General IIR optical filter design for WDM applications usingall-pass filters

A general design algorithm is presented for infinite impulse response (IIR) bandpass and arbitrary magnitude response filters that use optical all-pass filters as building blocks. Examples are given for an IIR multichannel frequency selector, an amplifier gain equalizer, a linear square-magnitude response, and a multi-level response. Major advantages are the efficiency of the IIR filter compared to finite impulse response (FIR) filters, the simplicity of the optical architecture, and its tolerance for loss. A reduced set of unique operating states is discussed for implementing a reconfigurable multichannel selection filter     

Design and performance of adaptive systems based on structured stochastic optimization strategies

The theory and design of linear adaptive filters based on FIR filter structures is well developed and widely applied in practice. However, the same is not true for more general classes of adaptive systems such as linear infinite impulse response adaptive filters (MR) and nonlinear adaptive systems. This situation results because both linear IIR structures and nonlinear structures tend to produce multi-modal error surfaces for which stochastic gradient optimization strategies may fail to reach the global minimum. After briefly discussing the state of the art in linear adaptive filtering, the attention of this paper is turned to MR and nonlinear adaptive systems for potential use in echo cancellation, channel equalization, acoustic channel modeling, nonlinear prediction, and nonlinear system identification. Structured stochastic optimization algorithms that are effective on multimodal error surfaces are then introduced, with particular attention to the particle swarm optimization (PSO) technique. The PSO algorithm is demonstrated on some representative IIR and nonlinear filter structures, and both performance and computational complexity are analyzed for these types of nonlinear systems.     

A new adaptive recursive RLS-based fast-array IIR filter for active noise and vibration control systems

Infinite impulse response filters have not been used extensively in active noise and vibration control applications. The problems are mainly due to the multimodal error surface and instability of adaptive IIR filters used in such applications. Considering these, in this paper a new adaptive recursive RLS-based fast-array IIR filter for active noise and vibration control applications is proposed. At first an RLS-based adaptive IIR filter with computational complexity of order O(n²) is derived, and a sufficient condition for its stability is proposed by applying passivity theorem on the equivalent feedback representation of this adaptive algorithm. In the second step, to reduce the computational complexity of the algorithm to the order of O(n) as well as to improve its numerical stability, a fast array implementation of this adaptive IIR filter is derived. This is accomplished by extending the existing results of fast-array implementation of adaptive FIR filters to adaptive IIR filters. Comparison of the performance of the fast-array algorithm with that of Erikson’s FuLMS and SHARF algorithms confirms that the proposed algorithm has faster convergence rate and ability to reach a lower minimum mean square error which is of great importance in active noise and vibration control applications.     

Recursive weighted median filters admitting negative weights andtheir optimization

A recursive weighted median (RWM) filter structure admitting negative weights is introduced. Much like the sample median is analogous to the sample mean, the proposed class of RWM filters is analogous to the class of infinite impulse response (IIR) linear filters. RWM filters provide advantages over linear IIR filters, offering near perfect “stopband” characteristics and robustness against noise. Unlike linear IIR filters, RWM filters are always stable under the bounded-input bounded-output criterion, regardless of the values taken by the feedback filter weights. RWM filters also offer a number of advantages over their nonrecursive counterparts, including a significant reduction in computational complexity, increased robustness to noise, and the ability to model “resonant” or vibratory behavior. A novel “recursive decoupling” adaptive optimization algorithm for the design of this class of recursive WM filters is also introduced. Several properties of RWM filters are presented, and a number of simulations are included to illustrate the advantages of RWM filters over their nonrecursive counterparts and IIR linear filters     

