Optimal signal reconstruction in noisy filter bank systems:multirate Kalman synthesis filtering approach

A multirate Kalman synthesis filter is proposed in this paper to replace the conventional synthesis filters in a noisy filter bank system to achieve optimal reconstruction of the input signal. Based on an equivalent block representation of subband signals, a state-space model is introduced for an M-band filter bank system with subband noises. The composite effect of the input signal, analysis filter bank, decimators, and interpolators is represented by a multirate state-space model. The input signal is embedded in the state vector, and the corrupting noises in subband paths are generally considered as additive noises. Hence, the signal reconstruction problem in the M-band filter bank systems with subband noises becomes a state estimation procedure in the resultant multirate state-space model. The multirate Kalman filtering algorithm is then derived according to the multirate state-space model to achieve optimal signal reconstruction in noisy filter bank systems. Based on the optimal state estimation theory, the proposed multirate Kalman synthesis filter provides the minimum-variance reconstruction of the input signal. Two numerical examples are also included. The simulation results indicate that the performance improvement of signal reconstruction in noisy filter bank systems is remarkable     

基于自适应Kalman滤波的二维有噪子带信号恢复

基于子带信号的多通道表示（multichannel representation)和输入信号的动态特征,本文尝试推出了一种多分辨率状态空间模型,它与带相加子带噪声的滤波器组（Filter Bank)系统是等价的,于是使有噪子带信号的恢复可表述为相应多分辨率态空间模型的最优状态估计问题。进一步又利用信号的向量动态模型,发展了适于二维Kalman滤波的二维多分辨率状态空间模型,根据信号行为的分布,目标平面（object plane)可分割为不同的区域并用不同的向量动态模型来表征信号的非平衡分布,计算机数字仿真结果进一步证实了本文所提出了二维多分辨率Kalman滤波器性能的优越性。     

基于稀疏重构的色噪声背景下未知线谱信号估计

针对色噪声背景下的未知线谱信号估计问题,该文提出一种基于分子频带处理的稀疏重构类线谱估计方法。首先,利用多速率余弦调制滤波器组对观测信号进行子带分解,得到功率谱相对平坦的子带信号。之后,在每个子带信号上,利用基于迭代最小化的稀疏学习方法进行线谱估计,并将各子带上的线谱估计结果进行频域综合滤波以及门限判决等处理。最终得到色噪声背景下的线谱估计结果。理论推导及仿真实验表明所提方法在色噪声背景下具有较好的线谱估计性能。其能够有效地去除色噪声背景,同时保留稀疏重构类线谱估计方法所具有的高频率分辨力等优点。     

Mixed H2/H∞ filtering design inmultirate transmultiplexer systems: LMI approach

A mixed H₂/H_∞ filter design is proposed for multirate transmultiplexer systems with dispersive channel and additive noise. First, a multirate state-space representation is introduced for the transmultiplexer with the consideration of channel dispersion. Then, the problem of signal reconstruction can be regarded as a state estimation problem. In order to design an efficient separating filterbank for a transmultiplexer system with uncertain input signal and additive noise, the H_∞ filter is employed for robust signal reconstruction. The H₂ filter design is considered to be a suboptimal approach to achieve the optimal signal reconstruction in transmultiplexer system under unitary noise power. Finally, a mixed H₂/H_∞ filter is proposed to achieve a better signal reconstruction performance in transmultiplexer systems. These design problems can be transformed to solving the eigenvalue problems (EVP) under some linear matrix inequality (LMI) constraint. The LMI Matlab toolbox can be applied to efficiently solve the EVP by convex optimization technique     

Rational sampling filter banks based on IIR filters

The problem of splitting the spectrum of a digital signal by using nonuniform infinite impulse response (IIR) filter banks is addressed. Near perfect reconstruction (NPR) is considered. The method uses the modulation of different IIR prototypes. The cancellation of the main aliasing components constrains the prototypes to be dependent on each other. By using this approach, linear-phase prototypes are needed, and noncausal filtering is required. Numerical examples of filter bank design are given, and the computational complexity is compared with the finite impulse response (FIR) case     

