Table Of Content

(cid:0)(cid:2)(cid:3)(cid:2)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:2)(cid:3)(cid:9)(cid:5)(cid:6)(cid:7)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:15)(cid:2)(cid:9)(cid:3) A. Anand Kumar Principal K.L.U. College of Engineering K.L. University Vijayawada, Andhra Pradesh New Delhi-110001 2013 DIGITAL SIGNAL PROCESSING A. Anand Kumar © 2013 by PHI Learning Private Limited, New Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. ISBN-978-81-203-4620-8 The export rights of this book are vested solely with the publisher. Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus, New Delhi-110001 and Printed by Rajkamal Electric Press, Plot No. 2, Phase IV, HSIDC, Kundli-131028, Sonepat, Haryana. (cid:0)(cid:2)(cid:3)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:7)(cid:9) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:4)(cid:5)(cid:7)(cid:8)(cid:9)(cid:8)(cid:5)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:8)(cid:7)(cid:13) Professor, ECE Dept., A.U. College of Engineering, Visakhapatnam (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:4)(cid:11)(cid:4)(cid:5)(cid:14)(cid:13)(cid:15)(cid:16)(cid:8)(cid:17)(cid:18)(cid:7)(cid:19)(cid:20)(cid:21) Professor & HOD, EEE Dept., GITAM University, Visakhapatnam and (cid:0)(cid:2)(cid:3)(cid:4)(cid:5)(cid:3)(cid:10)(cid:11)(cid:12)(cid:12)(cid:3)(cid:10)(cid:13)(cid:9)(cid:14)(cid:11)(cid:8)(cid:9) (cid:10)(cid:3)(cid:4)(cid:5)(cid:11)(cid:7)(cid:13)(cid:5)(cid:6)(cid:22)(cid:15)(cid:10)(cid:7)(cid:18)(cid:5)(cid:11)(cid:8)(cid:19)(cid:21)(cid:8)(cid:15)(cid:8)(cid:7)(cid:8)(cid:21)(cid:8)(cid:15)(cid:8) Honorable Chancellor, K.L. University, Vijayawada, A.P. (cid:11)(cid:7)(cid:13)(cid:5)(cid:23)(cid:4)(cid:5)(cid:24)(cid:10)(cid:15)(cid:18)(cid:5)(cid:16)(cid:22)(cid:25)(cid:8)(cid:14)(cid:8)(cid:5)(cid:6)(cid:7)(cid:13)(cid:11)(cid:20)(cid:15)(cid:8) Chairman, Sasi Educational Society, Velivennu, W.G. District, A.P. (cid:0)(cid:2)(cid:3)(cid:4)(cid:5)(cid:3)(cid:4)(cid:6) Preface..................................................................................................................................................xv Symbols, Notations and Abbreviations .............................................................................................xix 1. Discrete-Time Signals and Systems............................................................................1–89 1.1 Introduction.........................................................................................................................1 1.2 Representation of Discrete-time Signals ...................................................................3 1.2.1 Graphical Representation................................................................................3 1.2.2 Functional Representation..............................................................................3 1.2.3 Tabular Representation...................................................................................4 1.2.4 Sequence Representation................................................................................4 1.3 Elementary Discrete-time Signals..............................................................................5 1.3.1 Unit Step Sequence ........................................................................................5 1.3.2 Unit Ramp Sequence......................................................................................6 1.3.3 Unit Parabolic Sequence................................................................................6 1.3.4 Unit Impulse Function or Unit Sample Sequence........................................7 1.3.5 Sinusoidal Sequence.......................................................................................8 1.3.6 Real Exponential Sequence............................................................................9 1.3.7 Complex Exponential Sequence ....................................................................9 1.4 Basic Operations on Sequences...............................................................................11 1.4.1 Time Shifting................................................................................................11 1.4.2 Time Reversal...............................................................................................12 1.4.3 Amplitude Scaling ........................................................................................14 1.4.4 Time Scaling.................................................................................................15 1.4.5 Signal Addition.............................................................................................16 1.4.6 Signal Multiplication....................................................................................16 1.5 Classification of Discrete-time Signals....................................................................17 1.5.1 Deterministic and Random Signals .............................................................18 1.5.2 Periodic and Non-periodic Sequences.........................................................18 v vi | Contents 1.5.3 Energy and Power Signals...........................................................................23 1.5.4 Causal and Non-causal Signals....................................................................28 1.5.5 Even and Odd Signals..................................................................................29 1.6 Classification of Discrete-time Systems..................................................................33 1.6.1 Static and Dynamic Systems .......................................................................34 1.6.2 Causal and Non-causal Systems..................................................................35 1.6.3 Linear and Non-linear Systems ...................................................................36 1.6.4 Shift-invariant and Shift-varying Systems ..................................................40 1.6.5 Stable and Unstable Systems.......................................................................49 1.6.6 FIR and IIR Systems....................................................................................64 1.6.7 Invertible and Non-invertible Systems........................................................65 1.7 Representation of an Arbitrary Sequence ...............................................................66 Short Questions with Answers............................................................................................68 Review Questions................................................................................................................73 Fill in the Blanks................................................................................................................73 Objective Type Questions...................................................................................................74 Problems..............................................................................................................................76 MATLAB Programs.............................................................................................................78 2. Discrete Convolution and Correlation...................................................................90–178 2.1 Introduction...............................................................................................................90 2.2 Impulse Response and Convolution Sum ...............................................................90 2.3 Analytical Evaluation of Discrete Convolution......................................................92 2.4 Convolution of Finite Sequences...........................................................................101 2.5 Methods to Compute the Convolution Sum of Two Sequences x(n) and h(n)....101 2.5.1 Method 1 Linear Convolution Using Graphical Method.........................101 2.5.2 Method 2 Linear Convolution Using Tabular Array ...............................101 2.5.3 Method 3 Linear Convolution Using Tabular Method ............................102 2.5.4 Method 4 Linear Convolution Using Matrices.........................................103 2.5.5 Method 5 Linear Convolution Using the Sum-by Column Method.......103 2.5.6 Method 6 Linear Convolution Using the Flip, Shift, Multiply, and Sum