Question

Suppose that the algebraic expression for the z-transform of x[n] is:
X(z) = (1 − 1/4 z ⁻²) / (1 + 1/4 z ⁻²)(1 + 543 z⁻¹ + 3/8 z⁻²)
How many different regions of convergence could correspond to X(z)?

Answer 1

Answer:

z >$\frac{3}{4}$

Explanation:

step 1:

Calculating the poles for the given equation

first we have to consider the first term of denominator,

1+$\frac{1}{4} z^{-2}$=0

$z^{2}$ =-$\frac{1}{4}$

z=±j× $\frac{1}{2}$

now consider the second term of denominator

1+$\frac{5}{4} z^{-1} } +\frac{3}{8} z^{-2}=0$

$z^{2} +\frac{5}{4}z^{1}+\frac{3}{8}=0$

$(z+\frac{1}{2}).(z+\frac{3}{4})=0$

$z=-\frac{1}{2} and \frac{-3}{4}$

step 2:

$z=\frac{-j}{2} and \frac{j}{2} and \frac{-1}{2} and \frac{-3}{4}$

then,

z =$\frac{1}{2} and \frac{3}{4}$

representing the ROC,

-0< z <$\frac{1}{2}$

$\frac{1}{2} <$ z <$\frac{3}{4}$

-z >$\frac{3}{4}$

result:

z >$\frac{3}{4}$

note:

Bold z represent Modulus