CT321 Digital Signal Processing Yash Vasavada Autumn 2016

CT-321 Digital Signal Processing Yash Vasavada Autumn 2016 DA-IICT Lecture 10 Z Transform 24 th August 2016

Review and Preview • Review of past lecture: – Properties of DTFT and LSI Systems – Z Transform • Preview of this lecture: – Z Transform • Reading Assignment – OS: Chapter 3 – PM: Chapter 3 Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 10: 24 th August 2016. 2

Z Transform • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 10: 24 th August 2016. 3

Review of Complex Arithmetic • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 10: 24 th August 2016. 4

Review of Complex Arithmetic • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 10: 24 th August 2016. 5

Review of Complex Arithmetic • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 10: 24 th August 2016. 6

An Aside: Origin of Complex Arithmetic • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 10: 24 th August 2016. 7

An Aside: Origin of Complex Arithmetic • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 10: 24 th August 2016. 8

How About Quadratic Equations • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 10: 24 th August 2016. 9

An Aside: Origin of Complex Arithmetic • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 10: 24 th August 2016. 10

Region of Convergence of Real-Valued Power Series • [Needham 97]: T. Needham, Visual Complex Analysis, Oxford Press, 1997 Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 10: 24 th August 2016. 11

Region of Convergence (ROC) of Real-Valued Power Series • We can apply this power series to evaluate two different functions: ROD Yash Vasavada DA-IICT. Autumn 2016 ROC CT-321. Lecture 10: 24 th August 2016. ROD 12

Z Transform and DTFT • Pictorial Relation between Z transform and DTFT Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 10: 24 th August 2016. 13

Z Transform and DTFT • Maroon: ROC of Z Transform Green: Region of Divergence (ROD) Dotted circle: Unit circle in Z plane (UC) Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 10: 24 th August 2016. 14

Possible Shapes of ROC of Z Transform • A B D C E Maroon: ROC of Z Transform Green: Region of Divergence (ROD) Dotted circle: Unit circle in Z plane (UC) Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 10: 24 th August 2016. 15

Poles and Zeros of Z Transform • An Example Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 10: 24 th August 2016. 16

Poles and Zeros of Z Transform • Poles and Zeros of an Example Z Transform d. B A Pole Zero Plot Side View A B Side View B d. B Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 10: 24 th August 2016. 17