CT321 Digital Signal Processing Yash Vasavada Autumn 2016

CT-321 Digital Signal Processing Yash Vasavada Autumn 2016 DA-IICT Lecture 7 LSI Systems and DTFT 17 th August 2016

Review and Preview • Review of past lecture: – Different ways of understanding the convolution operation • Preview of this lecture: – Convolution Operation – Relation to DTFT – Properties of DTFT and LSI Systems • Reading Assignment – OS, 3 rd Edition: Sections 1. 2 to 1. 4, and Sections 1. 6 to 1. 9 • Note: available in the library – PM: Sections 2. 1 to 2. 3 Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 7: 17 th August 2016. 2

Convolution Operation Prior lecture • Today’s lecture Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 7: 17 th August 2016. 3

Polynomial Multiplication • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 7: 17 th August 2016. 4

A Brief Detour of Linear Algebra and Dot Products • Un 2 D it le pl ngt an h c e ir cle in Angle between the two vectors Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 7: 17 th August 2016. 5

A Brief Detour of Linear Algebra and Dot Products • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 7: 17 th August 2016. 6

Convolution Operation as Sequential Dot Products • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 7: 17 th August 2016. 7

An Aside: Why the Word “Linear”? • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 7: 17 th August 2016. 8

Convolution Operation and DTFT • Formulation of output as the convolution operation, as we have seen earlier Rewriting the convolution operation by a change of variables Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 7: 17 th August 2016. 9

Eigenfunction of an LSI System and Its Frequency Response • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 7: 17 th August 2016. 10

Eigenfunction of LSI System • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 7: 17 th August 2016. 11

An LSI System Provides for Multiplication (e. g. , Windowing) in Frequency Domain • Output viewed as convolution in time domain Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 7: 17 th August 2016. 12

Properties of LSI System (or, equivalently, of Convolution Operation) • Commutative: • Distributive: • Associative: Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 7: 17 th August 2016. 13

Properties of DTFT • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 7: 17 th August 2016. 14

Properties of DTFT • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 7: 17 th August 2016. 15