Synthesis and performance of a new class of implementation forhigh-order recursive digital filters

The authors investigate a new class of approaches to the synthesis of high-order recursive digital filters. These are all characterised by partitioning the overall filter into a set of parallel subfilters, each of which is of moderate order. The subfilters may be any of the classical implementations: direct form, parallel, cascade or lattice. The authors investigate cascade and lattice implementation of the subfilters. The investigation includes selection of the number of subfilters to use, how to derive their coefficients and theoretical noise calculations. This is supported by simulation testing of a variety of real and synthetic filters using maximal length binary sequences (MLS) as input as this provides information on both the linear and nonlinear errors involved. It is found that particularly when filter order is 50 or more, the new structures have reduced quantisation noise and improved tolerance to quantisation than their classical counterparts     

The structure and design of realizable decision feedback equalizers for IIR channels with colored noise

A simple algorithm for optimizing decision feedback equalizers (DFEs) by minimizing the mean-square error (MSE) is presented. A complex baseband channel and correct past decisions are assumed. The dispersive channel may have infinite impulse response, and the noise may be colored. Consideration is given to optimal realizable (stable and finite-lag smoothing) forward and feedback filters in discrete time. They are parameterized as recursive filters. In the special case of transmission channels with finite impulse response and autoregressive noise, the minimum MSE can be attained with transversal feedback and forward filters. In general, the forward part should include a noise-whitening filter (the inverse noise model). The finite realizations of the filters are calculated using a polynomial equation approach to the linear quadratic optimization problem. The equalizer is optimized essentially by solving a system of linear equations Ax=B, where A contains transfer function coefficients from the channel and noise model. No calculation of correlations is required with this method. A simple expression for the minimal MSE is presented. The DFE is compared to MSE-optimal linear recursive equalizers. Expressions for the equalizer in the limiting case of infinite smoothing lags are also discussed.<>     

Optimal adaptive multiuser detection in unknown multipath channels

An optimal multiuser detector in the weighted least squares (WLS) sense is derived. This detector, which includes the maximum likelihood multiuser detector as a special case, consists of two parts: a bank of linear fractionally chip spaced minimum mean squared error (MMSE) filters, and a nonlinear WLS metric minimizer. It is shown that the symbol spaced samples at the output of the MMSE filter bank provide a set of sufficient statistics for WLS detection. The relationship between the taps of a centralized decision feedback detector and the MMSE filter bank is derived. It is proven that all the necessary parameters for implementing the WLS detector can be realized by adaptively training a centralized decision feedback detector. Therefore, the WLS detector achieves optimal joint synchronization and data detection even in the presence of colored noise, such as narrowband interference, without any a priori knowledge of the users' signatures, multipath channel taps or statistics of the colored noise. Significant features of the WLS detector are that: (1) the WLS detector is a generalization of the maximum likelihood multiuser detector that employs a bank of matched filters; (2) it is implemented adaptively; and (3) it has structural flexibility in terms of implementation complexity     

Noise Reduction in Two-Dimensional Recursive Digital Filters with Discrete Optimized Error Feedback Coefficients

This paper treats the minimization problem for roundoff noise in two-dimensional (2-D) recursive digital filters with an error feedback (EF). The EF is known as an available scheme for the reduction of roundoff noise, and this can be achieved by extracting a roundoff error from the quantizer and feeding it back to the adder. Since the EF circuit is an additional device of the digital filter, the bit length of the coefficients should be as short as possible for the fast arithmetic and low-cost realization. In this paper, we propose an algorithm to optimize the EF coefficients in a discrete space. The proposed algorithm is based on the branch and bound method by estimating the lower bound of the optimal solution, and the optimality of the obtained solution can be guaranteed. Finally, we?give?a?numerical example to demonstrate the effectiveness of the proposed method, and show the optimal solution can be found without using the round-robin algorithm.     

Bias removal in equation-error adaptive IIR filters

In the equation-error formulation of adaptive IIR filters, the estimated parameters contain bias when there is noise in the desired response. A method that can eliminate this bias is investigated. The idea is to maintain a quadratic constraint on the feedback coefficients so that the noise contributes only a constant term to the mean-square error. This term does not affect minimization and thus the bias is eliminated. A quadratically constrained stochastic gradient search method is applied for optimization and convergence behavior, when the noise is white, is analyzed. Adaptation of the feedback FIR filter in second-order cascade form, useful for stability monitoring, is also considered. When the noise is nonwhite, the technique requires an adaptive whitening filter. Simulation results are included to demonstrate the bias removal capability of the method, corroborate the theoretical developments, and compare with existing techniques     