Method........................................................................................103 2.6 Deconvolution .........................................................................................................122 2.6.1 Deconvolution using Z-transform..............................................................122 2.6.2 Deconvolution by Recursion......................................................................123 2.6.3 Deconvolution using Tabular Method.......................................................125 2.7 Interconnection of LTI Systems..............................................................................126 2.7.1 Parallel Connection of Systems.................................................................126 2.7.2 Cascade Connection of Systems................................................................127 2.8 Circular Shift and Circular Symmetry ..................................................................130 2.9 Periodic or Circular Convolution...........................................................................134 2.10 Methods of Performing Periodic or Circular Convolution ..................................134 2.10.1 Method 1 Graphical Method (Concentric Circle Method)......................135 2.10.2 Method 2 Circular Convolution Using Tabular Array ............................135 2.10.3 Method 3 Circular Convolution Using Matrices......................................136 Contents | vii 2.11 Linear (Regular) Convolution from Periodic Convolution ..................................148 2.12 Periodic Convolution from Linear Convolution...................................................150 2.13 Periodic Extension of Non-periodic Signals.........................................................152 2.14 System Response to Periodic Inputs .....................................................................153 2.15 Discrete Correlation................................................................................................156 2.15.1 Cross Correlation........................................................................................156 2.15.2 Autocorrelation ...........................................................................................157 2.15.3 Computation of Correlation.......................................................................158 2.15.4 Correlation of Power and Periodic Signals..............................................159 2.16 Periodic Discrete Correlation.................................................................................162 Short Questions with Answers..........................................................................................164 Review Questions..............................................................................................................167 Fill in the Blanks..............................................................................................................168 Objective Type Questions.................................................................................................169 Problems............................................................................................................................171 MATLAB Programs...........................................................................................................174 3. Z-Transforms ...........................................................................................................179–276 3.1 Introduction.............................................................................................................179 3.1.1 Advantages of Z-transform........................................................................180 3.2 Relation between Discrete-time Fourier Transform (DTFT) and Z-transform......180 3.3 Z-Transform and ROC of Finite Duration Sequences..........................................186 3.3.1 Right-sided Sequence.................................................................................186 3.3.2 Left-sided Sequence...................................................................................187 3.3.3 Two-sided Sequence...................................................................................187 3.4 Properties of ROC ..................................................................................................190 3.5 Properties of Z-transform.......................................................................................191 3.5.1 Linearity Property.......................................................................................191 3.5.2 Time Shifting Property ..............................................................................191 3.5.3 Multiplication by an Exponential Sequence Property..............................193 3.5.4 Time Reversal Property .............................................................................193 3.5.5 Time Expansion Property ..........................................................................194 3.5.6 Multiplication by n or Differentiation in z-domain Property..................195 3.5.7 Convolution Property.................................................................................196 3.5.8 The Multiplication Property or Complex Convolution Property.............197 3.5.9 Correlation Property...................................................................................197 3.5.10 Parseval’s Theorem or Relation or Property............................................198 3.5.11 Initial Value Theorem................................................................................199 3.5.12 Final Value Theorem .................................................................................200 3.6 Inverse Z-transform..................................................................................................211 3.6.1 Long Division Method...............................................................................212 3.6.2 Partial Fraction Expansion Method...........................................................221 3.6.3 Residue Method..........................................................................................226 3.6.4 Convolution Method...................................................................................229 viii | Contents 3.7 Transform Analysis of LTI Systems.....................................................................231 3.7.1 System Function and Impulse Response...................................................231 3.7.2 Relationship between Transfer Function and Difference Equation.........232 3.8 Stability and Causality............................................................................................232 3.9 Solution of Difference Equations Using Z-transforms.........................................248 3.10 Deconvolution Using Z-transform .........................................................................261 3.11 Relation between s–Plane and z–Plane.................................................................262 Short Questions with Answers..........................................................................................263 Review Questions..............................................................................................................265 Fill in the Blanks..............................................................................................................265 Objective Type Questions.................................................................................................266 Problems............................................................................................................................268 MATLAB Programs...........................................................................................................272 4. System Realization..................................................................................................277–357 4.1 Introduction.............................................................................................................277 4.2 Realization of Discrete-time Systems....................................................................277 4.3 Structures for Realization of IIR Systems............................................................280 4.3.1 Direct Form-I Structure..............................................................................281 