Reconstruction of band-limited periodic nonuniformly sampled signals through multirate filter banks

A band-limited signal can be recovered from its periodic nonuniformly spaced samples provided the average sampling rate is at least the Nyquist rate. A multirate filter bank structure is used to both model this nonuniform sampling (through the analysis bank) and reconstruct a uniformly sampled sequence (through the synthesis bank). Several techniques for modeling the nonuniform sampling are presented for various cases of sampling. Conditions on the filter bank structure are used to accurately reconstruct uniform samples of the input signal at the Nyquist rate. Several examples and simulation results are presented, with emphasis on forms of nonuniform sampling that may be useful in mixed-signal integrated circuits.     

Constrained State Estimation for Nonlinear Systems with Unknown Input

This paper extends the problem of state estimation for linear discrete-time systems with unknown input to the nonlinear systems. Based on physical consideration, the constraints of state are also considered. And the constraints which can improve the quality of estimation are imposed on individual updated sigma points as well as the updated state. The advantage of algorithm is that it is able to deal with arbitrary constraints on the states during the estimation procedure, Least-squares unbiased estimation algorithm can be used to obtain unknown input, and the unknown input which can be any signal affects both the system and the outputs. The state estimation problem is transformed into a standard Unscented Kalman filter problem which can easily be solved. Simulations are provided to demonstrate the effectiveness of the theoretical results.     

A unified approach to linear estimation problems for nonstationary processes

The linear least mean-square (LLMS) error estimation problem of a nonstationary signal corrupted by additive white noise is studied. The formulation of the problem is very general, in the sense that it deals with different estimation problems (smoothing, filtering, and prediction) involving correlation between the signal and the white noise and the possibility of estimating a linear operation (in quadratic mean) of the signal. The obtained solution is in the form of a suboptimum estimate and is derived by using the approximate series expansions for stochastic processes with the aim of solving the Wiener-Hopf equation in the general (nonstationary) case. The main characteristic of this new solution is that it can be computed efficiently using a recursive algorithm similar to the Kalman filter without requiring the signal to obey a state-space model.     

Joint state filtering and parameter estimation for linear stochastic time-delay systems

This paper presents the joint state filtering and parameter estimation problem for linear stochastic time-delay systems with unknown parameters. The original problem is reduced to the mean-square filtering problem for incompletely measured bilinear time-delay system states over linear observations. The unknown parameters are considered standard Wiener processes and incorporated as additional states in the extended state vector. To deal with the new filtering problem, the paper designs the mean-square finite-dimensional filter for incompletely measured bilinear time-delay system states over linear observations. A closed system of the filtering equations is then derived for a bilinear time-delay state over linear observations. Finally, the paper solves the original joint estimation problem. The obtained solution is based on the designed mean-square filter for incompletely measured bilinear time-delay states over linear observations, taking into account that the filter for the extended state vector also serves as the identifier for the unknown parameters. In the example, performance of the designed state filter and parameter identifier is verified for a linear time-delay system with an unknown multiplicative parameter over linear observations.     

Linear and quadratic estimation using uncertain observations from multiple sensors with correlated uncertainty

In this paper, filtering algorithms are derived for the least-squares linear and quadratic estimation problems in linear systems with uncertain observations coming from multiple sensors with different uncertainty characteristics. It is assumed that, at each sensor, the state is measured in the presence of additive white noise and that the Bernoulli random variables describing the uncertainty are correlated at consecutive sampling times but independent otherwise. The least-squares linear estimation problem is solved by using an innovation approach, and the quadratic estimation problem is reduced to a linear estimation one in a suitable augmented system. The performance of the linear and quadratic estimators is illustrated by a numerical simulation example wherein a scalar signal is estimated from correlated uncertain observations coming from two sensors with different uncertainty characteristics.     

