Showing papers on "Spectral density estimation published in 2007"

Periodically correlated random sequences : spectral theory and practice

Abstract: Preface. Acknowledgments. Glossary. 1. Introduction. 1.1 Summary. 1.2 Historical Notes. Problems. 2. Examples, Models and Simulations. 2.1 Examples and Models. 2.1.1 Random Periodic Sequences. 2.1.2 Sums of Periodic and Stationary Sequences. 2.1.3 Products of Scalar Periodic and Stationary Sequences. 2.1.4 Time Scale Modulation of Stationary Sequences. 2.1.5 Pulse Amplitude Modulation. 2.1.6 A More General Example. 2.1.7 Periodic Autoregressive Models. 2.1.8 Periodic Moving Average Models. 2.1.9 Periodically Perturbed Dynamical Systems. 2.2 Simulations. 2.2.1 Sums of Periodic and Stationary Sequences. 2.2.2 Products of Scalar Periodic and Stationary Sequences. 2.2.3 Time Scale Modulation of Stationary Sequences. 2.2.4 Pulse Amplitude Modulation. 2.2.5 Periodically Perturbed Logistic Maps. 2.2.6 Periodic Autoregressive Models. 2.2.7 Periodic Moving Average Models. Problems. 3. Review of Hilbert Spaces. 3.1 Vector Spaces. 3.2 Inner Product Spaces. 3.3 Hilbert Spaces. 3.4 Operators. 3.5 Projection Operators. 3.6 Spectral Theory of Unitary Operators. 3.6.1 Spectral Measures. 3.6.2 Spectral Integrals. 3.6.3 Spectral Theorems. Problems. 4. Stationary Random Sequences. 4.1 Univariate Spectral Theory. 4.1.1 Unitary Shift. 4.1.2 Spectral Representation. 4.1.3 Mean Ergodic Theorem. 4.1.4 Spectral Domain. 4.2 Univariate Prediction Theory. 4.2.1 Infinite Past, Regularity and Singularity. 4.2.2 Wold Decomposition. 4.2.3 Innovation Subspaces. 4.2.4 Spectral Theory and Prediction. 4.2.5 Finite Past Prediction. 4.3 Multivariate Spectral Theory. 4.3.1 Unitary Shift. 4.3.2 Spectral Representation. 4.3.3 Mean Ergodic Theorem. 4.3.4 Spectral Domain. 4.4 Multivariate Prediction Theory. 4.4.1 Infinite Past, Regularity and Singularity. 4.4.2 Wold Decomposition. 4.4.3 Innovations and Rank. 4.4.4 Regular Processes. 4.4.5 Infinite Past Prediction. 4.4.6 Spectral Theory and Rank. 4.4.7 Spectral Theory and Prediction. 4.4.8 Finite Past Prediction. Problems. 5. Harmonizable Sequences. 5.1 Vector Measure Integration. 5.2 Harmonizable Sequences. 5.3 Limit of Ergodic Average. 5.4 Linear Time Invariant Filters. Problems. 6. Fourier Theory of the Covariance. 6.1 Fourier Series Representation of the Covariance. 6.2 Harmonizability of R(s t). 6.2.1 Harmonizability of Xt. 6.4 Covariance and Spectra for Specific Cases. 6.4.1 PC White Noise. 6.4.2 Products of Scalar Periodic and Stationary Sequences. 6.5 Asymptotic Stationarity. 6.6 Lebesgue Decomposition of F. 6.7 The spectrum of mt. 6.8 Effects of Common Operations on PC Sequences. 6.8.1 Linear Time Invariant Filtering. 6.8.2 Differencing. 6.8.3 Random Shifts. 6.8.4 Sampling. 6.8.5 Bandshifting. 6.8.6 Periodically Time Varying (PTV) Filters. Problems. 7. Representations of PC Sequences. 7.1 The Unitary Operator of a PC Sequence. 7.2 Representations Based on the Unitary Operator. 7.2.1 Gladyshev Representation. 7.2.2 Another Representation of Gladyshev Type. 7.2.3 Time-dependent Spectral Representation. 7.2.4 Harmonizability Again. 7.2.5 Representation Based on Principal Components. 7.3 Mean Ergodic Theorem. 7.4 PC Sequences as Projections of Stationary Sequences. Problems. 8. Prediction of PC Sequences. 8.1 Wold Decomposition. 8.2 Innovations. 8.3 Periodic Autoregressions of Order 1. 8.4 Spectral Density of Regular PC Sequences. 8.4.1 Spectral Densities for PAR(1). 8.5 Least Mean Square Prediction. 8.5.1 Prediction Based on Infinite Past. 8.5.2 Prediction for a PAR(1) Sequence. 8.5.3 Finite Past Prediction. Problems. 9. Estimation of Mean and Covariance. 9.1 Estimation of mt : Theory. 9.2 Estimation of mt : Practice. Problems. 10. Spectral Estimation. 10.1 The Shifted Periodogram. 10.2 Consistent Estimators. 10.3 Asymptotic Normality. 10.4 Spectral Coherence 363. 10.4.1 Spectral Coherence for Known T. 10.4.2 Spectral Coherence for Unknown T. 10.5 Spectral Estimation : Practice. 10.5.1 Confidence Intervals. 10.5.2 Examples. 10.6 Effects of Discrete Spectral Components. 10.6.1 Removal of the Periodic Mean. 10.6.2 Testing for Additive Discrete Spectral Components. 10.6.3 Removal of Detected Components. Problems. 11. A Paradigm for Nonparametric Analysis of PC Time Series. 11.1 The Period T is Known. 11.2 The Period T is Unknown. References. Index.

