tailieunhanh - Lecture Digital signal processing: Lecture 3 - Zheng-Hua Tan

Chapter 3 introduce the z-transform. This chapter presents the following content: z-transform, properties of the ROC, inverse z-transform, properties of z-transform. | Digital Signal Processing Fall 2006 Lecture 3 The z-transform Zheng-Hua Tan Department of Electronic Systems Aalborg University Denmark zt@ 1 Digital Signal Processing III Zheng-Hua Tan 2006 Course at a glance MM3 MM9 MM10 MM7 MM8 ___________ 1 Mt 2 Digital Signal Processing III Zheng-Hua Tan 2006 AALBORG UNIVERSITY 1 Part I z-transform z-transform Properties of the ROC Inverse z-transform Properties of z-transform 3 Digital Signal Processing III Zheng-Hua Tan 2006 AALBORG UNIVERSITY Limitation of Fourier transform Fourier transform X e Y x n e -J n -w x n -1-Ị 1 4 X ej ejand Condition for the convergence of the infinite sum v e Y . n - ZL -x n e Y x n n -w n - n n - n If x n is absolutely summable its Fourier transform exists sufficient condition . Example x n anu n a 1 Xj V- 1 a 1 X ej - Y nỗ 2nk a 1 - 1 - 4 Digital Signal Processing III Zheng-Hua Tan 2006 AALBORG UNIVERSITY 2 z-transform Fourier transform z-transform X X X x n n -x X X z x n n -x Z x n o- X z The complex variable z in polar form z rejc X X X z X re x n reJa -n x n r x e n -X -X z r 1 X z X e 5 Digital Signal Processing III Zheng-Hua Tan 2006 z-plane z-transform is a function of a complex variable - using the complex z-plane Z-transform on unit circle - Fourier transform Linear frequency axis in Fourier transform Unit circle in z-transform periodicity in freq. of Fourier transform Figure The unit circle in the complex z-plane. 6 Digital Signal Processing III Zheng-Hua Tan 2006