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Chapter 7: The z-Transform

Chapter 7: The z-TransformChih-Wei LiuOutline Introduction The z- transform Properties of the Region of Convergence Properties of the z- transform Inversion of the z- transform The Transfer Function Causality and Stability Determining Frequency Response from Poles & Zeros Computational Structures for DT-LTI Systems The Unilateral z-Transform2 Introduction3 The z-transformprovides a broader characterization of discrete-time LTI systems and their interaction with signals than is possible with DTFT Signal that is not absolutely summable Two varieties of z- transform : Unilateral or one-sided Bilateral or two-sided The unilateral z- transform is for solving difference equations with initial conditions. The bilateral z- transform offers insight into the nature of system characteristics such as stability, causality, and frequency General Complex Exponential zn4 Complex exponential z= rej with magnitude r and angle znis an eigenfunction of the LTI systemexponentially damped cosine exponentially damped sine < 0Re{zn}: exponential damped cosineIm{zn}: exponential damped sine)sin()cos(njrnrznnn exponentially damped cosine r: d a m p i n g f a c t o r : sinusoidal frequencyEigenfunction Property of zn5 Transfer function H(z) is the eigenvalue of the eig

Convergence of Laplace Transform 7 z-transform is the DTFT of x[n]r n A necessary condition for convergence of the z-transform is the absolute summability of x[n]r n: The range of r for which the z-transform converges is termed the region of convergence (ROC). Convergence example: 1.

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