Average Sampling Rate Research Articles

Time-frequency spectral estimation of speech

In recent years there has been a growing interest amongst the speech research community into the use of spectral estimators which circumvent the traditional quasi-stationary assumption and provide greater time-frequency (t-f) resolution than conventional spectral estimators, such as the short time Fourier power spectrum (STFPS). One distribution in particular, the Wigner distribution (WD), has attracted considerable interest. However, experimental studies have indicated that, despite its improved t-f resolution, employing the WD as the front end of speech recognition system actually reduces recognition performance; only by explicitly re-introducing t-f smoothing into the WD are recognition rates improved. In this paper we provide an explanation for these findings. By treating the spectral estimation problem as one of optimization of a bias variance trade off, we show why additional t-f smoothing improves recognition rates, despite reducing the t-f resolution of the spectral estimator. A practical adaptive smoothing algorithm is presented, whicy attempts to match the degree of smoothing introduced into the WD with the time varying quasi-stationary regions within the speech waveform. The recognition performance of the resulting adaptively smoothed estimator is found to be comparable to that of conventional filterbank estimators, yet the average temporal sampling rate of the resulting spectral vectors is reduced by around a factor of 10.

Spectrophotometric determination of vanadium (V), (IV), and (III) with pyrogallol in a flow injection system

The direct determination of vanadium (V), (IV), and (III) by the flow injection procedure is based on spectrophotometric measurements with pyrogallol at pH 6. The effects of important parameters such as reagent concentration, flow rate, and sample volume are reported. Beer's law is obeyed up to 10 μg ml −1 of vanadium. The limit of detection was obtained as 40 ng ml −1 of the vanadium. The relative standard deviation for the determination of 8 μg ml −1 of vanadiumis 0.7%. The average sampling rate is 60 samples/h. The validity of the method is shown by the accurate and precise analysis of different synthetic samples.

Note on interpolational basis functions in the method of moments (EM scattering)

Efficient basis sets for the method of moments may be obtained using quasi-localized bandlimited interpolational functions that, broadly speaking, are defined relative to a mean sampling rate that is adjusted to curvature and proximity to edges, thus reflecting the local spatial-frequency bandwidth. Computed scattering data in a number of structures, including perfectly conducting circular and elliptical two-dimensional cylinders as well as a flat infinite strip, indicate that reasonable accuracy can be obtained with an average rate of between 2.5 and 3 basis functions per wavelength. Average sampling rates need not correspond strictly to the bandwidth of the basis functions, and there is considerable latitude with respect to random variation of sampling intervals. Although each basis function typically extends over several sample points, required integrals can be obtained speedily by the use of standard sampling-theoretical methods. >

Antialiased ray tracing by adaptive progressive refinement

We describe an antialiasing system for ray tracing based on adaptive progressive refinement. The goals of the system are to produce high quality antialiased images at a modest average sample rate, and to refine the image progressively so that the image is available in a usable form early and is refined gradually toward the final result.The method proceeds by adaptive stochastic sampling of the image plane, evaluation of the samples by ray tracing, and image reconstruction from the samples. Adaptive control of the sample generation process is driven by three basic goals: coverage of the image, location of features, and confidence in the values at a distinguished "pixel level" of resolution.A three-stage process of interpolation, filtering, and resampling is used to reconstruct a regular grid of display pixels. This reconstruction can be either batch or incremental.

Prototype for real-time adaptive sampling using the fan algorithm.

