Global Journal of Researches in Engineering: F Electrical and Electronics Engineering Volume 18 Issue 4 Version 1.0 Year 2018 Type: Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Online ISSN: 2249-4596 & Print ISSN: 0975-5861
Discrete-Time, Discrete-Frequency Reassignment Method By Daniel L. Stevens & Stephanie A. Schuckers Clarkson University Abstract- The reassignment method is a non-linear, postprocessin technique which cans improve the localization of a time-frequency distribution by moving its values according to a suitable vector field. The reassignment methodβs scheme assumes that the energy distribution in the time-frequency plane resembles a mass distribution and moves each value of the time-frequency plane located at a point ( , )to another point,( β, β ), which is the center of gravity of the energy distribution in the area of ( , ). The result is a focused representation with very high intensity [11]. During thisππππ researchππππ it was investigatedππππ ππππ and determined that the frequency reassignment corrections derived fromππππ ππ ππthe Flandrin reassignment method have undesired noise sensitivity at very small noise levels as well as undesired observed distortions. In order to address these issues, a novel approach was derived-the discrete-time, discrete-frequency formulation of frequency reassignment. It is shown that in noise-free tone scenarios, this novel approach eliminates ambiguity and provides less distortion than the Flandrin reassignment method.
Keywords: reassignment method, time-frequency distribution, discrete-time, discrete-frequency reassignment method. GJRE-F Classification: FOR Code: 290901
Discrete-TimeDiscrete-FrequencyReassignmentMethod
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Β© 2018 Daniel L. Stevens & Stephanie A. Schuckers. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Discrete-Time, Discrete-Frequency Reassignment Method
Daniel L. Stevens Ξ± & Stephanie A. Schuckers Ο
Abstract- The reassignment method is a non-linear, post- overcome in order to obtain time-frequency distributions processing technique which cans improve the localization of a that can be both easily read by non-experts and easily time-frequency distribution by moving its values according to a included in a signal processing application [5]. Inability suitable vector field. The reassignment methodβs scheme to obtain readable time-frequency distributions may lead assumes that the energy distribution in the time-frequency to inaccurate signal detection and metrics extraction.
plane resembles a mass distribution and moves each value of 201 the time-frequency plane located at a point ( , )to another The reassignment method is a post-processing technique aimed at improving the readability of time- point,( , ), which is the center of gravity of the energy Year distribution in the area of ( , ). The result ππisππ a focused frequency distributions [7].The reassignment method representationπποΏ½ πποΏ½ with very high intensity [11].Duringthis research has application to many scientific and engineering 1 it was investigated and determinedππ ππ that the frequency fields, including signal processing [16], biology [8], reassignment corrections derived from the Flandrin music [9], and mechanical engineering [1].The concept reassignment method have undesired noise sensitivity at very of time-frequency reassignment can be first traced back small noise levels as well as undesired observed distortions. In to Kodera in the 1970βs, and was introduced in an order to address these issues, a novel approach was derived - attempt to improve the spectrogram [13].The the discrete-time, discrete-frequency formulation of frequency reassignment. It is shown that in noise-free tone scenarios, this reassignment operations proposed by Kodera could not sue IV Version I IV Version sue novel approach eliminates ambiguity and provides less be applied to discrete short-time Fourier transform distortion than the Flandrin reassignment method. (STFT) data, because the partial derivatives that formed Keywords: reassignment method, time-frequency these operations could not be computed directly on data that was discrete in time and frequency [6]. It has distribution, discrete-time, discrete-frequency reassignment method. been suggested that this difficulty was a primary barrier to wider use of the reassignment method.
