Question

A cascade of three Linear Time Invariant systems is causal and unstable. From this, we conclude that
1. Each system in the cascade is individually causal and unstable
2. At least one system is unstable and at least one system is causal
3. At least one system is causal and all systems are unstable
4. The majority are unstable and the majority are causal

Answer 1

Correct Answer - Option 2 : At least one system is unstable and at least one system is causal

Let three LTI systems have response H₁ (z), H₂ (z) and H₃ (z) are cascaded  as shown fellow

\(\frac{i}{p} \to \boxed{{H_1}\left( z \right)} \to \boxed{{H_2}\left( z \right)} \to \boxed{{H_3}\left( z \right)} \to H\left( z \right)\)

Assume   H₁ (z) = z² +z¹+1 (non-casual)

H₂ (z) = z³ +z²+1 (non-casual)

Overall response of the system

H (z) = H₁(z) H₂(z) H₃(z)

= (z²+z¹+1) (z³+z²+1) H₃(z)

To make H(z) causal we have to take H₃(z) also causal.

Let H₃(z) =z^-6+z^-4+1

H(z) =(z²+z+1) (z³+z²+1) (z^-6+z^-4+1)

H(z) → causal

Similarly, to make H(z) unstable at least one of the system should be unstable.