Showing papers on "Spectral density estimation published in 1989"

The Fourier transform

Abstract: To calculate a transform, just listen. The ear automatically performs the calculation, which the intellect can execute only after years of mathematical education. The ear formulates a transform by converting sound-the waves of pressure traveling through time and the atmosphere-into a spectrum, a description of the sound as a series of volumes at distinct pitches. The brain turns this information into perceived sound. Similar operations can be done by mathematical methods on sound waves or virtually any other fluctuating phenomenon, from light waves to ocean tides to solar cycles. These mathematical tools can decompose functions representing such fluctuations into a set of sinusoidal components-undulating curves that vary from a maximum to a minimum and back, much like the heights of ocean waves. The Fourier transform is a function that describes the amplitude and phase of each sinusoid, which corresponds to a specific frequency. The Fourier transform has become a powerful tool in diverse fields of science. In some cases, the Fourier transform can provide a means of solving unwieldy equations that describe dynamic response to electricity, hear or light. In other cases, it can identify the regular contributions to a fluctuating signal, thereby helping to make sense of observations inmore » astronomy, medicine and chemistry.« less

An introduction to digital signal processing

Abstract: Signals and Systems Sampled Data and the Z Transform Sinusoidal Response of LSI Systems Couplets and Elementary Filters The Discrete Fourier Transform The Continuous Fourier Integral Transform Application of the Fourier Transform to Digital Signal Processing Digital Filter Design Inverse Filtering and Deconvolution Spectral Factorization Power Spectral Estimation Multidimensional DSP References

General phase modulation method for stored waveform inverse fourier transform excitation for fourier transform ion cyclotron resonance mass spectrometry

Abstract: A method for reducing the dynamic range of FT-ICR signal generated by the SWIFT technique includes the step of time shifting wave packets corresponding to segments of the Fourier spectral magnitude function to prevent coherent summing of the various frequency components of the excitation signal.

Synthesis of high rejection filters with the Fourier transform method

Abstract: Although in theory the Fourier transform method is valid only for small rejections, in practice it can be modified for the synthesis of high rejection filters with minimum transmittances as low as 10−4. Two new spectral functions are proposed for use in the Fourier transforms. An empirical procedure which is much faster than refinement is described for optimization of the spectral performance. The method and optimization are illustrated numerically for several different spectral shapes.

A tutorial overview of modern spectral estimation

Abstract: A summary of several modern spectral estimation methods is presented. Most of the methods can be explained in the context of parametric time-series modeling. A few methods involve nonparametric treatment. The techniques discussed include classical spectral estimation, autoregressive (maximum entropy), ARMA (autoregressive moving average), Prony, maximum-likelihood, Pisarenko, and MUSIC methods. Many of the techniques have fast computational algorithms, making them viable for real-time applications. The tutorial concludes with a commentary concerning current spectral estimation research. >

High-resolution velocity spectra using eigenstructure methods

Abstract: Stacking spectra provide maximum‐likelihood estimates for the stacking velocity, or for the ray parameter, of well separated reflections in additive white noise. However, the resolution of stacking spectra is limited by the aperture of the array and the frequency of the data. Despite these limitations, parametric spectral estimation methods achieve better resolution than does stacking. To improve resolution, the parametric methods introduce a parsimonious model for the spectrum of the data. In particular, when the data are modeled as the superposition of wavefronts, the properties of the eigenstructure of the data covariance matrix can be used to obtain high‐resolution spectra. The traditional stacking spectra can also be expressed as a function of the data covariance matrix and directly compared to the eigenstructure spectra. The superiority of the latter in separating closely interfering reflections is then apparent from a simple geometric interpretation. Eigenstructure methods were originally developed...

