Direct Spectral Estimation Research Articles

Optimized signal processing for microphone arrays containing continuously-scanning sensors

The paper presents guidelines for the optimal treatment of signals from microphone arrays that include continuously-scanning sensors. The signals from the scanning sensors are non-stationary due to the time-varying source-sensor distance and the traversing of an acoustic field with spatially-varying statistics. Quasi-stationarity has been sought in past studies by dividing the signals into blocks and applying a frequency-dependent window within each block. Optimal schedules versus frequency for block size and block overlap are proposed that enhance the spatial resolution while containing the computational cost. The guidelines are applied to the acoustic fields emitted by an impinging-jets source and a subsonic turbulent jet in isolation and integrated with a shielding plate. The noise source distributions are obtained through a direct spectral estimation approach in which the error between the modeled and measured far-field statistics is minimized. The ability of the continuous-scan paradigm, coupled with the proposed treatment of the signal, to provide high-definition noise source maps is demonstrated.

Bias correction for direct spectral estimation from irregularly sampled data including sampling schemes with correlation

The prediction and correction of systematic errors in direct spectral estimation from irregularly sampled data taken from a stochastic process is investigated. Different sampling schemes are investigated, which lead to such an irregular sampling of the observed process. Both kinds of sampling schemes are considered, stochastic sampling with non-equidistant sampling intervals from a continuous distribution and, on the other hand, nominally equidistant sampling with missing individual samples yielding a discrete distribution of sampling intervals. For both distributions of sampling intervals, continuous and discrete, different sampling rules are investigated. On the one hand, purely random and independent sampling times are considered. This is given only in those cases, where the occurrence of one sample at a certain time has no influence on other samples in the sequence. This excludes any preferred delay intervals or external selection processes, which introduce correlations between the sampling instances. On the other hand, sampling schemes with interdependency and thus correlation between the individual sampling instances are investigated. This is given whenever the occurrence of one sample in any way influences further sampling instances, e.g., any recovery times after one instance, any preferences of sampling intervals including, e.g., sampling jitter or any external source with correlation influencing the validity of samples. A bias-free estimation of the spectral content of the observed random process from such irregularly sampled data is the goal of this investigation.

TRIADS: A phase-resolving model for nonlinear shoaling of directional wave spectra

We investigate the performance of TRIADS, a numerical implementation of a phase-resolving, nonlinear, spectral model describing directional wave evolution in intermediate and shallow water. TRIADS simulations of shoaling waves generated by Hurricane Bill, 2009 are compared to directional spectral estimates based on observations collected at the Field Research Facility of the US Army Corps Of Engineers, at Duck, NC. Both the ability of the model to capture the processes essential to the nonlinear wave evolution, and the efficiency of the numerical implementations are analyzed and discussed.

Fuzzy time quantization and local normalization for the direct spectral estimation from laser Doppler velocimetry data

An adaptation of the fuzzy slotting method and the local normalization to the direct type of power spectral estimation are given. Using experimentally obtained data, it is shown that the two optional processing methods have equivalent influence on the estimated turbulence spectra, for the two processing types, the direct spectral estimation and the slotting technique. The question about the estimation quality of the fuzzy slotting and the local normalization is not addressed here. However, it is shown that the impact of fuzzy time quantization on the spectral density is that of a low-pass filter of a $$\text {sinc}^2$$ shape.

Corrections to the direct spectral estimation for laser Doppler data

An algorithm for estimating the power spectral density and the correlation function of laser Doppler-generated data sets is introduced. The algorithm is of the type of direct spectral estimators including weighting of individual velocity values to correct statistical biases caused by the correlation of instantaneous data rate and velocity values. It is extended by the forward–backward inter-arrival time weighting, the correction of the wraparound error, that of dead-time influences, and an error due to the removal of estimated block mean values. A temporal limitation of the correlation function as an alternative to the block averaging allows the block lengths to be chosen in a wide range with less necessities for compromises between systematic and random errors.

