A survey of numerical methods for nonlinear filtering problems

The main goal of filtering is to obtain, recursively in time, good estimates of the state of a stochastic dynamical system based on noisy partial observations of the same. In settings where the signal/observation dynamics are significantly nonlinear or the noise intensities are high, an extended Kalman filter (EKF), which is essentially a first order approximation to an infinite dimensional problem, can perform quite poorly: it may require very frequent re-initializations and in some situations may even diverge. The theory of nonlinear filtering addresses these difficulties by considering the evolution of the conditional distribution of the state of the system given all the available observations, in the space of probability measures. We survey a variety of numerical schemes that have been developed in the literature for approximating the conditional distribution described by such stochastic evolution equations; with a special emphasis on an important family of schemes known as the particle filters. A numerical study is presented to illustrate that in settings where the signal/observation dynamics are non linear a suitably chosen nonlinear scheme can drastically outperform the extended Kalman filter.     

A comparison of filter design structures for multi-channel acoustic communication systems

The application of inverse filter designs as a means of providing improved communication performance in acoustic environments is investigated. Tikhonov regularized inverse filters of channel transfer functions calculated in the frequency domain are used as a means of obtaining multi-channel filters. Three classifications of inverse filter structures have been considered using time-domain simulations. The performance of Tikhonov regularized inverse filters designed according to each of these classifications is compared with each other and against a filter design developed by Stojanovic [Stojanovic, M. (2005). "Retrofocusing techniques for high rate acoustic communications," J. Acoust. Soc. Am. 117, 1173-1185]. It is shown that the filter design developed by Stojanovic requires less regularization and outperforms the Tikhonov regularized inverse filter designs when communicating over a single channel. While the filter developed by Stojanovic is designed to use multiple transmitters to transmit to a single receiver, the filter was implemented in a multi-channel system and proposed to have a focusing similar to that obtained using time-reversal. It was found that for the scenario used in the simulation, the Tikhonov regularized inverse design for full multi-channel inversion achieved better focusing than the design by Stojanovic, where simulation results show 20 dB less cross-talk at the expense of around 2 dB loss in signal strength.     

Informationene

The flow rate of some liquids (water, heptane, toluene, xylene, dodecane) through Kapton nuclear track filters has been measured. The results can be interpreted with a modified Poiseuille formula. The influence of the viscosity of the liquids on their throughput through Kapton nuclear track filters has been determined. The purification of liquids with Kapton filters has been investigated. It is possible to measure the density of solid particles in deionized water by using Kapton filters. Consequently, nuclear track filters can remove solid particles from high-purity liquids. The soaking effect of some common liquid chemicals on Kapton filters has also been studied, no such soaking could be observed in most of the cases. Finally, these Kapton nuclear track filters are compared with a filter device from Hamamatsu (Japan).     

A shock-detecting sensor for filtering of high-order compact finite difference schemes

A new shock-detecting sensor for properly switching between a second-order and a higher-order filter is developed and assessed. The sensor is designed based on an order analysis. The nonlinear filter with the proposed sensor ensures damping of the high-frequency waves in smooth regions and at the same time removes the Gibbs oscillations around the discontinuities when using high-order compact finite difference schemes. In addition, a suitable scaling is proposed to have dissipation proportional to the shock strength and also to minimize the effects of the second-order filter on the very small scales. Several numerical experiments are carried out and the accuracy of the nonlinear filter with the proposed sensor is examined. In addition, some comparisons with other filters and sensors are made.     

Detection performance improvement of photon counting chirped amplitude modulation lidar with response probability correction

Geiger mode avalanche photodiode detector(Gm-APD) possesses the ultra-high sensitivity. Photon counting chirped amplitude modulation(PCCAM) light detection and ranging(lidar) uses the counting results of the returned signal detected by Gm-APD to mix with the reference signal, which makes PCCAM lidar capable of realizing the ultra-high sensitivity, and this is very important for detecting the remote and weak signal. However, Gm-APD is a nonlinear device, different from traditional linear detectors. Due to the nonlinear response of Gm-APD, the counting results of the returned signal detected by Gm-APD are different from those of both the original modulation signal and the reference signal. This will affect the mixing effect and thus degrade the detection performance of PCCAM lidar. In this paper, we propose a response probability correction method. First, the response probability correction model is established on the basis of Gm-APD Poisson probability response model. Then, the response probability correction model is used to adjust the original modulation signal that is used to drive laser, in order to make the counting results of the returned signal detected by Gm-APD better mix with the local reference signal in the same form. Through this method, the detection performance of PCCAM lidar is enhanced efficiently.     

Accurate quantification of (1)H spectra: from finite impulse response filter design for solvent suppression to parameter estimation.