A new method of spectral analysis and its application to the Earth's free oscillations: The “Sompi” method

Abstract: A new and powerful method of spectral analysis, which is named the “Sompi” method after a Japanese word, is introduced and applied to low-frequency seismograms. The Sompi method is based on an autoregressive (AR) process model, which is different from an ordinary prediction type AR model. In the ordinary AR model the AR coefficients predict the present observation from the past observations so that they are determined by minimizing the prediction error. The model implicitly assumes that the past observations are noise-free and that only the present observation consists of signal and noise. The method thus gives an unbiased estimate of the linear relationship among the past data and the present signal. In our AR model the AR coefficients extract signals from the time series so that they are determined by minimizing the extraction residual. The model assumes that each observation, either past or present, consists of signal and noise. The method thus gives an unbiased estimate of the linear relationship among the successive signals. Those who wish to make an unbiased spectral estimation for the signal must therefore minimize the extraction residual, rather than the prediction error. Minimization of the extraction residual leads to an eigenvalue problem of a non-Toeplitz matrix of autocovariance. The minimum eigenvalue yields the extraction residual which is an estimate of the noise power. The corresponding eigenvector constitutes the AR coefficients whose characteristic equation gives the complex frequencies (frequencies and decay or growth rates) of the signals extracted. The complex amplitudes (amplitudes and phases) are then determined through a least squares procedure. The Sompi method thus first retrieves medium-sensitive parameters, frequencies, and Q and then excitation-sensitive parameters, amplitudes, and phases. From the practical point of view it is not feasible to apply the Sompi method directly to such data as the Earth's free oscillations in which spectral peaks are densely and nonuniformly distributed over the entire frequency domain. We propose an algorithm of aliased sampling which enables us to apply the Sompi method to such data. In this algorithm the time series is first narrowly and sharply band-pass filtered. The resultant time series is then decimated at a rate corresponding to the bandwidth of the filter. The contributions of all the decimated time series, which may mutually lag in time, are stacked into the matrix elements of autocovariance for the eigenvalue problem. The Sompi method with this algorithm is tested against a synthetic seismogram to see the resolvability of the two modes closely spaced in frequency. The Sompi method is next applied to the seismograms of the 1977 Sumbawa Island earthquake recorded in the International Deployment of Accelerometers network. First, the two radial modes 0S0 and 1S0 are analyzed to examine the uniformity of their spectral parameters among different stations. Second, the fundamental multiplets 0Sl (l = 5–43) are analyzed to see how their initial amplitudes and phases oscillate with respect to l at one station. Third, the gravest mode 0S2 is analyzed in an attempt to resolve the five singlets with their Q values. Fourth, the analysis is made for the coupled multiplets 0Sl-0Tl+1 (l = 10–12) to observe their mutual repulsion in frequency and their mutual attraction in decay rate. Fifth, the core mode 7S3, a mode with most of its energy confined to the inner core, is detected. All five experiments demonstrate remarkable accuracy and resolvability of the Sompi method.

A real-time autoregressive spectrum analyzer for Doppler ultrasound signals

Abstract: A system based on a digital signal processor and a microcomputer has been programmed to estimate the maximum entropy autoregressive (AR) power spectrum of ultrasonic Doppler shift signals and display the results in the form of a sonogram in real-time on a computer screen. The system, which is based on a TMS 320C25 digital signal processor chip, calculates spectra with 128 frequency components from 64 samples of the Doppler signal. The samples are collected at a programmable rate of up to 40.96 kHz, and the computation of each spectrum takes typically 3.2 ms. The feasibility of on-line AR spectral estimation makes this type of analysis an attractive alternative to the more conventional fast Fourier transform approach to the analysis of Doppler ultrasound signals.

Noise reduction by using spectral decomposition and non-linear transformation

Abstract: A noise reduction system for enhancing noisy speech signals by performing a spectral decomposition on the signal, passing each spectral component through a non-linear stage which progressively attenuates lower intensity spectral components (uncorrelated noise) but passes higher intensity spectral components (correlated speech) relatively unattenuated, and reconstituting the signal. Frames of noisy signal are transformed into the frequency domain by an FFT (Fast-Fourier Transform) device, with windowing. Each transformed frame is then processed to effect a non-linear transfer characteristic, which is linear above a soft "knee" region, and rolls off below, and transformed back to a reconstituted time-domain signal with reduced noise by an IFFT (Inverse Fast Fourier Transform) device (with overlapping). A level control matches the signal to the characteristic. In further embodiments, the characteristic may vary between frequency bands, and may be matched to speech formants by tracking formants using an LSP (Linear Spectral Pairs) technique.

