Question

The natural response of an LTI system described by the difference equation

y(n) – 1.5 y(n – 1) + 0.5 y(n – 2) = x(n) is

1. y(n) = 0.5u(n) – 2(0.5)ⁿ u(n) 
2. y(n) = 0.5u(n) – (0.5)ⁿ u(n)
3. y(n) = 2u(n) – 0.5(0.5)ⁿ u(n)
4. y(n) = 2u(n) – (0.5)ⁿ u(n)

Answer 1

Correct Answer - Option 3 : y(n) = 2u(n) – 0.5(0.5)ⁿ u(n)

Concept:

Natural response:

• The response of a network without a source in it is called a natural response (NR) and it gives the transient response
• This response depends on the nature of the passive element and it is independent of the type of input.
• This part of the response is found and by solving the complementary function in the differential equation.

Forced response:

• The response of a network with a source present in it is called forced response (F.R) and it gives a steady-state response.
• This response is independent of the nature of the passive element, but it depends entirely on the type of input.
• This part of the solution is found out by finding the particular integral part (P.I) of the differential equations. 

Analysis:

y(n) – 1.5 y(n – 1) + 0.5 y(n – 2) = x(n) 

Y(z) - 1.5 z^-1 Y(z) + 0.5 z^-2 Y(z) = X(z)

\(Y(z) = \frac{X(z)}{1-1.5z^{-1} + 0.5 z^{-2}}\)

As initial conditions are not given it is not possible to find the Natural response from the given differential equation. 

None of the options is correct.