Region of convergence and stability
Assume that x(n) is a causal sequence which can be written as a sum of complex exponentials. This takes in a wide range of signals including sinusoids, exponentials, and products thereof. Let
By taking the z transform of x(n) gives
The area of convergence R is intersection of the areas of convergence for each exponential as given below:
As the ROC for translated exponential remains the same for the original exponential, all the right-sided sequences which are sums of translated exponentials have ROCs similar to which are expressed above.
By similar argument all the left-sided sequences expressible as the sum of translated complex exponentials have a ROC, L, which is given by L = {z: |z| < smallest of |bi|}
If we have a arrangement of right- and left-sided sequences, the corresponding ROC is intersection of R and L. Thus the total ROC becomes an annular region as shown below in the figure and given by
RTotal = R ∩ L = {z: Largest of |ai| < |z| < smallest of |bi|}
Stability of the system with an impulse response which means that the sum of translated right- and left-sided sequences is determined from the region of convergence. Suppose that h(n) is the unit sample response of a causal or non-causal linear shift-invariant system. Let ?[h(n)] = H(z), the so-called system function. Then:
Theorem A linear shift-invariant system with system function H(z) is BIBO stable if the ROC for H(z) has unit circle.
The theorem can be used to determine the stability for a given H(z) without obtaining the impulse response or checking the outputs for all the bounded input signals.
Illustration of stability and causality
For A system function with two poles at, say, z = 0.5, and z = 1.5, there are 3 possible regions of convergence.
(1) ROC is 0.5 < |z| < 1.5. The system is stable here since the unit circle is inside the region of convergence. Impulse response, h(n), is 2-sided, so the system is noncausal.
(2) ROC is |z| < 0.5. The system is not stable here. The impulse response, h(n), is left- sided, hence the system is noncausal.
(3) ROC is |z| > 1.5. The system is not stable here. The impulse response, h(n), is right- sided, hence the system may be causal.
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