A general weighted median filter structure admitting negativeweights

Weighted median smoothers, which were introduced by Edgemore in the context of least absolute regression over 100 years ago, have received considerable attention in signal processing during the past two decades. Although weighted median smoothers offer advantages over traditional linear finite impulse response (FIR) filters, it is shown in this paper that they lack the flexibility to adequately address a number of signal processing problems. In fact, weighted median smoothers are analogous to normalized FIR linear filters constrained to have only positive weights. It is also shown that much like the mean is generalized to the rich class of linear FIR filters, the median can be generalized to a richer class of filters admitting positive and negative weights. The generalization follows naturally and is surprisingly simple. In order to analyze and design this class of filters, a new threshold decomposition theory admitting real-valued input signals is developed. The new threshold decomposition framework is then used to develop fast adaptive algorithms to optimally design the real-valued filter coefficients. The new weighted median filter formulation leads to significantly more powerful estimators capable of effectively addressing a number of fundamental problems in signal processing that could not adequately be addressed by prior weighted median smoother structures     

Stack filters, stack smoothers, and mirrored thresholddecomposition

Stack smoothers have received considerable attention in signal processing in the past decade. Stack smoothers define a large class of nonlinear smoothers based on positive Boolean functions (PBF) applied in the binary domain of threshold decomposition. Although stack smoothers can offer some advantages over traditional linear FIR filters, they are in essence smoothers lacking the flexibility to adequately address a number of signal processing problems that require bandpass or highpass filtering characteristics. In this paper, mirrored threshold decomposition is introduced, which, together with the associated binary PBF, define the significantly richer class of stack filters. Using threshold logic representation, a number of properties of stack filters are derived. Notably, stack filters defined in the binary domain of mirrored threshold decomposition require the use of double weighting of each sample in the integer domain. The class of recursive stack filters and the corresponding recursive weighted median (RWM) filters in the integer domain admitting negative weights are introduced. The new stack filter formulation leads to a more powerful class of estimators capable of effectively addressing a number of fundamental problems in signal processing that could not adequately be addressed by prior stack smoother structures     

Spectral design of weighted median Filters:A general iterative approach

A new design strategy for weighted median (WM) filters admitting real and complex valued weights is presented. The algorithms are derived from Mallows theory for nonlinear selection type smoothers, which states that the closest linear filter to a selection type smoother in the mean square error sense is the one having as coefficients the sample selection probabilities (SSPs) of the smoother. The new design method overcomes the severe limitations of previous approaches that require the construction of high order polynomial functions and high dimensional matrices. As such, previous approaches could only provide solutions for filters of very small sizes. The proposed method is based on a new closed-form function used to derive the SSPs of any WM smoother. This function allows for an iterative approach to WM filter design from the spectral profile of a linear filter. This method is initially applied to solve the median filter design problem in the real domain, and then, it is extended to the complex domain. The final optimization algorithm allows the design of robust weighted median filters of arbitrary size based on linear filters having arbitrary spectral characteristics.     

Permutation weighted order statistic filter lattices

We introduce and analyze a new class of nonlinear filters called permutation weighted order statistic (PWOS) filters. These filters extend the concept of weighted order statistic (WOS) filters, in which filter weights associated with the input samples are used to replicate the corresponding samples, and an order statistic is chosen as the filter output. PWOS filters replicate each input sample according to weights determined by the temporal-order and rank-order of samples within a window. Hence, PWOS filters are in essence time-varying WOS filters. By varying the amount of temporal-rank order information used in selecting the output for a given observation window size, we obtain a wide range of filters that are shown to comprise a complete lattice structure. At the simplest level in the lattice, PWOS filters reduce to the well-known WOS filter, but for higher levels in the lattice, the obtained selection filters can model complex nonlinear systems and signal distortions. It is shown that PWOS filters are realizable by a N! piecewise linear threshold logic gate where the coefficients within each partition can be easily optimized using stack filter theory. Simulations are included to show the advantages of PWOS filters for the processing of image and video signals.     

