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Note for Digital Signal Processing - DSP By Kalpana

  • Digital Signal Processing - DSP
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Digital Signal Processing About the Tutorial Digital Signal Processing is an important branch of Electronics and Telecommunication engineering that deals with the improvisation of reliability and accuracy of the digital communication by employing multiple techniques. This tutorial explains the basic concepts of digital signal processing in a simple and easy-to-understand manner. Audience This tutorial is meant for the students of E&TC, Electrical and Computer Science engineering. In addition, it should be useful for any enthusiastic reader who would like to understand more about various signals, systems, and the methods to process a digital signal. Prerequisites Digital signal processing deals with the signal phenomenon. Along with it, in this tutorial, we have shown the filter design using the concept of DSP. This tutorial has a good balance between theory and mathematical rigor. Before proceeding with this tutorial, the readers are expected to have a basic understanding of discrete mathematical structures. Disclaimer & Copyright © Copyright 2015 by Tutorials Point (I) Pvt. Ltd. All the content and graphics published in this e-book are the property of Tutorials Point (I) Pvt. Ltd. The user of this e-book is prohibited to reuse, retain, copy, distribute, or republish any contents or a part of contents of this e-book in any manner without written consent of the publisher. We strive to update the contents of our website and tutorials as timely and as precisely as possible, however, the contents may contain inaccuracies or errors. Tutorials Point (I) Pvt. Ltd. provides no guarantee regarding the accuracy, timeliness, or completeness of our website or its contents including this tutorial. If you discover any errors on our website or in this tutorial, please notify us at contact@tutorialspoint.com i

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Digital Signal Processing Table of Contents About the Tutorial ............................................................................................................................................ i Audience ........................................................................................................................................................... i Prerequisites ..................................................................................................................................................... i Disclaimer & Copyright ..................................................................................................................................... i Table of Contents ............................................................................................................................................ ii PART 1 – INTRODUCTION TO SIGNALS .................................................................................... 1 1. Signals – Definition ................................................................................................................................... 2 Definition ......................................................................................................................................................... 2 Continuous Time Signals.................................................................................................................................. 2 Discrete Time signals ....................................................................................................................................... 2 2. Basic CT Signals ......................................................................................................................................... 4 Unit Impulse or Delta Function ....................................................................................................................... 4 Unit Step Signal ............................................................................................................................................... 4 Ramp Signal ..................................................................................................................................................... 5 Parabolic Signal................................................................................................................................................ 5 Signum Function .............................................................................................................................................. 6 Sinc Function ................................................................................................................................................... 6 Sinusoidal Signal .............................................................................................................................................. 7 Rectangular Function....................................................................................................................................... 7 Triangular Pulse Signal ..................................................................................................................................... 8 3. Basic DT Signals......................................................................................................................................... 9 Unit Impulse Sequence .................................................................................................................................... 9 Unit Step Signal ............................................................................................................................................... 9 Unit Ramp Function ......................................................................................................................................... 9 Parabolic Function ......................................................................................................................................... 10 Sinusoidal Signal ............................................................................................................................................ 10 4. Classification of CT Signals ...................................................................................................................... 12 Even and Odd Signals .................................................................................................................................... 12 Periodic and Non- Periodic Signals ................................................................................................................ 14 Energy and Power Signals .............................................................................................................................. 15 Solved Examples ............................................................................................................................................ 16 5. Classification of DT Signals ...................................................................................................................... 18 Even and Odd Signals .................................................................................................................................... 18 Periodic and Non-Periodic Signals ................................................................................................................. 19 Energy and Power Signals .............................................................................................................................. 19 6. Miscellaneous Signals ............................................................................................................................. 21 Conjugate Signals .......................................................................................................................................... 21 Conjugate Anti-Symmetric Signals................................................................................................................. 21 Half Wave Symmetric Signals ........................................................................................................................ 22 Orthogonal Signal .......................................................................................................................................... 22 ii

