CT321 Digital Signal Processing Yash Vasavada Autumn 2016

CT-321 Digital Signal Processing Yash Vasavada Autumn 2016 DA-IICT Lecture 4 LSI Systems and DTFT 9 th August 2016

Review and Preview • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 4: 9 th August 2016. 2

Preview and Reading Assignment • Preview of this lecture: – Linear Time Invariant (LTI) Systems – Relation to DTFT • 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 4: 9 th August 2016. 3

Review of Past Lecture: Rectangular Window Function • Rectangular pulse is a key signal of frequent application in theoretical DSP studies • To see why, consider a hypothetical signal that we have received, that has an undesired component mixed in with the desired signal. We want to get rid of the unwanted component. • One solution is to multiply this received signal by a rectangular pulse of unity amplitude and of sufficient width centered at the desired signal’s location. – This will zero out the unwanted component and leave the desired part unchanged. – This operation is also called rectangular windowing • Notice that this mixture of desired and unwanted signals can occur either in time domain or in frequency domain • In time domain, it’s not hard to envision implementation of the multiplication operation between the received signal and the rectangular pulse. However, how to do so in the frequency domain? 1 Unwanted 0 Yash Vasavada Desired Time or Frequency DA-IICT. Autumn 2016 CT-321. Lecture 4: 9 th August 2016. 4

Review of Past Lecture: Averaging of Multiple Samples of Complex Phasor • • Surprisingly quite similar to the Fourier Transform of the rectangular pulse in continuous time domain! • Why? Now solve using this expression: Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 4: 9 th August 2016. 5

Discrete Time Fourier Transform • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 4: 9 th August 2016. 6

Recap • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 4: 9 th August 2016. 7

DTFT of the Rectangular Pulse • Frequency domain representation becomes localized (is narrower) as the time domain signal spreads out (becomes wider) Time Domain Rectangular Pulses DTFT (Frequency Domain Representation of Rectangular Pulses) – Same as the observation we have made earlier in context of C-T signals • Can DTFT be reversed? – i. e. , can the discrete time samples be recovered from continuousfrequency DTFT? – If so, what is likely to be this “inverse” DTFT of rectangular function in the frequency domain? Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 4: 9 th August 2016. 8

Inverse DTFT • Time Domain: Discrete Samples Yash Vasavada DA-IICT. Autumn 2016 Frequency Domain (Continuous and Periodic) CT-321. Lecture 4: 9 th August 2016. 9

Why DTFT is Important? • Recall our conundrum earlier: how to multiply any signal with another signal (say, a rectangular window function) in frequency domain? • Armed with the tool of DTFT, we are now in a position to begin to solve this puzzle Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 4: 9 th August 2016. 10

Digital Signal Processing Systems • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 4: 9 th August 2016. 11

Digital Signal Processing Systems: Some Key Properties • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 4: 9 th August 2016. 12

Digital Signal Processing Systems: Some Key Properties • Consider the two systems A and B shown below Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 4: 9 th August 2016. 13

Digital Signal Processing Systems: Some Key Properties • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 4: 9 th August 2016. 14

Digital Signal Processing Systems: Some Key Properties • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 4: 9 th August 2016. 15

Impulse Response of Linear Shift Invariant Systems • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 4: 9 th August 2016. 16

Output of Linear Shift Invariant Systems as a Function of Impulse Response and the Input • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 4: 9 th August 2016. 17

Output of Linear Shift Invariant Systems as a Function of Impulse Response and the Input • Yash Vasavada DA-IICT. Autumn 2016 CT-321. Lecture 4: 9 th August 2016. 18