Assertion (A): An LTI discrete system represented by the difference eq...
Explanation: Difference equation is y (n+2)-5 y (n+1) + 6 y (n) =x (n).
Taking z-transform, H (z) =1/ (z-2) (z-3).
The characteristic equation has roots z =2, 3. Since, the characteristic equation has roots outside the unit circle, hence the system is unstable.
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Assertion (A): An LTI discrete system represented by the difference eq...
Assertion (A): An LTI discrete system represented by the difference equation y(n-2) - 5y(n-1) + 6y(n) = x(n) is unstable.
Reason (R): A system is unstable if the roots of the characteristic equation lie outside the unit circle.
Explanation:
To determine the stability of the LTI discrete system represented by the given difference equation, we need to analyze its characteristic equation.
The characteristic equation of the system can be obtained by setting y(n) = 0 and solving for the values of z, where z is the complex variable representing the z-transform domain.
The given difference equation is:
y(n-2) - 5y(n-1) + 6y(n) = x(n)
Let's rewrite this equation in terms of the shift operator z:
z^2Y(z) - 5zY(z) + 6Y(z) = X(z)
Dividing both sides by Y(z), we get:
z^2 - 5z + 6 = X(z)/Y(z)
The characteristic equation is obtained by setting the numerator X(z) equal to zero:
z^2 - 5z + 6 = 0
Now, let's solve this quadratic equation to find the roots of the characteristic equation:
(z - 3)(z - 2) = 0
The roots are z = 3 and z = 2.
Stability Analysis:
For a discrete LTI system to be stable, all the roots of the characteristic equation should lie inside the unit circle in the z-plane.
In the given case, the roots of the characteristic equation are z = 3 and z = 2.
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Conclusion:
Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
The stability of the LTI discrete system represented by the given difference equation can be determined by analyzing the roots of the characteristic equation, and in this case, the roots lie inside the unit circle, indicating stability.
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