ON THE MINIMUM‐LENGTH PROPERTY OF ONE‐SIDED SIGNALS

Geophysics ◽  
1973 ◽  
Vol 38 (4) ◽  
pp. 657-672 ◽  
Author(s):  
A. J. Berkhout
Keyword(s):  
Amplitude Spectrum ◽  
Phase Spectrum ◽  
Signal Duration ◽  
Complex Conjugate ◽  
Minimum Length ◽  
Time Interval ◽  
Minimum Phase ◽  
Inverse Filtering ◽  
Complex Signals ◽  
Phase Property

Apart from a visual inspection, the length property of a signal is mostly judged by its amplitude spectrum (e.g., bandwidth or “roughness” of the spectrum). However, due to the important influence of the phase spectrum. a sensible yardstick for signal length should consider both amplitude and phase. In addition, a complete investigation should also include evaluation of the length of the (partly) deconvolved signal. In this paper, signal duration and signal length are introduced as two different concepts. Unlike signal duration, giving the minimum time interval outside of which the signal equals zero, signal length gives some information as to how the signal energy is distributed within this interval. An objective yardstick for signal length is proposed, and it is shown how signal length depends on both the amplitude and phase spectrum. The minimum‐length signal is introduced and it is shown that for one‐sided signals the minimum signal‐length property implies the minimum‐phase property and vice‐versa. The efficiency of causal least‐square inverse filtering on minimum‐length and nonminimum‐length signals is evaluated. An investigation scheme is given which enables the user to determine whether a one‐sided signal has the minimum‐length (minimum‐phase) property. The theory is applicable to complex signals. For real signals the * (symbol for complex conjugate) and absolute value marks may be omitted: [Formula: see text] and [Formula: see text].

Air‐gun signatures and the minimum‐phase assumption

Geophysics ◽  
1992 ◽  
Vol 57 (2) ◽  
pp. 263-271 ◽  
Author(s):  
Neil D. Hargreaves
Keyword(s):  
Phase Error ◽  
Amplitude Spectrum ◽  
Phase Spectrum ◽  
Time Shift ◽  
Minimum Phase ◽  
Limited Bandwidth ◽  
Phase Errors ◽  
Phase Rotation ◽  
Air Gun ◽  
The Hilbert Transform

The air‐gun array signature is close to minimum‐phase as a function of continuous time, in the sense that for processing purposes its phase spectrum can be derived from the Hilbert transform of the logarithm of its amplitude spectrum. This phase spectrum is different, however, from the minimum‐phase spectrum that is estimated by spiking deconvolution for a sampled and time‐windowed version of the signature. As a consequence, there can be large phase errors when spiking deconvolution is applied to an air‐gun signature or to a recording instrument response. The errors can be shown to consist primarily of a time shift and, at least visually over a limited bandwidth, a phase rotation of the output wavelet. The time shift is introduced by time sampling, while the phase rotation is caused by the spectral smoothing generated by time windowing. If the seismic wavelet as a whole, and not just the air‐gun signature, is minimum‐phase, then the total residual phase error after spiking deconvolution, including also the error due to data noise, can also be shown to be close to a time shift and a phase rotation. This may be physical justification for the phase rotation schemes that are often successful in matching seismic data and well‐log synthetics. The minimum‐phase assumption can be used for statistical air‐gun array signature deconvolution, providing that a limited amount of deterministic information (the instrument slopes and the source and receiver depths in the approach used here) is available to guide the process in those areas of the spectrum that are critical to the phase computation. Date examples show that, with care, almost identical results can then be obtained from either purely statistical deconvolution or deterministic deconvolution plus statistical deconvolution of multiples and ghosting.

THE APPLICATION OF HOMOMORPHIC DECONVOLUTION TO SHALLOW‐WATER MARINE SEISMOLOGY—PART I: MODELS

Geophysics ◽  
1974 ◽  
Vol 39 (4) ◽  
pp. 401-416 ◽  
Author(s):  
Paul L. Stoffa ◽  
Peter Buhl ◽  
George M. Bryan
Keyword(s):  
Shallow Water ◽  
Water Column ◽  
Amplitude Spectrum ◽  
Phase Spectrum ◽  
Phase Curve ◽  
Minimum Phase ◽  
Cepstrum Analysis ◽  
Infinite Extent ◽  
Complex Cepstrum ◽  
The Time Domain

The complex cepstrum is investigated mathematically and through models for functions of interest in shallow‐water marine seismology. Association of the slowly varying components of the phase spectrum with the source replaces the usual minimum‐phase assumption. This is analogous to the usual treatment of the amplitude spectrum. Complex cepstrum expressions are developed for an arbitrary (but minimum‐phase) reflector series water‐column multiple generator, and simplified bubble‐pulse oscillation. While the complex cepstrum of all functions is of infinite extent, removing only the first n nonzero complex‐cepstrum contributions of a decaying, impulsive, periodic time function (such as the water‐column multiple generator) serves to eliminate the first n multiples entirely in the time domain and reduces the remaining multiples to at most 1/(n+1) of their original value. A new method of computing the continuous, ramp‐free phase spectrum required for complex‐cepstrum analysis is developed on the basis of the derivative of the phase curve.