Fast, Accurate Frequency Estimators [DSP Tips & Tricks]

Abstract: This article presents computationally simple algorithms that provide substantial refinement of the frequency estimation of tones based on DFT samples without the need for increasing the DFT size. When estimating the frequency of a tone, the idea is to estimate the frequency of the spectral peak based on three DFT samples is discussed

Minimum-Variance Multitaper Spectral Estimation on the Sphere

Abstract: We develop a method to estimate the power spectrum of a stochastic process on the sphere from data of limited geographical coverage. Our approach can be interpreted either as estimating the global power spectrum of a stationary process when only a portion of the data are available for analysis, or estimating the power spectrum from local data under the assumption that the data are locally stationary in a specified region. Restricting a global function to a spatial subdomain—whether by necessity or by design—is a windowing operation, and an equation like a convolution in the spectral domain relates the expected value of the windowed power spectrum to the underlying global power spectrum and the known power spectrum of the localization window. The best windows for the purpose of localized spectral analysis have their energy concentrated in the region of interest while possessing the smallest effective bandwidth as possible. Solving an optimization problem in the sense of Slepian (1960) yields a family of orthogonal windows of diminishing spatiospectral localization, the best concentrated of which we propose to use to form a weighted multitaper spectrum estimate in the sense of Thomson (1982). Such an estimate is both more representative of the target region and reduces the estimation variance when compared to estimates formed by any single bandlimited window. We describe how the weights applied to the individual spectral estimates in forming the multitaper estimate can be chosen such that the variance of the estimate is minimized.

Direction of Arrival Estimation Using the Parameterized Spatial Correlation Matrix

Abstract: The estimation of the direction-of-arrival (DOA) of one or more acoustic sources is an area that has generated much interest in recent years, with applications like automatic video camera steering and multiparty stereophonic teleconferencing entering the market. DOA estimation algorithms are hindered by the effects of background noise and reverberation. Methods based on the time-differences-of-arrival (TDOA) are commonly used to determine the azimuth angle of arrival of an acoustic source. TDOA-based methods compute each relative delay using only two microphones, even though additional microphones are usually available. This paper deals with DOA estimation based on spatial spectral estimation, and establishes the parameterized spatial correlation matrix as the framework for this class of DOA estimators. This matrix jointly takes into account all pairs of microphones, and is at the heart of several broadband spatial spectral estimators, including steered-response power (SRP) algorithms. This paper reviews and evaluates these broadband spatial spectral estimators, comparing their performance to TDOA-based locators. In addition, an eigenanalysis of the parameterized spatial correlation matrix is performed and reveals that such analysis allows one to estimate the channel attenuation from factors such as uncalibrated microphones. This estimate generalizes the broadband minimum variance spatial spectral estimator to more general signal models. A DOA estimator based on the multichannel cross correlation coefficient (MCCC) is also proposed. The performance of all proposed algorithms is included in the evaluation. It is shown that adding extra microphones helps combat the effects of background noise and reverberation. Furthermore, the link between accurate spatial spectral estimation and corresponding DOA estimation is investigated. The application of the minimum variance and MCCC methods to the spatial spectral estimation problem leads to better resolution than that of the commonly used fixed-weighted SRP spectrum. However, this increased spatial spectral resolution does not always translate to more accurate DOA estimation

