Order-recursive spectrum estimation

The use of a multiple-pole filter in the time-average estimation of the autocorrelation allows the power spectrum estimates to be recursive in the order of multiplicity of the filter pole. The recursive generation of the estimates from various filter orders provides the flexibility to select the estimator of interest in terms of the variance and spectral and temporal resolution     

A Physically Constrained Maximum-Likelihood Method for Snapshot-Deficient Adaptive Array Processing

This paper presents a physically constrained maximum-likelihood (PCML) method for spatial covariance matrix and power spectral density estimation as a reduced-rank adaptive array processing algorithm. The physical constraints of propagating energy imposed by the wave equation and the statistical nature of the snapshots are exploited to estimate the ldquotruerdquo maximum-likelihood covariance matrix that is full rank and physically realizable. The resultant matrix may then be used in adaptive processing for interference cancellation and improved power estimation in nonstationary environments where the amount of available data is limited. Minimum variance distortionless response (MVDR) power estimates are computed for a given environment at different levels of snapshot support using the PCML method and several other reduced-rank techniques. The MVDR power estimates from the PCML method are shown to have less bias and lower standard deviation at a given level of snapshot support than any of the other reduced-rank methods used. Furthermore, the estimated power spectral density from the PCML method is shown to offer better low-level source detection than the MVDR power estimates.     

Two-dimensional spectrum estimation using noncausal autoregressive models

Two-dimensional (2-D) spectrum estimation from raw data is of interest in signal and image processing. A parametric technique for spectrum estimation using 2-D noncausal autoregressive (NCAR) models is given. The NCAR models characterize the statistical dependency of the observation at location s on its neighbors in all directions. This modeling assumption reduces the spectrum estimation problem to two subproblems: the choice of appropriate structure of the NCAR model and the estimation of parameters in NCAR models. By assuming that the true structure of the NCAR model is known, we first analyze the existence and uniqueness of Gaussian maximum likelihood (GML) estimates of NCAR model parameters. Due to the noncausal nature of the models, the computation of GML estimates is burdensome. By assuming specific boundary conditions, computationally tractable expressions are obtained for the likelihood function. Expressions for the asymptotic covariance matrix of the GML estimates as well as the simultaneous confidence bands for the estimated spectrum using GML estimates are derived. Finally, the usefulness of the method is illustrated by computer simulation results.     

An adaptive ARMA four-line lattice filter for spectral estimationwith frequency weighting

A method for designing an adaptive four-line lattice filter which can perform frequency-weighting spectral estimation, which provides more accurate spectral estimation for some frequency bands than for others, is proposed. Using a suitable frequency-weighting function, denoted as an ARMA (autoregressive moving-average) model, an estimated spectrum is obtained by arbitrarily weighing some frequency bands more heavily than others. if the frequency-weighting function has the property of a low-pass filter, the spectrum of the reference model can be estimated accurately with a reduced ARMA order in the low-frequency band. Spectra of time-varying models can be estimated with an exponentially weighted sliding window, and the input signal of the reference model can be estimated by assumption. The order-update and the time-update recursive formulas and the frequency-weighting method for the filter are described. The algorithm is verified by experimental results     

Two-dimensional linear prediction and spectral estimation on apolar raster

A zero-mean homogeneous random field is defined on a discrete polar raster. Given sample values inside a disk of finite radius, the authors wish to estimate the field's power spectral density using linear prediction. Issues arising include estimation of covariance lags and extendibility of a finite set of lag estimates into a positive semidefinite covariance extension (required for a meaningful spectral density). The authors give a generalized autocorrelation procedure that guarantees positive semidefinite covariance estimates. It first interpolates the data using Gaussians, computes its Radon transform, and applies familiar 1D techniques to each slice. Some numerical examples are provided to justify the validity of the proposed procedure. The authors also propose a correlation-matching covariance extension procedure that uses the Radon transform to extend a given set of covariance lags to the entire plane, when this is possible, and discuss circumstances for which this is impossible     

