Finite Wordlength Properties Research Articles

Design of Finite Word Length Linear-Phase FIR Filters in the Logarithmic Number System Domain

Logarithmic number system (LNS) is an attractive alternative to realize finite-length impulse response filters because of multiplication in the linear domain being only addition in the logarithmic domain. In the literature, linear coefficients are directly replaced by the logarithmic equivalent. In this paper, an approach to directly optimize the finite word length coefficients in the LNS domain is proposed. This branch and bound algorithm is implemented based on LNS integers and several different branching strategies are proposed and evaluated. Optimal coefficients in the minimax sense are obtained and compared with the traditional finite word length representation in the linear domain as well as using rounding. Results show that the proposed method naturally provides smaller approximation error compared to rounding. Furthermore, they provide insights into finite word length properties of FIR filters coefficients in the LNS domain and show that LNS FIR filters typically provide a better approximation error compared to a standard FIR filter.

A Generalized Lattice Filter for Finite Wordlength Implementation With Reduced Number of Multipliers

The excellent finite wordlength (FWL) property of lattice digital filters is well known. The four-multiplier normalized lattice, with signal power at all delay elements normalized to unity, has particular advantage in its overflow property. However, when used to implement an Nth-order digital filter, the normalized lattice implementation requires 5N+1 multipliers. There exists another lattice structure with excellent FWL property called the injected numerator lattice structure. In this paper, we combine the injected numerator lattice and tapped numerator lattice to form a new hybrid lattice structure, which is not only canonic in the number of multipliers resulting in a significant reduction in overall implementation cost but also exhibits much better FWL properties than the normalized lattice structure. An improved “peakedness” measure is also introduced for application where the input signal has a strong time varying sinusoidal component. The new structure requires a few additional adders; it can be used to implement any causal and stable z-transform transfer function. Two numerical examples are presented to demonstrate the performance of the proposed structure.

Design of optimal digital lattice filter structures based on genetic algorithm

In this paper, a new class of lattice-based digital filter structures is derived. The optimum structure problem is formulated in terms of minimizing the signal power ratio with respect to the two sets of free parameters in the proposed structures. An efficient genetic algorithm is proposed to solve the optimum structure problem with the constraint on the structure parameters to be represented in signed power-of-two format. Two design examples are given, in which the optimized structures show excellent finite wordlength properties such as very low parameter sensitivity and very uniform signal powers across signal nodes and outperform the classical lattice structures.

On the optimal design of IIR filter with its state-space realization

The most widely used digital filters are finite impulse response (FIR) filters that are usually implemented with the transversal structures. For FIR filter, the output signal is a linear combination of filter coefficients that produces a quadratic function (mean-square-error) with a distinctive optimal operation point. FIR filter alternatively be realized for obtaining improvements in comparison of transversal filter structure for speed of convergence, computational complexity and finite word length properties. IIR filtering techniques exhibit reliable alternative for traditional FIR filtering. The principal advantage of IIR filters is lesser parameterization to achieve at par performance of FIR filters. In addition, the pole-zero structures ease their modeling in physical systems. On the other hand, the tradeoff for the attributes have few drawbacks inherent to IIR filtering with recursive structure such as filter stability, convergence to bias and/or local minima and for very slow convergence. This research paper proposes a new structure and realization of IIR filter. Special consideration is given to the issue of optimal state-space realization and newpole modulus based stability is derived.

On the Optimal Design of IIR Filter with its State-space Realization

The most widely used digital filters are finite impulse response (FIR) filters that are usually implemented with the transversal structures. For FIR filter, the output signal is a linear combination of filter coefficients that produces a quadratic function (mean-square-error) with a distinctive optimal operation point. FIR filter alternatively be realized for obtaining improvements in comparison of transversal filter structure for speed of convergence, computational complexity and finite word length properties. IIR filtering techniques exhibit reliable alternative for traditional FIR filtering. The principal advantage of IIR filters is lesser parameterization to achieve at par performance of FIR filters. In addition, the pole-zero structures ease their modeling in physical systems. On the other hand, the tradeoff for the attributes have few drawbacks inherent to IIR filtering with recursive structure such as filter stability, convergence to bias and/or local minima and for very slow convergence. This research paper proposes a new structure and realization of IIR filter. Special consideration is given to the issue of optimal state-space realization and newpole modulus based stability is derived.