Spectral estimation for geophysical time-series with inconvenient gaps

SUMMARY The power of spectral estimation as a tool for studying geophysical processes is often limited by short records or breaks in available time-series. Direct spectral estimation using multitaper techniques designed to reduce variance and minimize leakage can help alleviate the first problem. For records with gaps, systematic interpolation or averaging of multitaper spectra derived from record fragments may prove adequate in some cases, but can be cumbersome to implement. Alternatively, multitapers can be modified for use in direct spectral estimation with intermittently sampled data. However, their performance has not been adequately studied. We investigate reliability and resolution of techniques that adapt prolate and minimum bias (MB) multitapers to accommodate the longest breaks in sampling, comparing the tapering functions (referred to as PRG or MBG tapers) with the standard prolate and MB tapers used for complete data series, and with the section-averaging approach. Using a synthetic data set, we test both jackknife and bootstrap methods to calculate confidence intervals for PRG and MBG multitaper spectral estimates and find the jackknife is both more accurate and faster to compute. To implement these techniques for a variety of data sets, we provide an algorithm that allows the user to balance judicious interpolation against the use of suitably adapted tapers, providing empirical measures of both bias and frequency resolution for candidate sets of tapers. These techniques are tested on diverse geophysical data sets: a record of change in the length of day, a model of the external dipole part of the geomagnetic field produced by the magnetospheric ring current, and a 12 Myr long irregularly sampled relative geomagnetic palaeointensity record with pernicious gaps. We conclude that both PRG and MBG tapers generally perform as well as, or better than, an optimized form of the commonly used section averaging approach. The greatest improvements seem to occur when the gap structure creates data segments of very unequal lengths. Ease of computation and more robust behaviour can make MBG tapers a better choice than PRG except when very fine-scale frequency resolution is required. These techniques could readily be applied for cross-spectral and transfer function estimation and are a useful addition to the geophysical toolbox.

Changing spectrum of geomagnetic intensity variations in a fragmented 12 My sediment record from the Oligocene

Time series of relative geomagnetic paleointensity variations derived from marine sediments can be calibrated using absolute data derived from igneous materials. The resulting records may be suitable for spectral analysis of geomagnetic dipole variations. This work re-evaluates the 12 My (22.74–34.77 Ma) sediment record from Deep Sea Drilling Project Leg 73, Site 522, that is a key data set for determining the paleomagnetic power spectrum in the frequency range 1–100 My −1. The 12 My record is marred by uneven sampling, with the interval between samples ranging from 1 to 640 ky, and contains several gaps that are considered too long to interpolate. The relative intensity data are calibrated using 129 globally distributed absolute paleointensity data from the same time interval. The power spectrum of the resulting time series is estimated using direct multi-taper spectral estimation with prolate data tapers adapted to deal with missing sections in the time series. The longest record available for analysis is thereby extended from 5.3 to 12 My. The new paleomagnetic power spectrum confirms the presence of a broad spectral peak at around 8 My −1 for the early Oligocene and uncovers a peak around 2.5 My −1 in the late Oligocene. Both peaks may be linked to tiny wiggles in marine magnetic anomalies. The new analysis unambiguously verifies that there is lower overall power in the younger part of the record, where the reversal process appears to dominate the power spectrum of the paleosecular variation. A comparison of the late Oligocene spectrum with that of PADM2M, a model of paleomagnetic axial dipole variations for 0–2 Ma, reveals some broad similarities; both time periods have similar power levels and a reversal rate of 4 My −1. During the early Oligocene the reversal rate is about a factor of two lower, the field strength is higher, and the secular variation is stronger, suggesting that a strong magnetic field inhibits reversals but produces more variability in field strength.

Imaging of directional distributed noise sources

This study relates to the acoustic imaging of noise sources that are distributed and strongly directional, such as in turbulent jets. The goal is to generate high-resolution noise source maps with self-consistency, i.e., their integration over the extent of the noise source region gives the far-field pressure auto-spectrum for a particular emission direction. Self-consistency is possible by including a directivity factor in the formulation of the source cross-spectral density. The resulting source distribution is based on the complex coherence, rather than the cross-spectrum, of the measured acoustic field. For jet noise, whose spectral nature changes with emission angle, it is necessary to conduct the measurements with a narrow-aperture array. Three coherence-based imaging methods were applied to a Mach 0.9 turbulent jet: delay-and-sum beamforming; deconvolution of the beamformer output; and direct spectral estimation that relies on minimizing the difference between the measured and modeled coherences of the acoustic field. The delay-and-sum beamforming generates noise source maps with strong spatial distortions and sidelobes. Deconvolution leads to a five-fold improvement in spatial resolution and significantly reduces the intensity of the sidelobes. The direct spectral estimation produces maps very similar to those obtained by deconvolution. The coherence-based noise source maps, obtained by deconvolution or direct spectral estimation, are similar at small and large observation angles relative to the jet axis.

