Theory of Quadratic TFDs

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Theory of Quadratic TFDs

Show simple item record Boashash, B 2012-02-21T17:04:15Z 2012-02-21T17:04:15Z 2003
dc.identifier.citation Time-Frequency Signal Analysis & Processing: A Comprehensive Reference, Elsevier Science, Oxford, 2003, Chapter 3, pages 59-82 en_US
dc.identifier.isbn 0080443354
dc.identifier.isbn 9780080443355
dc.description This manuscript shows that some form of reduced interference quadratic TFDs (RIDs) is to be preferred over the WVD; and that the design of RIDs is best undertaken by designing the desired kernel filter in the ambiguity domain, and using Fourier transforms to see the effects in the time-lag and time-frequency domains; with this knowledge, special-purpose quadratic TFDs can be easily designed to meet the specifications of particular applications. (Additional details can be found in the other chapters of the comprehensive book on Time-Frequency Signal Analysis and Processing (see In addition, the most recent upgrade of the original software package that calculates Time-Frequency Distributions and Instantaneous Frequency estimators can be downloaded from the web site: This was the first software developed in the field, and it was first released publicly in 1987 at the 1st ISSPA conference held in Brisbane, Australia, and then continuously updated). en_US
dc.description.abstract The quadratic time-frequency distributions (TFDs) introduced in the last chapter represent the majority of the methods used in practical applications that deal with non-stationary signals. In this chapter, which completes the introductory tutorial, we show that these particular quadratic TFDs belong to a general class of TFDs whose design follows a common procedure, and whose properties are governed by common laws. This quadratic class may be considered as the class of smoothed Wigner-Ville distributions (WVDs), where the “smoothing”is described in the (t, f) domain by convolution with a “time-frequency kernel” function (t, f), and in other domains by multiplication and/or convolution with various transforms of (t, f). The generalized approach allows the definition of new TFDs that are better adapted to particular signal types, using a simple and systematic procedure as opposed to the ad hoc methods of Chapter 2. The first section (3.1) extends Section 2.1 by enumerating in detail the key properties and limitations of theWVD. Thus it motivates the introduction of general quadratic TFDs (Section 3.2) and prepares for the discussion of their properties (Section 3.3). In Section 3.2, using Fourier transforms from lag to frequency and from time to Doppler (frequency shift), the quadratic TFDs and their kernels are formulated in four different but related two-dimensional domains. One of these, namely the Doppler-lag ( , ) domain, leads to the definition of the “ambiguity function” and allows the smoothing of the WVD to be understood as a filtering operation. In Section 3.3, the list of properties of the WVD is supplemented by mentioning some desirable TFD properties not shared by the WVD. The various TFD properties are then expressed in terms of constraints on the kernel, so that TFD design is reduced to kernel design. Three tables are provided showing the kernel properties equivalent to various TFD properties, the kernels of numerous popular TFDs in the various two-dimensional domains, and the properties of those same TFDs. en_US
dc.language.iso en en_US
dc.publisher Elsevier en_US
dc.subject smoothed Wigner-Ville distributions en_US
dc.subject WVD en_US
dc.subject quadratic TFDs en_US
dc.subject Doppler-Lag domain en_US
dc.subject Ambiguity function en_US
dc.subject kernel constraints en_US
dc.subject kernel design en_US
dc.subject time-frequency shift invariance en_US
dc.subject instantaneous frequency en_US
dc.subject time delay en_US
dc.subject time-frequency support en_US
dc.subject time-frequency inverse en_US
dc.subject bilinear TFDs en_US
dc.subject artifacts en_US
dc.subject cross-terms en_US
dc.subject inner artifacts en_US
dc.subject windowed WVD en_US
dc.subject outer artifacts en_US
dc.subject cross-WVD en_US
dc.subject cross-terms suppression en_US
dc.subject masked WVD en_US
dc.subject reduced-interference-distribution en_US
dc.subject RID en_US
dc.subject symmetrical ambiguity function en_US
dc.subject time-lag kernel en_US
dc.subject Doppler-lag kernel en_US
dc.subject time-frequency kernel en_US
dc.subject time-frequency formulation en_US
dc.subject quadratic TFD class en_US
dc.subject product kernel en_US
dc.subject Doppler-independent kernel en_US
dc.subject lag-independent kernel en_US
dc.subject time-frequency energy concentration en_US
dc.subject time-frequency resolution en_US
dc.subject IF-TD visualization en_US
dc.subject butterfly function en_US
dc.subject cone-shaped kernel en_US
dc.subject Born-Jordan TFD en_US
dc.subject ZAM TFD en_US
dc.subject DI kernel en_US
dc.subject LI kernel en_US
dc.subject inner artifacts en_US
dc.subject outer artifacts en_US
dc.subject Modified B distribution en_US
dc.subject MBD en_US
dc.subject time-frequency design en_US
dc.subject data-dependent TFDs en_US
dc.subject time-frequency tutorial en_US
dc.title Theory of Quadratic TFDs en_US
dc.type Book chapter en_US

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