A constrained notch Fourier transform

The paper presents a new sliding algorithm for estimating the amplitude and phase of the Fourier coefficients of noise corrupted harmonic signals given a priori knowledge of the signal frequencies. The proposed method is similar in principle to the notch Fourier transform (NFT) technique suggested by Tadokoro et al. [1987] except that it employs an infinite impulse response (IIR) rather than a finite impulse response (FIR) notch filter parameterization. This modification provides bandwidth controlled bandpass (BP) filters whose center frequencies are equally spaced in the frequency spectrum. In this sense, the proposed technique can be regarded as a constrained notch Fourier transform (CNFT). Sliding algorithms have been derived for both the NFT and CNFT for the purpose of estimating the Fourier coefficients of the sinusoidal components. The paper also proposes a similar algorithm to the CNFT for the signals containing sinusoids at arbitrary known frequencies. The main feature of the modified CNFT is that it uses second-order IIR BP filters whose bandwidth and center frequency can be adjusted independently. The bandwidth control aspect provides the user with an efficient means of achieving the required resolution as well as reducing spectral leakage. In general, the proposed approach leads to considerable reduction in terms of computational burden and memory storage     

Least squares acceleration filtering for the estimation of signalderivatives and sharpness at extrema [and application to biologicalsignals]

A family of finite impulse-response (FIR) filters is derived which estimate the second derivative or "acceleration" of a digitized signal. The acceleration is obtained from parabolas that are continuously fit to the signal using a least squares optimization criterion. A closed-form solution for the filter coefficients is obtained. The general approach is computationally simple, can be performed in real-time, and is robust in the presence of noise. An important application of the method, that of measuring sharpness in biologic signals, is presented using the electroencephalogram (EEG) and electrocardiogram (EKG) signals as examples. Furthermore, the design method is extended to derive FIR filters for estimating derivatives of arbitrary order in digital signals of biologic or other origins.     

Detection of harmonic sets

We examine the detection problem of signals with narrowband, harmonically related components received by a passive sensor array. We investigate detector structures based on the Fourier method. The harmonic detector estimates the total signal power by combining the DFT coefficients from harmonic frequency bins. This power estimate is normalized by the estimated background noise power and then compared to a threshold. We investigate two harmonic detector structures: one that operates with coherent, correlated signals and the other with uncorrelated harmonic signals. We derive statistical laws governing both detector structures that facilitate setting a power threshold for a given probability of false alarm; and present upper- and lower-bounds for the probability of detection. The results developed and presented demonstrate the inherent advantage of the harmonic detector. At operating conditions characterized by low signal-to-noise power ratio values the harmonic detector exhibits enhanced detection performance by combining the estimated signal power from harmonic frequency bins. We generalize results from single-bin and harmonic detector structures and present them as special cases of a unifying framework     

FFT Performance in the Presence of Noise

Contamination of signals by noise degrades the performance of fast Fourier transform (FFT) analysis of biological systems. Thus, we have developed simulation techniques to investigate the effects of noise on FFT computations. Continuous and discrete representations of forcing-noise and response-noise signals are derived. The FFT is used to estimate magnitude and phase of noisy digital signals constructed using the discrete representations. The estimates are then compared to the estimates obtained from the noise-free digital signals. The following factors are shown to have an important influence on estimation accuracy: the inclusion of noninteger as well as integer harmonic noise, the signal series length, the relative noise-to-forcing and response signal magnitude ratios, and the degree of noise signal cross correlation between the forcing and response signals. We demonstrate that the FFT estimation accuracy of magnitude and phase is similar for integer and noninteger noise harmonics, it varies directly with signal series length, and inversely with the noise-to-forcing and response signal magnitude ratios.     

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.     

