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Linear Convolution Using the DFT 183 Solution Clearly, the linear convolution x 3(n) isstill the same. x 3(n) ={1, 1, −1, −2, −1, 1, 1} When N =6,weobtain a 6-point sequence. x 4(n) =x 1(n) 6○ x 2(n) ={2, 1, −1, −2, −1, 1} Therefore e(n) ={2, 1, −1, −2, −1, 1}−{1, 1, −1, −2, −1, 1} , 0 ≤ n ≤ 5 = {1, 0, 0, 0, 0, 0} = x 3(n +6) as expected. When N =5,weobtain a 5-point sequence, x 4(n) =x 1(n) 5○ x 2(n) ={2, 2, −1, −2, −1} and e(n) ={2, 2, −1, −2, −1}−{1, 1, −1, −2, −1} , 0 ≤ n ≤ 4 = {1, 1, 0, 0, 0} = x 3(n +5) Finally, when N =4,weobtain a 4-point sequence, x 4(n) =x 1(n) 4○ x 2(n) ={0, 2, 0, −2} and e(n) ={0, 2, 0, −2}−{1, 1, −1, −2} , 0 ≤ n ≤ 3 = {−1, 1, 1, 0} = x 3(n +4) The last case of N =4also provides the following useful observation. Observation: When N = max(N 1,N 2)ischosen for circular convolution, then the first (M − 1) samples are in error (i.e., different from the linear convolution), where M = min(N 1,N 2). This result is useful in implementing long convolutions in the form of block processing. □ 5.5.2 BLOCK CONVOLUTIONS When we want to filter an input sequence that is being received continuously, such as a speech signal from a microphone, then for practical purposes we can think of this sequence as an infinite-length sequence. If we want to implement this filtering operation as an FIR filter in which the linear convolution is computed using the DFT, then we experience some practical problems. We will have to compute a large DFT, which is generally impractical. Furthermore, output samples are not available until all input samples are processed. This introduces an unacceptably large Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
184 Chapter 5 THE DISCRETE FOURIER TRANSFORM amount of delay. Therefore we have to segment the infinite-length input sequence into smaller sections (or blocks), process each section using the DFT, and finally assemble the output sequence from the outputs of each section. This procedure is called a block convolution (or block processing) operation. Let us assume that the sequence x(n) issectioned into N-point sequences and that the impulse response of the filter is an M-point sequence, where M
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Digital Signal Processing Using MAT
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Contents PREFACE xi 1 INTRODUCTION
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CONTENTS vii 6 IMPLEMENTATION OF DI
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CONTENTS ix 11.3 Suppression of Nar
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Preface From the beginning of the 1
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PREFACE xiii implementation. The di
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PREFACE xv ACKNOWLEDGMENTS We are i
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CHAPTER 1 Introduction During the p
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Overview of Digital Signal Processi
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A Brief Introduction to MATLAB 5 It
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A Brief Introduction to MATLAB 7 Va
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A Brief Introduction to MATLAB 9 us
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A Brief Introduction to MATLAB 11 o
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A Brief Introduction to MATLAB 13 o
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A Brief Introduction to MATLAB 15 T
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Applications of Digital Signal Proc
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Applications of Digital Signal Proc
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Brief Overview of the Book 21 In al
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Discrete-time Signals 23 In MATLAB
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Discrete-time Signals 25 For exampl
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Discrete-time Signals 27 n = min(mi
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Discrete-time Signals 29 Solution a
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Discrete-time Signals 31 Sequence i
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Discrete-time Signals 33 The quanti
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Discrete-time Signals 35 Rectangula
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Discrete Systems 37 In DSP we will
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Discrete Systems 39 every input seq
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Convolution 41 2 Input Sequence 1.5
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Convolution 43 x(k) and h(k) x(k) a
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Convolution 45 Hence y(n) ={6, 31,
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Difference Equations 47 Note that t
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Difference Equations 49 Solution Fr
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Difference Equations 51 8 Output Se
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Problems 53 IIR filter If the impul
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Problems 55 1. Modify the evenodd f
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Problems 57 2. x(n) ={1, 1, 0, 1, 1
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CHAPTER 3 The Discrete-time Fourier
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The Discrete-time Fourier Transform
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The Properties of the DTFT 67 TABLE
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The Properties of the DTFT 69 8. Mu
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The Properties of the DTFT 71 60 Ma
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The Properties of the DTFT 73 2 Rea
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The Frequency Domain Representation
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Sampling and Reconstruction of Anal
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Problems 97 % check >> error = max(
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Problems 99 Remember that the above
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Problems 101 P3.16 Foralinear, shif
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CHAPTER 4 The z-Transform In Chapte
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The Bilateral z-Transform 105 Im{z}
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Important Properties of the z-Trans
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Inversion of the z-Transform 113 wh
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Inversion of the z-Transform 115 be
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Inversion of the z-Transform 117 He
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System Representation in the z-Doma
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Solutions of the Difference Equatio
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Solutions of the Difference Equatio
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Page 184 and 185: Properties of the Discrete Fourier
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Page 208 and 209: The Fast Fourier Transform 189 Usin
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Page 212 and 213: The Fast Fourier Transform 193 Fina
Page 214 and 215: The Fast Fourier Transform 195 Let
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Page 222 and 223: Problems 203 determine the impulse
Page 224 and 225: Problems 205 and use this property.
Page 226 and 227: Problems 207 function x3 = circonvf
Page 228 and 229: Problems 209 where l is an integer.