Design of linear combination of weighted medians

This paper introduces a novel nonlinear filtering structure: the linear combination of weighted medians (LCWM). The proposed filtering scheme is modeled on the structure and design procedure of the linear-phase FIR highpass (HP) filter in that the linear-phase FIR HP filter can be obtained by changing the sign of the filter coefficients of the FIR lowpass (LP) filter in the odd positions. The HP filter can be represented as the difference between two LP subfilters that have all positive coefficients. This representation of the FIR HP filter is analogous to the difference of estimates (DoE) such as the difference of medians (DoM). The DoM is essentially a nonlinear HP filter that is commonly used in edge detection. Based on this observation, we introduce a class of LCWM filters whose output is given by a linear combination of weighted medians of the input sequence. We propose a method of designing the 1-D and 2-D LCWM filters satisfying required frequency specifications. The proposed method adopts a transformation from the FIR filter to the LCWM filter. We show that the proposed LCWM filter can offer various frequency filtering characteristics including “LP,” “bandpass (BP),” and “HP” responses     

Stack filters and selection probabilities

Based on the fact that the output of a given stack filter can be determined if the ranks of the samples in the input window are known and that this output always equals one of the samples in the input window, rank and sample selection probabilities are defined. The output distribution of the stack filter of size N with independent identically distributed (i.i.d.) inputs can be expressed as a weighted sum of the ith, i=1, 2, ..., N order statistics, where the rank selection probabilities are the weights. The sample selection probabilities equal the impulse response coefficients of a finite impulse response (FIR) filter whose output spectrum is closest, of all linear filters, to that of the stack filter for i.i.d. Gaussian inputs. Results are also derived for correlated inputs. Robustness and detail preserving properties of stack filters are related to the selection probabilities. Other statistical properties are also derived. Finally, methods to compute the selection probabilities of the stack filter from its positive Boolean function and the selection probabilities of the weighted median filter from its weights are given in detail     

Order-recursive FIR smoothers

This paper introduces order-recursive FIR smoothers and shows that order-recursive FIR filters are special forms that occur when no future data values are used to estimate the signal. The formulation leads naturally to generalizations of the concepts of prediction-error basis and Cholesky factorization which are well known in FIR filter design     

Fast adaptation and performance characteristics of FIR-WOS hybridfilters

Fast adaptive algorithms are developed for training weighted order statistic (WOS) filters and FIR-WOS hybrid (FWH) filters under the mean absolute error (MAE) criterion. These algorithms are based on the threshold decomposition of real-valued signals introduced in this paper. With this method an N-length WOS filter can be implemented by thresholding the input signals at most N times independent of the accuracy used. Beside saving in computations, the proposed algorithms can be applied to process arbitrary real-valued signals directly. Performance characteristics of FWH filters in 1-D and 2-D signal restoration are investigated through computer simulations. We show that both in restoration of signals containing edges and in the case of heavy tailed nonGaussian noise, considerable improvement in performance can be achieved with FWH filters over WOS filters, Ll filters, and adaptive linear filters. Two new FWH filter design strategies are found for removal of impulsive noise and for restoration of a square wave, respectively     

Design of Gaussian Approximate Filter and Smoother for Nonlinear Systems with Correlated Noises at One Epoch Apart

In this study, the authors investigate the filtering and smoothing problems of nonlinear systems with correlated noises at one epoch apart. A pseudomeasurement equation is firstly reconstructed with a corresponding pseudomeasurement noise, which is no longer correlated with the process noise. Based on the reconstructed measurement model, new Gaussian approximate (GA) filter and smoother are derived, from which Kalman filter and smoother can be obtained for linear systems. For nonlinear systems, different GA filters and smoothers can be developed through utilizing different numerical methods for computing Gaussian-weighted integrals involved in the proposed solution. Numerical examples concerning univariate nonstationary growth model, passive ranging problem, and target tracking show the efficiency of the proposed filtering and smoothing methods for nonlinear systems with correlated noises at one epoch apart.     