A scheme for accurate quantification of (1)H spectra is presented. The method uses maximum-phase finite impulse response (FIR) filters for solvent suppression and an iterative nonlinear least-squares (NLLS) algorithm for parameter estimation. The estimation algorithm takes the filter influence on the metabolites of interest into account and can thereby correctly incorporate a large variety of prior knowledge into the estimation phase. The FIR filter is designed in such a way that no distortion of the important initial samples is introduced. The FIR filter method is compared numerically with the HSVD method for water signal removal in a number of examples. The results show that the FIR method, using an automatic filter design scheme, slightly outperforms the HSVD method in most cases. The good performance and ease of use of the FIR filter method combined with its low computational complexity motivate the use of the proposed method.     

Time-domain noise reduction based on an orthogonal decomposition for desired signal extraction

This paper addresses the problem of noise reduction in the time domain where the clean speech sample at every time instant is estimated by filtering a vector of the noisy speech signal. Such a clean speech estimate consists of both the filtered speech and residual noise (filtered noise) as the noisy vector is the sum of the clean speech and noise vectors. Traditionally, the filtered speech is treated as the desired signal after noise reduction. This paper proposes to decompose the clean speech vector into two orthogonal components: one is correlated and the other is uncorrelated with the current clean speech sample. While the correlated component helps estimate the clean speech, it is shown that the uncorrelated component interferes with the estimation, just as the additive noise. Based on this orthogonal decomposition, the paper presents a way to define the error signal and cost functions and addresses the issue of how to design different optimal noise reduction filters by optimizing these cost functions. Specifically, it discusses how to design the maximum SNR filter, the Wiener filter, the minimum variance distortionless response (MVDR) filter, the tradeoff filter, and the linearly constrained minimum variance (LCMV) filter. It demonstrates that the maximum SNR, Wiener, MVDR, and tradeoff filters are identical up to a scaling factor. It also shows from the orthogonal decomposition that many performance measures can be defined, which seem to be more appropriate than the traditional ones for the evaluation of the noise reduction filters.     

滤片对能量色散X射线荧光光谱方法测定轻基体中痕量重金属Cd元素的作用

用能量色散X射线荧光光谱方法检测轻基体中重金属元素时,提高峰背比是一大难题,而设置滤片是解决这个难题的一种办法。该研究先对初级滤片(分为一类初级滤片和二类初级滤片)和次级滤片(平衡滤片)的选择(材料元素及厚度)作了理论分析,然后通过MC软件进行蒙特卡罗进行分别模拟,比较设置前后的峰背比,验证理论分析的准确性。一类初级滤片模拟设置了Fe和Cu两种材料,结果表明设置后的峰背比较设置前的1.36均提高了两倍左右;二类初级滤片模拟设置了Te和Ba两种材料,设置后的最大峰背比分别为14.88和13.58,均比比设置前提高了10倍左右;平衡滤片则是先后设置Rh和Ru两个滤片,两者计数相减即得到平衡通带,结果可见通带内特征峰透过率为83.1%,而通带外的背景值降低到几乎为0,使得峰背比大大增加了。经过比较,三种滤片的设置都有效地提高了峰背比,其中平衡滤片提高的倍数最大,二类初级滤片次之,最小为一类初级滤片。理论上三种滤片是可以同时设置以更大地提高峰背比的,但由于轻基体中Cd元素含量很小,而低计数的结果会大大增加放射性统计涨落误差,因此,在实际应用中,还应考虑到这一点,以决定是否进行两两同时设置或者三者同时设置。研究创新之处在于除了传统的两种滤片外,还提出了二类初级滤片的运用。     

Dynamics of pulse propagation in phase-shifted fibre Bragg gratings: a numerical investigation

Fibre Bragg gratings (FBGs) are an integral component of optical fibre based communication systems for signal processing. Uniform FBGs and phase-shifted FBGs find major applications as add-drop filters in dense wavelength division multiplexing (DWDM) systems where these are used to filter out channels and to shape the signal spectrum. We investigate numerically the spatiotemporal deformation of pulses as they propagate through the gratings. In particular, the effect on the pulse dynamics of phase shift or defect present in the gratings, often introduced intentionally in a grating to generate a narrow transmission spike within the stopband, is studied in time domain. This study is of significance since the deformation in the pulse in time limits the maximum bit rate achievable, and hence the pulse duration and shape need to be analysed and controlled accurately. A time domain numerical tool based on the Transmission Line Modelling (TLM) method is developed to study the pulse dynamics. The results obtained give useful information about the signal dynamics in gratings with defects, including the signal oscillations trapped in the defect, to be considered in applications such as while designing spectral filters.     