Wiener estimation method in estimating of spectral reflectance from RGB images

Abstract: Color is one of the most important features in digital images. The representation of color in digital form with a three-component image (RGB) is not very accurate, hence the use of a multiple-component spectral image is justified. At the moment, acquiring a spectral image is not as easy and as fast as acquiring a conventional three-component image. One answer to this problem is to use a regular digital RGB camera and estimate its RGB image into a spectral image by the Wiener estimation method, which is based on the use of a priori knowledge. In this paper, the Wiener estimation method is used to estimate the spectra of icons. The experimental results of the spectral estimation are presented.

An introduction to FFT and time domain windows

Abstract: This paper includes a brief tutorial on digital spectrum analysis and FFT-related issues to form spectral estimates on digitized signals. Some review of the DFT has been presented, and some discussion on the computational advantages of the FFT calculation has also been presented. Finally, the main considerations on windowing and window characteristics have been briefly discussed.

Ocean Wave Spectral Estimation Using Vessel Wave Frequency Motions

Abstract: Wave spectra are estimated from wave frequency motions of a vessel at zero or low advance speed. Minimization of a cost functional that indicates how well the estimated spectrum results in the measured motion spectra was based on sequential quadratic programming and a genetic algorithm. Two procedures have been developed and applied to numerically simulated motions of a 59 m length offshore supply vessel.

Semi-parametric estimation of shifts

Abstract: We observe a large number of functions differing from each other only by a translation parameter. While the main pattern is unknown, we propose to estimate the shift parameters using $M$-estimators. Fourier transform enables to transform this statistical problem into a semi-parametric framework. We study the convergence of the estimator and provide its asymptotic behavior. Moreover, we use the method in the applied case of velocity curve forecasting.

Multiscale windowed Fourier transform for phase extraction of fringe patterns

Abstract: A multiscale windowed Fourier transform for phase extraction of fringe patterns is presented. A local stationary length of signal is used to control the window width of a windowed Fourier transform automatically, which is measured by an instantaneous frequency gradient. The instantaneous frequency of the fringe pattern is obtained by detecting the ridge of the wavelet transform. The numerical simulation and experiment have proved the validity of this method. The combination of the windowed Fourier transform and the wavelet transform makes the extracted phase more precise than other methods.

EMG pattern classification using spectral estimation and neural network

Abstract: In this paper, we propose a method of pattern recognition of EMG signals of hand gesture using spectral estimation and neural network. Proposed system is composed of the Yule-Walker algorithm and the LVQ. The use of the Yule-Walker algorithm is to estimates the power spectral density (PSD) of the signal. The spectral estimate returned is the magnitude squared frequency response of AR model. A fine tuning step will then be incorporated to improve the accuracy of the classification by way of the LVQ. We describe in detail the experiment conducted to verify the usefulness of the proposed method for EMG pattern classification of hand gesture.

Spectral estimation in unevenly sampled space of periodically expressed microarray time series data

