Sensitivity reduction in a digital lattice filter

A modification is proposed in the design of the Gray-Markel lattice filter. When a k-parameter is close to 1 in magnitude, this modification reduces the sensitivity with respect to the parameter. Both the number of arithmetic operations and the roundoff noise are identical to those of the original filter.     

Robust relative stability of time-invariant and time-varyinglattice filters

We consider the relative stability of time-invariant and time-varying unnormalized lattice filters. First, we consider a set of lattice filters whose reflection parameters α_i obey |α_i|⩽δ_i and provide necessary and sufficient conditions on the δ_i that guarantee that each time-invariant lattice in the set has poles inside a circle of prescribed radius 1/ρ<1, i.e., is relatively stable with degree of stability ln ρ. We also show that the relative stability of the whole family is equivalent to the relative stability of a single filter obtained by fixing each α_i to δ_i and can be checked with only the real poles of this filter. Counterexamples are given to show that a number of properties that hold for stability of LTI Lattices do not apply to relative stability verification. Second, we give a diagonal Lyapunov matrix that is useful in checking the above pole condition. Finally, we consider the time-varying problem where the reflection coefficients vary in a region where the frozen transfer functions have poles with magnitude less than 1/ρ and provide bounds on their rate of variations that ensure that the zero input state solution of the time-varying lattice decays exponentially at a rate faster than 1/ρ₁>1/ρ     

Adaptive Laguerre-lattice filters

Adaptive Laguerre-based filters provide an attractive alternative to adaptive FIR filters in the sense that they require fewer parameters to model a linear time-invariant system with a long impulse response. We present an adaptive Laguerre-lattice structure that combines the desirable features of the Laguerre structure (i.e., guaranteed stability, unique global minimum, and small number of parameters M for a prescribed level of modeling error) with the numerical robustness and low computational complexity of adaptive FIR lattice structures. The proposed configuration is based on an extension to the IIR case of the FIR lattice filter; it is a cascade of identical sections but with a single-pole all-pass filter replacing the delay element used in the conventional (FIR) lattice filter. We utilize this structure to obtain computationally efficient adaptive algorithms (O(M) computations per time instant). Our adaptive Laguerre-lattice filter is an extension of the gradient adaptive lattice (GAL) technique, and it demonstrates the same desirable properties, namely, (1) excellent steady-state behavior, (2) relatively fast initial convergence (comparable with that of an RLS algorithm for Laguerre structure), and good numerical stability. Simulation results indicate that for systems with poles close to the unit circle, where an (adaptive) FIR model of very high order would be required to meet a prescribed modeling error, an adaptive Laguerre-lattice model of relatively low order achieves the prescribed bound after just a few updates of the recursions in the adaptive algorithm     

One-multiplier adaptive lattice algorithm for recursive filters

The formulation of one-multiplier lattice structures of the Gray-Markel type forinfinite impulse response filters is reviewed. Several special cases of this formulation — including the well-known Gray-Markel normalized lattice—are presented as scaled polynomial versions of the two-multiplier lattice. A new adaptive algorithm is presented for updating the parameters of one-multiplier lattice structured recursive filters. The LMS-based algorithm requires fewer computations than earlier reported algorithms [1]–[4].The research reported here was conducted with funds administered by the Naval Postgraduate School research council.     

