Finite Word Length Research Articles

Synthesis of Recursive Digital Filters with Finite Word Length: Problems and their Solutions

Synthesis of Recursive Digital Filters with Finite Word Length: Problems and their Solutions

A Modified Fixed-Point Chirp Scaling Algorithm Based on Updating Phase Factors Regionally for Spaceborne SAR Real-Time Imaging

To realize real-time imaging for the spaceborne synthetic aperture radar (SAR) using the chirp scaling (CS) algorithm, high generation rates of phase factors and heavy computation loads are unavoidable. For example, updating phase factors continuously in the 2-D time or frequency space is difficult for generating phase factors for real-time imaging. To solve this problem, a modified CS algorithm based on updating phase factors regionally is proposed. Moreover, conventional floating-point arithmetic is unaffordable for real-time imaging onboard. Therefore, fixed-point processing is adopted. In the proposed imaging algorithm, the phase factors are updated only at some specified times and frequencies, and fixed-point operation is adopted. Furthermore, based on the paired echo theory and the finite word length error mathematical model, the effect of the proposed imaging algorithm on the SAR image quality is studied. Moreover, based on the above analysis and the experiments of Chinese HJ-1C and GF-3 spaceborne SAR practical measured data, a hardware system including digital signal processor and field-programmable gate array boards is constructed.

High‐throughput and compact reconfigurable architectures for recursive filters

This study presents various high-throughput reconfigurable architectures and their hardware implementation characteristics for infinite impulse response (IIR) filters. It is known that finite word length effects can be alleviated, to some extent, by realising a high-order IIR filter using the cascade of lower-order filter sections. The authors utilise the cascade structure of the first-order and second-order filters and apply a set of optimisation techniques, such as cutset retiming, look-ahead transformations, and interleaving for high-throughput realisations of IIR filters. Since the cascade structure may require a relatively large number of computational resources and storage elements, which depends linearly on the number of sections, the authors also present a filter processor architecture for the compact implementation of IIR filters. Filter architectures are developed in the fully parameterisable fixed-point representation and verified against their synthesisable Verilog descriptions. The authors present the implementation results of the transformed filter architectures on a Xilinx Virtex-7 field-programmable gate array (FPGA). To the best of their knowledge, this is the first presentation of various high-throughput and compact IIR filter architectures and their FPGA implementation characteristics.

A metaheuristic optimization approach to discretize the fractional order Laplacian operator without employing a discretization operator

A single-step procedure to obtain first and higher order discrete-time models in terms of infinite impulse response templates of the fractional order Laplacian operator sα, where 0<α<1, is proposed in this paper. The Moth Flame Optimization (MFO) algorithm based rational approximations are generated using a discretization operator-free method. Solution accuracy and the convergence performance of MFO are extensively compared with several other advanced evolutionary algorithms. Simulations justify the improved modelling accuracy of the proposed models over the recently published designs. The effects due to the finite word length leading to truncated filter coefficients are also considered, and the design stability robustness is demonstrated. The efficacy of the proposed model as a fractional order proportional-derivative controller is also validated.

New Criterion for l2–l∞ Stability of Interfered Fixed-Point State-Space Digital Filters with Quantization/Overflow Nonlinearities

The problem of l2–l∞ stability and disturbance attenuation performance analysis of fixed-point state-space digital filters with external disturbance and finite wordlength nonlinearities is studied in this paper. The finite wordlength nonlinearities are composite nonlinearities representing concatenations of quantization and overflow correction employed in practice. Using sector-based characterization of the finite wordlength nonlinearities, a new l2–l∞ stability criterion for the reduction of the effects of external disturbance to a given level and to confirm the nonexistence of limit cycles in the absence of external interference is established. Numerical examples are given to illustrate the efficacy of the proposed criterion.

A Lattice Basis Reduction Approach for the Design of Finite Wordlength FIR Filters

Many applications of finite impulse response (FIR) digital filters impose strict format constraints on the filter coefficients. Such requirements increase the complexity of determining optimal designs for the problem at hand. We introduce a fast and efficient method, based on the computation of good nodes for polynomial interpolation and Euclidean lattice basis reduction. Experiments show that it returns quasi-optimal finite wordlength FIR filters; compared to previous approaches it also scales remarkably well (length 125 filters are treated in $ 9 s). It also proves useful for accelerating the determination of optimal finite wordlength FIR filters.

Robust Stabilization of Uncertain 2-D Discrete Delayed Systems

This paper is concerned with the stabilization of a class of two-dimensional discrete state-delayed systems described by the Fornasini–Marchesini second local state-space model subject to polytopic uncertainties. A delay-dependent stability criterion is derived for the existence of a memoryless state feedback controller such that the closed-loop system is robustly stable. To match practical issues the influence of finite wordlength nonlinearities in control implementation, delays and parameter uncertainties belonging to a known convex polytope are also discussed. The presented criteria are obtained by employing parameter-dependent Lyapunov–Krasovskii function and slack variables that augments the searching space allowing less conservative solutions. Finally, examples are provided to show the effectiveness of the proposal.