I. Introduction The next major step forward for the olume XVIII Is V
reassignment method was many years later when )
ilinear time-frequency distributions offer a wide range of methods designed for the analysis of several papers were written by Auger and Flandrin [2], ( non-stationary signals. Nevertheless, a critical [4] in which reassignment equations were derived not B only for the spectrogram, but also for a number of other point of these methods is their readability [15], which means both a good concentration of the signal time-frequency and time-scale distributions. components along with few misleading interference The spectrogram can be defined as a two- terms. A lack of readability, which is a known deficiency dimensional convolution of the WVD of the signal by the in the classical time-frequency analysis techniques (e.g. WVD of the analysis window, as in equation (1): Wigner-Ville distribution (WVD), spectrogram), must be +
( , ; ) = β ( , ) ( , ) (1) Researches in Engineering
π₯π₯ π₯π₯ β
ππ π‘π‘ ππ β οΏ½ ππ π π ππ ππ π‘π‘ β π π ππ β ππ ππππ ππππ of nal The distribution reduces the interferenceββ terms distributed around ( , ), which is the geometrical center of the signalβs WVD, but at the expense of time and of this domain. Therefore, their average should not be frequency localization. However, a closer look at assigned at this point,π‘π‘ ππ but rather at the center of gravity
equation 1 shows that ( , ) delimits a time- of this domain, which is much more representative of the obal Jour l frequency domain at the vicinity of the ( , ) point, inside local energy distribution of the signal [3]. Reasoning G β which a weighted averageππ ofπ‘π‘ βtheπ π signalβsππ βππ WVD values is with a mechanical analogy, the local energy distribution performed. The key point of the reassignmentπ‘π‘ ππ principle ( , ) ( , ) (as a function of ) can is that these values have no reason to be symmetrically be considered as a mass distribution, and it is much β π₯π₯ ππmoreπ‘π‘ βaccurate π π ππ β ππ ππto π π assignππ the total massπ π ππππ ππ(i.e.ππ the Author Ξ±: Air Force Research Laboratory Rome, NY 13441. spectrogram value) to the center of gravity of the e-mail: [email protected] domain rather than to its geometrical center. Another Author Ο: Department of Electrical and Computer Engineering, Clarkson University. Potsdam, NY 13699. way to look at it is this: the total mass of an object is e-mail: [email protected] assigned to its geometrical center, an arbitrary point
Β©2018 Global Journals Discrete-Time, Discrete-Frequency Reassignment Method
which except in the very specific case of a This is exactly how the reassignment method homogeneous distribution, has no reason to suit the proceeds: it moves each value of the spectrogram actual distribution. A much more meaningful choice is computed at any point ( , ) to another point ( , ) which to assign the total mass of an object, as well as the is the center of gravity of the signal energy distribution spectrogram value, to the center of gravity of their around ( , )[12](see equationsπ‘π‘ ππ (2) and (3)): π‘π‘Μ ππΜ respective distribution [5]. + π‘π‘ ππ ( , ) ( , ) ( ; , ) = +β (2) β( , ) π₯π₯( , ) β¬ββ π π ππ π‘π‘ β π π ππ β ππ ππ π π ππ ππππ ππππ β π‘π‘Μ π₯π₯ π‘π‘ ππ + β¬ββ ππβ π‘π‘ (β π π ππ, β ππ ππ) π₯π₯ π π ( ππ, ππ)ππ ππππ ( ; , ) = +β (3) β( , ) π₯π₯( , ) β¬ββ ππ ππ π‘π‘ β π π ππ β ππ ππ π π ππ ππππ ππππ ππΜ π₯π₯ π‘π‘ ππ β and thus leads to a reassigned spectrogram (equationββ β (4)), whose valueπ₯π₯ at any point ( , ) is the sum of all the 201 β¬ ππ π‘π‘ β π π ππ β ππ ππ π π ππ ππππ ππππ spectrogram values reassigned to this point: β² β² π‘π‘ ππ Year +
( ) 2 β ( , ; ) = ( , ; ) ( ; , ) ( ; , ) (4) ππ β² β² β² β² πππ₯π₯ π‘π‘ ππ β οΏ½ πππ₯π₯ π‘π‘ ππ β πΏπΏοΏ½π‘π‘ β π‘π‘Μ π₯π₯ π‘π‘ ππ οΏ½πΏπΏ οΏ½ππ β ππΜ π₯π₯ π‘π‘ ππ οΏ½ ππππ ππππ One of the most interesting propertiesββ of this Reassigned spectrograms are therefore very new distribution is that it also uses the phase easy to implement, and do not require a drastic increase information of the STFT, and not only its squared in computational complexity. modulus as in the spectrogram. It uses this information Since time-frequency reassignment is not a from the phase spectrum to sharpen the amplitude bilinear operation, it does not permit a stable sue IV Version I IV Version sue estimates in time and frequency. This can be seen from reconstruction of the signal. In addition, once the phase the following expressions of the reassignment operators: information has been used to reassign the amplitude ( , ; ) coefficients, it is no longer available for use in ( ; , ) = (5) reconstruction. This is perhaps why the reassignment ππΞ¦π₯π₯ π‘π‘ ππ β method has received limited attention from engineers, π‘π‘Μ π₯π₯ π‘π‘ ππ β ( , ; ) ( ) and why its greatest potential may be where olume XVIII Is olume ; , = + ππππ (6)
V π₯π₯ reconstruction is not necessary, that is, where signal
) Where ( , ; ) is the phaseππΞ¦ π‘π‘ ππof βthe STFT of : Μ