Orthogonal and Biorthogonal FIR Hexagonal Filter Banks With Sixfold Symmetry

Recently, hexagonal image processing has attracted attention. The hexagonal lattice has several advantages in comparison with the rectangular lattice, the conventionally used lattice for image sampling and processing. For example, a hexagonal lattice needs less sampling points; it has better consistent connectivity; it has higher symmetry; and its structure is plausible to human vision systems. The multiresolution analysis method has been used for hexagonal image processing. Since the hexagonal lattice has high degree of symmetry, it is desirable that the hexagonal filter banks designed for multiresolution hexagonal image processing also have high order of symmetry, which is pertinent to the symmetry structure of the hexagonal lattice. The orthogonal or prefect reconstruction (PR) hexagonal filter banks that are available in the literature have only threefold symmetry. In this paper, we investigate the construction of orthogonal and PR finite impulse response (FIR) hexagonal filter banks with sixfold symmetry. We obtain block structures of 7-size refinement (seven-channel two-dimensional) orthogonal and PR FIR hexagonal filter banks with sixfold rotational symmetry. $sqrt{7}$-refinement orthogonal and biorthogonal wavelets based on these block structures are constructed. In this paper, we also consider FIR hexagonal filter banks with axial (line) symmetry, and we present a block structure of FIR hexagonal filter banks with pseudo-sixfold axial symmetry.      

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.     

一种新型低复杂度的IIR格型滤波器

格型结构具有良好的抗有限字长效应能力而被应用于很多实时性较高的场合.归一化格型结构具备正交的特性,然而,对于一个N阶数字滤波器,它需要5N+1个乘法器.为了降低其实现复杂度,并保留正交的特性,本文基于单,双,三乘法器格型单元导出了其对应的新型格型单元和格型结构.仿真结果表明,当采用单乘法器新型格型单元时,新结构抵抗有限字长误差的能力与归一化格型结构非常接近,但由于其只需2N+1个乘法器,新结构在实时信号处理中具有更大的实用价值.     

FIR数字滤波器的简化格型实现

格型滤波器在数值计算性能和结构的模块化等方面都优于直接型,但实现起来较复杂。本文提出一种实现PIR数字滤波器的简化格型结构,它所需的硬件设备量只有常规格型实现时的一半,与直接型实现时相当。文中给出了从直接型到简化格型的综合算法。并对线性相位FIR数字滤波器的格型综合算法作了简化。用实例演示了本文的主要结论。     

Minimal realization for lifted periodic filters

Practical applications in signals and systems often deal with lifted models of periodic digital filters, to which well-known methods and computing software for linear time-invariant (LTI) systems can be applied in the analysis and synthesis of periodic filters. When the design of a periodic filter is based on the lifted model, the resultant filter is usually in the lifted LTI form and needs to be further converted into the periodic form for implementational and computational purposes. In this paper, we present a computational procedure for obtaining a minimal realization of periodic digital filters from the fifted model.This work was supported by the Australian Research Council.     

自适应格型IIR滤波器的进化学习算法

本文提出了自适应全零点格型ＩＩＲ滤波器结构及进化学习算法。全零点格型结构的自适应ＩＩＲ滤波器（ＡＩＩＲＦ）具有结构简单,计算量少等优点．新的滤波器结构和进化学习算法相结合,不仅解决了ＡＩＩＲＦ的稳定性问题,而且滤波器参数能够以很大的概率快速收敛于全局最优解,同时允许大动态范围的输入信号,尤其对高阶、极点靠近单位圆的ＡＩＩＲＦ。计算机仿真结果验证了该滤波器的有效性．     

Representations of Linear Dual-Rate System Via Single SISO LTI Filter, Conventional Sampler and Block Sampler

In this brief, it is proved that a linear dual-rate system can be represented via a series cascade of: 1) a conventional expander, a single-input single-output (SISO) linear time-invariant (LTI) filter and a block decimator, or 2) a block expander, an SISO LTI filter and a conventional decimator. Hence, incompatible nonuniform filter banks could achieve perfect reconstruction via LTI filters, conventional samplers and block samplers without expanding the input-output dimension of a subsystem of linear dual-rate systems or converting the nonuniform filter banks to uniform filter banks. The main advantage of the proposed representations is to avoid complicated design of the circuit layout caused by connecting subsystems with large input-output dimension or a lot of subsystems together.     