Design of High-Order Extrapolated Impulse Response FIR Filters with Signed Powers-of-Two Coefficients

Expensive multiplication operations can be replaced by simpler additions and hardwired shifters so as to reduce power consumption and area size, if the coefficients of a digital filter are signed power-of-two (SPT). As a consequence, FIR digital filters with SPT coefficients have been widely studied in the last three decades. However, most approaches for the design of FIR filters with SPT coefficients focus on filters with length less than 100. These approaches are not suitable for the design of high-order filters because they require excessive computation time. In this paper, an approach for the design of high-order filters with SPT coefficients is proposed. It is a two-step approach. Firstly, the design of an extrapolated impulse response (EIR) filter is formulated as a standard second-order cone programming (SOCP) problem with an additional coefficient sensitivity constraint for optimizing its finite word-length effect. Secondly, the obtained continuous coefficients are quantized into SPT coefficients by recasting the filter-design problem into a weighted least squares (WLS) sequential quadratic programming relaxation (SQPR) problem. To further reduce implementation complexity, a graph-based common subexpression elimination (CSE) algorithm is utilized to extract common subexpressions between SPT coefficients. Simulation results show that the proposed method can effectively and efficiently design high-order SPT filters, including Hilbert transformers and half-band filters with SPT coefficients. Experiment results indicate that 0.81N∼0.29N adders are required for 18-bit N-order FIR filters (N=335∼3261) to meet the given magnitude response specifications.     

Roundoff Noise Minimization for 2-D State-Space Digital Filters Using Joint Optimization of Error Feedback and Realization

The joint optimization problem of error feedback and realization for two-dimensional (2-D) state-space digital filters to minimize the effects of roundoff noise at the filter output subject to$L_2$-norm dynamic-range scaling constraints is investigated. It is shown that the problem can be converted into an unconstrained optimization problem by using linear-algebraic techniques. The unconstrained optimization problem at hand is then solved iteratively by applying an efficient quasi-Newton algorithm with closed-form formulas for key gradient evaluation. Analytical details are given as to how the proposed technique can be applied to the cases where the error-feedback matrix is a general, block-diagonal, diagonal, or block-scalar matrix. A case study is presented to illustrate the utility of the proposed technique.     

一种新型的高阶时域有限差分方法

相比于传统高阶时域有限差分算法(FDTD)而言,该文提出了一种改进的高阶FDTD的优化方法,该算法基于安培环路定律,通过计算机技术寻找到一组最优的系数使得FDTD方法的全局色散误差达到最小,通过不同分辨率下的点源辐射模拟证明了该方法在较低分辨率的情况下仍然具有极低的相位误差,对于解决电大尺寸结构建模中的数值色散等问题提供了有效的解决方案。     

Minimax estimation of unknown deterministic signals in colorednoise

The estimation of a deterministic signal corrupted by random noise is considered. The strategy is to find a linear noncausal estimator which minimizes the maximum mean square error over an a priori set of signals. This signal set is specified in terms of frequency/energy constraints via the discrete Fourier transform. Exact filter expressions are given for the case of additive white noise. For the case of additive colored noise possessing a continuous power spectral density, a suboptimal filter is derived whose asymptotic performance is optimal. Asymptotic expressions for the minimax estimator error are developed for both cases. The minimax filter is applied to random data and is shown to solve asymptotically a certain worst-case Wiener filter problem     

A Direct Design of Oversampled Perfect Reconstruction FIR Filter Banks

We address a problem to find optimal synthesis filters of oversampled uniform finite-impulse-response (FIR) filter banks (FBs) yielding perfect reconstruction (PR), when we are given an analysis FB, in the case where all the filters have the same length that is twice a factor of downsampling. We show that in this class of FBs, a synthesis FB that achieves PR can be found in closed form with elementary matrix operations, unlike conventional design methods with numerical optimization. This framework allows filter coefficients to be complex as well as real. Due to the extra degrees of freedom in a synthesis FB provided by oversampling, we can determine optimal coefficients of synthesis filters that meet certain criteria. We introduce in this paper two criteria: variance of additive noise and stopband attenuation. We show theoretical results of optimal synthesis filters that minimize these criteria and design examples of oversampled linear-phase FIR FBs and DFT-modulated FBs. Moreover, we discuss applications to signal reconstruction from incomplete channel data in transmission and inverse transform of windowed discrete Fourier transform with 50% overlapping.     

