--- 1 ----
y = F_sim_system1(y_init,u_signal)

u_signal is a N \times 1 vector of the input signal samples, and y_init
is the initial condition of the state (a scalar). The sample length N is arbitrary. 
The output y will have length N, with y(1) = y_init and y(k+1) = a y(k) + b u(k) + noise


--- 2-----
y = F_sim_system2(y_init,u_signal)

same as F_sim_system1 except the parameters a and b are different


--- 3 ---
y = F_sim_system3(u_signal)

same as before, except that you can't specify the initial condition. 
This corresponds to the "laboratory condition" where the initial condition 
of the system is not under your control. The output vector y now has dimension N, 
containing y[2] to y[N+1], where N is the length of 'u_signal'.