Compute the transfer function if the impulse response of an LTI systems is h(n) = 0.5ⁿ u(n)

Question

Question

This question was previously asked in

HPCL Engineer Instrumentation 11 Aug 2021 Official Paper

Answer 1

Concept:

The definition of z-transform is given by,

$X (z) = \sum_{n = - \infty}^{\infty} x (n) z^{- n}$

If x(n) = an u(n)

then, X(z) = $= \frac{z}{z - α}$

Calculation:

Given signal, h(n) = 0.5n u(n)

The z-transform of the above-given signal is given by

$H (z) = \frac{z}{(z - 0.5)}; | z | > 0.5$

Note: The z-transform of x(n) = -an u(-n – 1) is given by

$= \frac{1}{1 - a z^{- 1}} = \frac{z}{z - a}; R O C : | z | < | a |$