Fast algorithms for weighted myriad computation by fixed-pointsearch

This paper develops fast algorithms to compute the output of the weighted myriad filter. Myriad filters form a large and important class of nonlinear filters for robust non-Gaussian signal processing and communications in impulsive noise environments. Just as the weighted mean and the weighted median are optimized for the Gaussian and Laplacian distributions, respectively, the weighted myriad is based on the class of α-stable distributions, which can accurately model impulsive processes. The weighted myriad is an M-estimator that is defined in an implicit manner; no closed-form expression exists for it, and its direct computation is a nontrivial and prohibitively expensive task. In this paper, the weighted myriad is formulated as one of the fixed points of a certain mapping. An iterative algorithm is proposed to compute these fixed points, and its convergence is proved rigorously. Using these fixed point iterations, fast algorithms are developed for the weighted myriad. Numerical simulations demonstrate that these algorithms compute the weighted myriad with a high degree of accuracy at a relatively low computational cost     

Programmable discrete-time type I and type II FIR filter design on the memristor crossbar structure

A new method for effective realization of discrete-time type I and type II FIR filters on the memristor crossbar structure is developed in this paper. For this purpose, first the analog input signal (to be filtered using the discrete-time filter) is discretized using the classical switched-capacitor circuit and then all of the required delayed samples of this discrete-time signal are generated using the circuit designed for this purpose. Next, the weighted sum of these delayed samples of the original discrete-time signal (which forms the output of the FIR filter under consideration) is produced using the memristor crossbar structure. The proposed structure for FIR filter design is, compared to classical methods, advantageous in the way that it does not need any processors or A/D converter. Moreover, it is fully implemented using analog devices and consequently free of round-off error. Another related contribution of this paper is the circuit proposed for automatic tuning the memristance of the given memristor to the desired value with a high accuracy. Four numerical examples, including the application of the proposed FIR filter for demodulation of AM signals, are studied and HSPICE simulations are presented.     

Hybrid Polynomial Filters for Gaussian and Non-Gaussian Noise Environments

Traditional polynomial filtering theory, based on linear combinations of polynomial terms, is able to approximate important classes of nonlinear systems. The linear combination of polynomial terms, however, yields poor performance in environments characterized by Gaussian and heavy tailed distributions. Weighted median and weighted myriad filters, in contrast, are well known for their outlier suppression and detail preservation properties. It is shown here that the weighted median and weighted myriad methodologies are naturally extended to the polynomial sample case, yielding hybrid filter structures that exploits the higher-order statistics of the observed samples while simultaneously being robust to outliers for both Gaussian and heavy-tailed distributions environments. Moreover, the introduced hybrid polynomial filter classes are well motivated by analysis of cross and square term statistics of Gaussian and heavy-tailed distributions. A presented asymptotic tail mass analysis shows that polynomial terms, both under Gaussian and heavy-tailed noise statistics, have heavier tails than the observed samples, indicating that robust combination methods should be utilized to avoid undue influence of outliers. Further analysis shows weighted median processing of polynomial terms for the Gaussian noise case, and weighted median and weighted myriad processing of cross and square terms, respectively, for the heavy-tailed noise case, are justified from a maximum likelihood perspective. Filters parameter optimization procedures are also presented. Finally, the effectiveness of hybrid filters is demonstrated through simulations that include temporal, spectrum, and bispectrum analysis     

Affine order-statistic filters: “medianization” oflinear FIR filters

This paper introduces a novel, data-adaptive filtering framework: affine order-statistic filters. Affine order-statistic filters operate effectively on a wide range of signal statistics, are sensitive to the dispersion of the observed data, and are therefore particularly useful in the processing of nonstationary signals. These properties result from the introduction of a tunable affinity function that measures the affinity, or closeness, of observation samples in their natural order to their corresponding order statistics. The obtained affinity measures are utilized to control the influence of individual samples in the filtering process. Depending on the spread of the affinity function, which is controlled by a single parameter γ, affine order-statistic filters operate effectively in various environments ranging from Gaussian to impulsive. The class of affine order-statistic filters subsumes the family of weighted order-statistic (WOS) affine filters and the class of FIR affine filters. We focus on a representative of the WOS affine filter class-the median affine filter-whose behavior can be tuned from that of a linear FIR filter to that of a robust median filter by narrowing the affinity function to a process referred to as medianization. The superior performance of affine order-statistic filters is demonstrated in two applications     

