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qr code vb.net library Fig. 352. Unit circle. in Visual Studio .NET
Fig. 352. Unit circle. Recognizing USS Code 128 In .NET Framework Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in VS .NET applications. Draw USS Code 128 In VS .NET Using Barcode generator for .NET Control to generate, create Code 128 Code Set A image in VS .NET applications. Fig. 353
Recognizing Code 128 Code Set B In .NET Framework Using Barcode reader for .NET framework Control to read, scan read, scan image in .NET applications. Print Bar Code In VS .NET Using Barcode creator for .NET framework Control to generate, create bar code image in Visual Studio .NET applications. CHAPTER 13 The Digital Processor
Scanning Bar Code In .NET Framework Using Barcode reader for VS .NET Control to read, scan read, scan image in .NET framework applications. Code 128C Printer In Visual C# Using Barcode maker for VS .NET Control to generate, create Code 128C image in VS .NET applications. 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 Paint Code 128 In VS .NET Using Barcode maker for ASP.NET Control to generate, create Code 128 image in ASP.NET applications. Painting USS Code 128 In VB.NET Using Barcode printer for Visual Studio .NET Control to generate, create Code 128 Code Set B image in Visual Studio .NET applications. 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 rstorder 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 rstorder 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 rstorder recursive processor is stable only if the solution to eq. (597) lies within the unit circle. What we have just found, for the basic rstorder 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 zplane. 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 secondorder recursive processor, as follows. For the secondorder 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 secondorder 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 Drawing Code 128A In Visual Studio .NET Using Barcode printer for .NET framework Control to generate, create Code 128 image in .NET framework applications. Creating Matrix Barcode In Visual Studio .NET Using Barcode generator for .NET framework Control to generate, create Matrix Barcode image in .NET framework applications. which, in this form, shows that a secondorder 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 Paint Bar Code In .NET Using Barcode drawer for .NET Control to generate, create barcode image in .NET applications. Printing USD  8 In VS .NET Using Barcode generator for Visual Studio .NET Control to generate, create Code 11 image in VS .NET applications. 601 Paint Barcode In None Using Barcode drawer for Excel Control to generate, create barcode image in Office Excel applications. Barcode Printer In ObjectiveC Using Barcode creator for iPhone Control to generate, create barcode image in iPhone applications. 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: Bar Code Generation In None Using Barcode generation for Software Control to generate, create barcode image in Software applications. European Article Number 13 Drawer In VB.NET Using Barcode drawer for Visual Studio .NET Control to generate, create GS1  13 image in Visual Studio .NET applications. CHAPTER 13 The Digital Processor
Printing GS1  12 In Java Using Barcode encoder for Java Control to generate, create GS1  12 image in Java applications. Recognizing EAN13 In .NET Using Barcode decoder for .NET Control to read, scan read, scan image in Visual Studio .NET applications. Thus we have the important fact that a secondorder 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 zplane. Now let s consider a thirdorder 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 thirdorder 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 thirdorder (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 2D Barcode Printer In Visual Basic .NET Using Barcode generation for VS .NET Control to generate, create 2D Barcode image in .NET applications. European Article Number 13 Maker In VS .NET Using Barcode encoder for Reporting Service Control to generate, create EAN / UCC  13 image in Reporting Service applications. 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 fourthorder recursive processor and list the possible combinations of real and complex poles that might exist. Problem 309 Repeat problem 308 for a fthorder (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.

