One of the most popular time–frequency representations is certainly the Wigner distribution. To reduce the interferences coming from its quadratic nature, several related distributions have been proposed, among which is the so-called Born–Jordan distribution. It is well known that in the Born–Jordan distribution the ghost frequencies are in fact damped quite well, and the noise is in general reduced. However, the horizontal and vertical directions escape from this general smoothing effect, so that the interferences arranged along these directions are in general kept. Whereas these features are graphically evident on examples and heuristically well understood in the engineering community, there is no at present mathematical explanation of these phenomena, valid for general signals in and, more in general, in the space of temperate distributions. In the present note we provide such a rigorous study using the notion of wave-front set of a distribution. We use techniques from Time–frequency Analysis, such as the modulation and Wiener amalgam spaces, and also results of microlocal regularity of linear partial differential operators
On the reduction of the interferences in the Born-Jordan distribution / Cordero, Elena; De Gosson, Maurice; Nicola, Fabio. - In: APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS. - ISSN 1063-5203. - STAMPA. - 44:2(2018), pp. 230-245. [10.1016/j.acha.2016.04.007]
On the reduction of the interferences in the Born-Jordan distribution
NICOLA, FABIO
2018
Abstract
One of the most popular time–frequency representations is certainly the Wigner distribution. To reduce the interferences coming from its quadratic nature, several related distributions have been proposed, among which is the so-called Born–Jordan distribution. It is well known that in the Born–Jordan distribution the ghost frequencies are in fact damped quite well, and the noise is in general reduced. However, the horizontal and vertical directions escape from this general smoothing effect, so that the interferences arranged along these directions are in general kept. Whereas these features are graphically evident on examples and heuristically well understood in the engineering community, there is no at present mathematical explanation of these phenomena, valid for general signals in and, more in general, in the space of temperate distributions. In the present note we provide such a rigorous study using the notion of wave-front set of a distribution. We use techniques from Time–frequency Analysis, such as the modulation and Wiener amalgam spaces, and also results of microlocal regularity of linear partial differential operatorsFile | Dimensione | Formato | |
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https://hdl.handle.net/11583/2627731
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