A discrete problem on d-dimensional symmetric weighted median filters

Using a tool-energy of d  -dimensional signal sequences taking values 1 or -1-1, we have completely solved an interesting discrete mathematical problem on d-dimensional symmetric weighted median filters, i.e., when a d-dimensional symmetric weighted median filter is applied iteratively to a real signal sequence, then the limit of the iterates with even index as well as the limit of the iterates with odd index both converge.