Question

For a signal $x\left(n\right)=2^{n}u\left(n\right)-4^{n}u(-n-1)$ the ROC is defined as a<|z|< b. The correct statement(s) is /are

• A. The system is stable
• B. Z -transform of signal is $\frac{z}{z-2}+\frac{z}{z-4}$
• C. The value of a + b = 2
  
• D. The initial value of theorem is not applicable.
  

Answer 1

Answer: (B), (D)

The initial value of theorem is not applicable.


(b, c)

ROC is signals is 2<|z|<4

Hence a =2
b = 4
a+b = 6

(b) Initial value theroem is applicable only for causal type signals.
(d) As ROC does not contain unit circle thus system is unstable.

$\left(c\right)a^{n}u\left(n\right)\stackrel{Z.T.}{⟷}\frac{z}{z-a}ROC:|z|>a$

$-a^{n}u(-n-1)\stackrel{Z.T.}{⟷}\frac{z}{z-a}ROC:|z|>a$

So z-transform of total signal for given ROC is

$\frac{z}{z-2}+\frac{z}{z-4}$