自适应噪声抑制器的设计和Simulink验证

在测井信号的处理中,经常会遇到非平稳噪声环境下的信号检测问题,此时很难用经典的滤波器系数固定的FIR滤波器或IIR滤波器来解决噪声背景下的信号提取问题。本文首先介绍了一种系数可变的FIR滤波器实现一种不需要参考噪声的自适应噪声抑制器的基本原理,然后在此基础上阐述了在Simulink环境中建模的具体方法,最后使用该模型对一个非平稳有色噪声信号进行抑制,仿真结果表明在不需要参考噪声源时通过该自适应噪声抑制器同样可以获得比较好的噪声抑制效果。     

The “Harmonic” Rejecting Correlation Function

A classical problem in signal processing is accurate estimation of fundamental frequency/periodicity of periodic signals at low SNR. Typically, researchers address the estimation problem, assuming that the signal environment is a sum of sinusoids in white Gaussian noise. If the signals and noise are pulsed, the situation is much more complex since normal FFT based methods result in spectra which are sums of harmonic structures. Sorting radar signals can be especially difficult since there may be many pulsed signals present in a low SNR impulsive noise environment. In this paper, a method equivalent to integration along a hyperbola on the Wigner distribution is presented. This transform, which is closely related to both the Fourier transform and the correlation function, has the property that a periodic signal produces an expected non-zero complex-valued bulge at only the fundamental. The phase, magnitude and position of the correlation bulge are sufficient to characterize the time-domain pulse train. Finally, a simple super-resolution method is presented which may be used to refine the fundamental frequency/period estimate.     

基于欠奈奎斯特采样的超宽带信号总体最小二乘重建算法

该文针对超宽带无线通信中需要设计高速模数转换器的问题,提出了一种欠奈奎斯特采样方法,该方法所要求的采样率仅与信号新息率相关,低于奈奎斯特率1个数量级。基于欠采样得到的离散时间超宽带信号,从理论上推导出信号的傅里叶频谱表达式,由此给出了一种总体最小二乘参数估计算法,能够准确地估计出冲激串信号的幅度和时移;通过将估计出的冲激串信号与高斯单脉冲波形卷积,完成超宽带信号的波形重建。仿真和实验结果表明,该文算法能够准确地重建原始超宽带信号,且算法性能优于现有的零化滤波重建算法。     

Gaussian Noise Filtering from ECG by Wiener Filter and Ensemble Empirical Mode Decomposition

Empirical mode decomposition (EMD) is a powerful algorithm that decomposes signals as a set of intrinsic mode function (IMF) based on the signal complexity. In this study, partial reconstruction of IMF acting as a filter was used for noise reduction in ECG. An improved algorithm, ensemble EMD (EEMD), was used for the first time to improve the noise-filtering performance, based on the mode-mixing reduction between near IMF scales. Both standard ECG templates derived from simulator and Arrhythmia ECG database were used as ECG signal, while Gaussian white noise was used as noise source. Mean square error (MSE) between the reconstructed ECG and original ECG was used as the filter performance indicator. FIR Wiener filter was also used to compare the filtering performance with EEMD. Experimental result showed that EEMD had better noise-filtering performance than EMD and FIR Wiener filter. The average MSE ratios of EEMD to EMD and FIR Wiener filter were 0.71 and 0.61, respectively. Thus, this study investigated an ECG noise-filtering procedure based on EEMD. Also, the optimal added noise power and trial number for EEMD was also examined.     

Blind identification with periodic modulation: a time-domain approach

We propose a method for blind identification of finite impulse response (FIR) channels with periodic modulation. The time-domain formulation in terms of block signals is simple compared with existing frequency-domain formulations. It is shown that the linear equations relating the products of channel coefficients and the autocorrelation matrix of the received signal can be further arranged into decoupled groups. The arrangement reduces computations and improves accuracy of the solution; it also leads to very simple identifiability conditions and a very natural formulation of the optimal modulating sequence selection problem. The proposed optimal selection minimizes the effects of channel noise and error in autocorrelation matrix estimation; it results in a consistent channel estimate when the channel noise is white. Simulation results show that the method yields good performance. It compares favorably with an existing subspace modulation-induced-cyclostationarity method, and it is robust with respect to channel order overestimation. The effect of modulation period and threshold of the modulating sequence are also discussed.     

Sidelobe reduction via adaptive FIR filtering in SAR imagery.

