Two discrete-time systems are connected in cascade to form a new system T as shown in F...

Two discrete-time systems are connected in cascade to form a new system T as shown in Fig. P2.8. Prove or disprove the following statements.

(a) If are linear, then T is linear (i.e., the cascade connection of two linear systems is linear).

(b) If are time invariant, then T is time invariant.

(c) If are causal, then T is causal.

(d) If are linear and time invariant, the same holds for T .

(e) If are linear and time invariant, then interchanging their order does not change the system T .

(f) As in part (e) except that T₁ , T₂ are now time varying. (flint Use an example.)

(g) If are nonlinear, then T is nonlinear.

(h) If are stable, then T is stable.

(i) Show by an example that the inverses of parts (c) and (h) do not hold in general.