An adaptive Kalman filter for ECG signal enhancement

The ongoing trend of ECG monitoring techniques to become more ambulatory and less obtrusive generally comes at the expense of decreased signal quality. To enhance this quality, consecutive ECG complexes can be averaged triggered on the heartbeat, exploiting the quasi-periodicity of the ECG. However, this averaging constitutes a tradeoff between improvement of the SNR and loss of clinically relevant physiological signal dynamics. Using a bayesian framework, in this paper, a sequential averaging filter is developed that, in essence, adaptively varies the number of complexes included in the averaging based on the characteristics of the ECG signal. The filter has the form of an adaptive Kalman filter. The adaptive estimation of the process and measurement noise covariances is performed by maximizing the bayesian evidence function of the sequential ECG estimation and by exploiting the spatial correlation between several simultaneously recorded ECG signals, respectively. The noise covariance estimates thus obtained render the filter capable of ascribing more weight to newly arriving data when these data contain morphological variability, and of reducing this weight in cases of no morphological variability. The filter is evaluated by applying it to a variety of ECG signals. To gauge the relevance of the adaptive noise-covariance estimation, the performance of the filter is compared to that of a Kalman filter with fixed, (a posteriori) optimized noise covariance. This comparison demonstrates that, without using a priori knowledge on signal characteristics, the filter with adaptive noise estimation performs similar to the filter with optimized fixed noise covariance, favoring the adaptive filter in cases where no a priori information is available or where signal characteristics are expected to fluctuate.     

An exact forward-backward maximum likelihood autoregressiveparameter estimation method

A method for obtaining an exact maximum likelihood estimate (MLE) of the autoregressive (AR) parameters is proposed. The method is called the forward-backward maximum likelihood algorithm. Based on a new form of the log likelihood function for a Gaussian AR process, an iterative maximization is used to obtain an MLE of the inverse covariance matrix. The AR parameters are then determined via the normal equations. Experimental results comparing the new method with other popular AR spectrum estimation methods indicate the new method achieves low bias and low variance AR parameter estimates comparable with the existing methods     

基于稀疏恢复的自适应角度多普勒补偿方法

自适应角度多普勒补偿方法是补偿机载非正侧阵雷达杂波距离相关性的有效手段,该方法解决了角度多普勒补偿方法处理性能受系统误差影响较大的问题,可以实现杂波距离相关性的自适应补偿。但该方法利用子孔径平滑估计杂波协方差矩阵,存在孔径损失,系统自由度下降,估计准确度下降。此外,该方法还存在着运算量大的问题,不利于实时实现。本文提出了一种基于稀疏恢复的自适应角度多普勒补偿方法,并利用正交投影逼近子空间追踪方法对估计杂波协方差矩阵进行特征分解。与常规自适应角度多普勒补偿方法相比,本文方法运算量显著降低,杂波补偿性能更优。理论分析和仿真结果验证了该方法的有效性。     

Weighted subspace methods and spatial smoothing: analysis andcomparison

The effect of using a spatially smoothed forward-backward covariance matrix on the performance of weighted eigen-based state space methods/ESPRIT, and weighted MUSIC for direction-of-arrival (DOA) estimation is analyzed. Expressions for the mean-squared error in the estimates of the signal zeros and the DOA estimates, along with some general properties of the estimates and optimal weighting matrices, are derived. A key result is that optimally weighted MUSIC and weighted state-space methods/ESPRIT have identical asymptotic performance. Moreover, by properly choosing the number of subarrays, the performance of unweighted state space methods can be significantly improved. It is also shown that the mean-squared error in the DOA estimates is independent of the exact distribution of the source amplitudes. This results in a unified framework for dealing with DOA estimation using a uniformly spaced linear sensor array and the time series frequency estimation problems     

Ocean wave extraction from RADARSAT synthetic aperture radarinter-look image cross-spectra

This study is concerned with the extraction of directional ocean wave spectra from synthetic aperture radar (SAR) image spectra. The statistical estimation problem underlying the wave-SAR inverse problem is examined in detail in order to properly quantify the wave information content of SAR. As a concrete focus, a data set is considered comprising six RADARSAT SAR images co-located with a directional wave buoy off the east coast of Canada. These SAR data are transformed into inter-look image cross-spectra based on two looks at the same ocean scene separated by 0.4 s. The general problem of wave extraction from SAR is cast in terms of a statistical estimation problem that includes the observed SAR spectra, the wave-SAR transform, and prior spectral wave information. The central role of the weighting functions (inverse of the error covariances) is demonstrated, as well as the consequence of approximate (based on the quasilinear wave-SAR transform) versus exact linearizations on the convergence properties of the algorithm. Error estimates are derived and discussed. This statistical framework is applied to the extraction of spectral wave information from observed RADARSAT SAR image cross-spectra. A modified wave-SAR transform is used to account for case-specific geophysical and imaging effects. Analysis of the residual error of simulated and observed SAR spectra motivates a canonical form for the SAR observation error covariance. Wave estimates are then extracted from the SAR spectra, including wavenumber dependent error estimates and explicit identification of spectral null spaces where the SAR contains no wave information. Band-limited SAR wave information is also combined with prior (buoy) spectral wave estimates through parameterization of the wave spectral shape and use of regularization     

