If there are complex conjugate poles on the imaginary axis (+jw and -jw) for a continuous system, what would be the location of the poles of the equivalent discrete time system?
In Analog domain our range of vision for frequency is 0 to infinity that is why our j omega axis in s-plane is a straight line starting from 0 to infinity. In digital domain our range of vision is within 0 to 2pi ( due to inherent nature of sampling and periodicity in frequency). Any digital frequency higher than 2pi will be folded back to a lower frequency between 0 to 2pi( Aliasing) . This is the reason of having a unit circle in Z plane.
So by sampling we map the entire j-omega axis in s-plane to unit circle in Z-plane and remember Z plane is a continuous complex plane.
Every complex conjugate pole in S-plane has a corresponding complex conjugate pole on the unit circle in Z-plane. The location where it exactly map depends on the sampling frequency and the pole location in the S-plane.
IF YOU WANT MORE MATHEMATICAL RIGOROUS ANALYSIS JUST LET ME KNOW
That woulb be complex conjugate poles on the unit circle (i.e with modulus equal to one).
If you want to known where will be located the poles of the corresponding discrete system, the exact position, I think that it is not possible because the discrete model depends on the sample time and the algorithm and method for discretization. Nevertheless, I you want to known if the system is stable, then the poles must be located in the unitary circle with center in zero.
In Analog domain our range of vision for frequency is 0 to infinity that is why our j omega axis in s-plane is a straight line starting from 0 to infinity. In digital domain our range of vision is within 0 to 2pi ( due to inherent nature of sampling and periodicity in frequency). Any digital frequency higher than 2pi will be folded back to a lower frequency between 0 to 2pi( Aliasing) . This is the reason of having a unit circle in Z plane.
So by sampling we map the entire j-omega axis in s-plane to unit circle in Z-plane and remember Z plane is a continuous complex plane.
Every complex conjugate pole in S-plane has a corresponding complex conjugate pole on the unit circle in Z-plane. The location where it exactly map depends on the sampling frequency and the pole location in the S-plane.
IF YOU WANT MORE MATHEMATICAL RIGOROUS ANALYSIS JUST LET ME KNOW
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