Question

What is the zero-input response of the system described by the homogenous second order equation y(n)-3y(n-1)-4y(n-2)=0 if the initial conditions are y(-1)=5 and y(-2)=0?

(a) (-1)^n-1 + (4)^n-2

(b) (-1)^n+1 + (4)^n+2

(c) (-1)^n+1 + (4)^n-2

(d) None of the mentioned

I had been asked this question in a job interview.

My query is from Discrete Time Systems Described by Difference Equations topic in division Discrete Time Signals and Systems of Digital Signal Processing

Answer 1

Correct choice is (b) (-1)^n+1 + (4)^n+2

To elaborate: Given difference equation is y(n)-3y(n-1)-4y(n-2)=0—-(1)

Let y(n)=λ^n

Substituting y(n) in the given equation

=> λ^n – 3λ^n-1 – 4λ^n-2 = 0

=> λ^n-2(λ^2 – 3λ – 4) = 0

the roots of the above equation are λ=-1,4

Therefore, general form of the solution of the homogenous equation is

yh(n)=C1 λ1^n+C2 λ2^n

=C1(-1)^n+C2(4)^n—-(2)

The zero-input response of the system can be calculated from the homogenous solution by evaluating the constants in the above equation, given the initial conditions y(-1) and y(-2).

From the given equation (1)

y(0)=3y(-1)+4y(-2)

y(1)=3y(0)+4y(-1)

=3[3y(-1)+4y(-2)]+4y(-1)

=13y(-1)+12y(-2)

From the equation (2)

y(0)=C1+C2 and

y(1)=C1(-1)+C2(4)=-C1+4C2

By equating these two set of relations, we have

C1+C2=3y(-1)+4y(-2)=15

-C1+4C2=13y(-1)+12y(-2)=65

On solving the above two equations we get C1=-1 and C2=16

Therefore the zero-input response is Yzi(n) = (-1)^n+1 + (4)^n+2.