Correction of non-linearities in detectors in fourier transform spectroscopy

Abstract: An interferogram is formed as in the prior art by dividing a beam of radiation from the source into two beams and interfering these beams so as to form an interferogram on the detector. A Fourier transform is then made of this interferogram. This transform has a signal spectrum above the cutoff frequency of the detector; and because of non-linearities in the detector and in the electronic signal processing circuitry, this transform also has a spectrum below the cutoff frequency of the detector. In accordance with the invention, two correction factors are calculated from this Fourier transform and these correction factors are then used to calculate a corrected interferogram. The first correction factor is evaluated by determining from the portion of the spectrum below the cutoff frequency a valve for the spectral signal at zero frequency. In addition, the integral of the square of the spectrum signal above the cutoff frequency is determined and the correction factor is found by dividing the signal at zero frequency by the integral of the square of the spectrum above the cutoff. The second correction factor is a function of the first correction factor and the integral of the spectrum signal above cutoff. These two corrections factors are then used in calculating a second order approximation to a corrected interferogram. Finally to produce the corrected Fourier transform, a Fourier transformation is made.

The effects of subtractive-type speech enhancement/noise reduction algorithms on parameter estimation for improved recognition and coding in high noise environments

Abstract: The authors present the results of a study designed to investigate the effects of subtractive-type noise reduction algorithms on LPC-based spectral parameter estimation as related to the performance of speech processors operating with input SNRs of 15 dB and below. Subtractive noise preprocessing greatly improves the SNR, but system performance improvement is not commensurate. LPC spectral estimation is affected by the character of the residual noise which exhibits greater variance and spectral granularity than the original broadband noise. The study shows that removing less than the full amount of noise and whitening it improves spectral estimation and speech device performance. Techniques and performance results are presented. >

Estimation of Fourier coefficients

Abstract: It is shown that the maximum-likelihood estimation or robust estimation of the Fourier coefficients may be preferable to Fourier transformation if the noise contains outliers or is otherwise not normally distributed. The reason is that, in that case, these estimators produce Fourier coefficient estimates and, therefore, system parameter estimates having a smaller variance. >

Spectral estimation using a log-distance error criterion applied to speech recognition

Abstract: A novel algorithm is presented for the estimation of a signal in noise. The distortion criterion used is based on the distance between log spectra. In many signal-processing applications, such as speech recognition, log spectra are much closer to the parameters used in a discriminator than power spectra. Therefore, it is believed that this spectral estimation technique should lead to better results than previously developed techniques such as spectral subtraction. The present technique performed better than spectral subtraction in noise immunity experiments on the IBM isolated word speech-recognition system, although at the expense of additional computational requirements. >

Biases and variances of several FFT spectral estimators as a function of noise type and number of samples

Abstract: The biases and the variances are theoretically and experimentally investigated of fast Fourier transform (FFT) spectral estimates with different windows (data tapers) when used to analyze power-law noise types f/sup 0/, f/sup -2/, f/sup -3/ and f/sup -4/. The experimental techniques introduced here permit one to analyze the performance of virtually any window for any power-law noise. This makes it possible to determine the level of a particular noise type to a specified statistical accuracy for a particular window. >

Standard deviation of instantaneous frequency

Abstract: Explicit expressions are derived for the standard deviation of instantaneous frequency (local bandwidth) at a particular time using the short-time Fourier transform and the general class of bilinear distributions. Examples are given. Application to the characterization and description of a multicomponent signal is discussed. The theorem of Fink (1966) and Mandel (1974), which states that the bandwidth of a signal is always greater than the global deviation of the derivative of the phase from the average frequency, is generalized for the short-time Fourier transform. The difference in the two standard deviations is found to be precisely the average of the local bandwidth as calculated with the bilinear distributions. >

