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

z >
`(3)/(4)`

Step-by-step explanation:

step 1:

Calculating the poles for the given equation

first we have to consider the first term of denominator,

1+
`(1)/(4) z^(-2)`=0

`z^(2)` =-
`(1)/(4)`

z=±j×
`(1)/(2)`

now consider the second term of denominator

1+
`(5)/(4) z^(-1) } +(3)/(8) z^(-2)=0`

`z^(2) +(5)/(4)z^(1)+(3)/(8)=0`

`(z+(1)/(2)).(z+(3)/(4))=0`

`z=-(1)/(2) and (-3)/(4)`

step 2:

`z=(-j)/(2) and (j)/(2) and (-1)/(2) and (-3)/(4)`

then,

z =
`(1)/(2) and (3)/(4)`

representing the ROC,

-0< z <
`(1)/(2)`

`(1)/(2) <` z <
`(3)/(4)`

-z >
`(3)/(4)`

result:

z >
`(3)/(4)`

note:

Bold z represent Modulus