Noise and Group Delay in Active Filters

The output noise spectrum and the group delay of active filters are related. For any state-space realization of any transfer function without finite transmission zeros, there is a combination of noise transfer functions that is equal to the group delay. For any transfer function without transmission zeros, it is possible to generate a filter for which the output noise spectrum is an exact multiple of the group delay. This relationship allows for quick noise estimates, and gives insight into the noise behavior of filters.     

Bias and variability of diagnostic spectral parameters extractedfrom closing sounds produced by bioprosthetic valves implanted in themitral position

A method is proposed to estimate the bias and variability of eight diagnostic spectral parameters extracted from mitral closing sounds produced by bioprosthetic heart valves. These spectral parameters are: the frequency of the dominant (F1) and second dominant (F2) spectral peaks, the highest frequency of the spectrum found at -3 dB (F-3), -10 dB (F-10) and -20 dB (F-20) below the highest peak, the relative integrated area above -20 dB of the dominant peak (RIA20), the bandwidth at -3 dB of the dominant spectral peak (BW3), and the ratio of F1 divided by BW3 (Q1). The bias and variability of four spectral techniques were obtained by comparing parameters extracted from each technique with the parameters of a spectral "standard." This "standard" consisted of 19 normal mitral sound spectra computed analytically by evaluating the Z transform of a sum of decaying sinusoids on the unit circle. Truncation of the synthesized mitral signals and addition of random noise were used to simulate the physiological characteristics of the closing sounds. Results show that the fast Fourier transform method with rectangular window provides the best estimates of F1 and Q1, that the Steiglitz-McBride method with maximum entropy (pole-zero modeling with four poles and four zeros) can best evaluate F2, F-20, RIA20 and BW3, and that the all-pole modeling with covariance method (16 poles) is best suited to compute F-3. It was also shown that both the all-pole modeling and the Steiglitz-McBride methods can be used to estimate F-10. It is concluded that a single algorithm would not provide the best estimates of all spectral parameters.     

Nonparametric estimation of mean Doppler and spectral width

This paper proposes a new nonparametric method for estimation of spectral moments of a zero-mean Gaussian process immersed in additive white Gaussian noise. Although the technique is valid for any order moment, particular attention is given to the mean Doppler (first moment) and to the spectral width (square root of the centered second spectral moment). By assuming that the power spectral density (PSD) of the underlying process is bandlimited, the maximum-likelihood estimates of its spectral moments are derived. A suboptimal estimate based on the sample covariance is also studied. Both methods are robust in the sense that they do not rely on any assumption concerning the PSD (besides being bandlimited). Under weak conditions, the set of estimates based on sample covariance is unbiased and strongly consistent. Compared with the classical pulse pair and the periodogram-based estimators, the proposed methods exhibit better statistical properties for asymmetric spectra and/or spectra with large spectral widths, while involving a computational burden of the same order     

用于多路MCP-EBPSK信号解调的冲击滤波器组设计

修正的随机极性连续相位扩展的二元相移键控调制（MCP-EBPSK）通过随机化调制指数的符号,并加入功率谱调节系数,进一步降低了连续相位扩展的二元相移键控(CP-EBPSK)调制信号功率谱中的线谱分量,使得功率谱占用带宽更窄,信息传输更加的高效高速。多载波作为高频谱利用率的复用调制方式,与MCP-EBPSK结合势必会带来更高的系统性能,因此本文对用于解调单路MCP-EBPSK信号的冲击滤波器进行初步改进,通过添加陷波零点来抑制旁路干扰,设计出带陷波的冲击滤波器组。引入量子粒子群优化算法对加入陷波的冲击滤波器组进行优化得到滤波器组系数,仿真显示即使时频混叠的子载波间不满足正交关系,利用各冲击滤波器中心频率处极陡峭的陷波选频特性依然可以实现各子载波的正确解调。因此, 设计的冲击滤波器组可以用于子载波无保护间隔的多路MCP-EBPSK信号解调。      

