anti-causal. The ROC is usually chosen to include the imaginary axis since it is important for most practical systems to have BIBO stability. Example This system has no (finite) zeros and two poles: and The pole-zero plot would be: Notice that these two poles are complex conjugates, which is the necessary and sufficient condition to have real-valued coefficients in the differential equation representing the system. Discrete-time systems In general, a rational transfer function for a discrete-time LTI system has the form: where is the order of the numerator polynomial, is the m-th coefficient of the numerator polynomial, is the order of the denominator polynomial, and is the n-th coefficient of the denominator polynomial. Either M or N or both may be zero. Poles and zeros such that are the zeros of the system such that are the poles of the system. Region of convergence The region of convergence (ROC) for a given DT transfer function is a disk or annulus which contains no poles. In general, the ROC is not unique, and the particular ROC in any given case depends on whether the system is causal or anti-causal. If the ROC includes the unit circle, then the system is bounded-input, bounded-output (BIBO) stable. If the ROC extends outward from the pole with the largest (but not infinite) magnitude, then the system has a right-sided impulse response. If the ROC extends outward from the pole with the largest magnitude and there is no pole at infinity, then the system is causal. If the ROC extends inward from the pole with the smallest (nonzero) magnitude, then the system is anti-causal. The ROC is usually chosen to include the unit circle since it is important for most practical systems to have BIBO stability. Example If and are completely factored, their solution can be easily plotted in the z-plane. For example, given the following transfer function: The only (finite) zero is located at: , and the two poles are located at: , where j is the imaginary unit. The pole–zero plot would be: See also Root locus Laplace transform Z-transform Rational function Bibliography Signal processing