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Thus we have the important fact that a second-order recursive processor will always possess TWO POLES, g and h, in which either (a) g and h are both real numbers, or (b) g and h are conjugate complex numbers. However, regardless of whether we have case (a) or case (b) for a given processor, for stability both poles must lie within the unit circle on the z-plane. Now let s consider a third-order recursive processor (meaning the use of three delays). For this case q 3 in eq. (597), which thus, for this case, becomes z 3 a 1 z 2 a2 z a3 0 which, theoretically, it s always possible to factor into the form z f z g z h 0 showing that a third-order recursive processor possesses THREE POLES, denoted here by f , g, and h. Since complex poles can exist only in conjugate form it follows that the possibilities for a third-order (recursive) processor are (a) three real poles, or (b) one real and one pair of conjugate poles. Again, for stability it s necessary that all three poles lie within the unit circle. To continue on, it s a fundamental fact that any algebraic equation of the form of eq. (597), in which the highest power of the unknown, z, is an integer, q, can always be factored into the form z h1 z h2 z h3 z hq 0 602

remove any leading or trailing space characters. Similar to comments, extraneous space characters will trip up comparisons, so they are removed. Finally, the disallow path is added to the list of disallow paths. After the disallow path list has been created, it is added to the disallow list cache, as shown here:

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which clearly shows that any such equation has q solutions * which we re denoting here by h1 ; h2 ; h3 ; . . . hq . You may have noticed that, so far, we ve not said much about eq. (596). As we already know, solutions to eq. (596) are called zeros because these are the values of z for which H z 0. For now, however, let us just note that certain procedures do exist that require the use of both the zeros and the poles of a processor. Problem 308 Write the equation for nding the poles of a fourth-order recursive processor and list the possible combinations of real and complex poles that might exist. Problem 309 Repeat problem 308 for a fth-order (recursive) processor. Note: The following problems will call for a certain amount of factoring. In some cases this can be done by direct inspection, while in other cases you may wish to review the standard quadratic formula found in note 1 in the Appendix.

* In the general language of algebra the solutions are said to be the roots of the equation, but in the speci c application here it s customary to call the roots poles.

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Problem 310 Find the zeros and poles, given the transfer function H z 4z 9 z z 9 z2 5z 7

Problem 311 The following are transfer functions for certain DT processors. Determine, in each case, whether the processor is stable or unstable. (a) H z (b) H z (c) H z z z 0:46 z 0:22 z2 z 0:61 z2 1:6z 0:48 z 1 1:2z 2 1 1:37z 1 0:305z 2

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