Very Robust Low Complexity Lattice Filters

In this paper, a novel lattice filter structure is derived by combining the “tapped numerator” and “injected numerator” lattices [Y. C. Lim, “On the Synthesis of IIR Digital Filters Derived From Single Channel AR Lattice Network,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-32, no. 4, pp. 741-749, August 1984]. The injector coefficients are optimized for finite wordlength performance while the tapped coefficients synthesize the transfer function of the filters. For an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> th-order digital filter, the proposed “tapped numerator” and “injected numerator” hybrid lattice structure, after optimization, has only 2 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> +1 multipliers. It is not only canonic in the number of multipliers but also possesses much improved finite wordlength properties such as very low parameter sensitivity and very uniform signal powers across signal nodes. The excellent finite wordlength performance of the proposed structure is demonstrated with a design example and compared with those of traditional structures, including several well-known lattice structures.

FRM-Based FIR Filters With Optimum Finite Word-Length Performance

It is well known that filters designed using the frequency response masking (FRM) technique have very sparse coefficients. The number of nontrivial coefficients of a digital filter designed using the FRM technique is only a very small fraction of that of a minimax optimum design meeting the same set of specifications. A digital filter designed using FRM technique is a network of several subfilters. Several methods have been developed for optimizing the subfilters. The earliest method optimizes the subfilters separately and produces a network of subfilters with excellent finite word-length performance. Subsequent techniques optimize the subfilters jointly and produce filters with significantly smaller numbers of nontrivial coefficients. Unfortunately, these joint optimization techniques, that optimize only the overall frequency response characteristics, may produce filters with undesirable finite word-length properties. The design of FRM-based filters that simultaneously optimizes the frequency response and finite word-length properties had not been reported in the literatures. In this paper, we develop several new optimization approaches that include the finite word-length properties of the overall filter into the optimization process. These new approaches produce filters with excellent finite word-length performance with almost no degradation in frequency response performance

On Roundoff Errors in Block-Floating-Point Arithmetic

A special case of floating point data representation is block floating point format where a block of operands are forced to have a joint exponent term. This paper deals with the finite wordlength properties of this data format. The theoretical errors associated with the error model for block floating point quantization process is investigated with the help of error distribution functions. A fast and easy approximation formula for calculating signal-to-noise ratio in quantization to block floating point format is derived. This representation is found to be efficient compared to the fixed point and floating point format.

Two-dimensional delta-operator formulated discrete-time systems: state-space realization and its coefficient sensitivity properties

Delta-operator based implementation of one-dimensional (1-D) discrete-time systems has been the focus of considerable research activity because of its superior finite wordlength properties and possibility of addressing both continuous- and discrete-time systems in a unified manner. We investigate its corresponding two-dimensional (2-D) implementation by introducing the delta-operator based counterpart to the Roesser (1975) local state-space model, the corresponding gramians, and the notion of a balanced realization. The computation of the latter and coefficient sensitivity properties of the resulting implementations are also studied.

New class of discrete-time models for continuous-time systems

Digital computing in estimation, control or signal processing for continuous-time systems requires the use of discrete-time models. While conventional difference equation or z-transfer function models are widely popular, a class of methods exists that uses discrete approximations of continuous signals and operators, retaining the continuous-time parameters. Some important advantages of this class have been demonstrated in the contexts of parameter estimation, adaptive control and controller design. This paper proposes a new class of discrete-time models that originates from the z transfer function but which is close to continuous-time models in structure and parameters, thereby retaining its advantageous features. The recently proposed ‘delta’ model is seen to be a member of this class. The interrelations among various digital model types are brought out. Better sensitivity properties over z transfer function models are established. Finite word length properties of these models vis-a-vis the z-transfer fu...

Design of narrow-band FIR bandpass digital filters with reduced arithmetic complexity

A computationally efficient realization of a symmetrical bandpass FIR filter is derived. The realization is composed of two branches, each consisting of a cascade of two FIR sections. In each branch, the first FIR section has a sparse impulse response with only every L th sample being nonzero. The second section generates the remaining samples via interpolation. The method is applicable to both linear and nonlinear phase cases. Approximate expressions for the optimal value of L and the corresponding number of multipliers and adders required in the overall realization are derived. In narrow-band implementations, the number of multipliers and adders is approximately proportional to 1/\sqrt{{\Delta}F} , where {\Delta}F is the desired relative transition bandwidth. Typically, the required number of multipliers and adders is 1/2 to 1/4 th of those in the conventional linear or nonlinear phase direct-form implementations. The total number of delays is only slightly larger than that required in the conventional implementations. The structure allows simple tuning of the center frequency of the bandpass filter and has good finite wordlength properties.