Question

Use Z transforms to solve the second-order recurrence relation xₖ₊₂​−xk​=1 where x₀​=0 and x₁​=−1.

Answer 1

The question involves solving a second-order recurrence relation using Z transforms. The solution requires converting the relation into its Z transform, applying initial conditions, solving for the transformed variable, and then finding the inverse Z transform.

Step-by-step explanation:

The student is asking to solve a second-order recurrence relation xk+2 - xk = 1 with initial conditions x0 = 0 and x1 = -1 using Z transforms. To address this, we will convert the recurrence relation into its Z transform equivalent, apply initial conditions, solve for X(z), and then apply inverse Z transform to find the solution in the time domain.

To apply Z transform, assume the Z transform of xk is X(z). Then, we can write the Z transform of the recurrence relation as:

Z2X(z) - Zx0 - zx1 - X(z) = (z/z - 1)

With initial conditions we have:

Z2X(z) - Z(0) - z(-1) - X(z) = (z/z - 1)

Simplify and solve for X(z) and then apply the inverse Z transform to find the solution for xk.