Minimax deconvolution design of multirate systems with channelnoises: a unified approach

The conventional signal reconstruction problem of multirate systems with channel noises can be cast as a robust multirate deconvolution design problem. We investigate a unified minimax approach for the robust deconvolution design of multirate systems. We discuss two typical multirate systems: the multirate filter bank system and the transmultiplexer system. We consider transmission noises resulting from quantization coding errors or external noises. The deconvolution filters for these systems that we derive are all IIR filters. The keypoint is converting the original robust deconvolution design problem to an equivalent minimax matching problem via polyphase decomposition and noble identities. Then, in spite of the presence of input signals and channel noises, we can solve this minimax matching problem by an optimization technique. The proposed method can be interpreted as designing an optimal multirate deconvolution filter such that the worst-case multirate system reconstruction error is minimized over all possible inputs and noises from the energy perspective. Therefore, our proposed design method is more robust than the conventional design method for multirate systems in the presence of uncertain input signals and channel noises. We present several numerical examples that show the good performance of our design method     

Adaptive blind calibration of timing offset and gain mismatch for two-channel time-interleaved ADCs

In this paper, we describe a blind calibration method for gain and timing mismatches in a two-channel time-interleaved low-pass analog-to-digital converters (ADC). The method requires that the input signal should be slightly oversampled. This ensures that there exists a frequency band around the zero frequency where the Fourier transforms of the ADC subchannels are alias free. Low-pass filtering the ADC subchannels to this alias-free band reduces the blind calibration problem to a conventional gain and time delay estimation problem for an unknown signal in noise. An adaptive filtering structure with three fixed FIR filters and two adaptive gain and delay parameters is employed to achieve the calibration. A convergence analysis is presented for the blind calibration technique. Numerical simulations for a bandlimited white noise input and for inputs containing several sinusoidal components demonstrate the effectiveness of the proposed method.     

基于新息的自适应增量Kalman滤波器

在一定环境条件下,当系统的量测方程没有进行验证或校准时,使用该量测方程往往会产生未知的系统误差,从而导致较大的滤波误差。增量方程的引入可以有效解决欠观测系统的状态估计问题。该文考虑带未知噪声统计的线性离散增量系统,首先提出一种基于新息的噪声统计估计算法。可以得到系统噪声统计的无偏估计。进而,提出一种新的增量系统自适应Kalman滤波算法。相比已有的自适应增量滤波算法,该文所提算法得到的状态估计精度更高。两个仿真实例证明了其有效性和可行性。     

Nonuniform filter banks: a reconstruction and design theory

A general procedure for the design of analysis-synthesis systems based on nonuniform filter banks is described. The procedure is based on a time-domain analysis of nonuniform systems, which results in a set of conditions for the exact reconstruction of the input signal at the output. These conditions are used as part of a powerful iterative algorithm for designing finite impulse response (FIR) filter banks with an arbitrary nonuniform frequency resolution. This new framework permits the design of systems with arbitrary rational decimation rates in different bands. Systems based on maximally or nonmaximally decimated filter banks, on low and minimum delay systems, and on block decimators are also among the systems that can be designed using this method     

Maneuvering Target Tracking with High-Order Correlated Noise – A Multirate Kalman Filtering Approach

A multirate Kalman filtering algorithm for target tracking with high-ordercorrelatednoise is proposed. The measurement signal is first split into subbands usinga filter bank.Then, the correlated noise in each subband is modeled using a first-order ARprocess and the AR parametersare identified online. Finally, a multirate Kalman reconstruction filter isused to obtain the state estimate.This method can be directly incorporated into the IMM algorithm, resulting inan effective tracking scheme.Simulations show that the new multirate processing scheme can significantlyimprove tracking performance.     

基于EEMD算法在信号去噪中的应用

为了抑制经验模态分解中出现的端点效应和模态混叠现象,利用白噪声辅助数据分析方法——集合经验模态分解构造一个自适应滤波器组,对原信号进行各级滤波,最终得到纯净的信号.然后与小波阈值去噪方法进行比较,通过仿真可以看出,集合经验模态分解构造的滤波器组滤波效果比较理想.     