A multiscale relaxation algorithm for SNR maximization innonorthogonal subband coding

Develops a technique for improving the applicability of complete, nonorthogonal, multiresolution transforms to image coding. As is well known, the L(2) norm of the quantization errors is not preserved by nonorthogonal transforms, so the L(2) reconstruction error may be unacceptably large. However, given the quantizers and synthesis filters, the authors show that this artifact can be eliminated by formulating the coding problem as that of minimizing the L(2) reconstruction error over the set of possible encoded images. With this new formulation, the coding problem becomes a high-dimensional, discrete optimization problem and features a coupling between the redundancy-removing and quantization operations. A practical solution to the optimization problem is presented in the form of a multiscale relaxation algorithm, using inter- and intrascale quantization noise feedback filters. Bounds on the coding gain over the standard coding technique are derived. A simple extension of the algorithm allows for the use of a weighted L(2) error criterion and deadband (non-MMSE) quantizers. Experiments using biorthogonal spline filter banks demonstrate appreciable SNR gains over the standard coding technique, and comparable visual improvements.     

Optimal design of two-channel nonuniform-division FIR filter bankswith -1, 0, and +1 coefficients

This paper deals with the optimal design of two-channel nonuniform-division filter (NDF) banks whose linear-phase FIR analysis and synthesis filters have coefficients constrained to -1, 0, and +1 only. Utilizing an approximation scheme and a weighted least squares algorithm, we present a method to design a two-channel NDF bank with continuous coefficients under each of two design criteria, namely, least-squares reconstruction error and stopband response for analysis filters and equiripple reconstruction error and least-squares stopband response for analysis filters. It is shown that the optimal filter coefficients can be obtained by solving only linear equations. In conjunction with the proposed filter structure, a method is then presented to obtain the desired design result with filter coefficients constrained to -1, 0, and +1 only. The effectiveness of the proposed design technique is demonstrated by several simulation examples     

Design for 2-D error feedback networks with identical coefficient sets

In realization of recursive digital filters with fixed point arithmetic, an error caused by roundoff arises. It is known that the level of the roundoff noise of an IIR filter tends to be high when the poles are close to the unit circle. Error feedback (EF) is an effective method to reduce the roundoff noise. It is desirable to design an EF network using as few parameters as possible in order to keep computational costs low. In this paper, we propose a method for designing a 2D EF network with identical coefficient sets. That is, the EF coefficients are divided into several subsets such that all the elements within each set have the same absolute value. In order to optimize the coefficient sets, we propose an algorithm by using the genetic algorithm. In the numerical example, we demonstrate the effectiveness of the proposed method.     

Design of high-order active filters in integrated circuits

This correspondence considers the design of high-order filters using follow-the-leader feedback configurations and active-R biquadratic blocks. Statistical sensitivities of various second-order active-R and active-RC sections are compared, and experimental results of a sixth-order Butterworth bandpass filter are also included. The FLF configuration and versatile active-R blocks are also suitable for fully integrated filters.     

Synthèse des filtres numériques non récursifs à coefficients quantifiés

The design of finite impulse response linear phase digital filters in direct and cascade structures is investigated with regard to coefficient wordlength and bitmultiplier product. Solving the continuous approximation problem by minimising the statistical wordlength of the coefficients provides a good insight for comparing the structures and chosing the optimal filter order. A comparison of the structures, for various tolerance schemes, shows that the direct form is generally superior with regard to the bitmultiplier product.     

Optimal a posteriori time domain filter for average evokedpotentials

Evoked potentials measured with scalp electrodes are often described as a deterministic process corrupted by unrelated noise. The common procedure to determine the signal is to average N repetitive measurements. By obtaining additional information from the N measurements, signal detection can be improved. An algorithm that estimates the signal autocorrelation from N given measurements is proposed. The estimator is consistent and unbiased, and its variance tends to 0 as o(N). Two filters that are applied to the average response are introduced. Both depend on the estimation of the signal and the noise autocorrelations. One filter is based on the assumption that the average response is a stationary process. The second filter coefficients are obtained by minimizing the mean squared error (MSE) of an optimal filter of a nonstationary process applied on a single sweep. When a small number of sweeps are averaged, the stationary assumption is adequate, and the MSE of the stationary optimal filter is two to five times less than the MSE of the average response. When a large number of measurements are considered, the error in estimating the autocorrelations decreases. In this case, applying the optimal filter for a nonstationary process leads to a significant improvement in the signal estimation.     

