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|z|>4 will include the values between 4 and 5, not satisfying |z|>5. Now, we need the RoC to satisfy both conditions. The first component is RSS which means the limits will be |z|>4 and the second component is LSS which means the limits will be |z|4 and |z|>5 Z-plane of ration and RSS signal The z transform of the individual components will be and. The RoC will have combined properties of both poles and fulfill the limits of both.Ĭonditions: More than one component in x and both are rational, which means it has two poles. Property 8: The ROC is bounded by poles if x is rational, which means it would extend toĮxplanation: If the values of x were rational and bounded by poles, the RoC is extended to infinity. The RSS would point outside the circle, and the LSS would point inside the circle, which means they would cancel out and the RoC would be the ring around the radius at modulus of z equal to r. However, the denominator would change based on the z value. The z transform of both components of x(n) would be the same. Let us look at an example combining an RSS and an LSS. Property 7: If x is both causal and non-causal, the ROC will be a ring.Įxplanation: If the n values fall on both sides of the x-axis, the ROC will be a ring around the radius in the z-plane. Hence, the system is non-causal and unstable. Z=1/4 falls within the unit circle but z=-1.25 falls outside the unit circle Ĭomplete the square in the denominator to get the poles Property 6: The ROC includes 0 if the signal x is non-causalĮxplanation: When the aforementioned conditions are not met, the ROC would include 0 instead of. Since both the poles fall inside the unit circle, the system is causal and stable.
Prove of bibo stability condition series#
However, the Z-transform will exist only for those values of Z, which if put in this series results in a finite value. You will remember that the limits of the summation were from -∞ to +∞. The Region of Convergence maps all the values for which the transform converges to a finite value.The particular signals that tend to meet at a point are the signals that lie in the Region of Convergence(RoC).
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