Optimal weighted median filtering under structural constraints

A new expression for the output moments of weighted median filtered data is derived. The noise attenuation capability of a weighted median filter can now be assessed using the L-vector and M-vector parameters in the new expression. The second major contribution of the paper is the development of a new optimality theory for weighted median filters. This theory is based on the new expression for the output moments, and combines the noise attenuation and some structural constraints on the filter's behavior. In certain special cases, the optimal weighted median filter can be obtained by merely solving a set of linear inequalities. This leads in some cases to closed form solutions for optimal weighted median filters. Some applications of the theory developed in this paper, in 1-D signal processing and image processing are discussed. Throughout the analysis, some striking similarities are pointed out between linear FIR filters and weighted median filters     

基于FPGA的FIR滤波器设计

数字滤波器在数字信号处理中占有很重要的地位,该文介绍了FIR滤波器的两种实现算法：乘累加算法和优化的分布式算法,其中分布式算法作为优化算法进行研究。其次,根据FIR滤波器理论,采用线性相位结构优化滤波器的设计。并给出了FIR滤波器的模块划分和FIR滤波器的主要模块的实现,最后对FIR滤波器进行了系统仿真和验证。     

Filter design for MIMO sampling and reconstruction

We address the problem of finite impulse response (FIR) filter design for uniform multiple-input multiple-output (MIMO) sampling. This scheme encompasses Papoulis' generalized sampling and several nonuniform sampling schemes as special cases. The input signals are modeled as either continuous-time or discrete-time multiband input signals, with different band structures. We present conditions on the channel and the sampling rate that allow perfect inversion of the channel. Additionally, we provide a stronger set of conditions under which the reconstruction filters can be chosen to have frequency responses that are continuous. We also provide conditions for the existence of FIR perfect reconstruction filters, and when such do not exist, we address the optimal approximation of the ideal filters using FIR filters and a minmax l/sub 2/ end-to-end distortion criterion. The design problem is then reduced to a standard semi-infinite linear program. An example design of FIR reconstruction filters is given.     

Median power and median correlation theory

We show that the maximum likelihood (ML) estimate of location under the Laplacian model, which forms the basis for weighted median filters, can be generalized to correlation estimates based on weighted medians. Much like linear sample correlations, the resultant median correlation estimates have a surprisingly simple structure. Unlike linear correlations, median correlations are robust to data contamination. Notably, weights in this framework do not assume fixed values as with weighted median filters but take on random values determined by the underlying data itself. The underlying parameters associated with the sample median correlations are obtained, leading to well-defined expressions that can be used in subspace-based signal processing algorithms. The properties of median correlations are illustrated through a number of simulations where the MUltiple SIgnal Classification (MUSIC) algorithm is applied on linear and median sample correlation matrices for real-valued frequency estimation applications. This paper thus unveils new and powerful capabilities of weighted medians for use in modern signal processing applications.     

Performance analysis of LMS adaptive prediction filters

The conditions required to implement real-time adaptive prediction filters that provide nearly optimal performance in realistic input conditions are delineated. The effects of signal bandwidth, input signal-to-noise ratio (SNR), noise correlation, and noise nonstationarity are explicitly considered. Analytical modeling, Monte Carlo simulations and experimental results obtained using a hardware implementation are utilized to provide performance bounds for specified input conditions. It is shown that there is a nonlinear degradation in the signal processing gain as a function of the input SNR that results from the statistical properties of the adaptive filter weights. The stochastic properties of the filter weights ensure that the performance of the adaptive filter is bounded by that of the optimal matched filter for known stationary input conditions     

Perfect-reconstruction filter banks having linear-phase FIR filterswith equiripple response

The design of equiripple linear-phase analysis and synthesis FIR filters of two-channel perfect-reconstruction (PR) filter banks is formulated as the minimization of a weighted peak-error under both linear inequality (arising from the desired responses of the analysis filters) and nonlinear equality (PR) constraints. The effectiveness of a proposed method to solve the design problem (a modified dual-affine scaling variant of Karmarkar's (1989) algorithm and an approximation scheme) is illustrated through several design examples     

基于FIR与相位补偿IIR滤波器的雷达回波信号处理

FIR与IIR频率选择滤波器的设计,被广泛应用于数字信号处理领域之中。文章以雷达回波信号的数字处理为例,首先分别设计FIR,IIR滤波器完成了对信号特定频率分量的滤除。进而,针对IIR滤波器的非线性相位,基于最优化设计全通系统实现了相位补偿,并对FIR,IIR滤波器进行了综合比较。