Design and analysis of vector color error diffusion halftoningsystems

Traditional error diffusion halftoning is a high quality method for producing binary images from digital grayscale images. Error diffusion shapes the quantization noise power into the high frequency regions where the human eye is the least sensitive. Error diffusion may be extended to color images by using error filters with matrix-valued coefficients to take into account the correlation among color planes. For vector color error diffusion, we propose three contributions. First, we analyze vector color error diffusion based on a new matrix gain model for the quantizer, which linearizes vector error diffusion. The model predicts the key characteristics of color error diffusion, esp. image sharpening and noise shaping. The proposed model includes linear gain models for the quantizer by Ardalan and Paulos (1987) and by Kite et al. (1997) as special cases. Second, based on our model, we optimize the noise shaping behavior of color error diffusion by designing error filters that are optimum with respect to any given linear spatially-invariant model of the human visual system. Our approach allows the error filter to have matrix-valued coefficients and diffuse quantization error across color channels in an opponent color representation. Thus, the noise is shaped into frequency regions of reduced human color sensitivity. To obtain the optimal filter, we derive a matrix version of the Yule-Walker equations which we solve by using a gradient descent algorithm. Finally, we show that the vector error filter has a parallel implementation as a polyphase filterbank.     

Perfect reconstruction versus MMSE filter banks in source coding

Classically, the filter banks (FBs) used in source coding schemes have been chosen to possess the perfect reconstruction (PR) property or to be maximally selective quadrature mirror filters (QMFs). This paper puts this choice back into question and solves the problem of minimizing the reconstruction distortion, which, in the most general case, is the sum of two terms: a first one due to the non-PR property of the FB and the other being due to signal quantization in the subbands. The resulting filter banks are called minimum mean square error (MMSE) filter banks. Several quantization noise models are considered. First, under the classical white noise assumption, the optimal positive bit rate allocation in any filter bank (possibly nonorthogonal) is expressed analytically, and an efficient optimization method of the MMSE filter banks is derived. Then, it is shown that while in a PR FB, the improvement brought by an accurate noise model over the classical white noise one is noticeable, it is not the case for the MMSE FB. The optimization of the synthesis filters is also performed for two measures of the bit rate: the classical one, which is defined for uniform scalar quantization, and the order-one entropy measure. Finally, the comparison of rate-distortion curves (where the distortion is minimized for a given bit rate budget) enables us to quantify the SNR improvement brought by MMSE solutions     

L(p) norm design of stack filters.

Addresses the problem of designing optimal stack filters by employing an L(p) norm of the error between the desired signal and the estimated one. It is shown that the L(p) norm can be expressed as a linear function of the decision errors at the binary levels of the filter. Thus, an L(p)-optimal stack filter can be determined as the solution of a linear program. The conventional design of using the mean absolute error (MAE), therefore, becomes a special ease of the general L(p) norm-based design developed here. Other special cases of the proposed approach, of particular interest in signal processing, are the problems of optimal mean square error (p=2) and minimax (p-->infinity) stack filtering. Since an Linfinity optimization is a combinatorial problem, with its complexity increasing faster than exponentially with the filter size, the proposed L(p ) norm approach to stack filter design offers an additional benefit of a sound mathematical framework to obtain a practical engineering approximation to the solution of the minimax optimization problem. The conventional MAE design of an important subclass of stack filters, the weighted order statistic filters, is also extended to the L(p) norm-based design. By considering a typical application of restoring images corrupted with impulsive noise, several design examples are presented, to illustrate the performance of the L(p)-optimal stack filters with different values of p. Simulation results show that the L(p)-optimal stack filters with p=/>2 provide a better performance in terms of their capability in removing impulsive noise, compared to that achieved by using the conventional minimum MAE stack filters.     

Adaptive recovery of a chirped sinusoid in noise. II. Performanceof the LMS algorithm

For pt.I see ibid., vol.39, no. 3, p.583-94 (1991). The authors present a methodology for evaluating the tracking behavior of the least-mean square (LMS) algorithm for the nontrivial case of recovering a chirped sinusoid in additive noise. A complete closed-form analysis of the LMS tracking properties for a nonstationary inverse system modeling problem is also presented. The mean-square error (MSE) performance of the LMS algorithm is calculated as a function of the various system parameters. The misadjustment or residual of the adaptive filter output is the excess MSE as compared to the optimal filter for the problem. It is caused by three errors in the adaptive weight vector: the mean lag error between the (time-varying mean) weight and the time-varying optimal weight; the fluctuations of the lag error; and the noise misadjustment which is due to the output noise. These results are important because they represent a precise analysis of a nonstationary deterministic inverse modeling system problem with the input being a colored signal. The results are in agreement with the form of the upper bounds for the misadjustment provided by E. Eweda and O. Macchi (1985) for the deterministic nonstationarity