scholarly journals Grey Spectrum Analysis of Air Quality Index and Housing Price in Handan

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-6 ◽  
Author(s):  
Kai Zhang ◽  
Yan Chen ◽  
Lifeng Wu
Keyword(s):  
Air Quality ◽  
Quality Index ◽  
Amplitude Spectrum ◽  
Housing Price ◽  
Lag Time ◽  
Phase Spectrum ◽  
Air Quality Index ◽  
The Relationship ◽  
Grey Phase ◽  
Main Change

To analyze the relationship between air quality index (AQI) and housing price, six relationship indexes between air quality index and housing price were calculated using grey spectrum theory, specifically grey association spectrum, grey cospectrum, grey amplitude spectrum, grey phase spectrum, grey lag time length, and grey condense spectrum. Three main change periods were extracted. There was a negative correction between the air quality and the housing price in Handan. The results provide a basis for the government’s measures to prevent haze.

scholarly journals Image Saliency Prediction in Transformed Domain: A Deep Complex Neural Network Method

2019 ◽  
Vol 33 ◽  
pp. 8521-8528 ◽  
Author(s):  
Lai Jiang ◽  
Zhe Wang ◽  
Mai Xu ◽  
Zulin Wang
Keyword(s):  
Neural Network ◽  
Amplitude Spectrum ◽  
Ground Truth ◽  
Phase Spectrum ◽  
Saliency Maps ◽  
Image Saliency ◽  
Saliency Prediction ◽  
Network Method ◽  
Salient Regions ◽  
Robust To Noise

The transformed domain fearures of images show effectiveness in distinguishing salient and non-salient regions. In this paper, we propose a novel deep complex neural network, named SalDCNN, to predict image saliency by learning features in both pixel and transformed domains. Before proposing Sal-DCNN, we analyze the saliency cues encoded in discrete Fourier transform (DFT) domain. Consequently, we have the following findings: 1) the phase spectrum encodes most saliency cues; 2) a certain pattern of the amplitude spectrum is important for saliency prediction; 3) the transformed domain spectrum is robust to noise and down-sampling for saliency prediction. According to these findings, we develop the structure of SalDCNN, including two main stages: the complex dense encoder and three-stream multi-domain decoder. Given the new SalDCNN structure, the saliency maps can be predicted under the supervision of ground-truth fixation maps in both pixel and transformed domains. Finally, the experimental results show that our Sal-DCNN method outperforms other 8 state-of-theart methods for image saliency prediction on 3 databases.

Predictive deconvolution and the zero‐phase source

Geophysics ◽  
1984 ◽  
Vol 49 (4) ◽  
pp. 379-397 ◽  
Author(s):  
Bruce Gibson ◽  
Ken Larner
Keyword(s):  
White Noise ◽  
Field Data ◽  
Synthetic Data ◽  
Phase Correction ◽  
Phase Spectrum ◽  
Signal Spectrum ◽  
Minimum Phase ◽  
Predictive Deconvolution ◽  
Different Levels ◽  
Zero Phase

Predictive deconvolution is commonly applied to seismic data generated with a Vibroseisr® source. Unfortunately, when this process invokes a minimum‐phase assumption, the phase of the resulting trace will not be correct. Nonetheless, spiking deconvolution is an attractive process because it restores attenuated higher frequencies, thus increasing resolution. For detailed stratigraphic analyses, however, it is desirable that the phase of the data be treated properly as well. The most common solution is to apply a phase‐shifting filter that corrects for errors attributable to a zero‐phase source. The phase correction is given by the minimum‐phase spectrum of the correlated Vibroseis wavelet. Because no minimum‐phase spectrum truly exists for this bandlimited wavelet, white noise is added to its amplitude spectrum in order to design the phase‐correction filter. Different levels of white noise, however, produce markedly different results when field data sections are filtered. A simple argument suggests that the amount of white noise used should match that added in designing the (minimum‐phase) spiking deconvolution operator. This choice, however, also produces inconsistent results; field data again show that the phase treatment is sensitive to the amount of added white noise. Synthetic data tests show that the standard phase‐correction procedure breaks down when earth attenuation is severe. Deterministically reducing the earth‐filter effects before deconvolution improved the resulting phase treatment for the synthetic data. After application of the inverse attenuation filter to the field data, however, phase differences again remain for different levels of added white noise. These inconsistencies are attributable to the phase action of spiking deconvolution. This action is dependent upon the shape of the signal spectrum as well as the spectral shape and level of contaminating noise. Thus, in practice the proper treatment of phase in data-dependent processing requires extensive knowledge of the spectral characteristics of both signal and noise. With such knowledge, one could apply deterministic techniques that either eliminate the need for statistical deconvolution or condition the data so as to satisfy better the statistical model assumed in data‐dependent processing.

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