Correct Answer - Option 3 : All the poles must lie within |s| = 1.
Concept:
Continuous-time signal: A signal of continuous amplitude and time is known as a continuous-time signal or an analog signal.
Discrete-time signal: Unlike a continuous-time signal, a discrete-time signal is not a function of a continuous argument; however, it may have been obtained by sampling from a continuous-time signal. When a discrete-time signal is obtained by sampling a sequence at uniformly spaced times, it has an associated sampling rate.
Analysis:
We will check each option to find out the incorrect one.
All the poles of the system must lie on the left side of the jω axis.
- The system is stable if and only if all the roots lie on the left side of the S plane or jω axis. Also, we can say all the poles have a negative real part.
Example:
If ROC is Re[s]>-a, the system h(t) =e-at u(t) is causal.
Its Laplace transform \({\rm{H}}\left( {\rm{s}} \right) = \frac{1}{{{\rm{a}} + {\rm{s}}}}\)
If a<0, i.e., imaginary axis Re[s] =0 can be included in the ROC, the system is stable.
Zeros of the system can lie anywhere on the plane.
When zeros are in the right half of the S plane or outside the unit circle in the Z- plane, it does not cause the system to be unstable.
All the poles of the system must lie within |s|=1
- The system is stable if and only if the ROC of the system function H(z) includes the unit circle |z|=1 and not |s|=1.
- The system is stable in the case of rational system function H(z) if and only if all the poles of H(z) lie inside the unit circle, they must all have a magnitude smaller than unity.
All the roots of the characteristics equation must lie on the left side of the jω axis.
For stability, all the roots of the characteristics equation must lie on the left side of the s-plane, even if one root lies on the right side the system becomes unstable.
Conclusion:
So, we can say options a, b and d are satisfying property of a continuous-time casual and stable system.
Option c is correct for a discrete-time casual and stable LTI system.