4.3.2 Direct Form-II Structure............................................................................283 4.3.3 Transposed Form Structure Realization of IIR System...........................287 4.3.4 Cascade Form Realization .........................................................................288 4.3.5 Parallel Form Realization ..........................................................................290 4.3.6 Lattice Structure Realization of IIR Systems...........................................291 4.3.7 Ladder Structure Realization of IIR Systems...........................................294 4.4 Structures for Realization of FIR Systems ...........................................................324 4.4.1 Direct Form Realization of FIR System...................................................326 4.4.2 Transposed Form Structure Realization of FIR System..........................327 4.4.3 Cascade Form Structure Realization of FIR System...............................328 4.4.4 Lattice Structure Realization of FIR Systems..........................................330 4.4.5 Linear Phase Realizations..........................................................................333 Short Questions with Answers..........................................................................................342 Review Questions..............................................................................................................345 Fill in the Blanks..............................................................................................................345 Objective Type Questions.................................................................................................346 Problems............................................................................................................................347 MATLAB Programs...........................................................................................................350 5. Discrete-time Fourier Transform .........................................................................358–411 5.1 Introduction.............................................................................................................358 5.2 Discrete-time Fourier Transform (DTFT) .............................................................358 5.3 Existence of DTFT.................................................................................................359 5.4 Relation between Z-transform and Fourier Transform.........................................359 5.5 Inverse Discrete-time Fourier Transform..............................................................368 Contents | ix 5.6 Properties of Discrete-time Fourier Transform.....................................................370 5.6.1 Linearity Property.......................................................................................370 5.6.2 Periodicity Property....................................................................................371 5.6.3 Time Shifting Property ..............................................................................371 5.6.4 Frequency Shifting Property......................................................................371 5.6.5 Time Reversal Property .............................................................................372 5.6.6 Differentiation in the Frequency Domain Property .................................372 5.6.7 Time Convolution Property .......................................................................373 5.6.8 Frequency Convolution Property...............................................................373 5.6.9 The Correlation Theorem...........................................................................374 5.6.10 The Modulation Theorem..........................................................................374 5.6.11 Parseval’s Theorem....................................................................................374 5.6.12 Symmetry Properties..................................................................................375 5.7 Transfer Function....................................................................................................383 5.8 Frequency Response of Discrete-time Systems....................................................384 Short Questions with Answers..........................................................................................393 Review Questions..............................................................................................................395 Fill in the Blanks..............................................................................................................395 Objective Type Questions.................................................................................................396 Problems............................................................................................................................398 MATLAB Programs...........................................................................................................399 6. Discrete Fourier Series (DFS) and Discrete Fourier Transform (DFT) ........412–478 6.1 Introduction.............................................................................................................412 6.2 Discrete Fourier Series...........................................................................................413 6.2.1 Exponential Form of Discrete Fourier Series...........................................413 6.2.2 Trigonometric Form of Discrete Fourier Series.......................................414 6.2.3 Relationships between the Exponential and Trigonometric Forms of Discrete Fourier Series...............................................................................415 6.3 Properties of DFS...................................................................................................418 6.3.1 Linearity......................................................................................................418 6.3.2 Time Shifting..............................................................................................418 6.3.3 Symmetry Property.....................................................................................418 6.3.4 Periodic Convolution..................................................................................419 6.4 Relation between DFT and Z-transform ...............................................................419 6.5 Comparison between DTFT and DFT...................................................................426 6.6 A Slightly Faster Method for Computing DFT Values.......................................427 6.7 Matrix Formulation of the DFT and IDFT...........................................................428 6.8 The IDFT from the Matrix Form..........................................................................428 6.9 Using the DFT to Find the IDFT..........................................................................429 6.10 Properties of DFT...................................................................................................431 6.10.1 Periodicity...................................................................................................432 6.10.2 Linearity......................................................................................................432 6.10.3 DFT of Even and Odd Sequences............................................................432 x | Contents 6.10.4 Time Reversal of the Sequence ................................................................433 6.10.5 Circular Frequency Shift............................................................................433 6.10.6 Complex Conjugate Property.....................................................................434 6.10.7 DFT of Delayed Sequence (Circular time shift of a sequence) .............434 6.10.8 DFT of Real Valued Sequence .................................................................435 6.10.9 Multiplication of Two Sequences..............................................................435 6.10.10 Circular Convolution of Two Sequences..................................................436 6.10.11 Parseval’s Theorem....................................................................................437 6.10.12 Circular