Question

Consider an LTI system with a system function

\(H\left( z \right) = \frac{1}{{1 - \frac{1}{4}{z^{ - 1}}}}\)

Its difference equation will be:

1. \(y\left( n \right) - \frac{1}{2}y\left( {n - 1} \right) = x\left( n \right)\)
2. \(y\left( n \right) - \frac{1}{4}y\left( {n - 1} \right) = x\left( n \right)\)
3. \(y\left( n \right) + \frac{1}{2}y\left( {n - 1} \right) = x\left( n \right)\)
4. \(y\left( n \right) - \frac{1}{4}y\left( {n + 1} \right) = x\left( n \right)\)

Answer 1

Correct Answer - Option 2 : \(y\left( n \right) - \frac{1}{4}y\left( {n - 1} \right) = x\left( n \right)\)

Given system function,

\(H\left( z \right) = \frac{1}{{1 - \frac{1}{4}{z^{ - 1}}}}\)

\(\Rightarrow \frac{{Y\left( z \right)}}{{X\left( z \right)}} = \frac{1}{{1 - \frac{1}{4}{z^{ - 1}}}}\)

\(\Rightarrow Y\left( z \right) - \frac{1}{4}{z^{ - 1}}y\left( z \right) = \;X\left( z \right)\)

By applying inverse z-transform,

\(\Rightarrow y\left( n \right) - \frac{1}{4}y\left( {n - 1} \right) = x\left( n \right)\)