Low-sensitivity Digital Filters Research Articles

Evolutionary synthesis of low-sensitivity digital filters using adjacency matrix

An evolutionary synthesis method to generate digital filters with low coefficient sensitivity is presented. The method uses a chromosome coding based on the graph adjacency matrix representation. It is shown that the proposed chromosome representation enables to easily verify and avoid the generation of topologically invalid and non-computable individuals during the evolutionary process. The efficiency of the proposed algorithm is tested in the synthesis of two low-pass digital filters and the results are compared with other examples found in the literature.

Synthesis of low-sensitivity orthogonal digital filters

Synthesis of low-sensitivity orthogonal digital filters

Synthesis of low‐sensitivity orthogonal digital filters

A novel method is proposed for the synthesis of low-sensitivity digital filters meeting any prescribed transfer function. The method is based on extracting an orthogonal matrix from the filter state matrix, resulting in structures that need n Givens rotations and at most n + 1 multipliers. Thus the proposed realization is canonic in the sense that it has the same degrees of freedom as the original transfer function. It is also shown that when the filter transfer function is a reciprocal reactant bounded function, it can be decomposed into allpass functions that need only Givens rotations for their realization. As the basic module in the realization is the Givens rotation, the CORDIC computation algorithm can be applied directly. This means considerable savings in computation time and complexity. It also results in structures that are less sensitive to the effects of finite word length. Illustrative examples, including the design and synthesis of linear phase selective filters, are given to show the extremely low sensitivity with respect to finite word length of the resulting realizations when compared with other methods.

Improved LBR two-pair extraction methods for low sensitivity digital filter realizations

In this paper, we present two improved lossless bounded-real (LBR) two-pair extraction methods using an allpass input function, which significantly reduce roundoff errors during the extraction process and reduce the amount of computation nearly to a half. Both methods are based on the induced allpass transfer function from the given BR function, which is obtained by replacing the value of the terminating multiplier with 1 or -1. The first method is an LBR two-pair extraction based directly on the allpass input function induced from the given input transfer function. The second extraction method is based on two new real polynomials obtained by further decomposing the induced allpass input function. Even with two polynomials, the involved polynomial division and parameter calculation are simpler in the second method. Consequently, numerical errors are further reduced in the second method. The benefits of employing the two improved methods are confirmed through simulation examples.

Two-type-interlaced structure and LBR test for low-sensitivity digital filter realization

The authors present a new digital filter structure which shows a low-sensitivity property. It is a cascade structure consisting of Type 1 and Type 2 LBR two-pairs interlacing each other, and is called a two-type-interlaced (TTI) structure. In addition, a test procedure called an LBR test is introduced as a means to determine the LBR-ness of the quantized TTI structure by checking the LBR-ness of the residual Type 1 two-pair in the associated Type 3 re-assembled structure. Computer simulation confirms that this new structure exhibits a much improved sensitivity characteristic over the other existing structures.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Adaptive bilinear digital filters

How adaptive algorithms can be applied to low-sensitivity digital filter structures is shown. The considered structure is derived by a bilinear transformation of an analog doubly resistively terminated ladder filter. A fifth-order IIR low-pass filter is used as example. The adaptive algorithm is derived using standard stochastic approximation arguments and both gradient and Newton updating equations are presented. The adaptive bilinear filter is used in system identification simulations and compared to a direct-form IIR filter. It is demonstrated that the adaptive bilinear filter, although not unconditionally stable, has excellent stability properties. It is also shown that the adaptive algorithm can be used as a tool for synthesis of pseudopassive and hence numerically robust fixed filters. The adaptive algorithm is used to minimize a cost function chosen as a weighted difference between the desired transfer function and the adaptive transfer function.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Letter to the editor design of low-sensitivity quarter-plane recursive 2d digital filters

A new technique is given for designing two-dimensional quarter-plane recursive digital filters. These filters have low sensitivity in the passband. an implementation scheme using complex multipliers is also given for these filters.

