Finite Word Length Arithmetic Research Articles

Generalized Dissipativity Analysis of Digital Filters With Finite-Wordlength Arithmetic

This brief investigates the generalized dissipativity of digital filters with finite-wordlength arithmetic. First, a new sufficient condition is proposed to guarantee the generalized dissipativity of single digital filters with finite-wordlength arithmetic. Unlike the existing works in the literature, this brief presents criteria for ${H}_{\infty}$ performance, $l_2-l_{\infty}$ performance, passivity, and $(\mathcal{Q},\mathcal{S},\mathcal{R})\mbox{-}\alpha$ -dissipativity in a unified framework. Based on this result, we examine the generalized dissipativity of feedback interconnected digital filters as well as the asymptotic stability of unforced feedback interconnected digital filters. A numerical example demonstrates the effectiveness of the obtained approach.

Finite Wordlength Recursive Sliding-DFT for Phase Measurement

This paper proposes a modified recursive sliding DFT to measure the phase of a single-tone. The modification is to provide a self error-cancelling mechanism so that it can significantly reduce the numerical error, which is generally introduced and accumulated when a recursive algorithm is implemented in finite wordlength arithmetic. The phase measurement error is analytically derived to suggest optimized distributions of quantization bits. The analytic derivation and the robustness of the algorithm are also verified by computer simulations. It shows that the maximum phase error of less than <TEX>$5{\times}10^{-2}$</TEX> radian is obtained even when the algorithm is coarsely implemented with 4-bit wordlength twiddle factors.

Regular orthonormal and biorthogonal wavelet filters

New methods for half-band filter design are developed, which structurally incorporate the regularity constraint into the design procedure. The first method results in half-band filters which do not have strictly positive frequency response. It is suitable for one class of biorthogonal filter banks. The second method results in half-band filters with strictly positive frequency response from which orthonormal wavelet filters can be obtained by spectral factorization. All filters are regular and have sharp transition bands. A new factorization of the first polyphase component of every half-band filter is found. This factorization suggests an efficient implementation of biorthogonal filter banks of the type considered. The implementation not only preserves the perfect-reconstruction property assuming finite word-length arithmetic, but has reduced coefficient sensitivity compared with the direct form. The properties of the resulting scaling functions and wavelets are summarized.

Elimination of limit cycles due to two's complement quantization in normal form digital filters

Normal form digital filters are attractive due to their desirable properties when implemented in finite wordlength arithmetic. These filters are free from all overflow limit cycles and quantization limit cycles when magnitude truncation is used, However, when two's complement truncation (TCT) quantization is used, limit cycles can still exist. In this paper, it is shown that when block structures are used, normal form digital filters can be made free of limit cycles due to TCT quantization. It is shown that this can be done with a small block size. An algorithm is also presented to find the minimum block size required for a given filter. Some examples are given to illustrate the results.

Robust perturbation algorithms for adaptive antenna arrays

The generation of a perturbation sequence for an adaptive beamformer is described. This perturbation sequence permits simultaneous adaption and reception by use of weight perturbations that do not obstruct the look direction constraint. It is shown that this sequence is generally shorter in length than previously described sequences and offers scope for computational savings through reduction of the number of projection operations required. Convergence in the mean of the resulting adaptive algorithm is demonstrated. Experiments conducted using a four-element linear array operating in the MW RF range have confirmed that the predicted results are achievable under the nonideal conditions of quantized array weights and finite word length arithmetic. >

Computational noise effects on adaptive filter algorithms

Finite word length arithmetic roundoff noise in adaptive filter algorithms results in statistical variations in the filter weight vector about the infinite precision arithmetic weight vector. These roundoff errors may be modeled as a statistically non stationary driving noise affecting weight mean and covariance convergence. Mean and covariance expressions and bounds are desired for word lengths in fixed-point arithmetic by making use of multiplication roundoff error models. The adaptive filter algorithms consist of the LMS algorithm, the Widrow-Hoff LMS algorithm, pilot-vector algorithm and clipped vector algorithm. All of these algorithms can be implemented on-line and real-time. However, only the behavior of the LMS algorithm is reported here. The implementation of the adaptive filter algorithms in finite word length arithmetic is most evident in minicomputer, microprocessor, and dedicated digital signal processors for on-line real-time signal identification and parameter estimation in many disciplines. Radar signal processing, adaptive beam forming, acoustic signal identification, communication channel enhancement have a definite need for advanced filtering concepts. Our adaptive algorithms are typically employed in these filter configurations. These filters can also be employed in phase distortion equalizers. A particular advantage of these filters is that they can be trained to equalize a variety of distortions. Should a particular distortion scenario change in time, the filters can be made to easily adapt to the new problem.