Abstract: Periodogram analysis of time-series is widespread in biology. A new challenge for analyzing the microarray time series data is to identify genes that are periodically expressed. Such challenge occurs due to the fact that the observed time series usually exhibit non-idealities, such as noise, short length, and unevenly sampled time points. Most methods used in the literature operate on evenly sampled time series and are not suitable for unevenly sampled time series. For evenly sampled data, methods based on the classical Fourier periodogram are often used to detect periodically expressed gene. Recently, the Lomb-Scargle algorithm has been applied to unevenly sampled gene expression data for spectral estimation. However, since the Lomb-Scargle method assumes that there is a single stationary sinusoid wave with infinite support, it introduces spurious periodic components in the periodogram for data with a finite length. In this paper, we propose a new spectral estimation algorithm for unevenly sampled gene expression data. The new method is based on signal reconstruction in a shift-invariant signal space, where a direct spectral estimation procedure is developed using the B-spline basis. Experiments on simulated noisy gene expression profiles show that our algorithm is superior to the Lomb-Scargle algorithm and the classical Fourier periodogram based method in detecting periodically expressed genes. We have applied our algorithm to the Plasmodium falciparum and Yeast gene expression data and the results show that the algorithm is able to detect biologically meaningful periodically expressed genes. We have proposed an effective method for identifying periodic genes in unevenly sampled space of microarray time series gene expression data. The method can also be used as an effective tool for gene expression time series interpolation or resampling.

Spectral color imaging system for estimating spectral reflectance of paint

Abstract: In this paper, the analysis methods used for developing imaging systems estimating spectral reflectance are considered. The chosen system incorporates an estimation technique for spectral reflectance. Several traditional and machine learning estimation techniques are compared for this purpose. The accuracy of spectral estimation with this system and each estimation technique is evaluated and the system's performance is presented.

Instantaneous Spectral Analysis: Time-Frequency Mapping via Wavelet Matching with Application to Contaminated-Site Characterization by 3D GPR

Abstract: Spectral decomposition, by which a time series is transformed from the 1D time/amplitude domain to the 2D time/spectrum domain, has become a popular and useful tool in seismic exploration for hydrocarbons. The windowed, or short-time Fourier transform (STFT) was one early approach to computing the time-frequency (t-f) distribution. This method relies on the user selecting a fixed time window, then computing the Fourier spectrum within the time window while sliding the window along the length of the trace. The primary limitation of the STFT is the fixed window which prevents either time localization of high frequency components (if a long window is used) or spectral resolution of the low-frequency components (if a short window is used).

Analyzing Chaotic Spectra of DC–DC Converters Using the Prony Method

Abstract: When dc-dc converters operate in chaotic modes, they can generate spread spectra, which are useful for reducing the electromagnetic interference (EMI). Conventionally, the Fast Fourier Transform (FFT) is used to analyze the spectra. However, it is not applicable to the inner-harmonics, the nonintegral multiples of the fundamental frequency, which is a prominent feature of chaotic signals. In this brief, the Prony method is suggested for chaotic spectral estimation of dc-dc converters. Numerical simulations show its advantages over the traditional FFT

Coding of sparse digital media spectral data

Abstract: An audio encoder/decoder provides efficient compression of spectral transform coefficient data characterized by sparse spectral peaks. The audio encoder/decoder applies a temporal prediction of the frequency position of spectral peaks. The spectral peaks in the transform coefficients that are predicted from those in a preceding transform coding block are encoded as a shift in frequency position from the previous transform coding block and two non-zero coefficient levels. The prediction may avoid coding very large zero-level transform coefficient runs as compared to conventional run length coding. For spectral peaks not predicted from those in a preceding transform coding block, the spectral peaks are encoded as a value trio of a length of a run of zero-level spectral transform coefficients, and two non-zero coefficient levels.

Least Mean Square Method for LVDT Signal Processing

Abstract: This paper describes an innovative approach for position estimation using traditional displacement inductive sensors such as linear variable transducers. In addition, the same algorithm offers an evaluation of velocity and acceleration. However, the same approach can be applied to any other kind of alternating-current-excited sensor. The proposed method overcomes the typical limit of traditional processing techniques as coherent demodulation or spectral analysis by implementing a least mean squares estimation of the variables of interest. A working prototype has been designed around a low-cost digital signal processor from Texas Instruments Inc., and an estimation time on the order of 1 ms has been obtained. In static conditions, the resolution is about 0.01% of the full scale of the considered sensor, which is on the same order as the one obtained with spectral estimation. In dynamic conditions, simulations show a performance improvement in position and velocity estimation with a sensible root mean square error (RMSE) reduction. The experimental results in dynamic conditions are difficult to quantify, owing to noise, even if the performances are better than with traditional methods.