The paper describes a class of adaptive weighting functions that greatly reduce sidelobes, interference, and noise in Fourier transform data. By restricting the class of adaptive weighting functions, the adaptively weighted Fourier transform data can be represented as the convolution of the unweighted Fourier transform with a data adaptive FIR filter where one selects the FIR filter coefficients to maximize signal-to-interference ratio. This adaptive sidelobe reduction (ASR) procedure is analogous to Capon's (1969) minimum variance method (MVM) of adaptive spectral estimation. Unlike MVM, which provides a statistical estimate of the real-valued power spectral density, thereby estimating noise level and improving resolution, ASR provides a single-realization complex-valued estimate of the Fourier transform that suppresses sidelobes and noise. Further, the computational complexity of ASR is dramatically lower than that of MVM, which is critical for large multidimensional problems such as synthetic aperture radar (SAR) image formation. ASR performance characteristics can be varied through the choice of filter order, l(1)- or l(2)-norm filter vector constraints and a separable or nonseparable multidimensional implementation. The author compares simulated point scattering SAR imagery produced by the ASR, MVM, and MUSIC algorithms and illustrates ASR performance on three sets of collected SAR imagery.     

Nonparametric estimation of instantaneous frequency

The local polynomial approximation of time-varying phase is used in order to estimate the instantaneous frequency and its derivatives for a complex-valued harmonic signal given by discrete-time observations with a noise. The considered estimators are high-order nonparametric generalizations of the short-time Fourier transform and the Wigner-Ville distribution. The asymptotic variance and bias of the estimates are obtained     

Spectrum estimation of short-time stationary signals in additivenoise and channel distortion

In this work, spectrum estimation of a short-time stationary signal that is degraded by both channel distortion and additive noise is addressed. A maximum likelihood estimation (MLE) algorithm is developed to jointly identify the degradation system and estimate short-time signal spectra. The source signal is assumed to be generated by a hidden Markov model (HMM) with state-dependent short-time spectral distributions described by mixtures of Gaussian densities. The distortion channel is linear time-invariant, and the noise is Gaussian. The algorithm is derived by using the principle of expectation-maximization (EM), where the unknown parameters of channel and noise are estimated iteratively, and the short-time signal power spectra are obtained from the posterior sufficient statistics of the source signal. Other spectral representation parameters, such as autoregressive model parameters or cepstral parameters, are obtained by minimum mean-squared error (MMSE) estimation from the power spectral estimates. The estimation algorithm was evaluated on simulated signals at the signal-to-noise ratios (SNRs) of 20 dB down to 0 dB, where it produced convergent estimation and significantly reduced spectral distortion     

Modeling, analysis, and optimum design of quantized M-band filterbanks

This paper provides a rigorous modeling and analysis of quantization effects in M-band subband codecs, followed by optimal filter bank design and compensation. The codec is represented by a polyphase decomposition of the analysis/synthesis filter banks and an embedded nonlinear gain-plus-additive noise model for the pdf-optimized scalar quantizers. We construct an equivalent time-invariant but nonlinear structure operating at the slow clock rate that allows us to compute the exact expression for the mean square quantization error in the reconstructed output. This error is shown to consist of two components: a distortion component and a dominant random noise component uncorrelated with the input signal. We determine the optimal paraunitary and biorthogonal FIR filter coefficients, compensators, and integer bit allocation to minimize this MSE subject to the constraints of filter length, average bit rate, and perfect reconstruction (PR) in the absence of quantizers. The biorthogonal filter bank results in a smaller MSE but the filter coefficients are very sensitive to signal statistics and to average bit constraints. By comparison, the paraunitary structure is much more robust. We also show that the null-compensated design that eliminates the distortion component is more robust than the optimally-compensated case that minimizes the total MSE, but only at nominal conditions. Both modeling and optimal design are validated by simulation in the two-channel case     

Convolution-Based Trigonometric Interpolation of Band-Limited Signals

This paper presents a method to obtain a trigonometric polynomial that accurately interpolates a given band-limited signal from a finite sequence of samples. The polynomial delivers accurate approximations in the range covered by the sequence, except for a short frame close to the range limits. Besides, its accuracy increases exponentially with the frame width. The method is based on using a band-limited window in order to reduce the truncation error of a convolution series. It is shown that the polynomial can be efficiently constructed and evaluated using algorithms designed for the discrete Fourier transform (DFT). Specifically, two basic procedures are presented, one based on the fast Fourier transform (FFT), and another based on a recursive update algorithm for the short-time FFT. The paper contains three applications. The first is a variable fractional delay (VFD) filter, which consists of a short-time FFT combined with the evaluation of a trigonometric polynomial. This filter has low complexity and can be implemented using CORDIC rotations. The second is the interpolation of nonuniform Fourier summations, where the proposed method eliminates the need to interpolate any kernel sample. Finally, the third can be viewed as a generalization of the FFT convolution algorithm and makes it possible to interpolate the output of an finite-impulse-response (FIR) filter efficiently.      