Синтез рекурсивных цифровых фильтров с конечной длиной слова: проблемы и их решения

Синтез рекурсивных цифровых фильтров с конечной длиной слова: проблемы и их решения

A Novel Synchronization Scheme for Impulsive UWB-based PLC Systems

This paper focuses on a low-cost synchronization technique based on power criteria for impulsive ultra wideband scheme applied to power line communication systems. The advantage offered by this technique is that the knowledge of both channel pulse response and noise information are not required. To assess the proposed technique, we discuss performance results obtained with typical in-home, outdoor, and low-voltage power line channels covering the frequency band between 1.7 and 50 MHz and additive noises modeled as white Gaussian and impulsive Gaussian. The results show that the proposed technique achieves the best performance in terms of detection probability, false alarm probability, and bit error rate in comparison with previous synchronization techniques, which were design to work with the aforementioned data communication system. Besides, implementation analysis on field programmable gate array shows that the proposed technique demands less hardware resource utilization than previous techniques and its finite word length implementation with at least 16 bits word length results in negligible performance degradation.

Power Allocation in Mobile Cellular Communication Under Multiplicative Noise and Interference Uncertainty

In this paper, we study the problem of closed-loop power allocation in mobile cellular-communication networks subject to multiplicative uncertainty. Multiple-access interference, nonuniform quantization, and finite wordlength representations are all characterized by multiplicative uncertainty models in the forward and feedback paths of closed-loop power allocation strategies. The quality of service (QoS) in the cellular network is evaluated with respect to signal to interference-noise ratio after the estimation process, where the uplink channel (mobile users to base station) is addressed under arbitrary round-trip delays per user. The multiplicative uncertainty terms are modeled by a stochastic framework, where the mean-square small gain theorem is employed to evaluate robust stability. Two inequalities are then derived to check internal stability with respect to the norms of the multiplicative uncertainty terms, and the sensitivity and its complementary transfer functions of the closed-loop system. The distributed power control scheme that maximizes the stability upper bounds is derived that depends on the round-trip delay per user, and whose structure enables simplifying the inequalities to evaluate robust stability. By establishing a numerical relation between the size of the interference and the multiplicative noise, a unique stability bound is derived for closed-loop power allocation. A numerical evaluation was carried out in a code-division multiple-access cellular network with multiple base stations, and under distinct QoS requirements.

CORDIC‐based Hann windowed sliding DFT architecture for real‐time spectrum analysis with bounded error‐accumulation

This study presents a COordinate rotation DIgital computer (CORDIC)-based novel architecture combining the sliding discrete Fourier transform (DFT) with Hann windowing to reduce the leakage effect of the DFT spectrum. The proposed architecture also presents a refreshing approach to minimise error due to the finite word-length in the output windowed spectrum compared with the existing method. The architecture can also be extended for other high order generalised cosine windows such as Blackman, Blackman-Harris, and flat-top. The post-route results on the Virtex-6 FPGA as well as the physical ASIC post-layout results are also presented for the proposed architecture.

Criteria for stability of uncertain discrete-time systems with time-varying delays and finite wordlength nonlinearities

This paper considers the problem of global asymptotic stability of a class of uncertain discrete-time systems under the influence of finite wordlength nonlinearities (quantization and/or overflow) and time-varying delays. The parameter uncertainties are assumed to be norm-bounded. Utilizing the concept of a Wirtinger-based inequality and a reciprocally convex method, two delay-dependent stability criteria are presented. The selection of the criteria depends on the type of the nonlinearities, that is, a combination of quantization and overflow or saturation overflow nonlinearities involved in the present systems. The approach presented in this paper yields less conservative results and reduces the computational burden as compared to previously reported criteria. Numerical examples are given to illustrate the effectiveness of the presented approach.

Formal Non-Fragile Stability Verification of Digital Control Systems with Uncertainty

A verification methodology is described and evaluated to formally determine uncertain linear systems stability in digital controllers with considerations to the implementation aspects. In particular, this methodology is combined with the digital-system verifier (DSVerifier), which is a verification tool that employs Bounded Model Checking based on Satisfiability Modulo Theories to check the stability of digital control systems with uncertainty. DSVerifier determines the control system stability, considering all the plant interval variation set, together with the Finite Word-length (FWL) effects in the digital controller implementation; DSVerifier checks the robust non-fragile stability of a given closed-loop system. The proposed methodology and respective tool are evaluated considering non-fragile control examples from literature. Experimental results show that the approach used in this study is able to foresee fragility problems in robust controllers, which could be overlooked by other existing approaches due to underestimating of FWL effects.

Variable state-space digital filters using series approximations

This paper proposes new state-space formulations of IIR Variable Digital Filters (VDFs) based on frequency transformations. The existing frequency transformation-based state-space VDFs require restrictions on the transfer functions, state-space representations, and tuning characteristics. On the other hand, the proposed method is free from such restrictions. We achieve this goal by applying series approximations to the conventional state-space formulations of frequency transformations. This approach also allows us to realize the proposed VDFs without complicated computations such as the inverse matrix and the square root. Furthermore, the proposed VDFs show high accuracy with respect to the finite wordlength effects as well as the approximation errors.

