Question

Analyse Unilateral Z transform to generate the output of a system characterized by the difference equation as y[n]+1/2y[n−1]+1/4y[n−2]=0 with initial conditions of y[−1]=1 and y[−2]=1.

a) Express the given difference equation in Z-transform notation.
b) Apply the Z-transform to the difference equation.
c) Solve for the Z-transform of the output, Y(z).
d) Use inverse Z-transform to find the time-domain sequence y[n].

Answer 1

To determine y[n] for the given system, express the difference equation in Z-transform notation, apply the Z-transform considering initial conditions, solve for the Z-transform of the output Y(z), and then apply the inverse Z-transform to find y[n].

Step-by-step explanation:

To analyze the Unilateral Z transform to generate the output of a system characterized by the given difference equation, we will follow these steps:

1. Express the given difference equation in Z-transform notation.
2. Apply the Z-transform to the difference equation.
3. Solve for the Z-transform of the output, Y(z).
4. Use inverse Z-transform to find the time-domain sequence y[n].

(a) In Z-transform notation, the given difference equation is:

Y(z) + \frac{1}{2}z^{-1}Y(z) + \frac{1}{4}z^{-2}Y(z) = 0

(b) Applying the Z-transform, we consider the initial conditions y[-1] = 1 and y[-2] = 1, resulting in:

Y(z) + \frac{1}{2}z^{-1}[(Y(z) - y[-1]) + \frac{1}{4}z^{-2}[(Y(z) - y[-1]z^{-1} - y[-2])] = 0

(c) Solving the equation, the Z-transform of the output Y(z) is found by isolating Y(z) and substituting the initial conditions.

(d) The inverse Z-transform is then applied to Y(z) to obtain y[n], the time-domain sequence.