Optimal FIR pre- and postfilters for decimation and interpolationof random signals

A technique for the design of finite-impulse-response (FIR) filters for decimation and interpolation in multirate systems is introduced. FIR prefilters and postfilters that jointly minimize a frequency-weighted mean-square error between the original and reconstructed signals can be designed. There is no need for ideal filter prototypes: the optimal pre-postfilter pair is determined from the signal and noise spectra and the up-sampling and down-sampling factors. Some examples of image and speech processing show that the mean square optimal filter pair leads to typical SNR improvements of 2-6 dB, in comparison to other commonly used filters     

混沌信号的自适应阈值同步挤压小波变换消噪

针对同步挤压小波变换（SST）消噪过程中仅使用单一阈值的不足,对SST消噪时的幅度阈值进行了改进,提出了一种基于SST分层阈值的混沌信号消噪方法.首先,根据信号和噪声经SST分解后系数的分布模型,推导SST混沌去噪时幅度阈值权系数的均方误差计算公式;进而,根据均方误差最小准则,计算幅度阈值权系数的最优取值;最后,根据最优阈值权系数和噪声标准差,确定SST混沌去噪时的分层阈值.利用模拟混沌信号和实测月太阳黑子信号对所提方法进行了实验分析,实验结果表明,本文方法可较好地滤除混沌信号中的噪声,同时原始信号的内在混沌特性也能得到较大程度的恢复.与小波阈值法和集合经验模态分解（EEMD）消噪法相比,可获得更好的消噪效果.     

一种基于结构优化的FIR滤波器舍入噪声性能改进方案

针对滤波器组系统硬件实现时原型滤波器的有限字长效应问题,该文研究如何改善FIR原型滤波器由信号量化引起的舍入噪声,即降低舍入噪声增益,提出一种FIR滤波器优化结构。通过分析舍入噪声来源,利用多项式参数化方法对舍入噪声增益表达式进行推导。仿真实例证明,在不同字长约束条件下所提结构滤波器的幅频相频响应与理想状态基本吻合;通过与现有算法对比,所提结构具有较小的舍入噪声增益。     

Optimization of filter banks using cyclostationary spectralanalysis

This paper proposes a design method of optimal biorthogonal FIR filter banks that minimize the time-averaged mean squared error (TAMSE) when the high-frequency subband signal is dropped. To study filter banks from a statistical point of view, cyclostationary spectral analysis is used since the output of the filter bank for a wide-sense stationary input is cyclostationary. First, the cyclic spectral density of the output signal is derived, and an expression for the TAMSE is presented. Then, optimal filter banks are given by minimizing the TAMSE with respect to the coefficients of the filters under the biorthogonality condition. By imposing the additional constraints on the coefficients, the optimal biorthogonal linear phase filter bank can be obtained     

基于子空间旋转不变性的加性有色噪声中谐波频率的估计

噪声中的谐波恢复问题是信号处理领域的一个典型问题,在众多领域中有着广泛的应用。本文主要研究加性有色噪声中谐波频率的估计问题,提出了一种基于子空间旋转不变性的谐波频率的高分辨率估计方法。利用观测信号的自协方差函数构造了一个协方差矩阵,通过对协方差矩阵的特征值进行理论分析,结合子空间旋转不变性,得到了加性有色噪声中谐波的频率和协方差矩阵之间的一种内在联系。利用这个性质可以估计加性有色噪声中谐波的频率。本文方法对于有色噪声的模型无任何假设,而且对于噪声的分布也没有限制,对于高斯和非高斯有色噪声都适用。仿真实验验证了本文所提算法的有效性。