The Power Law of Sensory Adaptation: Simulation by a Model of Excitability in Spider Mechanoreceptor Neurons

The power law of sensory adaptation was introduced more than 50 years ago. It is characterized by action potential adaptation that follows fractional powers of time or frequency, rather than exponential decays and corresponding frequency responses. Power law adaptation describes the responses of a range of vertebrate and invertebrate sensory receptors to deterministic stimuli, such as steps or sinusoids, and to random (white noise) stimulation. Hypotheses about the physical basis of power law adaptation have existed since its discovery. Its cause remains enigmatic, but the site of power law adaptation has been located in the conversion of receptor potentials into action potentials in some preparations. Here, we used pseudorandom noise stimulation and direct spectral estimation to show that simulations containing only two voltage activated currents can reproduce the power law adaptation in two types of spider mechanoreceptors. Identical simulations were previously used to explain the different responses of these two types of sensory neurons to step inputs. We conclude that power law adaptation results during action potential encoding by nonlinear combination of a small number of activation and inactivation processes with different exponential time constants.

Spectral Analysis as a Tool for Financial Policy: An Analysis of the Short-End of the British Term Structure

In this paper, we show how to derive the spectra and cross-spectra of economic time series from an underlying econometric or VAR model. This allows us to conduct a proper frequency analysis evaluation of economic and financial variables on a reduced sample of data, without it being ruled out by the large sample requirements of direct spectral estimation. We show, in particular, how this can be done for time-varying models and time-varying spectra. We use our techniques to show how the behaviour of British interest rates changed during and following the ERM crisis of 1992/3.

On the Efficiency of Capital Markets ? A Dynamic Analysis of the Term Structure of Interest Rates in Britain

This paper investigates the uncovered interest parity hypothesis (UIP) in an unusual way. A common prediction from theoretical models is that inefficient capital markets cause greater volatility of the observed time series. We provide empirical evidence on the efficiency of capital markets using a time domain approach. From this model we derive the dynamic properties in the time and frequency domain. Furthermore, we show how to conduct a frequency analysis on a reduced sample of data, without the latter being ruled out by the large sample requirements of direct spectral estimation. We also show, in particular, how this can be done for time-varying models resulting in time-varying spectra. We use our techniques to examine the changing stability of the relationship between British, German, and US interest rates during and following the ERM crisis of 1992/3. It turns out that UIP does not hold and that the risk premium plays a crucial role in the agents' behaviour.

Online analysis of intracranial pressure waves.

Brain pressure waves, particularly B-waves, as they were described by Lundberg contain valuable diagnostic information about the patient's status. So far, this information cannot be used on a daily routine basis because manual analysis of the recorded ICP data is very time consuming. Furthermore, the analysis is only retrospective, after incidents of interest occurred. To overcome these drawbacks, an automated B-wave detection system has been developed. Using a direct spectral estimation method (so called ARMA approach) the most significant spectral components and their amplitudes in the ICP signal are identified. The frequency with the biggest amplitude inside the B-wave band is a reliable indicator of B-wave activity and can be displayed on-line on a bedside computer.

Frequency response functions and information capacities of paired spider mechanoreceptor neurons.

Pseudorandom white-noise stimulation followed by direct spectral estimation was used to obtain linear frequency response and coherence functions from paired, but dynamically different, spider mechanosensory neurons. The dynamic properties of the two neuron types were similar with either mechanical or electrical stimulation, showing that action potential encoding dominates the dynamics. Phase-lag data indicated that action potential initiation occurs more rapidly during mechanical stimulation, probably in the distal sensory dendrites. Total information capacity, calculated from coherence, as well as information per action potential, were both similar in the two types of neurons, and similar to the few available estimates from other spiking neurons. However, information capacity and information per action potential both depended strongly on neuronal firing rate, which has not been reported before.

A Spectral Analysis of the Short-End of the British Term Structure

In this paper, we show how to derive the spectra and cross-spectra of economic times series from the dynamic structure of an underlying econometric model. This allows us to conduct a proper frequency analysis of economic and financial variables on a limited sample of data, without it being ruled out by the large sample requirements of direct spectral estimation. We show, in particular, how this can be done for time-varying models and time-varying spectra. To demonstrate our technique in action, we use it to analyse the behaviour of British interest rates during and following the ERM crisis of 1992/3.

