Question
Download Solution PDFA system is defined by the difference equation
y(n) = 1.8y(n - 1) - 0.72y (n - 2) + x (n) + 0.5x (n - 1), then the system is
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
For a linear time-invariant (LTI) discrete-time system, stability is determined by the location of the poles (roots of the characteristic equation). The system is:
- Stable if all poles lie inside the unit circle (|z| < 1).
- Unstable if any pole lies outside the unit circle (|z| > 1).
- Marginally stable if poles lie on the unit circle and are non-repeating.
Given:
Difference equation:
y(n) = 1.8y(n - 1) - 0.72y (n - 2) + x (n) + 0.5x (n - 1)
Characteristic equation (from homogeneous part):
\(H(z): z^2 - 1.8z + 0.72 = 0 \)
Calculation:
Use the quadratic formula to find roots:
\(\frac{1.8 \pm \sqrt{(1.8)^2 - 4 \cdot 0.72}}{2}\)
\(z = \frac{1.8 \pm 0.6}{2} ⇒ z = 1.2\)Since one root is greater than 1, the system is unstable.
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