ELEC212 Chapter 3 Z-Transform

ELEC212 Chapter 3 Z-Transform - 3.1 z-Transform The...

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ELEC 212 Chapter 3: Z-Transform Prof. Matthew R. McKay 2 09-10 Fall ELEC 212 : Matthew McKay z-Transform 3.1 z -Transform The z -transform X ( z ) of a sequence x [ n ] is defined as where z is a continuous complex variable . Moreover generally, we can express the complex variable z in polar form as where r = | z |> 0 is the magnitude and is the angle of z z -transform gives In particular, when r = 1 , then and the above expression becomes () [] −∞ = = = n n z n x n x Z z X } { ω j re z = n n n j n r n x DTFT e r n x z X −∞ = = = j e z = [] [] j n n j e z e X n x DTFT e n x z X j = = = −∞ = = 3 09-10 Fall ELEC 212 : Matthew McKay z-Transform 3.1 z -Transform The relationship between X ( z ) and X ( e j ω ) can also be illustrated in the z -plane (a real-imaginary plane for complex number). The z -transform evaluated on the unit circle (as ω varies from 0 to 2 π ) corresponds to the DTFT of x [ n ]. 4 09-10 Fall ELEC 212 : Matthew McKay z-Transform 3.1 z -Transform Main advantage of using z -transform over DTFT: z -transform can encompass a broader class of signals than DTFT • Recall the discussion in Chapter 2, a sufficient condition for convergence of the DTFT is: • E.g. The DTFT of does not exist since it is unbounded in the negative direction: < −∞ = −∞ = n n n j j n x e n x e X ] 1 [ 5 . 0 = n u n x n n -5 -4 -3 -2 -1 0 -32 -16 -8 -4 -2
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5 09-10 Fall ELEC 212 : Matthew McKay z-Transform 3.1 z -Transform z -transform can encompass a broader class of signals than DTFT • On the other hand, the z -transform exists if • We can choose a region of convergence (ROC) for z such that z -transform converges • E.g. Consider again . Although the DTFT does not exist, the z -transform exists (converges) if (see below) () [] < = −∞ = −∞ = n n n n j n j r n x e r n x re X z X ω ) ( ] 1 [ 5 . 0 = n u n x n ( ) ( ) 1 1 1 1 1 1 1 5 . 0 1 1 5 . 0 1 5 . 0 5 . 0 5 . 0 ] 1 [ 5 . 0 ) ( = −∞ = −∞ = = = = = = z z z z z z n u z X m m n n n n n 1 5 . 0 1 < z Magnitude must be < 1 for series convergence Î | z | < 1/2 Useful geometric series: 1 , 1 1 < = = r r r r k k 6 09-10 Fall ELEC 212 : Matthew McKay z-Transform 3.1 z -Transform Other advantages of using z -transform: – More convenient notation than DTFT when dealing with analytical problems – Convenient block diagram representation for implementation of practical systems. Eg., a unit delay system is expressed as: – Useful for determining and solving difference equations for discrete-time systems j e z z -1 ] 1 [ ] [ = n x n y ] [ n x 7 09-10 Fall ELEC 212 : Matthew McKay z-Transform Region of Convergence (ROC) Since z can be any point on the z -plane, generally there exists some z which makes X ( z ) not converge The set of z values for which X ( z ) converges is called the region of convergence (ROC).
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ELEC212 Chapter 3 Z-Transform - 3.1 z-Transform The...

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