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Digital Signal Processing PART 2 – OPERATIONS ON SIGNALS ....................................................................................... 24 7. Shifting ................................................................................................................................................... 25 Time Shifting .................................................................................................................................................. 25 Amplitude Shifting ......................................................................................................................................... 26 8. Scaling .................................................................................................................................................... 28 Time Scaling ................................................................................................................................................... 28 Amplitude Scaling .......................................................................................................................................... 29 9. Reversal .................................................................................................................................................. 31 Time Reversal ................................................................................................................................................ 31 Amplitude Reversal ....................................................................................................................................... 31 10. Differentiation ........................................................................................................................................ 32 Differentiation ............................................................................................................................................... 32 11. Integration .............................................................................................................................................. 34 12. Convolution ............................................................................................................................................ 35 Steps for convolution .................................................................................................................................... 35 Properties of Convolution ............................................................................................................................. 36 Solved Examples ............................................................................................................................................ 37 PART 3: BASIC SYSTEM PROPERTIES ..................................................................................... 39 13. Static Systems ......................................................................................................................................... 40 14. Dynamic Systems .................................................................................................................................... 41 15. Causal Systems ....................................................................................................................................... 42 16. Non-Causal Systems ................................................................................................................................ 43 17. Anti-Causal Systems ................................................................................................................................ 44 18. Linear Systems ........................................................................................................................................ 45 19. Non-Linear Systems ................................................................................................................................ 47 20. Time-Invariant Systems........................................................................................................................... 48 21. Time-Variant Systems ............................................................................................................................. 49 22. Stable Systems ........................................................................................................................................ 50 23. Unstable Systems.................................................................................................................................... 51 24. Solved Examples ..................................................................................................................................... 52 iii

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Digital Signal Processing PART 4 – Z-TRANSFORM ............................................................................................................. 53 25. Introduction ............................................................................................................................................ 54 Region of Convergence .................................................................................................................................. 54 Signals and their ROC .................................................................................................................................... 55 26. Properties of Z-Transform ....................................................................................................................... 56 Linearity ......................................................................................................................................................... 56 Time Shifting .................................................................................................................................................. 56 Time Scaling ................................................................................................................................................... 57 Successive Differentiation ............................................................................................................................. 58 Convolution ................................................................................................................................................... 59 Initial Value Theorem .................................................................................................................................... 60 Final Value Theorem ...................................................................................................................................... 60 Differentiation in Frequency ......................................................................................................................... 61 Multiplication in Time ................................................................................................................................... 62 Conjugation in Time....................................................................................................................................... 62 27. Existence of Z-Transform ........................................................................................................................ 63 Z -Transform for Causal System ..................................................................................................................... 64 Z-transform for Anti-causal System ............................................................................................................... 64 28. Inverse Z-Transform ................................................................................................................................ 65 Methods to Find Inverse Z-Transform ........................................................................................................... 65 Long Division Method .................................................................................................................................... 65 Partial Fraction Expansion Method ............................................................................................................... 66 Residue or Contour Integral Method ............................................................................................................ 66 29. Solved Examples ..................................................................................................................................... 67 PART 5 – DISCRETE FOURIER TRANSFORM ......................................................................... 69 30. Introduction ............................................................................................................................................ 70 Frequency Domain Sampling ......................................................................................................................... 70 Properties of DFT ........................................................................................................................................... 71 31. Discrete Time Frequency Transform ....................................................................................................... 73 Properties of DTFT ......................................................................................................................................... 73 Twiddle Factor ............................................................................................................................................... 74 Linear Transformation ................................................................................................................................... 74 Circular Symmetry ......................................................................................................................................... 75 Some Other Important Properties................................................................................................................. 77 32. CIRCULAR CONVOLUTION ....................................................................................................................... 79 Methods of Circular Convolution .................................................................................................................. 79 33. DFT in Linear Filtering ............................................................................................................................. 81 34. Sectional Convolution ............................................................................................................................. 82 Overlap Save Method .................................................................................................................................... 82 Overlap Add Method ..................................................................................................................................... 83 iv

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