The autocorrelation function extrapolation and equivalence of determinant and entropy maximizations

This paper uses the theoretical properties of the autocorrelation matrices to approach the autocorrelation function extrapolation method based on maximizing the determinant of the autocorrelation matrix. Thus it arrives at the same results as appearing in the literature, while employing less complicated matrix algebra and considering the determinant calculus by matrix partitioning. Then the results obtained in this paper are compared with the maximum entropy spectral estimation method, and the equivalence of the maximum determinant and maximum entropy approaches is shown. Hence, a further understanding of the method is provided and it is deduced that the maximum determinant extrapolation algorithm, may also be used for direct spectral estimation in the maximum entropy method; i.e. without extrapolating the autocorrelation function.

Airborne measurements of the ocean's Ku‐band radar cross‐section at low incidence angles

Abstract Ocean backscatter data obtained with a Ku‐band airborne radar are presented along with coincident altimeter and directional wave spectral estimates. These data were collected using one sensor, NASA's radar ocean wave spectrometer (ROWS). The measurements are compared with an electromagnetic scattering model for perfectly conducting Gaussian random surfaces. The normalized radar cross‐section (NRCS) data cover those incidence angles (0–20°) where both quasi‐specular and Bragg scattering mechanisms are expected. Under certain conditions, identification and separation of these two mechanisms is possible. The scanning radar allows observations of the azimuthal variations in NRCS that are at times indicative of short‐scale wave generation in the wind direction.

Fine‐scale measurements of the vertical ambient noise field

The fine‐scale structure of the directional ambient noise field is investigated utilizing a large‐aperture (900‐m) vertical array. Frequency and directional spectral estimates are calculated during the passage of a local storm, providing a detailed study of the ambient noise field temporal and spatial characteristics at low frequencies (15–130 Hz) as wind speed increases from 2 to 12 m/s over a 21‐h period. Spectral levels of horizontal beams reflect the variability of discrete distant sources. Spectral levels of beams directed toward the surface and the bottom display a threshold‐type behavior, suggesting the abrupt onset of a source mechanism such as breaking waves.

Measuring the vertical directional spectra caused by sea surface sound

The objective of the measurements discussed here is the development of a database of directional spectra obtained in an acoustically isolated area (Tongue of the Ocean, the Bahamas) where the acoustic ambient is principally controlled by the local (observable) sea surface conditions. The purpose of the database is the identification of a model of the space‐time statistics of sea surface sound. A wideband (40‐ to 4000‐Hz) vertical acoustic linear antenna system was used. Data logging took place in a remote autonomously operating subsurface buoy system. Recording restrictions limited each of the seven octavely nested apertures of the sensor system to four wavelengths. However, quite adequate spatial resolution of the directional spectral estimates was obtained using a physically based parametric model of the random process. The results of the initial 2‐month deployment (wind speed range of 2–16 m/s) are discussed here. The measured directional spectra are displayed as a function of frequency, elevation angle, and surface friction velocity. The sea surface sound source structure is discussed.

Field intercomparison of nearshore directional wave sensors

Five measurement strategies (four in situ, one remote) for estimating directional wave spectra were intercompared in a 1980 experiment at the Coastal Engineering Research Center's Field Research Facility in Duck, NC. The systems included two pressure sensor/biaxial current meter combinations (different manufacturers), a triaxial acoustic current meter, an SXY gauge (square array of four pressure sensors), and a shore-based imaging radar. A detailed error analysis suggests sources for differences in estimated wave spectra from the different instruments; in general, they intercompare favorably. The major deviation among in situ gauges was associated with the triaxial acoustic current meter. Reliance on a vertical velocity measurement (instead of a direct pressure or sea-surface elevation measurement) can contribute additional uncertainty in directional spectral estimates. The imaging radar was successful in distinguishing multiple wave trains at the same frequency, which was not possible with the simple spectral estimation analysis applied to in situ data. However, the radar is not useful in providing accurate estimates of spectral density, nor in distinguishing multiple wave trains of different frequencies coming from the same direction. Selection of a measurement strategy for a particular need depends on the precise data requirements for that application. Although the five tested intercompared well, in practice not all are equally suitable for every application.