1/31/08
difference equations: often described in two ways: advance operator notation and delay operator notation

advance operator:(Fig1)
for this system to be causal, m<=k, otherwise, output would depend on future inputs
delay operator:(Fig2)
automatically causal
either can possibly make math easier
delay more intuitive
could be recursive or nonrecursive depending on dependence on previous outputs
recursive:(Fig3)
nonrecursive:(Fig4)

solution of diff equations can be done by a process called recursion
an nth order recursive DE is given in delay notation as (Fig5)
begin by solving for y at n=0. thuse we need a set of init conds ex (Fig6) we also need the vals for the inputs for n<0 (Fig6)

solve the first order DE by recursion for n=1,2,3. assume zero init cond and x[n] = nu[n] (Fig7)
seen unstable after more plots

solve the second order DE by recursion for n=1,2,3. assume a 0 init cond and x[n] = u(n)e^(-n) (Fig8)

an example of converting between advance order and delay order (Fig9)