A low delay, variable resolution, perfect reconstruction spectral analysis-synthesis system for speech enhancement

Abstract: The choice of the window function and window length in short time analysis-synthesis (AS) systems based on the discrete Fourier transform (DFT) has to balance conflicting requirements: Long windows provide high spectral resolution while short windows allow for high temporal resolution. Furthermore, for many applications a low algorithmic delay is desirable. Therefore, long standard windows such as the Hann or Hamming windows cannot be used. In this contribution we propose a novel AS system which achieves perfect reconstruction (PR) and a low delay by using a variable length analysis window and a relatively short synthesis window. The variable length analysis windows allow a spectral analysis that is adapted to the signals span of stationarity. The AS windows are designed such that they can be switched at any time instant without violating PR. We show that the spectral representation of typical speech data is improved as compared to AS systems with standard windows.

Knock Signal Conditioning Using the Discrete Fourier Transform and Variable Detection Window Length

Abstract: As the speed and memory capabilities of microprocessors have increased, it has become more popular for the signal conditioning of knock sensor outputs to be performed entirely within the microprocessor. One method of this signal conditioning process utilizes the Discrete Fourier Transform (DFT). It is common for systems that use this method to limit the knock detection window to one length across all RPM and load points to reduce computation and memory constraints on the processor. This paper will summarize this method and explain a further enhancement, variable window length, that may become more popular as processor capabilities increase.

Recursive and fast recursive capon spectral estimators

Abstract: The Capon algorithm, which was originally proposed for wavenumber estimation in array signal processing, has become a powerful tool for spectral analysis. Over several decades, a significant amount of research attention has been devoted to the estimation of the Capon spectrum. Most of the developed algorithms thus far, however, rely on the direct computation of the inverse of the input correlation (or covariance) matrix, which can be computationally very expensive particularly when the dimension of the matrix is large. This paper deals with fast and efficient algorithms in computing the Capon spectrum. Inspired from the recursive idea established in adaptive signal processing theory, we first derive a recursive Capon algorithm. This new algorithm does not require an explicit matrix inversion, and hence it is more efficient to implement than the direct-inverse approach. We then develop a fast version of the recursive algorithm based on techniques used in fast recursive least-squares adaptive algorithms. This new fast algorithm can further reduce the complexity of the recursive Capon algorithm by an order of magnitude. Although our focus is on the Capon spectral estimation, the ideas shown in this paper can also be generalized and applied to other applications. To illustrate this, we will show how to apply the recursive idea to the estimation of the magnitude squared coherence function, which plays an important role for problems like time-delay estimation, signal-to-noise ratio estimation, and doubletalk detection in echo cancellation.

Techniques to generate constant envelope multicarrier transmission for wireless networks

Abstract: Various example embodiments are disclosed herein. According to an example embodiment, an apparatus for use in a wireless transmitter may include a continuous phase modulation (CPM) sample generator configured to generate a group of constant envelope CPM modulated signal samples, a Fourier transform block configured to perform a Fourier transform on the group of constant envelope signal samples to generate an initial group of Fourier coefficients, a zero insertion block configured to generate an expanded group of Fourier coefficients by inserting one or more zeros in the initial group of Fourier coefficients, and an inverse Fourier transform block configured to perform an inverse Fourier transform on the expanded group of Fourier coefficients to generate a group of constant envelope time-domain samples and to map the constant envelope time-domain samples onto a group of orthogonal subcarriers for transmission.

Time-frequency representation based chirp-like signal analysis using multiple level crossings

Abstract: The instantaneous frequency of a chirp-like signal varies with time. The sampling density of a signal should preferably correspond to its local bandwidth. The paper discusses the level-crossing sampling principle, which provides the capability for performing such an analog-to-digital conversion. The properties of level-crossing sampling are considered. As the captured samples are spaced non-uniformly, appropriate digital signal processing is required. The chirp-like signal is characterized by time-frequency representation. Short-time Fourier transform and Wigner-Ville distribution are inspected and enhanced to make them applicable to signal processing in the level-crossing sampling case. Additional benefits are obtained if the combined approach is used — artifacts are removed and localization of the chirp-like signal is improved. The simulation results are demonstrated.