Correlation....................................................................................437 6.11 Methods of Performing Linear Convolution.........................................................444 6.11.1 Linear Convolution Using DFT ................................................................444 6.12 Methods of Performing Circular Convolution......................................................449 6.12.1 Circular Convolution Using DFT and IDFT............................................449 6.13 Convolution of Long Sequences (Sectioned Convolutions)................................452 6.13.1 Overlap-add Method...................................................................................453 6.13.2 Overlap-save Method .................................................................................453 Short Questions with Answers..........................................................................................460 Review Questions..............................................................................................................462 Fill in the Blanks..............................................................................................................463 Objective Type Questions.................................................................................................464 Problems............................................................................................................................466 MATLAB Programs...........................................................................................................468 7. Fast Fourier Transform.........................................................................................479–547 7.1 Introduction ..............................................................................................................479 7.2 Fast Fourier Transform............................................................................................479 7.3 Decimation in Time (DIT) Radix-2 FFT ...............................................................481 7.4 The 8-point DFT Using Radix-2 DIT FFT............................................................485 7.4.1 Butterfly Diagram.......................................................................................490 7.5 Decimation in Frequency (DIF) Radix-2 FFT.......................................................490 7.6 The 8-point DFT Using Radix-2 DIF FFT............................................................494 7.6.1 Computation of IDFT through FFT..........................................................499 7.7 FFT Algorithms for N a Composite Number ........................................................522 7.7.1 Radix-3 FFT ...............................................................................................523 7.7.2 Radix-4 FFT ...............................................................................................523 Short Questions with Answers..........................................................................................534 Review Questions..............................................................................................................540 Fill in the Blanks..............................................................................................................540 Objective Type Questions.................................................................................................541 Problems............................................................................................................................542 MATLAB Programs...........................................................................................................544 Contents | xi 8. Infinite-duration Impulse Response (IIR) Filters...............................................548–650 8.1 Introduction.............................................................................................................548 8.2 Requirements for Transformation..........................................................................548 8.3 Design of IIR Filter by Approximation of Derivatives .......................................551 8.4 Design of IIR Filter by Impulse Invariant Transformation .................................556 8.5 Design of IIR Filter by the Bilinear Transformation Method.............................565 8.6 Specifications of the Low-pass Filter....................................................................575 8.7 Design of Low-pass Digital Butterworth Filter....................................................576 8.8 Design of Low-pass Chebyshev Filter..................................................................599 8.9 Inverse Chebyshev Filters......................................................................................618 8.10 Elliptic Filters..........................................................................................................620 8.11 Frequency Transformation......................................................................................620 8.11.1 Analog Frequency Transformation............................................................621 8.11.2 Digital Frequency Transformation.............................................................623 Short Questions with Answers..........................................................................................624 Review Questions..............................................................................................................632 Fill in the Blanks..............................................................................................................633 Objective Type Questions.................................................................................................634 Problems............................................................................................................................637 MATLAB Programs...........................................................................................................639 9. FIR Filters................................................................................................................651–745 9.1 Introduction.............................................................................................................651 9.2 Characteristics of FIR Filters with Linear Phase .................................................652 9.3 Frequency Response of Linear Phase FIR Filters................................................657 9.3.1 Frequency Response of Linear Phase FIR Filter when Impulse Response is Symmetrical and N is Odd...................................................657 9.3.2 Frequency Response of Linear Phase FIR Filter when Impulse Response is Symmetrical and N is Even..................................................659 9.3.3 Frequency Response of Linear Phase FIR Filter when Impulse Response is Antisymmetric and N is Odd ...............................................662 9.3.4 Frequency Response of Linear Phase FIR Filter when Impulse Response is Antisymmetric and N is Even..............................................664 9.4 Design Techniques for FIR Filters........................................................................667 9.5 Fourier Series Method of Design of FIR Filters..................................................668 9.6 Design of FIR Filters Using Windows .................................................................674 9.6.1 Rectangular Window..................................................................................674 9.6.2 Triangular or Bartlett Window..................................................................678 9.6.3 Raised Cosine Window..............................................................................678 9.6.4 Hanning Window........................................................................................678 9.6.5 Hamming Window .....................................................................................679 9.6.6 Blackman Window.....................................................................................680 9.6.7 Kaiser Window...........................................................................................703 9.7 Design of FIR Filters by Frequency Sampling Technique..................................713

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Digital Signal Processing PDF