A general class of nonlinear normalized adaptive filteringalgorithms

The normalized least mean square (NLMS) algorithm is an important variant of the classical LMS algorithm for adaptive linear filtering. It possesses many advantages over the LMS algorithm, including having a faster convergence and providing for an automatic time-varying choice of the LMS stepsize parameter that affects the stability, steady-state mean square error (MSE), and convergence speed of the algorithm. An auxiliary fixed step-size that is often introduced in the NLMS algorithm has the advantage that its stability region (step-size range for algorithm stability) is independent of the signal statistics. In this paper, we generalize the NLMS algorithm by deriving a class of nonlinear normalized LMS-type (NLMS-type) algorithms that are applicable to a wide variety of nonlinear filter structures. We obtain a general nonlinear NLMS-type algorithm by choosing an optimal time-varying step-size that minimizes the next-step MSE at each iteration of the general nonlinear LMS-type algorithm. As in the linear case, we introduce a dimensionless auxiliary step-size whose stability range is independent of the signal statistics. The stability region could therefore be determined empirically for any given nonlinear filter type. We present computer simulations of these algorithms for two specific nonlinear filter structures: Volterra filters and the previously proposed class of Myriad filters. These simulations indicate that the NLMS-type algorithms, in general, converge faster than their LMS-type counterparts     

Performance analysis of the sign algorithm for a constrained adaptive IIR notch filter

Adaptive infinite impulse response (IIR) notch filters are very attractive in terms of their reasonable performances and low computational requirements. Generally, it is very difficult to assess their performances analytically due to their IIR nature. This paper analyzes in detail the steady-state performance of the sign algorithm (SA) for a well-known adaptive IIR notch filter with constrained poles and zeros. Slow adaptation and Gaussianity of the notch filter output are assumed for the sake of analysis. Two difference equations are first established for the convergences in the mean and mean square in the vicinity of the steady state of the algorithm. Steady-state estimation error or bias and mean square error (MSE) of the SA are then derived in closed forms. A coarse stability bound is also derived for the algorithm. Theory-based comparison between the algorithm and the plain gradient (PG) algorithm is done in some detail. Extensive simulations are conducted to demonstrate the validity of the analytical results for both slow and relatively fast adaptations.     

Low-power digital filtering using approximate processing

We present an algorithmic approach to the design of low-power frequency-selective digital filters based on the concepts of adaptive filtering and approximate processing. The proposed approach uses a feedback mechanism in conjunction with well-known implementation structures for finite impulse response (FIR) and infinite impulse response (IIR) digital filters. Our algorithm is designed to reduce the total switched capacitance by dynamically varying the filter order based on signal statistics. A factor of 10 reduction in power consumption over fixed-order filters is demonstrated for the filtering of speech signals     

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     

Suppression of transients in variable recursive digital filterswith a novel and efficient cancellation method

A new method for suppressing transients in recursive infinite impulse response (IIR) digital filters is proposed. The technique is based on modifying the state (delay) variables of the filter when coefficients are changed so that the filter enters a new state smoothly without transient attacks, as originally proposed by Zetterberg and Zhang (1988). In this correspondence, we modify the Zetterberg-Zhang algorithm to render it feasible for efficient implementation. We define a mean square error (MSE) measure for transients and determine the optimal transient suppressor to cancel the transients down to a desired level at the minimum complexity of implementation. The application of the method to all-pole and direct-form II (DF II) IIR filter sections is studied in detail. Time-varying recursive filtering with transient elimination is illustrated for tunable fractional delay filters and variable-bandwidth lowpass filters     

Blind adaptation of stable discrete-time IIR filters in state-space form

Blind deconvolution consists of extracting a source sequence and impulse response of a linear system from their convolution. In the presence of system zeros close to the unit circle, which give rise to very long impulse responses, infinite-impulse-response (IIR) adaptive structures are of use, whose adaptation should be carefully designed in order to guarantee stability. In this paper, we propose a blind-type discrete-time IIR adaptive filter structure realized in state-space form that, with a suitable parameterization of its coefficients, remains stable. The theory is first developed for a two-pole filter, whose numerical behavior is investigated via numerical experiments. The proposed structure/adaptation theory is then extended to a multipole structure realized as a cascade of two-pole filters. Computer-based experiments are proposed and discussed, which aim at illustrating the behavior of the filter cascade on several cases of study. The numerical results obtained show the proposed filters remain stable during adaptation and provide satisfactory deconvolution results.     

