Question 3.SGPYQ.23: A cascade of three linear-time invariant systems is causal a......
A cascade of three linear-time invariant systems is causal and unstable. From this, we conclude that
(a) each system in the cascade is individually causal and unstable.
(b) at least one system is unstable and at least one system is causal.
(c) at least one system is causal and all systems are unstable.
(d) the majority are unstable and the majority are causal.
For a cascaded system
Input \rightarrow \boxed{H_1(z)}\rightarrow \boxed{H_2(z)} \rightarrow \boxed{H_3(z)} \rightarrow H(z)
For non-causal system, let
\begin{aligned} & H_1(z)=z^2+z^1+1 \\ & H_2(z)=z^3+z^2+1 \end{aligned}
Therefore,
H(z)=H_1(z) H_2(z) H_3(z)
=\left(z^2+z+1\right)\left(z^3+z^2+1\right) H_3(z) (i)
As, H(z) is causal and H_3(z) is causal, so
H_3(z)=z^{-6}+z^{-4}+1
Substituting in Eq. (1), we have H(z) is causal
H(z)=\left(z^2+z+1\right)\left(z^3+z^2+1\right)\left(z^{-6}+z^{-4}+1\right)
Hence, H(z)will be unstable, when any one of the system is unstable.