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(cid:0)(cid:2)(cid:3)(cid:2)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:2)(cid:3)(cid:9)(cid:5)(cid:6)(cid:7)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:15)(cid:2)(cid:9)(cid:3) A. Anand Kumar Principal K.L.U. College of Engineering K.L. University Vijayawada, Andhra Pradesh New Delhi-110001 2013 DIGITAL SIGNAL PROCESSING A. Anand Kumar © 2013 by PHI Learning Private Limited, New Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. ISBN-978-81-203-4620-8 The export rights of this book are vested solely with the publisher. Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus, New Delhi-110001 and Printed by Rajkamal Electric Press, Plot No. 2, Phase IV, HSIDC, Kundli-131028, Sonepat, Haryana. (cid:0)(cid:2)(cid:3)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:7)(cid:9) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:4)(cid:5)(cid:7)(cid:8)(cid:9)(cid:8)(cid:5)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:8)(cid:7)(cid:13) Professor, ECE Dept., A.U. College of Engineering, Visakhapatnam (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:4)(cid:11)(cid:4)(cid:5)(cid:14)(cid:13)(cid:15)(cid:16)(cid:8)(cid:17)(cid:18)(cid:7)(cid:19)(cid:20)(cid:21) Professor & HOD, EEE Dept., GITAM University, Visakhapatnam and (cid:0)(cid:2)(cid:3)(cid:4)(cid:5)(cid:3)(cid:10)(cid:11)(cid:12)(cid:12)(cid:3)(cid:10)(cid:13)(cid:9)(cid:14)(cid:11)(cid:8)(cid:9) (cid:10)(cid:3)(cid:4)(cid:5)(cid:11)(cid:7)(cid:13)(cid:5)(cid:6)(cid:22)(cid:15)(cid:10)(cid:7)(cid:18)(cid:5)(cid:11)(cid:8)(cid:19)(cid:21)(cid:8)(cid:15)(cid:8)(cid:7)(cid:8)(cid:21)(cid:8)(cid:15)(cid:8) Honorable Chancellor, K.L. University, Vijayawada, A.P. (cid:11)(cid:7)(cid:13)(cid:5)(cid:23)(cid:4)(cid:5)(cid:24)(cid:10)(cid:15)(cid:18)(cid:5)(cid:16)(cid:22)(cid:25)(cid:8)(cid:14)(cid:8)(cid:5)(cid:6)(cid:7)(cid:13)(cid:11)(cid:20)(cid:15)(cid:8) Chairman, Sasi Educational Society, Velivennu, W.G. District, A.P. (cid:0)(cid:2)(cid:3)(cid:4)(cid:5)(cid:3)(cid:4)(cid:6) Preface..................................................................................................................................................xv Symbols, Notations and Abbreviations .............................................................................................xix 1. Discrete-Time Signals and Systems............................................................................1–89 1.1 Introduction.........................................................................................................................1 1.2 Representation of Discrete-time Signals ...................................................................3 1.2.1 Graphical Representation................................................................................3 1.2.2 Functional Representation..............................................................................3 1.2.3 Tabular Representation...................................................................................4 1.2.4 Sequence Representation................................................................................4 1.3 Elementary Discrete-time Signals..............................................................................5 1.3.1 Unit Step Sequence ........................................................................................5 1.3.2 Unit Ramp Sequence......................................................................................6 1.3.3 Unit Parabolic Sequence................................................................................6 1.3.4 Unit Impulse Function or Unit Sample Sequence........................................7 1.3.5 Sinusoidal Sequence.......................................................................................8 1.3.6 Real Exponential Sequence............................................................................9 1.3.7 Complex Exponential Sequence ....................................................................9 1.4 Basic Operations on Sequences...............................................................................11 1.4.1 Time Shifting................................................................................................11 1.4.2 Time Reversal...............................................................................................12 1.4.3 Amplitude Scaling ........................................................................................14 1.4.4 Time Scaling.................................................................................................15 1.4.5 Signal Addition.............................................................................................16 1.4.6 Signal Multiplication....................................................................................16 1.5 Classification of Discrete-time Signals....................................................................17 1.5.1 Deterministic and Random Signals .............................................................18 1.5.2 Periodic and Non-periodic Sequences.........................................................18 v vi | Contents 1.5.3 Energy and Power Signals...........................................................................23 1.5.4 Causal and Non-causal Signals....................................................................28 1.5.5 Even and Odd Signals..................................................................................29 1.6 Classification of Discrete-time Systems..................................................................33 1.6.1 Static and Dynamic Systems .......................................................................34 1.6.2 Causal and Non-causal Systems..................................................................35 1.6.3 Linear and Non-linear Systems ...................................................................36 1.6.4 Shift-invariant and Shift-varying Systems ..................................................40 1.6.5 Stable and Unstable Systems.......................................................................49 1.6.6 FIR and IIR Systems....................................................................................64 1.6.7 Invertible and Non-invertible Systems........................................................65 1.7 Representation of an Arbitrary Sequence ...............................................................66 Short Questions with Answers............................................................................................68 Review Questions................................................................................................................73 Fill in the Blanks................................................................................................................73 Objective Type Questions...................................................................................................74 Problems..............................................................................................................................76 MATLAB Programs.............................................................................................................78 2. Discrete Convolution and Correlation...................................................................90–178 2.1 Introduction...............................................................................................................90 2.2 Impulse Response and Convolution Sum ...............................................................90 2.3 Analytical Evaluation of Discrete Convolution......................................................92 2.4 Convolution of Finite Sequences...........................................................................101 2.5 Methods to Compute the Convolution Sum of Two Sequences x(n) and h(n)....101 2.5.1 Method 1 Linear Convolution Using Graphical Method.........................101 2.5.2 Method 2 Linear Convolution Using Tabular Array ...............................101 2.5.3 Method 3 Linear Convolution Using Tabular Method ............................102 2.5.4 Method 4 Linear Convolution Using Matrices.........................................103 2.5.5 Method 5 Linear Convolution Using the Sum-by Column Method.......103 2.5.6 Method 6 Linear Convolution Using the