Fig: summative kernel

Signal sampling or signal acquisition modelling

Signal acquisition is just obtaining a value of the signal at a given location x, known as the position of the sensor. This can be rudely modeled by the convolution of the underlying signal with the Dirac distribution at the location x. In order to move nearer the modeling to the reality of the sensor acquisition, integrating the signal over a neighborhood (an interval) of x seems to be more appropriate. Actually this 2nd modeling is performed by the convolution of the underlying signal with the characteristic function of the neighborhood of x. A 3rd modeling proposal is to integrate the signal over a weighted neighborhood, i.e. convolving the underlying signal with a summative kernel representing a weighted neighborhood of x. In that case, the chosen summative kernel is nothing else than the impulse response of the sensor. The acquired value is subject to uncertainty, the sensor is not perfect. As already mentionned, the acquisition is modeled by the convolution of the real physical signal with the summative kernel modeling the sensor. This model is nothing else than the expected value of the underlying signal in the translated neighborhood

Summative kernels and uncertainty

It is worth noting that the definition of a summative kernel coincides with the definition of a probability distribution of a random variable. Interpreting a summative kernel, used in signal sampling or acquisition modeling, as being a model of an uncertain phenomenon is sensible. Indeed, the value acquired by the sensor can be interpreted as a realization of a random variable whose probability distribution is the impulse response of the sensor. However, this approach does not incorporate imprecision. It is supposed that we have a perfect knowledge of the sensor behavior. After a thorough survey of uncertainty theories taking into account imprecision, I decided to replace summative kernels by another kind of neighborhood : the maxitive kernels.