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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 9 13 Copyright © Orchard Publications The Z Transform of Common Discrete Time Functions (9.36) Subtracting (9.36) from (9.35), we obtain or (9.37) for To determine from , we examine the behavior of the term in the numerator of (9.37). We write the terms and in polar form, that is, and (9.38) From (9.38) we observe that, for the values of for which , the magnitude of the com- plex number as and therefore, (9.39) for For the values of for which , the magnitude of the complex number becomes unbounded as , and therefore, is unbounded for . In summary, the transform converges to the complex number for , and diverges for . Also, since then, implies that , while implies and thus, az 1 F k z ( ) az 1 a 2 z 2 a 3 z 3 a k z k + + + + = F k z ( ) az 1 F k z ( ) 1 a k z k = F k z ( ) 1 a k z k 1 az 1 --------------------- 1 az 1 ( ) k 1 az 1 -------------------------- = = az 1 1 F z ( ) F k z ( ) az 1 ( ) k az 1 az 1 ( ) k az 1 az 1 e j θ = az 1 ( ) k az 1 k e jk θ = z az 1 1 < az 1 ( ) k 0 k F z ( ) F k z ( ) k lim 1 1 az 1 ------------------ z z a ---------- = = = az 1 1 < z az 1 1 > az 1 ( ) k k F z ( ) F k z ( ) k lim = az 1 1 > F z ( ) az 1 ( ) n n 0 = = z z a ( ) az 1 1 < az 1 1 > az 1 a z -- a z ----- = = az 1 1 < z a > az 1 1 > z a <

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Chapter 9 Discrete Time Systems and the Z Transform 9 14 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications (9.40) The regions of convergence and divergence for the sequence of (9.40) are shown in Figure 9.2. Figure 9.2. Regions of convergence and divergence for the geometric sequence To determine whether the circumference of the circle, where |, lies in the region of con- vergence or divergence, we evaluate the sequence at . Then, (9.41) We see that this sequence becomes unbounded as , and therefore, the circumference of the circle lies in the region of divergence. 9.3.2 The Transform of the Discrete Time Unit Step Function The definition and the waveform of the discrete time unit step function are as shown in Fig- ure 9.3. Figure 9.3. The discrete unit step function From the definition of the Z transform, Z a n u 0 n [ ] { } a n z n n 0 = z z a ---------- for z a > unbounded for z a < = = |a| Re[z] Im[z] Region of Divergence Region of Convergence F z ( ) F z ( ) z z a ---------- = a n z a = F k z ( ) z a = F k z ( ) a n z n n 0 = k 1 1 az
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