# qr code vb.net library Fig. 352. Unit circle. in Visual Studio .NET Create Code 128B in Visual Studio .NET Fig. 352. Unit circle.

Fig. 352. Unit circle.
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Fig. 353
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CHAPTER 13 The Digital Processor
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As can be seen from the enlarged version in Fig. 353, the equation of the unit circle can be written in either the rectangular or the exponential form; thus z x jy cos  j sin   j 598
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Now consider the basic recursive equation, eq. (597). Note, carefully, that the notation used in eq. (597) refers to the notation used in the recursive portion of Fig. 346. Now consider the fundamental rst-order processor of Fig. 348. With regard to eq. (597), note that here q 1, with all the a coe cients equal to zero EXCEPT for a1 ; thus, for Fig. 348, eq. (597) becomes z a1 0 showing that a rst-order recursive processor has just ONE POLE located at z a1 . However, as we already know, Fig. 348 is stable only if a1 is less than 1. Thus it is true that a rst-order recursive processor is stable only if the solution to eq. (597) lies within the unit circle. What we have just found, for the basic rst-order recursive processor, can be extended to ANY ORDER of such processors; that is, any recursive processor is stable only if all the poles of its transfer function H z lie within the unit circle on the z-plane. The poles of a given H z may be all real numbers, or all complex numbers, or a combination of real and complex numbers. HOWEVER, it s an important fact that COMPLEX POLES can occur only in the form of CONJUGATE PAIRS of complex numbers; thus, if c jd is a pole of H z , then c jd is also a pole, and vice versa. By way of an explanation, let s rst consider the case of a second-order recursive processor, as follows. For the second-order case, in eq. (597) we would have q 2, with all a coe cients equal to zero EXCEPT for a1 and a2 ; thus, for a second-order recursive processor eq. (597) becomes z 2 a1 z a2 0 which, if we wish, can be put in the factored form z g z h 0 600 599
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which, in this form, shows that a second-order recursive processor will have TWO POLES, one at z g, the other at z h. If, now, we multiply as indicated in eq. (600), we have that z2 g h z gh 0 and upon comparing this last result with eq. (599) we see that ' a1 g h a2 gh
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601
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Now, in an actual processor the a1 and a2 coe cients will always be real numbers. However, even though a1 and a2 are themselves real numbers, the two poles, g and h, can be either two real numbers or two CONJUGATE complex numbers. To show this, suppose that g and h are two conjugate complex numbers, g c jd and h c jd. Then, by eq. (601), we have a1 c jd c jd 2c; a real number and a2 c jd c jd c2 d 2 ; a real number:
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CHAPTER 13 The Digital Processor
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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
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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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