A lower bound of the necessary bit length of memory for approximation of discrete time systems

In this paper, we deal with approximation problem of discrete time linear systems by bit memory systems. The bit memory systems are operators from analog inputs to discretized outputs and their memories are elements of discrete numbers. Their dynamics is represented by the time evolution of the bit memories and output equations. In this paper, we consider the minimization problem of bit length of the memories with which the bit memory systems attain a given output error bound for the original systems. We at first show that this problem can be followed by an optimal quantization problem of the state space of the original systems and give a lower bound of the minimal bit length of memory by the properties of the original systems in the case of nth order systems. Moreover, we also give the exact value of the minimal bit length of memory in the case 1st order systems. This is the first result on the minimal bit length of memory in the approximation problem.