MULTISTAGE ADAPTIVE HIGHER-ORDER NONLINEAR FINITE IMPULSE RESPONSE FILTERS FOR CHAOTIC TIME SERIES PREDICTIONS

A multistage adaptive higher-order nonlinear finite impulse response (MAHONFIR) filter is proposed to predict chaotic time series. Using this approach, we may readily derive the decoupled parallel algorithm for the adaptation of the coefficients of the MAHONFIR filter, to guarantee a more rapid convergence of the adaptive weights to their optimal values. Numerical simulation results show that the MAHONFIR filters proposed here illustrate a very good performance for making an adaptive prediction of chaotic time series.     

Extracting hidden weak sinusoidal signal with short duration from noisy data: Analytical theory and computational realization

Signal detection is both a fundamental topic of data science and a great challenge for practical engineering. One of the canonical tasks widely investigated is detecting a sinusoidal signal of known frequency ω with time duration T :I(t) = A cos ω t + Γ(t), embedded within a stationary noisy data. The most direct, and also believed to be the most efficient,method is to compute the Fourier spectral power at ω : B = 2 T T0 I(t) ei ω tdt. Whether one can out-perform the linear Fourier approach by any other nonlinear processing has attracted great interests but so far without a consensus. Neither a rigorous analytic theory has been offered. We revisit the problem of weak signal, strong noise, and finite data length T = O(1), and propose a signal detection method based on resonant filtering. While we show that the linear approach of resonant filters yield a same signal detection efficiency in the limit of T →∞, for finite time length T = O(1), our method can improve the signal detection due to the highly nonlinear interactions between various characteristics of a resonant filter in finite time with respect to transient evolution. At the optimal match between the input I(t), the control parameters, and the initial preparation of the filter state, its performance exceeds the above threshold B considerably. Our results are based on a rigorous analysis of Gaussian processes and the conclusions are supported by numerical computations.     

Accurate Quantification of H Spectra: From Finite Impulse Response Filter Design for Solvent Suppression to Parameter Estimation

A scheme for accurate quantification of ¹H spectra is presented. The method uses maximum-phase finite impulse response (FIR) filters for solvent suppression and an iterative nonlinear least-squares (NLLS) algorithm for parameter estimation. The estimation algorithm takes the filter influence on the metabolites of interest into account and can thereby correctly incorporate a large variety of prior knowledge into the estimation phase. The FIR filter is designed in such a way that no distortion of the important initial samples is introduced. The FIR filter method is compared numerically with the HSVD method for water signal removal in a number of examples. The results show that the FIR method, using an automatic filter design scheme, slightly outperforms the HSVD method in most cases. The good performance and ease of use of the FIR filter method combined with its low computational complexity motivate the use of the proposed method.     

Normalized LMS digital filter for chromatic dispersion equalization in 112-Gbit/s PDM-QPSK coherent optical transmission system

High bit rates optical communication systems pose the challenge of their tolerance to linear and non-linear fiber impairments. Coherent optical receivers using digital signal processing techniques can mitigate the fiber impairments in the optical transmission system, including the chromatic dispersion equalization with digital filters. In this paper, an adaptive finite impulse response filter employing normalized least mean square algorithm is developed for compensating the chromatic dispersion in a 112-Gbit/s polarization division multiplexed quadrature phase shift keying coherent communication system, which is established in the VPI simulation platform. The principle of the adaptive normalized least mean square algorithm for signal equalization is analyzed theoretically, and at the meanwhile, the taps number and the tap weights in the adaptive finite impulse response filter for compensating a certain fiber chromatic dispersion are also investigated by numerical simulation. The chromatic dispersion compensation performance of the adaptive filter is analyzed by evaluating the behavior of the bit-error-rate versus the optical signal-to-noise ratio, and the compensation results are also compared with other present digital filters.     

The intensity detection of single-photon detectors based on photon counting probability density statistics

Single-photon detectors possess the ultra-high sensitivity, but they cannot directly respond to signal intensity. Conventional methods adopt sampling gates with fixed width and count the triggered number of sampling gates, which is capable of obtaining photon counting probability to estimate the echo signal intensity. In this paper, we not only count the number of triggered sampling gates, but also record the triggered time position of photon counting pulses. The photon counting probability density distribution is obtained through the statistics of a series of the triggered time positions. Then Minimum Variance Unbiased Estimation(MVUE) method is used to estimate the echo signal intensity. Compared with conventional methods, this method can improve the estimation accuracy of echo signal intensity due to the acquisition of more detected information. Finally, a proof-of-principle laboratory system is established. The estimation accuracy of echo signal intensity is discussed and a high accuracy intensity image is acquired under low-light level environments.     

