Implementing IIR filter using FSM
Use feedback from output to input to perpetuate the output of the state machine.  You don't need an infinite number of states to produce an infinitely long output sequence.  For example, a state machine with a single memory element and feedback can be used to implement a first order IIR filter.  An example of such a system has the recursion relationship y[n] = 0.5y[n-1] + x[n] where y is the output x is the input and 0.5y[n-1] is the past output scaled, fedback and summed with the input x[n].