Flip, Shift, Multiply, and Sum Method........................................................................................103 2.6 Deconvolution .........................................................................................................122 2.6.1 Deconvolution using Z-transform..............................................................122 2.6.2 Deconvolution by Recursion......................................................................123 2.6.3 Deconvolution using Tabular Method.......................................................125 2.7 Interconnection of LTI Systems..............................................................................126 2.7.1 Parallel Connection of Systems.................................................................126 2.7.2 Cascade Connection of Systems................................................................127 2.8 Circular Shift and Circular Symmetry ..................................................................130 2.9 Periodic or Circular Convolution...........................................................................134 2.10 Methods of Performing Periodic or Circular Convolution ..................................134 2.10.1 Method 1 Graphical Method (Concentric Circle Method)......................135 2.10.2 Method 2 Circular Convolution Using Tabular Array ............................135 2.10.3 Method 3 Circular Convolution Using Matrices......................................136 Contents | vii 2.11 Linear (Regular) Convolution from Periodic Convolution ..................................148 2.12 Periodic Convolution from Linear Convolution...................................................150 2.13 Periodic Extension of Non-periodic Signals.........................................................152 2.14 System Response to Periodic Inputs .....................................................................153 2.15 Discrete Correlation................................................................................................156 2.15.1 Cross Correlation........................................................................................156 2.15.2 Autocorrelation ...........................................................................................157 2.15.3 Computation of Correlation.......................................................................158 2.15.4 Correlation of Power and Periodic Signals..............................................159 2.16 Periodic Discrete Correlation.................................................................................162 Short Questions with Answers..........................................................................................164 Review Questions..............................................................................................................167 Fill in the Blanks..............................................................................................................168 Objective Type Questions.................................................................................................169 Problems............................................................................................................................171 MATLAB Programs...........................................................................................................174 3. Z-Transforms ...........................................................................................................179–276 3.1 Introduction.............................................................................................................179 3.1.1 Advantages of Z-transform........................................................................180 3.2 Relation between Discrete-time Fourier Transform (DTFT) and Z-transform......180 3.3 Z-Transform and ROC of Finite Duration Sequences..........................................186 3.3.1 Right-sided Sequence.................................................................................186 3.3.2 Left-sided Sequence...................................................................................187 3.3.3 Two-sided Sequence...................................................................................187 3.4 Properties of ROC ..................................................................................................190 3.5 Properties of Z-transform.......................................................................................191 3.5.1 Linearity Property.......................................................................................191 3.5.2 Time Shifting Property ..............................................................................191 3.5.3 Multiplication by an Exponential Sequence Property..............................193 3.5.4 Time Reversal Property .............................................................................193 3.5.5 Time Expansion Property ..........................................................................194 3.5.6 Multiplication by n or Differentiation in z-domain Property..................195 3.5.7 Convolution Property.................................................................................196 3.5.8 The Multiplication Property or Complex Convolution Property.............197 3.5.9 Correlation Property...................................................................................197 3.5.10 Parseval’s Theorem or Relation or Property............................................198 3.5.11 Initial Value Theorem................................................................................199 3.5.12 Final Value Theorem .................................................................................200 3.6 Inverse Z-transform..................................................................................................211 3.6.1 Long Division Method...............................................................................212 3.6.2 Partial Fraction Expansion Method...........................................................221 3.6.3 Residue Method..........................................................................................226 3.6.4 Convolution Method...................................................................................229 viii | Contents 3.7 Transform Analysis of LTI Systems.....................................................................231 3.7.1 System Function and Impulse Response...................................................231 3.7.2 Relationship between Transfer Function and Difference Equation.........232 3.8 Stability and Causality............................................................................................232 3.9 Solution of Difference Equations Using Z-transforms.........................................248 3.10 Deconvolution Using Z-transform .........................................................................261 3.11 Relation between s–Plane and z–Plane.................................................................262 Short Questions with Answers..........................................................................................263 Review Questions..............................................................................................................265 Fill in the Blanks..............................................................................................................265 Objective Type Questions.................................................................................................266 Problems............................................................................................................................268 MATLAB Programs...........................................................................................................272 4. System Realization..................................................................................................277–357 4.1 Introduction.............................................................................................................277 4.2 Realization of Discrete-time Systems....................................................................277 4.3 Structures for Realization of IIR Systems............................................................280 4.3.1 Direct Form-I Structure..............................................................................281 