Singularity in signal theory

This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. A discrete-time random signal is singular if its values are singular random variables defined by a distribution function continuous but with a derivative equal to zero almost everywhere. Singular random signals can be obtained at the output of some linear filters when the input is a discrete-valued white noise. Sufficient conditions for singularity are established. In particular it is shown that if the poles of the filter are inside a circle called the circle of singularity and if the input is white and discrete-valued the output is singular. Computer experiments using histograms at different scales exhibit the structure of singular signals. The influence of input correlation is also analysed. It is shown that when the input is not white, but has a specific Markovian structure, the output can be singular. This is also verified by computer experiments. Finally, mixtures of singular and discrete-valued random signals are analysed.