Retrospective studies have established that the fan algorithm is an effective means of acquiring high-quality digital cardiac measurements. A prototype has now been constructed that samples adaptively in real time. The prototype uses two Texas Instruments TMS32010 signal processors and 79 additional TTL integrated circuits, at a component cost of approximately US$1000. The device acquires transient samples of the incoming analogue signal at up to 16000 samples s−1, selects samples from this input stream according to the fan algorithm, and transmits the samples and the time intervals between them at a fixed permanent rate. The permanent rate is between 100 and 400 samples s−1, as selected by the user. To respond to changing waveforms and repetition rates, while maintaining a fixed transmission rate, the prototype varies the fan tolerance. The lowest fan tolerance that will allow transmission of samples and time intervals at a rate not exceeding the permanent rate is used. If permanent samples at uniform intervals result in lower peak error than the fan samples, the prototype transmits the uniformly spaced samples instead of the fan samples. Results with test and real signals show that the device performs as designed and can accurately measure fast waveforms at low average sampling rates. For example, bipolar pulses 350 μs in duration were recorded with permanent rates of only 400 samples s−1, and cardiac signals were measured with 10–390 times lower peak error than would have resulted from uniform sampling at the same average rates.

Adaptive sampling of intracellular and extracellular cardiac potentials with the fan method.

Uniform sampling rates of 5000 samples s−1 and 15000 samples s−1 or more are required to accurately represent intracellular and extracellular cardiac electrograms. These high rates are necessary because of short intervals where rapid deflections occur; thus uniform sampling results in far too many samples from slow phases when voltages change little between consecutive samples. Adaptive sampling with the fan algorithm selects samples with an irregular temporal spacing that specifies each waveform with the minimum number of samples required for a given maximum error or tolerance e. This paper describes the performance of the fan as a function of tolerance on cardiac action potentials and electrograms simulated with the Beeler-Reuter model or measured directly from the heart, including examples showing signal quality, average sampling rate and intervals between samples. With low tolerances of ɛα = 100 μV (intracellular) or ɛα = 25 μV (extracellular), the average fan sampling rates were about 8000 samples s−1 for measured waveforms, and about 200 samples s−1 for simulated ones. With higher tolerances of ɛβ = 1 mV (intracellular) or ɛβ = 60 μV (extracellular) average fan sampling rates near 200 samples s−1 were found in all cases. The results suggest that good quality measurements of action potentials and electrograms can be obtained with remarkably low average sampling rates.

Flow-injection analysis with the iron-induced perbromate—iodide reaction: spectrophotometric determination of iron

Total iron is determined by a flow-injection spectrophotometric technique. The production of I − 3 from the iron(II)- or iron(III)-induced perbromate—iodide reaction is monitored at 353 nm. Calibration graphs are linear from 10 to 100 ng/ml with correlations up to 0.9998 and can be extended up to 10 μg/ml by appropriate adjustment of conditions. The average sampling rate is 30 samples/hr. Detection limits and relative standard deviations compare well with those of other FIA methods.

Evaluation of the fan method of adaptive sampling on human electrocardiograms.

The fan is a method of adaptive sampling that selects samples from electrocardiograms more rapidly during periods of rapid waveform change and more slowly otherwise. One attribute of the fan is the guarantee that the original waveform can be reconstructed within tolerance e. Many questions about the particulars of the fan's performance on human ECGs have been undocumented, e.g. what e choice leads to good quality recording, how does the choice of e affect visual quality, and what average sampling rates occur? The paper provides answers to these and other questions. It is based on retrospective analysis of 20700 human ECG waveforms from subjects of all ages. The results show, for example, that e=10μV leads to high quality waveforms sampled at an average rate of 266 samples s−1 with maximum errors only 1/24th the maximum errors using uniform sampling at 250 samples s−1, and that e=30μV leads to waveforms showing all deflections at an average rate of 45 samples s−1 with maximum errors only 1/57th of the maximum errors from uniform sampling at 45 samples s−1.

Multichannel Recovery of Quadrature Components of Bandpass Signals

The problem of sampling signals maintaining the theoretically minimum (average) sampling rate and allowing a separate interpolation is considered from a general point of view. The formulation will follow a recently published method for multichannel sampling; this way there is the advantage of working with functions depending only on the frequency and not on time and frequency as in other approaches. This simplifies the general expressions and the determination of separation conditions. Under the assumption of nonsingularity of certain matrices, we obtain the necessary and sufficient conditions for a separate interpolation and discuss them. Finally, an especially important case is considered as an example.