4.3.2 Direct Form-II Structure............................................................................283 4.3.3 Transposed Form Structure Realization of IIR System...........................287 4.3.4 Cascade Form Realization .........................................................................288 4.3.5 Parallel Form Realization ..........................................................................290 4.3.6 Lattice Structure Realization of IIR Systems...........................................291 4.3.7 Ladder Structure Realization of IIR Systems...........................................294 4.4 Structures for Realization of FIR Systems ...........................................................324 4.4.1 Direct Form Realization of FIR System...................................................326 4.4.2 Transposed Form Structure Realization of FIR System..........................327 4.4.3 Cascade Form Structure Realization of FIR System...............................328 4.4.4 Lattice Structure Realization of FIR Systems..........................................330 4.4.5 Linear Phase Realizations..........................................................................333 Short Questions with Answers..........................................................................................342 Review Questions..............................................................................................................345 Fill in the Blanks..............................................................................................................345 Objective Type Questions.................................................................................................346 Problems............................................................................................................................347 MATLAB Programs...........................................................................................................350 5. Discrete-time Fourier Transform .........................................................................358–411 5.1 Introduction.............................................................................................................358 5.2 Discrete-time Fourier Transform (DTFT) .............................................................358 5.3 Existence of DTFT.................................................................................................359 5.4 Relation between Z-transform and Fourier Transform.........................................359 5.5 Inverse Discrete-time Fourier Transform..............................................................368 Contents | ix 5.6 Properties of Discrete-time Fourier Transform.....................................................370 5.6.1 Linearity Property.......................................................................................370 5.6.2 Periodicity Property....................................................................................371 5.6.3 Time Shifting Property ..............................................................................371 5.6.4 Frequency Shifting Property......................................................................371 5.6.5 Time Reversal Property .............................................................................372 5.6.6 Differentiation in the Frequency Domain Property .................................372 5.6.7 Time Convolution Property .......................................................................373 5.6.8 Frequency Convolution Property...............................................................373 5.6.9 The Correlation Theorem...........................................................................374 5.6.10 The Modulation Theorem..........................................................................374 5.6.11 Parseval’s Theorem....................................................................................374 5.6.12 Symmetry Properties..................................................................................375 5.7 Transfer Function....................................................................................................383 5.8 Frequency Response of Discrete-time Systems....................................................384 Short Questions with Answers..........................................................................................393 Review Questions..............................................................................................................395 Fill in the Blanks..............................................................................................................395 Objective Type Questions.................................................................................................396 Problems............................................................................................................................398 MATLAB Programs...........................................................................................................399 6. Discrete Fourier Series (DFS) and Discrete Fourier Transform (DFT) ........412–478 6.1 Introduction.............................................................................................................412 6.2 Discrete Fourier Series...........................................................................................413 6.2.1 Exponential Form of Discrete Fourier Series...........................................413 6.2.2 Trigonometric Form of Discrete Fourier Series.......................................414 6.2.3 Relationships between the Exponential and Trigonometric Forms of Discrete Fourier Series...............................................................................415 6.3 Properties of DFS...................................................................................................418 6.3.1 Linearity......................................................................................................418 6.3.2 Time Shifting..............................................................................................418 6.3.3 Symmetry Property.....................................................................................418 6.3.4 Periodic Convolution..................................................................................419 6.4 Relation between DFT and Z-transform ...............................................................419 6.5 Comparison between DTFT and DFT...................................................................426 6.6 A Slightly Faster Method for Computing DFT Values.......................................427 6.7 Matrix Formulation of the DFT and IDFT...........................................................428 6.8 The IDFT from the Matrix Form..........................................................................428 6.9 Using the DFT to Find the IDFT..........................................................................429 6.10 Properties of DFT...................................................................................................431 6.10.1 Periodicity...................................................................................................432 6.10.2 Linearity......................................................................................................432 6.10.3 DFT of Even and Odd Sequences............................................................432 x | Contents 6.10.4 Time Reversal of the Sequence ................................................................433 6.10.5 Circular Frequency Shift............................................................................433 6.10.6 Complex Conjugate Property.....................................................................434 6.10.7 DFT of Delayed Sequence (Circular time shift of a sequence) .............434 6.10.8 DFT of Real Valued Sequence .................................................................435 6.10.9 Multiplication of Two Sequences..............................................................435 6.10.10 Circular Convolution of Two Sequences..................................................436 6.10.11 Parseval’s Theorem....................................................................................437 6.10.12 Circular