直接序列扩频通信中基于循环平稳特性的抗窄带干扰算法研究

从扩频通信接收机等效基带信号的模型出发，分析了接收信号中扩频信号分量和窄带干扰分量的循环平稳性以及接收信号的波特率谱相关性与采样之间的关系，并给出基于频移滤波器窄带干扰抑制接收机的自适应实现结构，理论分析和仿真结果表明，使用频移滤波器的接收机抗窄带干扰性能优于基于线性时不变滤波器的接收机。     

Quadratic detectors for energy estimation

The estimation of signal energy is an important part of physics and signal processing. A commonly used energy estimate in signal processing is instantaneous energy that is defined by the square of the signal magnitude at time t, i.e., |x(t)|². For a noisy signal, a standard energy detector, which consists of a linear time-invariant (LTI) filter followed by a magnitude-squared operator, is commonly used to reduce noise and extract signal energy in a certain frequency band. However, due to the temporal response of the LTI filtering, this energy estimate is smeared in time. In addition, it is unclear how this estimate relates to the physical energy in the system that produced the signal. e propose simple quadratic systems producing frequency-selective energy estimates and effective noise reduction with little or no smearing in time. We introduce the new concept of quadratic detectors, discuss desirable time and frequency resolution properties of a general quadratic detector, and study five different applications to demonstrate the simplicity of quadratic detector design and implementation     

Relationship between an LPTV system and the equivalent LTI MIMO structure

It is understood how to transform a linear periodically time-varying (LPTV) filter/ difference equation into an equivalent multiple-input/multiple-output (MIMO) structure, or transfer matrix, with linear time-invariant (LTI) elements, but no published method exists for the reverse operation. The paper presents a technique to transform from the LTI MIMO structure to the original single-input/single-output LPTV difference equation and discusses its implications. In addition, it is shown how this new result is then used to represent parallel and cascade connections of LPTV systems as single LPTV filters, implement order reduction of an LPTV difference equation, and finally obtain an LPTV difference equation representation that is equivalent to an LPTV state-space structure.     

Parametric cumulant based phase estimation of 1-D and 2-Dnonminimum phase systems by allpass filtering

This paper proposes a parametric cumulant-based phase-estimation method for one-dimensional (1-D) and two-dimensional (2-D) linear time-invariant (LTI) systems with only non-Gaussian measurements corrupted by additive Gaussian noise. The given measurements are processed by an optimum allpass filter such that a single Mth-order (M⩾3) cumulant of the allpass filter output is maximum in absolute value. It can be shown that the phase of the unknown system of interest is equal to the negative of the phase of the optimum allpass filter except for a linear phase term (a time delay). For the phase estimation of 1-D LTI systems, an iterative 1-D algorithm is proposed to find the optimum allpass filter modeled either by an autoregressive moving average (ARMA) model or by a Fourier series-based model. For the phase estimation of 2-D LTI systems, an iterative 2-D algorithm is proposed that only uses the Fourier series-based allpass model. A performance analysis is then presented for the proposed cumulant-based 1-D and 2-D phase estimation algorithms followed by some simulation results and experimental results with real speech data to justify their efficacy and the analytic results on their performance. Finally, the paper concludes with a discussion and some conclusions     

Adaptive lattice IIR filtering revisited: convergence issues andnew algorithms with improved stability properties

Several algorithms for adaptive IIR filters parameterized in lattice form can be found in the literature. The salient feature of these structures when compared with the direct form is that ensuring stability is extremely easy. On the other hand, while computing the gradient signals that drive the direct form update algorithms is straightforward, it is not so for the lattice algorithms. This has led to simplified lattice algorithms using gradient approximations. Although, in general, these simplified schemes present the same stationary points as the original algorithms, whether this is also true for convergent points has remained an open problem. This also applies to nongradient-based lattice algorithms such as hyperstability based and the Steiglitz-McBride algorithms. Here, we answer this question in the negative, by showing that for several adaptive lattice algorithms, there exist settings in which the stationary point corresponding to identification of the unknown system is not convergent. In addition, new lattice algorithms with properties are derived. They are based on the cascade lattice structure, which allows the derivation of sufficient conditions for local stability     

Analysis and design of active optical filter structures with two-port couplers

A signal processing approach to modeling, analyzing, and synthesizing a particular integrated photonic architecture of optical filters with tunable gains is presented. This particular architecture has two-port couplers and current-controlled semiconductor optical amplifiers (SOAs) fabricated on the same substrate. The device architecture forms a new lattice filter structure. Layer-peeling-type algorithms are developed for the analysis and synthesis of the device. The role of the adjustable gains in a lossless or lossy device is considered, and a novel stability algorithm for the filter structure is presented.     

