Consider the cascade of two LTI systems, see Figure 3.21, where the input of the cascade is z(t) and the output is y(t), while x(t)is the output of the first system and the input of the second system. The input to the cascaded systems is z(t) = (1 ˆ’ t) [u(t) ˆ’ u(t ˆ’ 1)].

(a) The input/output characterization of the first system is x(t) = dz(t)/dt. Find the corresponding output x(t)of the first system.

(b) It is known that for the second system when the input is δ(t)the corresponding output is e^ˆ’2tu(t), and that when the input is u(t)the output is 0.5(1 €“ e ^ˆ’2t) u(t). Use this information to calculate the output y(t)of the second system when the input is x(t)found above.

(c) Determine the ordinary differential equation corresponding to the cascaded system with input z(t)and output y(t).

Figure 3.21:

Transcribed Image Text:

x(t) System z(t) y(t) System