Signal abstractions in signal processing software

Abstract: The authors propose and formally define the concept of multilevel signal abstractions as an organizational principle for real-world signal processing software that requires both algorithmic and heuristic techniques. As an example, they have implemented a set of signal abstractions, the extended spectrum, for harmonic spectra. The extended spectrum is shown to be useful in a variety of problems associated with harmonic spectra. For example, the focus in spectral estimation is often on adjusting parameters to maximize the 'peakness' of harmonically related peaks. It is demonstrated that this can be conveniently performed by taking advantage of the multiple abstraction levels of the extended spectrum representation. The extended spectrum can also be used to represent explicitly the evolution of harmonic spectra over time. To illustrate this concept, the authors have implemented a helicopter pitch and power tracking system. >

A direct algorithm for computing 2-D AR power spectrum estimates

Abstract: An algorithm for computing the parameters in a 2-D autoregressive spectral estimate without prior estimation of the correlation is described. The algorithm utilizes the multichannel form of the Burg algorithm and the relation between multichannel and 2-D AR modeling. The procedure permits computation of the spectral matrix for several channels of 2-D data; models with support in different quadrants are combined to form the spectral estimate. >

Multichannel 2-D AR spectrum estimation

Abstract: Spectral estimation for multiple 2-D signals by autoregressive modeling is discussed. The procedure computes the entire spectral matrix of autospectra and cross spectra for the set of 2-D signals. Specific differences between AR models for this problem and those for lower dimensional problems are discussed. Experimental results are presented. >

A tracking FFT processor for pulsed Doppler analysis beyond the Nyquist limit (medical ultrasound)

Abstract: The processor is based on the observation that the frequency translation required to reconstruct properly an aliased spectrum can be achieved by means of a simple reordering of data provided by a digital fast Fourier transform (FFT) unit. The amount of reordering is automatically derived by the computed value of a spectral parameter, e.g. the mean frequency. The procedure has been tested by introducing some modifications at the output of an FFT unit included in a conventional pulsed Doppler system. The dynamic evolution of the full Doppler spectrum and related mean frequency can be followed in real time over an extended range. Results of in vitro and in vivo experiments, as well as quantitative measurements performed with test signals, are presented. >

Pitch synchronous spectral analysis scheme for voiced speech

Abstract: The problem of spectrum analysis of harmonic signals which are periodic or at least quasi-periodic, such as human voice, is addressed. An efficient and accurate pitch-synchronized spectral analysis scheme for obtaining the Fourier coefficients of a harmonic signal, sampled at an arbitrary rate above the Nyquist critical rate, is outlined. The pitch is derived from the sampled signal prior to the spectral analysis. The rationale behind the scheme is based on an interpolation of the signal with an upsampling rate that is synchronized with the pitch period of the signal. It is shown that the resulting unsampled sequence is aperiodic, but nevertheless can be decomposed into a periodic signal corrupted by a small, aperiodic, high-frequency noise. The fact that this noise is correlated with the signal is used to obtain a closed-form solution for the desired Fourier coefficients from the noisy values, using the computationally superior fast Fourier transform (FFT) algorithm. The accuracy of the scheme is demonstrated for synthetic speech for which the spectrum is known a priori. The results obtained for real speech signals show better consistency across adjacent frames as compared to conventional methods. >

Spectral estimation utilizing a minimum free energy method with recursive reflection coefficients

Abstract: A method for rapidly estimating power spectral density components ρ(f) in the spectrum of an input signal, by first digitizing the input signal over a selected time interval at a selected sample rate; computing an m-th order prediction error energy as an arithmetic mean of forward and backward prediction error energies; and then computing an m-th order prediction error power from a previous reflection coefficient Γ computation. A control parameter α is generated; using α and Γ, an m-th order entropy H and free energy F are then computed, from which is computed m-th order reflection coefficients as extremes of the m-th order Free energy. If the proper extremes are not found, new feedback for the (m+1)-st order solution is generated. If the proper extremes are found, the spectral components are computed and recorded.

