Question

consider an LTI system y(n)-1.5 y(n-1)+0.5y(n-2)=x(n) find the natural response of this system ​

Answer 1

Given:

LTI system: y(n)-1.5y(n-1)+0.5y(n-2)=x(n)

To Find:

Natural Response    

Solution:

The homogeneous equation ⇒

y(n)-1.5y(n-1)+0.5y(n-2)=0

let y(n)=λⁿ

The equation becomes,

⇒λⁿ - 1.5λⁿ⁻¹ + 0.5λⁿ⁻² = 0

⇒λⁿ⁻² (λ² - 1.5λ +0.5) = 0      [∵ λⁿ⁻² ≠ 0]

⇒λ² - 1.5λ - 0.5=0

⇒λ²- λ- 0.5λ +0.5=0

⇒λ(λ-1) -0.5(λ-1)=0

   ⇒λⁿ=0.5,1

Hence,

y(n)=C₁(0.5)ⁿ+C₂(1)ⁿ

Putting n=0 and n=1 in y(n), we get:

y(0)=C₁+C₂ ......(1)

y(1)=0.5C₁+C2 .......(2)

Since, no initial conditions are given, we assume:

y(0)=0 and y(1)=0

Putting the values of y(0) and y(1) in equations (1) and (2)

For n=0

   y(0)=1.5y(-1)+0.5y(-2)=0

⇒y(0)-1.5+0.5=0

⇒C₁+C₂=0.5 ........(3)

For n=1

   y(1)-1.5y(0)+0.5y(-1)=0

⇒y(1)-1.5(0.5)+0.5(1)=0

⇒0.5C₁+C₂=0.25....... (4)

Now, (3)-(4) we get:

0.5C₁=0.25

C₁=0.5

C₂=0

Hence,

y(n)= (0.5)ⁿ⁺¹   , n≥0

      =(0,5)ⁿ⁺¹u(n)   is the required natural response