A New Algorithm for Sparse Aperture Interpolation

Abstract: A new algorithm for filling sparse aperture synthetic aperture radar (SAR)/inverse SAR (ISAR) data, which applies for widely gapped apertures, is proposed in this letter. An Estimating Signal Parameter via Rotational Invariance Techniques(ESPRIT)-based parametric approach is first used to estimate the power distribution with the sparse data. With the estimated power spectrum as prior information, by minimizing a weighted norm as a constraint, the full aperture data can be estimated. Although the algorithm is proposed for the sparse aperture interpolation in SAR/ISAR, it can be applied to other gapped data spectral estimation problems as well. Both numerical and experimental examples are provided to demonstrate the performance of the proposed algorithm.

Direction of Arrival Estimation using Eigenanalysis of the Parameterized Spatial Correlation Matrix

Abstract: The estimation of the direction-of-arrival (DOA) of one or more acoustic sources is an area that has generated much interest in recent years, with applications like automatic video camera steering and multi-party stereophonic teleconferencing entering the market. Time-difference-of-arrival (TDOA) based methods compute each relative delay using only two microphones, even though additional microphones are usually available, and thus suffer from the effects of background noise and reverberation. This paper deals with DOA estimation based on spatial spectral estimation, and proposes a novel DOA estimator based on the eigenvalues of the parameterized spatial correlation matrix. Simulation results confirm the ability of the proposed method to provide reliable estimates even in heavily reverberant environments.

OFDM Symbol Synchronization Using Frequency Domain Pilots in Time Domain

Abstract: In this paper, we describe and evaluate a novel time synchronization algorithm for OFDM systems robust to channels with long echoes. Most of the OFDM based standards provide known pilots in the frequency domain for channel estimation purposes. The basic idea of our contribution consists of interpreting the frequency domain pilots in the time domain and using them to obtain a rough channel estimation prior to the receiver discrete Fourier transform (DFT). The rough channel estimation is then used to perform the fine time synchronization.

Improvement of the HHT method and application in weak vortex signal detection

Abstract: The vortex signal is greatly disturbed by noises from external interference when the meter works at low flow rate, which results in a limited measuring range for the flowmeter. In order to solve the problem, a new signal processing method based on the Hilbert–Huang transform (HHT) is proposed. With its good performance on local adaptability and time–frequency analysis, noises are removed by the empirical mode decomposition (EMD) and the residue components are analysed by the Hilbert transform; then instantaneous frequency distributions are achieved. When the probability density of a certain frequency component exceeds 5%, the sifting process will be terminated. Subsequently, the vortex frequency can be calculated from the last residue component. Experimental studies were carried out to compare the improved method with the classic method FFT at low flow rate. A better linearity and lower limit of measurement are achieved by the proposed method.

Estimation of Structured Gaussian Mixtures: The Inverse EM Algorithm

Abstract: This contribution is devoted to the estimation of the parameters of multivariate Gaussian mixture where the covariance matrices are constrained to have a linear structure such as Toeplitz, Hankel, or circular constraints. We propose a simple modification of the expectation-maximization (EM) algorithm to take into account the structure constraints. The basic modification consists of virtually updating the observed covariance matrices in a first stage. Then, in a second stage, the estimated covariances undergo the reversed updating. The proposed algorithm is called the inverse EM algorithm. The increasing property of the likelihood through the algorithm iterations is proved. The strict increasing for nonstationary points is proved as well. Numerical results are shown to corroborate the effectiveness of the proposed algorithm for the joint unsupervised classification and spectral estimation of stationary autoregressive time series.

Statistical study of generalized nonlinear phase step estimation methods in phase-shifting interferometry

Abstract: Signal processing methods based on maximum-likelihood theory, discrete chirp Fourier transform, and spectral estimation methods have enabled accurate measurement of phase in phase-shifting interferometry in the presence of nonlinear response of the piezoelectric transducer to the applied voltage. We present the statistical study of these generalized nonlinear phase step estimation methods to identify the best method by deriving the Cramer-Rao bound. We also address important aspects of these methods for implementation in practical applications and compare the performance of the best-identified method with other bench marking algorithms in the presence of harmonics and noise.