The pole-zero plots of three discrete time systems P, Q and R on the z-plane are shown below.

Which one of the following is TRUE about the frequency selectivity of these system?

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GATE EE 2017 Official Paper: Shift 2

Option 2 : All three are band-pass filters

CT 1: Ratio and Proportion

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Transfer function of P is,

\({H_p}\left( z \right) = \frac{{\left( {z - 1} \right)\left( {z + 1} \right)}}{{{z^2}}} = \frac{{{z^2} - 1}}{{{z^2}}}\)

At low frequency, i.e. at z = 1, H_{p}(z) = 0

At high frequency, i.e. at z = -1, H_{p}(z) = 0

So, it is a band pass filter.

Transfer function of Q is,

\({H_Q}\left( z \right) = \frac{{\left( {z - 1} \right)\left( {z + 1} \right)}}{{\left( {z - j0.5} \right)\left( {z + j0.5} \right)}} = \frac{{{z^2} - 1}}{{{z^2} + 0.25}}\)

At low frequency, i.e. at z = 1, H_{Q}(z) = 0

At high frequency, i.e. at z = -1, H_{Q}(z) = 0

So, it is a bandpass filter.

Transfer function of R is,

\({H_R}\left( z \right) = \frac{{\left( {z - 1} \right)\left( {z + 1} \right)}}{{\left( {z - j} \right)\left( {z + j} \right)}} = \frac{{{z^2} - 1}}{{{z^2} + 1}}\)

At low frequency, i.e. at z = 1, H_{R}(z) = 0

At high frequency, i.e. at z = -1, H_{R}(z) = 0