Recursive short-data-record estimation of AV and MMSE/MVDR linear filters for DS-CDMA antenna array systems

The presence of the desired signal during estimation of the minimum mean-square error (MMSE)/minimum-variance distortionless-response (MVDR) and auxiliary-vector (AV) filters under limited data support leads to significant signal-to-interference-plus-noise ratio (SINR) performance degradation. We quantify this observation in the context of direct-sequence code-division multiple-access (DS-CDMA) communications by deriving close approximations for the mean-square filter estimation error, the probability density function of the output SINR, and the probability density function of the symbol-error rate (SER) of the sample matrix inversion (SMI) receiver evaluated using both a desired-signal-"present" and desired-signal-"absent" input covariance matrix. To avoid such performance degradation, we propose a DS-CDMA receiver that utilizes a simple pilot-assisted algorithm that estimates and then subtracts the desired signal component from the received signal prior to filter estimation. Then, to accommodate decision-directed operation, we develop two recursive algorithms for the on-line estimation of the AV and MMSE/MVDR filter and we study their convergence properties. Finally, simulation studies illustrate the SER performance of the overall receiver structures.     

Special problems in the estimation of power-law spectra as appliedto topographical modeling

An increasing number of topographical studies find that natural surfaces possess power-law roughness spectra. Power-law spectra introduce unique difficulties in the spectral estimation process. The authors describe how an improper window choice allows leakage that yields a spectral estimate that is insensitive to the spectral slope. In addition, the commonly used Fourier-based spectral estimates have higher variances than other available estimators. Higher variance is particularly problematic when data records are short, as is often the case in remote sensing studies. The authors show that Capon's spectral estimator has less variance than Fourier-based estimators and measures the spectral slope more accurately. The authors also show how estimates of a 2D roughness spectrum can be obtained from estimates of the 1D spectrum for the isotropic power-law case     

Analysis of linear prediction, coding, and spectral estimation fromsubbands

Linear prediction schemes make a prediction xˆ_i of a data sample x_i using p previous samples. It has been shown by Woods and O'Neil (1986) as well as Pearlman (1991) that as the order of prediction p→∞, there is no gain to be obtained by coding subband samples. This paper deals with the less well understood theory of finite-order prediction and optimal coding from subbands which are generated by ideal (brickwall) filtering of a stationary Gaussian source. We first prove that pth-order prediction from subbands is superior to pth-order prediction in the fullband, when p is finite. This fact adduces that optimal vector p-tuple coding in the subbands is shown to offer quantifiable gains over optimal fullband p-tuple coding, again when p is finite. The properties of subband spectra are analyzed using the spectral flatness measure. These results are used to prove that subband DPCM provides a coding gain over fullband DPCM, for finite orders of prediction. In addition, the proofs provide means of quantifying the subband advantages in linear prediction, optimal coding, and DPCM coding in the form of gain formulas. Subband decomposition of a source is shown to result in a whitening of the composite subband spectrum. This implies that, for any stationary source, a pth-order prediction error filter (PEF) can be found that is better than the pth-order PEF obtained by solving the Yule-Walker equations resulting from the fullband data. We demonstrate the existence of such a “super-optimal” PEF and provide algorithmic approaches to obtaining this PEF. The equivalence of linear prediction and AR spectral estimation is then exploited to show theoretically, and with simulations, that AR spectral estimation from subbands offers a gain over fullband AR spectral estimation     

An algorithm for pole-zero system model order estimation

In system identification, estimates of the unknown system model orders are often required. An algorithm for estimating model orders is described that looks at input/output data covariance matrix eigenvectors. When model orders are overestimated, zeros appear in the noise subspace eigenvectors. The number of zeros present can be used to estimate model orders     

Inferring the eigenvalues of covariance matrices from limited,noisy data

The eigenvalue spectrum of covariance matrices is of central importance to a number of data analysis techniques. Usually, the sample covariance matrix is constructed from a limited number of noisy samples. We describe a method of inferring the true eigenvalue spectrum from the sample spectrum. Results of Silverstein (1986), which characterize the eigenvalue spectrum of the noise covariance matrix, and inequalities between the eigenvalues of Hermitian matrices are used to infer probability densities for the eigenvalues of the noise-free covariance matrix, using Bayesian inference. Posterior densities for each eigenvalue are obtained, which yield error estimates. The evidence framework gives estimates of the noise variance and permits model order selection by estimating the rank of the covariance matrix. The method is illustrated with numerical examples