Question

Which of the following statement(s) is/are correct about given z-transform of the system?

$H\left(z\right)=\frac{z-\alpha }{1-\beta z^{-1}}$

• A. $\text{The system H(z) acts a high pass filter if}\alpha =1and\beta =1$
• B. $\text{The system can not be causal}$
• C. $\text{The system H(z) acts as low pass filter if}\alpha =-1and\beta <1$
• D. $\text{The system H(z) can act as all pass filter for}\beta =\frac{1}{{\alpha }^{\ast }}.$

Answer 1

Answer: (B), (C), (D)

$\text{The system H(z) can act as all pass filter for}\beta =\frac{1}{{\alpha }^{\ast }}.$

For LPF:

$\underset{z\to 1}{lim}\frac{z+1}{1-\beta z^{-1}}=\frac{2}{1-\beta }\left(high\right)$

$\underset{z\to 1}{lim}\frac{z+1}{1-\beta z^{-1}}=\frac{0}{1+\beta }=0\left(low\right)Correct$

(b) For HPF

$\underset{z\to 1}{lim}\frac{z-1}{1-z^{-1}}=1\left(high\right)$

So system is not high pass filter

(c) $H\left(z\right)=\frac{z^2-z\alpha }{z-\beta }$

Since, order of numerator is greater than order of denominator. Therefore system can not be casual.

(d) For all pass filter,

$Pole=\frac{1}{Zer{o}^{\ast }}$