Hybrid adaptive control of continuous time systems

In this note a method is presented for doing discrete adaptive control of continuous time systems. Rather than make the entire compensator discrete, the compensator is separated into a continuous time adjustable control structure and a discrete time adaptation mechanism for updating the controller parameters. The discrete adaptation algorithm which randomly samples filtered versions of the plant input and output signals, and periodically updates the controller parameters, is shown to produce a stable closed loop system for any average sampling rate.

A simplified approach to optimum quadrature sampling

A deterministic bandpass signal x(t) bandlimited to ‖ ω ‖ ε I1≡(ω0−σ/2, ω0+σ/2) can be represented in the form x(t)=p1(t) cosω0′t−q1(t) sinω0′t, where p1, q1 are low-pass (the ’’in phase’’ and ’’quadrature’’) components bandlimited to ‖ ω ‖?σ/2+‖ ω0−ω0′ ‖ and ω0′ is an arbitrary frequency within the band I1. Quadrature sampling has as its aim the recovery of the low-pass components p1, q1 directly from samples of both the bandpass signal x(t) and its quarter-wavelength translation x(t−π/2ω0′), the samples being taken at a low-pass rate. Grace and Pitt [J. Acoust. Soc. Am. 44, 1453 (1968)] obtained a result for the case ω0′=ω0?σ requiring an overall or average sampling rate of (σ/π)(γ/[γ]) samples/s, where γ=ω0/σ and [⋅] denotes the greatest integer function. By reducing the problem to an application of the classical Shannon sampling theorem in a special case (2ω0/σ =integer) and then proceeding to the general case via an embedding technique, we arrive here at an improved average sampling rate of (2γ+1/[2γ+1])(σ/π) samples/s. The method, assuming only ω0?σ/2 and finite energy signals, provides a simpler and more accessible treatment of the quadrature sampling problem than the earlier techniques of Grace and Pitt, Kohlenberg [J. Appl. Phys. 24, 1432 (1953)], and the author [IEEE Trans. Aerosp. Electron. Syst. AES-15, 366 (1979)].

A Digital Power Spectrum Analyzer

In this paper we develop a new algorithm for a digital power spectrum analyzer. The advantage of this new algorithm is that it makes possible the lowering of the average sampling rate without incurring large errors.

The measurement of the spectra of highly turbulent flows by a randomly triggered pulsed-wire anemometer

The pulsed-wire anemometer has enabled velocity measurements to be made in a variety of unsteady turbulent flows. So far the use of the instrument has been confined to the determination of mean velocity, turbulence intensity and probability density. Here we show how spectral information can also be obtained. The instrument provides estimates of the velocity by measuring the time of flight of a heated flow tracer. Periodic samples of velocity can be generated by driving the anemometer with a regular train of pulses, but unfortunately it is not possible to pulse the heater at a rate greater than about 50 or so times per second, without risk of burning out the wire. This limits the spectral information that can be obtained to situations where all the energy occurs at frequencies below 25 Hz. The random sampling scheme used here avoids the aliasing problem inherent in periodic sampling and enables estimates of power spectral density to be formed up to frequencies many times the average sampling rate. This technique is used to obtain spectral estimates of the velocity fluctuations that arise at various locations in the wake of a flat plate.

Discrete-time spectral estimation of continuous-parameter processes -- A new consistent estimate

This paper presents a new estimation scheme for the spectral density function of a stationary time series from observations taken at discrete instants of time. The sampling instants are determined by a Poisson point process on the positive real line. Under weak smoothness conditions on the spectral density, asymptotic expressions for the bias and Variance are derived, and it is shown that the estimate is mean-square consistent for all positive values of the average sampling rate. The new estimate compares favorably with the classical continuous-time spectral estimates.

Interpolation and extrapolation in nonuniform sampling sequences with average sampling rates below the Nyquist rate

Coefficients for interpolation and extrapolation, including average sampling rates below the Nyquist rate, have been derived. The method is applicable for nonuniform patterns obtained through omission of samples in a uniform pattern with a sampling rate slightly higher than the Nyquist rate. The interpolations and extrapolations are valid for band-limited signals, except those theoretical signals that are zero valued at all sampling points.

