so that the whole rational transfer function approximates the desired magnitude response under the assumption that the denominator is fixed. This is done by solving a linear equation. The present technique has the advantage of always ensuring the stability of the resultant filter. Two examples are given to illustrate the utility of the proposed technique.

Design of two-dimensional recursive digital filters using mirror-image polynomials

In this paper, a computationally efficient technique is developed for the design of two-dimensional (2-D) recursive digital filters that meet simultaneously magnitude and group delay specifications. The denominator in the filter is chosen structurally stable so that the filter stability is always guaranteed. Alternatively, the numerator is expressed in terms of a 2-D mirror-image polynomial. This makes it possible to design the denominator and the numerator separately. As a result, the amount of calculations can be reduced to some extent and the convergence can also be improved. Finally, two examples are given to illustrate the utility of the proposed technique.

Digital lattice filters with reduced number of tap-multipliers

A synthesis of digital lattice filters with reduced number of tap multipliers is proposed. The resultant structures have about half as many tap-multipliers as Gray-Markel structures. This reduction is applicable when the numerator of a given transfer function is either a mirror image polynomial or an antimirror image polynomial.

Some design methods for two‐dimensional digital filters with complex coefficients in consideration of linear phase characteristics

AbstractIn designing a two‐dimensional digital filter with special consideration of the phase characteristics, particularly the group delay characteristics, a very high‐order filter is generally required. This paper extends the coefficients of the filter from real to complex and describes a design technique for comparatively low‐order, two‐dimensional, circularly symmetric filters with consideration of phase characteristics. First, the method of transforming a two‐dimensional input signal into an analytic signal is described and then the design technique of two kinds of complex coefficient filters which use the properties of analytic signals is discussed. In the first kind variable transformation to a 1‐D real coefficient FIR filter is used to obtain a 2‐D FIR filter with complex coefficients. It has the merits of permitting independent setting of the sharpness of cutoff characteristics and circular symmetry. In the other kind of complex coefficient filter a recursive digital filter is obtained by using an all‐pole filter having constant group delay in the passband of the transfer function and having a 2‐D complex coefficient mirror image polynomial. It always has a constant group delay unrelated to the values of the coefficients and the overall group delay is also constant. Moreover, in this paper, design examples based on these methods are given and the validity of our methods is shown.

A New Z Domain Continued Fraction Expansion

A new <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z</tex> domain continued fraction expansion is presented which proceeds in terms of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z - 1</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1 - z^{-1}</tex> factors. It is proved to be always convergent for polynomials whose roots all lie within the unit circle. The procedure involves a unique decomposition of the given polynomial into a mirror image polynomial (MIP) and an antimirror image polynomial (AMIP) whose degrees differ by unity. The ratio of these polynomials is shown to possess the properties of a digital reactance function. The continued fraction expansion developed here-due to the fact that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z - 1</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1 - z^{-1}</tex> are the inverse transmittances of digital accumulators, as well as of switched capacitor integrators-has application to the synthesis of digital and switched capacitor ladder filters.

Linear phase digital filter design

The results of an investigation of linear phase digital filters are presented. It was determined that stable filters of this type are necessarily nonrecursive. The corresponding transfer function is presented in a general form in terms of basic z-plane pole and zero configurations. The design of digital filters of this form is accomplished by approximating a desired frequency function by one of four types of Fourier series. The types of series correspond to the forms of the z transfer function polynomial, which is either a mirror-image polynomial or a negative mirror-image polynomial, and either even or odd order.