Let us consider a signal
where 
are known samples of the signal and the missing samples of the signal are filled with zeros.
Let us consider a simple smoothing filter:
From a conventional convolution of the signal with the filter, one would obtain a smoothed one where gaps of missing samples have been filled with the information available, that is:
The idea of normalized convolution is that of associating to each signal a certainty component expressing the level of confidence in the reliability of each measure of a signal. In the case of missing samples the certainty associated with them is equal to zero. Therefore we can express the certainty associated with signal f(t) as a map c(t) which has the same dimension as f(t) and is defined as:
It is easy to notice that the certainty map associated with the signal is nothing more than a map of the locations where samples are to be found.
Let us consider then the convolution of the certainty map c(t) with the smoothing filter g(t):
It is possible to use this second convolution as a weight for the first convolution, which will express the confidence in the results of the conventional convolution. The way to do this is to divide the first convolution by the second one obtaining an approximation of the original signal 
:
It is possible to notice that an approximation of the original signal has been obtained. It is not a perfect interpolation as not all the original sampling values have been obtained back, but only 
and 
. The missing samples have been approximated by nearest neighbouring interpolation or linear interpolation between neighbouring samples.