Correlation....................................................................................437 6.11 Methods of Performing Linear Convolution.........................................................444 6.11.1 Linear Convolution Using DFT ................................................................444 6.12 Methods of Performing Circular Convolution......................................................449 6.12.1 Circular Convolution Using DFT and IDFT............................................449 6.13 Convolution of Long Sequences (Sectioned Convolutions)................................452 6.13.1 Overlap-add Method...................................................................................453 6.13.2 Overlap-save Method .................................................................................453 Short Questions with Answers..........................................................................................460 Review Questions..............................................................................................................462 Fill in the Blanks..............................................................................................................463 Objective Type Questions.................................................................................................464 Problems............................................................................................................................466 MATLAB Programs...........................................................................................................468 7. Fast Fourier Transform.........................................................................................479–547 7.1 Introduction ..............................................................................................................479 7.2 Fast Fourier Transform............................................................................................479 7.3 Decimation in Time (DIT) Radix-2 FFT ...............................................................481 7.4 The 8-point DFT Using Radix-2 DIT FFT............................................................485 7.4.1 Butterfly Diagram.......................................................................................490 7.5 Decimation in Frequency (DIF) Radix-2 FFT.......................................................490 7.6 The 8-point DFT Using Radix-2 DIF FFT............................................................494 7.6.1 Computation of IDFT through FFT..........................................................499 7.7 FFT Algorithms for N a Composite Number ........................................................522 7.7.1 Radix-3 FFT ...............................................................................................523 7.7.2 Radix-4 FFT ...............................................................................................523 Short Questions with Answers..........................................................................................534 Review Questions..............................................................................................................540 Fill in the Blanks..............................................................................................................540 Objective Type Questions.................................................................................................541 Problems............................................................................................................................542 MATLAB Programs...........................................................................................................544 Contents | xi 8. Infinite-duration Impulse Response (IIR) Filters...............................................548–650 8.1 Introduction.............................................................................................................548 8.2 Requirements for Transformation..........................................................................548 8.3 Design of IIR Filter by Approximation of Derivatives .......................................551 8.4 Design of IIR Filter by Impulse Invariant Transformation .................................556 8.5 Design of IIR Filter by the Bilinear Transformation Method.............................565 8.6 Specifications of the Low-pass Filter....................................................................575 8.7 Design of Low-pass Digital Butterworth Filter....................................................576 8.8 Design of Low-pass Chebyshev Filter..................................................................599 8.9 Inverse Chebyshev Filters......................................................................................618 8.10 Elliptic Filters..........................................................................................................620 8.11 Frequency Transformation......................................................................................620 8.11.1 Analog Frequency Transformation............................................................621 8.11.2 Digital Frequency Transformation.............................................................623 Short Questions with Answers..........................................................................................624 Review Questions..............................................................................................................632 Fill in the Blanks..............................................................................................................633 Objective Type Questions.................................................................................................634 Problems............................................................................................................................637 MATLAB Programs...........................................................................................................639 9. FIR Filters................................................................................................................651–745 9.1 Introduction.............................................................................................................651 9.2 Characteristics of FIR Filters with Linear Phase .................................................652 9.3 Frequency Response of Linear Phase FIR Filters................................................657 9.3.1 Frequency Response of Linear Phase FIR Filter when Impulse Response is Symmetrical and N is Odd...................................................657 9.3.2 Frequency Response of Linear Phase FIR Filter when Impulse Response is Symmetrical and N is Even..................................................659 9.3.3 Frequency Response of Linear Phase FIR Filter when Impulse Response is Antisymmetric and N is Odd ...............................................662 9.3.4 Frequency Response of Linear Phase FIR Filter when Impulse Response is Antisymmetric and N is Even..............................................664 9.4 Design Techniques for FIR Filters........................................................................667 9.5 Fourier Series Method of Design of FIR Filters..................................................668 9.6 Design of FIR Filters Using Windows .................................................................674 9.6.1 Rectangular Window..................................................................................674 9.6.2 Triangular or Bartlett Window..................................................................678 9.6.3 Raised Cosine Window..............................................................................678 9.6.4 Hanning Window........................................................................................678 9.6.5 Hamming Window .....................................................................................679 9.6.6 Blackman Window.....................................................................................680 9.6.7 Kaiser Window...........................................................................................703 9.7 Design of FIR Filters by Frequency Sampling Technique..................................713

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