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3.1 definition 3.2 properties of ROC 3.3 the inverse z-transform 3.4 z-transform properties Chapter 3 the z-Transform

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3.1 definition ω j n n re z z n x z X = = −∞ = , ] [ ) ( Complex variable One motivation for introducing z-transform is that the Fourier transform doesn’t converge for all sequences and it is useful to have a generalization of the Fourier transform that encompasses a broader class of signals. A second advantages is that in analytical problems the z-form notation is often more convenient than the Fourier transform notation. The z-transform of a sequence x[n] is defined as
∑∑ −∞ = −∞ = < = nn n n z n x z n x | || ] [ | | ] [ | + < < Rx z Rx | | Figure 3.1 The region of convergence (ROC) are the set of values of z for which the z-transform converges. The condition for convergence of the z-transform is: So the convergence depends only on |z|. Thus if some values of z Say, z=z1, is in the ROC, then all values of z on the circle define By |z|=|z1| will also be in the ROC. As one consequence of this, the region of convergence will consist of a ring in the z-plane centered about the origin. This is expressed as

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) ( / ) ( ) ( z Q z P z X = Figure 3.2 When X(z) is a rational function inside the ROC the values of z for which P(z)=0 are called the zeros of X(z), and the values of z for which Q(z)=0 are called the poles of X(z). An example is shown in Figure 3.2, where a “o” denotes the zeros and a “x” denotes the poles. The ROC is bounded with the pole.
−∞ = = n n z n h z H ] [ ) ( ( ) ( ) ω j e z is that z j z X e X = = = , 1 | | | () [] [] [] n n n j n n n j r n x FT e r n x re n x z X −∞ = −∞ = = = = ) ( ] [

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## This note was uploaded on 02/29/2012 for the course EE 325 taught by Professor Lilili during the Fall '11 term at BYU.

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ç¬¬03ç«  Zå˜æ¢

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