Question

Determine the z-transform corresponding to the given pole-zero plot.

• A. \[ X(z)=G\frac{(z+r\cos\omega_0)} {(z-re^{j\omega_0})(z-re^{-j\omega_0})}, \qquad |z|<r \]
• B. \[ X(z)=G\frac{(z+r\cos\omega_0)} {(z+re^{j\omega_0})(z-re^{-j\omega_0})}, \qquad |z|<r \]
• C. \[ X(z)=G\frac{z(z-r\cos\omega_0)} {(z-re^{j\omega_0})(z-re^{-j\omega_0})}, \qquad |z|>r \]
• D. \[ X(z)=G\frac{z(z+r\cos\omega_0)} {(z+re^{j\omega_0})(z-re^{-j\omega_0})}, \qquad |z|>r \]

Hint:

For pole-zero plots: \[ X(z)=G\frac{\prod(z-\text{zero})} {\prod(z-\text{pole})} \] and the ROC is determined separately from the shaded region. \[ |z|>r \Rightarrow \text{Right-sided (causal) sequence} \] \[ |z|<r \Rightarrow \text{Left-sided (anti-causal) sequence} \]

Answer 1

The Correct Option is C