A new IIR adaptive echo canceler: GIVE

A novel IIR adaptive gradient instrumental variable echo canceler (GIVE) is presented. Its features include adaptive controllability during double-talk periods in acoustic conference systems; guarantee of global convergence; low computational cost (the same order as the IIR LMS algorithm of the equation error method); and flexible structures (parallel or series-parallel structures). We also show a convergence analysis for gradient adaptive algorithms including GIVE. Based on this analysis, the optimum stepsize for GIVE and three suboptimum algorithms are proposed to accelerate convergence and reduce misadjustment. In addition, a simple method that guarantees the stability of IIR filters and a configuration of GIVE applicable to closed loop systems are presented. These proposals are extensively studied by computer simulations     

秩排序非线性滤波器的自适应优化算法

在图像处理中，秩排序非线性滤波器具有非常重要的作用。本文从加权顺序统计滤波器的自适应优化算法出发，导出了加权中值滤波器、秩选择滤波器和置换滤波器等几种重要的秩排序非线性滤波器的自适应优化设计算法，这些算法采用迭代计算实现，并可以在MAE、MSE和／Z准则下得到最优解。模拟实验表明，这些算法具有非常好的收敛效果，并可以很好地用于类似图像的非平稳信号的处理。     

The gamma-filter-a new class of adaptive IIR filters withrestricted feedback

Generalized feedforward filters, a class of adaptive filters that combines attractive properties of finite impulse response (FIR) filters with some of the power of infinite impulse response (IIR) filters, are described. A particular case, the gamma filter, generalizes Widrow's adaptive transversal filter (adaline) to an infinite impulse response filter. Yet, the stability condition for the gamma filter is trivial, and LMS adaptation is of the same computational complexity as the conventional transversal filter structure. Preliminary results indicate that the gamma filter is more efficient than the adaptive transversal filter. The authors extend the Wiener-Kopf equation to the gamma filter and develop some analysis tools     

An algorithm for efficient, unbiased, equation-error infiniteimpulse response adaptive filtering

An algorithm for efficiently adjusting the coefficients of equation-error infinite impulse response (IIR) adaptive filters is described. Unlike the RLS algorithm, the proposed algorithm yields unbiased filter coefficients. Simulations involving the identification of unknown pole-zero systems demonstrate the algorithm's improved performance over the equation-error RLS algorithm     

Least-squares design of IIR filters with prescribed magnitude andphase responses and a pole radius constraint

This paper presents a method for the frequency domain design of infinite impulse response (IIR) digital filters. The proposed method designs filters approximating prescribed magnitude and phase responses. IIR filters of this kind can have approximately linear-phase responses in their passbands, or they can equalize magnitude and phase responses of given systems. In many cases, these filters can be implemented with less memory and with fewer computations per output sample than equivalent finite impulse response (FIR) digital filters. An important feature of the proposed method is the possibility to specify a maximum radius for the poles of the designed rational transfer function. Consequently, stability can be guaranteed, and undesired effects of implementations using fixed-point arithmetic can be alleviated by restricting the poles to keep a prescribed distance from the unit circle. This is achieved by applying Rouche's theorem in the proposed design algorithm. We motivate the use of IIR filters with an unequal number of poles and zeros outside the origin of the complex plane. In order to satisfy simultaneous specifications on magnitude and phase responses, it is advantageous to use IIR filters with only a few poles outside the origin of the z-plane and an arbitrary number of zeros. Filters of this type are a compromise between IIR filters with optimum magnitude responses and phase-approximating FIR filters. We use design examples to compare filters designed by the proposed method to those obtained by other methods. In addition, we compare the proposed general IIR filters with other popular more specialized structures such as FIR filters and cascaded systems consisting of frequency-selective IIR filters and phase-equalizing allpass filters     

An algorithm for efficient, unbiased, equation-error infiniteimpulse response adaptive filtering