Criterion for non‐existence of limit cycles in 2D state‐space digital filters described by the Fornasini–Marchesini second model with finite wordlength non‐linearities

This study investigates the problem of global asymptotic stability of fixed-point two-dimensional (2D) digital filters described by the Fornasini–Marchesini second local state-space model with the combined effects of quantisation and overflow non-linearities. Utilising a more precise characterisation of the non-linearities, an improved criterion for the limit cycle-free realisation of 2D digital filters is brought out. The criterion is compared with several previously reported criteria.

On the Extended Thin-Wire Kernel

A number of recent works have shown that moment-method solutions of Hallen’s and Pocklington’s equations with the approximate kernel present undesirable and unphysical oscillations near the ends of the antenna and possibly (depending on the feed model) near the driving point. Such oscillations have been associated with the nonsolvability of the underlying integral equations. They necessarily occur when the number of basis functions is sufficiently large and cannot be blamed on finite computer word length or matrix ill-conditioning effects. In this communication, we show that similar phenomena occur when one uses the extended thin-wire kernel, originally proposed by Poggio and Adams in the 1970s and currently used by the Numerical Electromagnetics Code, which is an antenna modeling software suite that has many variants (e.g., GNEC, EZNEC Pro, SuperNeC, MiniNEC). We discuss the nature and properties of the oscillations. In particular, we show that they are milder and less rapid than those with the approximate kernel. Our main tools of study are: 1) numerical investigations that carefully distinguish from effects due to roundoff and 2) an analytical study for the case of the antenna of infinite length. We finally apply the so-called effective-current method as a remedy to the issues arising from the said oscillations.

A Note on the Induced $${\varvec{l}}_\infty $$ l ∞ Stability of Fixed-Point Digital Filters Without Overflow Oscillations and Instability Due to Finite Wordlength Effects

A recently reported criterion due to Ahn and Lee (Adv Differ Equ 51:1---7, 2012) for the induced $${\varvec{l}}_\infty $$lź stability of fixed-point digital filters without overflow oscillations and instability due to finite wordlength effects is reviewed. This article points out that there is a technical error in the main result and at the same time presents an approach to correcting the main result. Moreover, a relaxed version of the criterion is made available. Illustrative examples are given to demonstrate the effectiveness of the proposed result.

Bounded model checking for fixed-point digital filters

Currently, digital filters are employed in a wide range of signal processing applications, using fixed- and floating-point processors. Regarding the former, some filter implementations may be highly prone to errors, due to problems related to finite word-length. In particular, signal processing modules may produce overflows and unwanted noise, which are caused by quantization and round-off effects, during accumulative-addition and multiplication operations. As a consequence, the system output may overflow or even keep oscillating, which compromises the expected system performance. The present paper addresses this problem and proposes a new methodology for verifying digital filters, called digital systems verifier, which is based on state-of-the-art bounded model checkers that support full C and employ solvers for boolean satisfiability and satisfiability modulo theories. In addition to verifying overflow and limit-cycle occurrences, the present approach can also check output errors and time constraints, based on discrete-time models implemented in C. Experiments conducted during this work show that the proposed methodology is effective, when finding realistic design errors with respect to fixed-point implementations of digital filters. Going further than previous attempts, the present results suggest that the proposed method, in addition to helping designers in determining the number of bits for fixed-point representations, can also aid in defining details about filter realization and structure.

Verification of fixed-point digital controllers using direct and delta forms realizations

The extensive use of fixed-point digital controllers demands a growing effort to prevent design errors that appear in the discrete-time domain. The present article describes a novel verification methodology, which employs bounded model checking (BMC) based on satisfiability modulo theories (SMT) to verify the occurrence of the design errors, because of the finite word-length (FWL) format, in fixed-point digital controllers. Here, the performance realizations of the digital controllers realizations that use delta operators are compared to those that use traditional direct forms. The experimental results show that the delta-form realization substantially reduces the digital controllers' fragility when compared to the direct-form realization. Additionally, the proposed methodology can be very effective and efficient to verify real-world digital controllers, where conclusive results are obtained in nearly 98 % of the benchmarks.

Two Universality Properties Associated with the Monkey Model of Zipf’s Law

The distribution of word probabilities in the monkey model of Zipf’s law is associated with two universality properties: (1) the exponent in the approximate power law approaches −1 as the alphabet size increases and the letter probabilities are specified as the spacings from a random division of the unit interval for any distribution with a bounded density function on [0,1] ; and (2), on a logarithmic scale the version of the model with a finite word length cutoff and unequal letter probabilities is approximately normally distributed in the part of the distribution away from the tails. The first property is proved using a remarkably general limit theorem from Shao and Hahn for the logarithm of sample spacings constructed on [0,1] and the second property follows from Anscombe’s central limit theorem for a random number of independent and identically distributed (i.i.d.) random variables. The finite word length model leads to a hybrid Zipf-lognormal mixture distribution closely related to work in other areas.