NOTESZtransform

# NOTESZtransform - Chapter 6 The Z-Transform The Z-transform...

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Chapter 6 - The Z-Transform The Z-transform is the Discrete-Time counterpart of the Laplace Trans- form. Laplace : G ( s ) = Z -∞ g ( t ) e - st dt Z : G ( z ) = X n = -∞ g [ n ] z - n It is Used in Digital Signal Processing Used to Deﬁne Frequency Response of Discrete-Time System. Used to Solve Diﬀerence Equations – use algebraic methods as we did for diﬀerential equations with Laplace Transforms; it is easier to solve the transformed equations since they are algebraic. We will see that 1. Lines on the s-plane map to circles on the z-plane. 2. Role of -axis is replaced by unit circle, so (a) The DT Fourier Transform exists for a signal if the ROC includes the unit circle. (b) A stable system must have an ROC that contains the unit circle. (c) A causal and stable system must have poles inside the unit circle. 1

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Aside: You can relate the Z transform and Laplace transform directly when you are dealing with sampled signals. First recall the deﬁnition of Laplace transform: Take a CT signal g ( t ) and sample it: g s ( t ) = g ( t ) X n = -∞ δ ( t - nT ) = X n = -∞ g ( nT ) δ ( t - nT ) The Laplace transform of the sampled signal is L [ g s ( t )] = Z -∞ " X n = -∞ g ( nT ) δ ( t - nT ) # e - st dt = X n = -∞ Z -∞ g ( nT ) δ ( t - nT ) e - st dt = X n = -∞ g ( nT ) e - snT . Let g [ n ] = g ( nT ) be the discrete representation and z = e sT , then G ( z ) = X n = -∞ g [ n ] z - n G ( z ) | z = e sT = X n = -∞ g [ n ] e - sTn = X n = -∞ g ( nT ) e - snT = L [ g s ( t )] Thus, the Z transform with z = e sT is the same as the Laplace transform of a sampled signal! Of course, if the signal is already discrete, the notion of sampling is unnecessary for understanding and using the Z transform. 2
G ( z ) = X n = -∞ g [ n ] z - n is the bilateral (2-sided) Z-transform. Its inverse Z-transform is deﬁned as: Z - 1 [ H ( z )] = h [ n ] = 1 2 πj I H ( z ) z n - 1 dz which is a counterclockwise contour integral along a closed path in the z - plane. We will see how to take inverse Z-transforms using tables and partial fraction expansion. Some textbooks work with the unilateral Z-transform: H u ( z ) = X n =0 h [ n ] z - n . IMPORTANT: We do not use this at all, but make sure you know the diﬀerence in case you come across it later. In this course, our focus is on the Bilateral Z-Transform, to which we simply refer as Z-transorm: G ( z ) = X n = -∞ g [ n ] z - n , deﬁned for 2-sided, anticausal, and causal signals (i.e. all signals). Note that, whereas for Laplace Transform we considered where the inte- gral converges, here we consider where the sum converges. 3

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NOTESZtransform - Chapter 6 The Z-Transform The Z-transform...

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