An algorithm for efficiently adjusting the coefficients of equation-error infinite impulse response (IIR) adaptive filters is described. Unlike the recursive least squares (RLS) algorithm, the proposed algorithm yields unbiased filter coefficients. Simulations involving the identification of unknown pole-zero systems demonstrate the algorithm's improved performance over the equation-error RLS algorithm     

Extended fast fixed-order RLS adaptive filters

The existing derivations of conventional fast RLS adaptive filters are intrinsically dependent on the shift structure in the input regression vectors. This structure arises when a tapped-delay line (FIR) filter is used as a modeling filter. We show, unlike what original derivations may suggest, that fast fixed-order RLS adaptive algorithms are not limited to FIR filter structures. We show that fast recursions in both explicit and array forms exist for more general data structures, such as orthonormally based models. One of the benefits of working with orthonormal bases is that fewer parameters can be used to model long impulse responses     

A structural view of asymptotic convergence speed of adaptive IIRfiltering algorithm. II. Finite precision implementation

For pt.I see ibid., vol.41, no.4, p.1493-1517, 1993. Finite precision (FP) implementation is the ultimately inevitable reality of all adaptive filters, including adaptive infinite impulse response (IIR) filters. This paper continues to examine the asymptotic convergence speed of adaptive IIR filters of various structures and algorithms, including the simple constant gain type and the Newton type, but under FP implementation. A stochastic differential equation (SDE) approach is used in the analysis. Such an approach not only greatly simplifies the FP analysis, which is traditionally very involved algebraically, but it also provides valuable information about the first-order as well as the second-order moments that (the latter) are not available using the ordinary differential equation (ODE) approach. The asymptotic convergence speed, as well as the convergent values, of the pertinent moments of FP errors are examined in terms of unknown system pole-zero locations. The adverse effects of lightly damped low-frequency (LDLF) poles resulting from fast sampling on the local transient and convergent behavior of various structures and algorithms are analyzed and compared. The new results agree with the existing ones when reduced to the finite impulse response (FIR) case. In particular, the explosive behavior of pertinent error variances of Newton-type IIR algorithms when the forgetting factor λ=1 is also concluded. Computer simulation verifies the predicted theoretical results     

Tracking properties of a gradient-based second-order adaptive IIRnotch filter with constrained poles and zeros

Gradient-type adaptive IIR notch filters have many attractive merits for various real-life applications since they require a small number of computations and yet demonstrate practical performance. However, it is generally quite difficult to assess their performance analytically. Their tracking properties, in particular, have not yet been investigated. In this paper, the tracking performance of a plain gradient (PG) algorithm is analyzed in detail for a second-order adaptive IIR notch filter with constrained poles and zeros, which takes a linear chirp signal as its input. First, two sets of difference equations for the frequency tracking error and mean square error (MSE) are established in the sense of convergence in the mean and convergence in the mean square, respectively. Closed-form expressions for the asymptotic tracking error and MSE are then derived from these difference equations. An optimum step-size parameter for the algorithm is also evaluated based on the minimization of the asymptotic tracking error or the tracking MSE. It is discovered that the asymptotic tracking error may be driven to zero for a positive chirp rate by selecting a proper step size, which is an interesting property for a real-valued adaptive filtering algorithm. Extensive simulations are performed to support the analytical findings     

Adaptive blind equalization using second- and higher orderstatistics

This paper presents two classes of adaptive blind algorithms based on second- and higher order statistics. The first class contains fast recursive algorithms whose cost functions involve second and third- or fourth-order cumulants. These algorithms are stochastic gradient-based but have structures similar to the fast transversal filters (FTF) algorithms. The second class is composed of two stages: the first stage uses a gradient adaptive lattice (GAL) while the second stage employs a higher order-cumulant (HOC) based least mean squares (LMS) filter. The computational loads for these algorithms are all linearly proportional to the number of taps used. Furthermore, the second class, as various numerical examples indicate, yields very fast convergence rates and low steady state mean square errors (MSE) and intersymbol interference (ISI). MSE convergence analyses for the proposed algorithms are also provided and compared with simulation results