New low-sensitivity and low-noise GIC digital filter structures

New digital filter structures based on the concept of generalized-immittance convertors (GICs) are proposed. These structures are shown to possess better sensitivity and noise properties than the already known GIC structures. A comparison study is carried out using different types of sensitivity and noise measures. Further, it is shown that the limit cycles under zero-input and forced response conditions can be eliminated in the proposed filters. With suitable modifications, these filters can be stabilized against constant-input limit cycles as well.

An alternative sensitivity measure for designing low-sensitivity digital biquads

A sensitivity measure is proposed for determining the optimum set of machine-representable multiplier constants in the derivation of low-sensitivity digital biquadratic filters. A comparison of this measure with two other sensitivity measures of this class is then undertaken using four different design examples. The four sixth-order filters considered are a Butterworth low-pass, a Chebyshev high-pass, an elliptic bandstop, and an elliptic bandpass filter. It is shown that the three sensitivity measures usually give slightly different low-sensitivity designs and that the new sensitivity measure often gives the best design. >

Bounded Complex and Lossless Bounded Complex Transfer Functions and their Applications

The concepts of structural boundedness and structural losslessness are known to be the keys behind the realization of low passband sensitivity digital filters implemented using real arithmetic. A generalization of these concepts leads to bounded complex and lossless bounded complex transfer functions, which are shown to play a similar role in the realization of low passband sensitivity digital filters with complex coefficients. Various filter realization strategies are pointed out, which involve the interconnection of complex lossless building blocks, thereby assuring the structural passivity property in the realization. Some design examples are included to verify the low sensitivity properties.

Delay replacement in direct form structures

A class of high-order low-sensitivity digital filter structures is generated by means of a direct transformation of a high-order direct-form (DF) structure. The proposed structures have low coefficient dynamic range, and exhibit low coefficient frequency-response sensitivity and roundoff noise characteristics, yet require only 3N multiplications per output sample as compared to 2N in the direct-form case. >

Low-sensitivity digital filters based on zero extraction

A method is proposed for synthetizing digital filters derived from the LBR two-pair extraction technique. The method forces the LBR two-pairs to have their transmission zeros coincident with those of the filter. The basic two-pair having transmission zeros on the unit circle is considered in detail with regard both to the realization problem and to the simplification of the synthesis procedure. In particular, explicit formulas are proposed for directly determining the representation of this two-pair. A simulation example illustrates the sensitivity performance of the resulting schemes.

On Complimentary and sensitivity of generalized wave digital filters

In recent literature, a general theory has been developed to design low-sensitivity digital filter structures in the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> -domain. These structures are two-ports with two inputs and two outputs and wave digital filters belong to this general class. This paper explores the significance of the para-unitary or complementary condition on generalized wave digital filters. Satisfying this condition allows, for example, the simultaneous generation of low- and high-pass filter outputs. There is discussion too of sensitivity and its significance to those generalized wave digital filter structures that do not satisfy the para-unitary condition.

Low sensitivity digital LDI ladder filters with elliptic magnitude response

A method of designing elliptic LDI [I] type switched-capacitor filters is known [2], [3]. However, this method cannot be extended directly to digital filter design and implementation. Preliminary results involving the design of a fifth-order digital LDI ladder filter have been proposed 141. In this contribution a complete design method for low sensitivity digital LDI ladder filters which have elliptic magnitude response is presented. A relationship between the bilinear and LDI transformed impedances is investigated. A precompensation procedure together with the bilinear transformation are applied to transform an elliptic analog ladder filter into a realizable digital LDI ladder filter. A voltage-current signal flowgraph ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V/I</tex> SFG) simulation method is adopted in this design procedure. The design procedure provides a method of realizing the bilinear transformed transfer function using the LDI as a building block. An example of fifth-order elliptic LDI low-pass filter design is given. The effects of multiplier coefficient quantization on both the digital elliptic LDI ladder filter and an equivalent wave digital ladder are compared.