A noise insensitive solution to an ambiguity problem in spectral estimation

Abstract: The instantaneous frequency measurement (IFM) receiver is capable of measuring the center frequency of single frequency pulses over a wide range (bandwidth) of center frequencies. Because of various constraints, the frequency resolution requirement results in long correlator delay times that reduce the single correlator bandwidth. A large bandwidth can be achieved only if two or more correlators are used. The problem of estimating frequency is then reduced to the simultaneous congruence problem of number theory. A design procedure is presented for solving the congruence problem for a given amount of noise protection, a stated frequency resolution, a minimum bandwidth, and a fixed level of precision (bits) in the IFM receiver. >

Superresolution techniques and ISAR imaging

Abstract: High-resolution optimal estimation techniques are applied to the problem of radar imaging of rotating objects, sometimes referred to as ISAR (inverse synthetic array radar) imaging. Typical digital range-Doppler processing operations are described, utilizing two spectral estimation techniques. Quality ISAR images have been obtained from such processing, and particular examples, which are based on simulated data generated from point-target models of rotating objects, are shown. The first example is a so-called merry-go-round of 24 point-targets, and the MLM (maximum-likelihood method) algorithm is utilized to process a 3-D range-Doppler image estimate. The second example is a rotating boom along which are located 15 point-targets including a doublet, a triplet, and a quadruplet cluster that require superresolution techniques to resolve in the Doppler domain. It is concluded that superresolution techniques offer a viable alternative to conventional DFT (discrete Fourier transform) ISAR image processing and should permit either higher resolution images from the same data samples or equal-quality images from significantly fewer data samples. >

Local Bandwidth And Optimal Windows For The Short Time Fourier Transform

Abstract: The standard deviation of instantaneous frequency (local bandwidth) is derived for the short time Fourier transform. This is done by calculating the local moments of frequency for a given time instants using the spectrogram as a joint time-frequency distribution. By minimizing the local bandwidth optimal windows are obtained. We show that amplitude modulation has a very significant effect on the optimum window. We also show that to obtain the highest possible resolution, divergent windows which non the less lead to convergent short time Fourier transforms, must sometimes be used. Series expansions for the estimated instantaneous frequency and local bandwidth are derived in terms of the derivatives pf the phase. The theorem of Ville, Mandel and Fink, relating the global bandwidth to the excursions of the instantaneous frequency, is generalized to the short time Fourier transform. The bandwidth and duration of the spectrogram are related to those of the signal and window and a local uncertainty relationship for the spectrogram is derived. Also, the concept of local duration for a particular frequency is introduced and explicit formulas are given.© (1989) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Instantaneous frequency and time-frequency distributions

Abstract: A discussion of instantaneous frequency from the point of view of joint time-frequency distributions is presented. From this perspective, instantaneous frequency is the average frequency at a particular time. This view forces the consideration of the standard deviation of instantaneous frequency at a given time. The authors consider these issues and argue that this approach is plausible and clarifies other issues, particularly the description of a multicomponent signal. >

Spectral estimation utilizing an autocorrelation-based minimum free energy method

Abstract: A method for rapidly estimating power spectral density components in the spectrum of an input signal, by digitizing the input signal over a selected time interval at a selected sample rate; and estimating an autocorrelation sequence for the digitized input signal before generating a solution b s 0 to the autocorrelation sequence in a Yule-Walker equation by use of Levinson recursion. After generating a control parameter (temperature) α, a non-linear MFE equation, ##EQU1## is solved with b s 0 as an initial solution. Then, power spectral density components ##EQU2## are generated and recorded as estimates of the input signal.

Frequency estimation system

Abstract: A signal of interest is periodically sampled and the samples decomposed into quadrature components. The signal's phase angle at each sample is computed from the ratio of the real to imaginary parts. A linear regression analysis is then performed on the sampled phase data to obtain a best estimate of the signal's phase rate of change. From this estimate, the signal frequency is determined. In superheterodye instruments, the frequency of an input signal can be deduced by analyzing an intermediate frequency signal using this technique and factoring out the frequency of intervening local oscillator(s).