Position sensing in brake-by-wire callipers using resolvers

Recent designs for brake-by-wire systems use "resolvers" to provide accurate and continuous measurements for the absolute position and speed of the rotor of the electric actuators in brake callipers (permanent magnet DC motors). Resolvers are absolute-angle transducers that are integrated with estimator modules called "angle tracking observer" and together they provide position and speed measurements. Current designs for angle-tracking observers are unstable in applications with high acceleration and/or speed. In this paper, we introduce a new angle-tracking observer in which a closed-loop linear time-invariant (LTI) observer is integrated with a quadrature encoder. Finite-gain stability of the proposed design and its robustness to three different kinds of parameter variations are proven based on theorems of input-output stability in nonlinear control theory. In our experiments, we examined the performance of our observer and two other methods (a well-known LTI observer and an extended Kalman filter) to estimate the position and speed of a brake-by-wire actuator. The results show that because of the very high speed and acceleration of the actuator in this application, the LTI observer and Kalman filter cannot track the rotor position and diverge. In contrast, with a properly designed open-loop transfer function and selecting a suitable switching threshold, our proposed angle-tracking observer is stable and highly accurate in a brake-by-wire application.     

Trapezoid 2D photonic crystal nanoring resonator-based channel drop filter for WDM systems

In the present work, a nanostructure of trapezoid photonic crystal ring resonator-based channel drop filter is designed for wavelength division multiplexing systems (WDM) to drop a channel at a center peak wavelength of 1543 nm. The proposed channel drop filter is composed of bus waveguide, drop waveguide, trapezoid nanoring resonator and reflector in a two-dimensional (2D) hexagonal lattice with circular rods arranged in air host. The trapezoid nanoring resonator is playing a very important role in WDM systems for dropping a single channel over a wide wavelength range. The photonic band gaps of perfect lattice structure and non-perfect lattice structure are absolutely calculated by plane wave expansion method. The functional properties of the designed filter are evaluated by finite difference time domain method (FDTD). The functional properties are center peak wavelength, dropping efficiency, passband width and quality factor. The FDTD method results show dropping efficiency is 100%, and quality factor is about 514.33 which are highly suitable for WDM systems. Further, lattice constant, inner and outer rod radius and refractive index difference of the structure are varied to tune the filter center peak wavelength and its corresponding functional parameters effects are investigated. The proposed nanoring resonator-based optical filter is ultra-compact size around \(14\, \upmu \hbox {m} \times 8.4\, \upmu \hbox {m}\); hence, it is extremely suitable for WDM-based photonic communication systems and photonic integrated circuits.     

Adaptive-FRESH Filters for Compensation of Cycle-Frequency Errors

This paper deals with the problem of uncertainties in the periodicities of linear almost-periodically time-variant (LAPTV) filters. These filters are usually implemented as a set of branches, each consisting of a frequency shifter followed by a linear time-invariant (LTI) filter. This implementation is also known as FRESH filters. This paper is motivated by the fact that, when there exist errors in the frequency shifts, the optimum set of LTI filters is obtained by canceling the outputs of the corresponding branches. The purpose of this paper is to analyze the nonstationary behavior of adaptive filters in order to mitigate this problem. Our results show that an adaptive filter can offset the errors in the frequency shifts. The reason is that the coefficients of the adaptive filter are updated so that the filter actually performs as a linear periodically time-variant filter for each branch. This allows to track the errors in the frequency shifts when the rate of convergence of the adaptive algorithm is suitably selected. An analytical study of the convergence is presented, which allows to compute the optimal rate of convergence and the mean squared-error attained by the adaptive filter.