A Consistent Estimate of the Spectrum by Random Sampling of the Time Series

A new estimation scheme of the spectral density function of a stationary Gaussian time series using observations at discrete times is presented. For a broad class of spectral densities, asymptotic expressions for the bias and variance of the estimate are derived. It is shown that the estimate is consistent even though the time series is not assumed to be band-limited. It is further shown that the consistency of the estimate holds for all positive values of the average sampling rate.

Adaptive Sampling of Speech Signals

The present work gives a proposed method of sampling speech signals with a nonuniform sampling rate according to the magnitude of the slope of the signal, provided that the average sampling rate satisfies Shannon's requirements. With this proposed method, the quality of the reconstructed speech signal is improved. Experimental results have been compared with those obtained mathematically, and satisfactory agreement is indicated.

A General Approach for Generating Natural Random Variables

Simulation, Monte Carlo method, and other forms of stochastic computation consume random numbers in vast quantity and at high speed. Pseudorandom numbers or numbers generated from natural random phenomena—shot noise, thermal noise, radio-active decay, etc.,—can be used. The advantages and disadvantages of the two methods are discussed; natural random numbers are concluded to be superior in both speed and generality. A generalized natural random number generator is proposed, circuit suggestions are given, the maximum average sampling rate is derived, and recommendations for insuring incoherent operation are made.

Alias-free randomly timed sampling of stochastic processes

The notion of alias-free sampling is generalized to apply to random processes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> sampled at random times <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t_n</tex> ; sampling is said to be alias free relative to a family of spectra if any spectrum of the family can be recovered by a linear operation on the correlation sequence <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{r(n)\}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r(n) = E[x(l_{m+n}) \overline{x(t_m)}]</tex> . The actual sampling times <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t_n</tex> need not be known to effect recovery of the spectrum of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> . Various alternative criteria for verifying alias-free sampling are developed. It is then shown that any spectrum whatsoever can be recovered if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{t_n\}</tex> is a Poisson point process on the positive (or negative) half-axis. A second example of alias-free sampling is provided for spectra on a finite interval by periodic sampling (for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t \leq t_o</tex> or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t \geq t_o</tex> ) in which samples are randomly independently skipped (expunged), such that the average sampling rate is an arbitrarily small fraction of the Nyquist rate. A third example shows that randomly jittered sampling at the Nyquist rate is alias free. Certain related open questions are discussed. These concern the practical problems involved in estimating a spectrum from imperfectly known <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{ r(n) \}</tex> .

Speech Processing by the Selective Amplitude Sampling System

Selective amplitude sampling is a speech processing technique in which the input speech waveform is sampled at the times of its maxima and minima. The original wave is reconstructed from its samples by means of a box-car circuit followed by suitable filtering. The maxima and minima are determined from the zero crossings of the differentiated speech wave. Center clipping is employed to provide amplitude discrimination and a reduction in the average sampling rate. The intelligibility of speech processed by the system is compared with that for clipped differentiated speech and the original speech. “Quality” tests attempted to measure the ability of listeners to distinguish between words processed by the selective amplitude sampling system and a system in which the original words were filtered to a pass band of 300 to 3000 cps. Distributions were obtained of the sampling role averaged over 1- and 10-sec intervals. Word intelligibility and sampling rate are also related to the degree of center clipping in the sampler. The degree of center clipping is measured in terms of an rms voice to-threshold ratio. The word intelligibility scores on the selective amplitude sampling system had a mean of 93.8% and a standard deviation of 4.3%. In addition, 97% of all the test scores were equivalent to 100% sentence intelligibility. The results of the “quality” tests indicate that at least 36% of the time, the listeners were unable to distinguish between the selective amplitude sampling system and the filtered system. The average sampling rate necessary for 100% sentence intelligibility is approximately 1600 samples/sec. This rate corresponds to an rms voice-to-threshold ratio of 7 db.