分数阶傅里叶域滤波器组的一般化设计方法

 分数阶傅里叶变换相对于传统的傅里叶变换具有灵活的时频分析特性,在最优分数阶傅里叶域进行滤波可以实现对某些非平稳信号的最优检测和参数估计以及对某些干扰和噪声的滤除.分数阶傅里叶域滤波器组理论的提出弥补了分数阶傅里叶域滤波不具备多尺度分析以及运算量过大的缺点,但现有的分数阶傅里叶域准确重建滤波器组设计方法不具备形式一般化的特点,很难满足很多实际工程的需要.本文从分数阶傅里叶域多抽样率信号处理基本理论和分数阶卷积定理出发,推导出了分数阶傅里叶域准确重建滤波器组的一般化设计方法,为分数阶傅里叶域滤波器组理论在实际工程中的推广应用奠定了理论基础.最后,仿真实验验证了本文所提分数阶傅里叶域滤波器组一般化设计方法的有效性.     

Nonideal sampling and interpolation from noisy observations in shift-invariant spaces

Digital analysis and processing of signals inherently relies on the existence of methods for reconstructing a continuous-time signal from a sequence of corrupted discrete-time samples. In this paper, a general formulation of this problem is developed that treats the interpolation problem from ideal, noisy samples, and the deconvolution problem in which the signal is filtered prior to sampling, in a unified way. The signal reconstruction is performed in a shift-invariant subspace spanned by the integer shifts of a generating function, where the expansion coefficients are obtained by processing the noisy samples with a digital correction filter. Several alternative approaches to designing the correction filter are suggested, which differ in their assumptions on the signal and noise. The classical deconvolution solutions (least-squares, Tikhonov, and Wiener) are adapted to our particular situation, and new methods that are optimal in a minimax sense are also proposed. The solutions often have a similar structure and can be computed simply and efficiently by digital filtering. Some concrete examples of reconstruction filters are presented, as well as simple guidelines for selecting the free parameters (e.g., regularization) of the various algorithms.     

加权融合鲁棒增量Kalman滤波器

在一定环境条件下,当系统的量测方程没有进行验证或校准时,使用该量测方程往往会产生未知的系统误差,从而导致较大的滤波误差。同样地,当系统的噪声方差不确定时,滤波的性能也将会变坏,甚至会引起滤波器发散。增量方程的引入可以有效消除系统的未知量测误差,从而带未知量测误差的欠观测系统的状态估计问题可以转换为增量系统的状态估计问题。该文考虑带未知量测误差和未知噪声方差的线性离散系统,首先提出一种基于增量方程的鲁棒增量Kalman滤波器。进而,基于线性最小方差最优融合准则,提出一种加权融合鲁棒增量Kalman滤波算法。仿真实例证明了所提算法的有效性和可行性。     

Iterative and sequential algorithms for multisensor signalenhancement

In problems of enhancing a desired signal in the presence of noise, multiple sensor measurements will typically have components from both the signal and the noise sources. When the systems that couple the signal and the noise to the sensors are unknown, the problem becomes one of joint signal estimation and system identification. The authors specifically consider the two-sensor signal enhancement problem in which the desired signal is modeled as a Gaussian autoregressive (AR) process, the noise is modeled as a white Gaussian process, and the coupling systems are modeled as linear time-invariant finite impulse response (FIR) filters. The main approach consists of modeling the observed signals as outputs of a stochastic dynamic linear system, and the authors apply the estimate-maximize (EM) algorithm for jointly estimating the desired signal, the coupling systems, and the unknown signal and noise spectral parameters. The resulting algorithm can be viewed as the time-domain version of the frequency-domain approach of Feder et al. (1989), where instead of the noncausal frequency-domain Wiener filter, the Kalman smoother is used. This approach leads naturally to a sequential/adaptive algorithm by replacing the Kalman smoother with the Kalman filter, and in place of successive iterations on each data block, the algorithm proceeds sequentially through the data with exponential weighting applied to allow adaption to nonstationary changes in the structure of the data. A computationally efficient implementation of the algorithm is developed. An expression for the log-likelihood gradient based on the Kalman smoother/filter output is also developed and used to incorporate efficient gradient-based algorithms in the estimation process