用于混沌时间序列自适应预测的一种少参数二阶Volterra滤波器

研究了二阶Volterra滤波器的一种乘积耦合近似实现结构及其非线性NLMS自适应算法,并用这种少参数二阶Volterra滤波器(RPSOVF)研究了一些混沌信号的非线性自适应预测性能.仿真研究结果表明:所给出的非线性NLMS自适应算法能够保证这种RPSOVF的稳定性和收敛性,且RPSOVF用这种非线性NLMS自适应算法能够自适应预测一些混沌时间序列. 关键词： 混沌 非线性自适应预测 Volterra滤波器 非线性NLMS自适应算法     

Analytical Study of Filters Dimensions and Spectral Thermal Effects on Optical Filters Operation Performance Characteristics

We study the different optical filters design considerations over wide range of the operation conditions in near infrared optical spectrum transmission region. There are many operating conditions parameters describing optical filter properties such as the amount absorbed electromagnetic radiation which depends on the operating signal wavelength; the amount of the absorbing material in the filter thickness; and the absorption coefficient of the material at that wavelength. Filter modulation depth, filtration signal quality, filter delay time, filter bit error rate, and filter correction are the major interesting performance parameters in the current study. Best candidate materials based optical filters are used which namely barium fluoride (BaF₂), and zinc selenide (ZnSe) and compared with published measured works. Our results are validated against published experimental studies and show a good agreement and matching.     

Engineering the nonlinear phase shift

We treat the nonlinear phase shift response in the weak perturbation limit as a linear digital filter that can be synthesized into the values of its poles and zeros and mapped onto an optical architecture. This procedure results in a significant enhancement in the nonlinear sensitivity with a response that is robust to frequency changes within the filter passband. A precompensation technique can be used to reduce distortions under strongly driven nonlinear operation to achieve a larger phase shift. We also show that nonlinear sensitivity improves with increasing filter group delay and can be increased within constant linear bandwidth by use of higher-order filters.     

Effects of signal delay on auditory filter shapes derived from psychophysical tuning curves and notched-noise data obtained in simultaneous masking

Psychophysical tuning curves (PTCs) measured in simultaneous masking usually sharpen as a short duration signal is moved from the onset to the temporal center of a longer duration masker. Filter shapes derived from notched-noise maskers have not consistently shown this effect. One possible explanation for this difference is that the signal level is fixed in the PTC paradigm, whereas the masker level is usually fixed in the notched-noise paradigm. In the present study, the signal level was fixed at 10 dB SL in both paradigms. The signal was 20 ms in duration, and presented at the onset or temporal center of the 400-ms masker. The masker was a pure tone presented in quiet (PTC) or in the presence of a pure-tone "restrictor" intended to limit off-frequency listening (PTCr), or it was a noise with a spectral notch placed symmetrically or asymmetrically about the 2-kHz signal frequency. Filter shapes were derived from the PTC, PTCr, and notched-noise data using the roex (p, w, t) model. The effects of signal delay and masking paradigm on filter bandwidth were analyzed with a two-factor repeated-measures ANOVA. There was a significant effect of signal delay (the filters sharpened with time) and masking paradigm (the filters derived from the notched-noise data were significantly wider than those derived from either of the PTC measurements, which did not differ from one another). Although the interaction between delay and paradigm was not significant, the filter derived from the notched-noise data sharpened more with time than did the other filters, and thus the bandwidth of the filters from the three paradigms were more similar at the longer delay than at the shorter delay. It is likely that the tuning-curve and notched-noise paradigms measure the same underlying filtering, but that various other factors contribute differentially to the derived filter shapes.     

Si PIN X‐ray photon counter

A Si PIN detector for visible light detection, instead of a Geiger‐Müller tube, is applied to X‐ray photon counting. We counted radiation from a checking source of a Geiger‐Müller counter with a Si PIN counter and with a Geiger‐Müller counter. White X‐ray of energy up to 20 keV emitted from a pyroelectric X‐ray emitter was also counted, and the Si PIN X‐ray counter showed a similar curve of count rate versus source distance in both measurements. Pulse counting was performed by spectroscopy circuits. An audio digitizer with computer software for signal processing was also used to simplify the photon counter. A plot of count rate versus time was obtained with this setup. With simple pulse counting circuits, Si PIN X‐ray counters have advantages such as compact structure, low cost and easy application. Copyright © 2011 John Wiley & Sons, Ltd.     

Design and development of far infrared (FIR) filters for space applications

Theoretical/experimental studies on design and development of FIR filters have been carried out. A simple design consisting of a staggered edge filter and a four cavity Fabry-Perot filter, has been developed, for the first time, which meets the required pass band characteristics anywhere in 2–20 μm and completely eliminates transmission in the rest of the infrared spectrum. A typical designed filter useful in 14–16 μm band has been realised experimentally. The resulting filters satisfy the stringent spectral and durability requirements laid down as per MIL-F-48616.