Passivity properties of low-sensitivity digital filter structures

A general representation of a class of low passband sensitivity digital filter structures is proposed. The proposed representation for a transfer function of order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> consists of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(N + 1)</tex> -pair memoryless system terminated at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> -pairs by delays. The <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(N + 1)</tex> -pair system contains only adders and multipliers, and is described by an orthogonal transfer matrix. The set of terminating delays can be looked upon as an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> -pair system with transfer matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z^{-1}{\bf 1}</tex> . Certain wave digital filter structures, Gray-Markel lattice structures and the coupled-form biquadratic section belong to the general form advanced here. Several properties satisfied in these special cases are derived in a unified manner using the generalized representation. Also, a quantization scheme that makes the structure free from zero-input limit cycles even under time-varying conditions is advanced, unifying similar such results independently reported for the above well-known structures.

Signal-processor-based digital filters having low sensitivity

New low-sensitivity second-order digital-filter structures are proposed which are suited for implementation on an integrated signal processor with an on-chip fast multiplier. The inherent parallelism in such a processor is utilised to design very low sensitivity digital filters without any remarkable decrease in the overall throughput.

Low passband sensitivity digital filters: A generalized viewpoint and synthesis procedures

The concepts of losslessness and maximum available power are basic to the low-sensitivity properties of doubly terminated lossless networks of the continuous-time domain. Based on similar concepts, we develop a new theory for low-sensitivity discrete-time filter structures. The mathematical setup for the development is the bounded-real property of transfer functions and matrices. Starting from this property, we derive procedures for the synthesis of any stable digital filter transfer function by means of a low-sensitivity structure. Most of the structures generated by this approach are interconnections of a basic building block called digital "two-pair," and each two-pair is characterized by a lossless bounded-real (LBR) transfer matrix. The theory and synthesis procedures also cover special cases such as wave digital filters, which are derived from continuous-time networks, and digital lattice structures, which are closely related to unit elements of distributed network theory.

Wave digital filters for second and fourth order basic reactance sections

AbstractOne way to realize a low coefficient sensitivity digital filter is to simulate a resistively terminated LC ladder network. This is called a wave digital filter. However, the transfer functions realized with a resistively terminated LC ladder network are limited. A resistively terminated reactance network realizing an arbitrary transfer function may be synthesized by cascading basic reactance sections. In this paper, we use Brune, Types C, E and D sections with 2‐wire lines as the models for the basic reactance section. These basic sections are represented using modal decomposition by equivalent circuits made of ideal transforming networks (including gyrators for nonreciprocal circuits) and uncoupled lossless lines. Then we derive basic wave digital filters of reciprocal and nonreciprocal second‐order sections and reciprocal fourth‐order sections. the digital filters realized from the basic wave digital filters in the present method do not correspond exactly to the original analog circuit as long as they are constructed with multipliers with finite word length; however, they become low coefficient sensitivity digital filters with as good performance as the wave digital ladder filters.

An approach to digital filter equalizing of the multiplier coefficients

AbstractThe application of digital two‐pair networks in the design of loco‐sensitivity digital filters is considered. Digital twopair networks using parallel connection and certain limited ladder‐type constructions have been proposed. In this paper, digital two‐pair networks using the cascade connection are discussed. In realizing digital two‐pair networks, if the given connection matrix (K‐matrix) is a general rational function matrix it is difficult to reduce the K‐matrix to product form. Therefore, when the K‐matrix is given as a polynomial matrix with determinant 1, the ladder‐type construction is proposed. As an example of the application of the digital two‐pair network, a low‐sensitivity digital filter is considered. A delay‐free loop problem has been noted. In our approach, this problem is solved by including the terminating part of the digital two‐pair network in the K‐matrix. Furthermore, in view of the fact that most multiplier coefficients of the realized lowsensitivity digital filters are almost equal, an approach using digital filter equalizing of the multiplier coefficients is also considered as a digital filter design method.

A new class of very low sensitivity and low roundoff noise recursive digital filter structures

A class of new structures is proposed, for the realization of the recursive digital filter with poles near the unit circle, by introducing the particular type of delay replacement scheme in the existing digital filter structure and then, inserting, delays at the. proper places to avoid the delay-free loops. By applying the proposed method to the second-order direct form structures, new structures are found with very low magnitude transfer function sensitivities with respect to multiplier coefficients and low roundoff noises. Some numerical results are also presented.