Page 230 and 231: Problems 211 2. From the following
Page 232 and 233: Basic Elements 213 characteristics
Page 234 and 235: IIR Filter Structures 215 FIGURE 6.
Page 236 and 237: IIR Filter Structures 217 FIGURE 6.
Page 238 and 239: IIR Filter Structures 219 for i = 1
Page 240 and 241: IIR Filter Structures 221 6.2.6 PAR
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Page 244 and 245: IIR Filter Structures 225 □ EXAMP
Page 246 and 247: IIR Filter Structures 227 a1 = 1.00
Page 248 and 249: FIR Filter Structures 229 FIGURE 6.
Page 250 and 251: FIR Filter Structures 231 FIGURE 6.
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FIR Filter Structures 233 FIGURE 6.
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FIR Filter Structures 235 6.3.8 MAT
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FIR Filter Structures 237 FIGURE 6.
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Lattice Filter Structures 239 1.000
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Lattice Filter Structures 241 There
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Lattice Filter Structures 243 FIGUR
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Lattice Filter Structures 245 FIGUR
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Lattice Filter Structures 247 funct
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Lattice Filter Structures 249 Solut
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Representation of Numbers 251 new f
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Representation of Numbers 253 rule
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Representation of Numbers 255 Solut
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Representation of Numbers 257 Ten
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Representation of Numbers 259 negat
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Representation of Numbers 261 Two
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Representation of Numbers 263 For t
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Representation of Numbers 265 IEEE
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The Process of Quantization and Err
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Quantization of Filter Coefficients
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Problems 289 z −1 x(n) K 2 2 0.5
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Problems 291 x(n) −1/2 1/2 −2/3
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Problems 293 0.5 1 −1 z −1 2 1
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Problems 295 1. Direct form 2. Line
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Problems 297 P6.24 MATLAB provides
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Problems 299 4. Explain why the mag
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Problems 301 P6.38 An IIR bandstop
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CHAPTER 7 FIR Filter Design We now
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Preliminaries 305 1 + δ 1 1 1 −
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Properties of Linear-phase FIR Filt
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Window Design Techniques 323 where
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Window Design Techniques 325 Rectan
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Window Design Techniques 327 of att
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Window Design Techniques 329 Hammin
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Window Design Techniques 331 where
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Window Design Techniques 333 To dis
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Window Design Techniques 335 Ideal
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Window Design Techniques 337 Soluti
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Window Design Techniques 339 The id
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Window Design Techniques 341 □ EX
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Window Design Techniques 343 Ideal
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Frequency Sampling Design Technique
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Optimal Equiripple Design Technique
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Problems 375 h(n) 0.8 0.6 0.4 0.2 0
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Problems 377 where coefficients ˜c
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Problems 379 Plot the impulse respo
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Problems 381 2.02 1.98 Hr(ω) 1 0 0
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Problems 383 P7.34 Design a minimum
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Problems 385 P7.38 Design a minimum
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Some Preliminaries 387 8.1 SOME PRE
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Some Preliminaries 389 8.1.2 PROPER
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Some Special Filter Types 391 The c
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Some Special Filter Types 393 Magni
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Some Special Filter Types 395 Imagi
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Some Special Filter Types 397 (a) F
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Some Special Filter Types 399 circl
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Characteristics of Prototype Analog
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Analog-to-Digital Filter Transforma
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Lowpass Filter Design Using MATLAB
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Frequency-band Transformations 449
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Problems 461 1 0.8913 Magnitude Res
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Problems 463 1. Determine the value
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Problems 465 function [b,a] =afd(ty
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Problems 467 function [b,a] = mzt(c
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Problems 469 P8.35 Design a highpas
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Problems 471 Using this function, d
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Problems 473 1 0.9 Equiripple |H (e
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Introduction 475 x a (t) ADC F s =
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Decimation by a Factor D 477 as h(n
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Decimation by a Factor D 479 given
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Decimation by a Factor D 481 |X(ω
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Decimation by a Factor D 483 The ou
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Decimation by a Factor D 485 % (c)
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Interpolation by a Factor I 487 Sol
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Interpolation by a Factor I 489 Ide
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Interpolation by a Factor I 491 % (
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Sampling Rate Conversion by a Ratio
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FIR Filter Designs for Sampling Rat
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FIR Filter Structures for Sampling
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Problems 531 P9.3 Consider a signal
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Problems 533 1. Compute and plot th
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Problems 535 3. Let x(n) =cos(0.3π
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Problems 537 3. Using the fir2 func
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Analysis of A/D Quantization Noise
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Round-off Effects in IIR Digital Fi
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Round-off Effects in FIR Digital Fi
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Problems 591 P10.6 Consider the 1st
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Problems 593 which is implemented i
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Applications in Adaptive Filtering
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LMS Algorithm for Coefficient Adjus
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System Identification or System Mod
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Suppression of Narrowband Interfere
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Adaptive Channel Equalization 603 T
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Adaptive Channel Equalization 605 F
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CHAPTER 12 Applications in Communic
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Pulse-Code Modulation 609 called a
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Differential PCM (DPCM) 611 the sig
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Differential PCM (DPCM) 613 digits
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Adaptive PCM and DPCM (ADPCM) 615 a
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Adaptive PCM and DPCM (ADPCM) 617 F
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Delta Modulation (DM) 619 FIGURE 12
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Delta Modulation (DM) 621 FIGURE 12
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Linear Predictive Coding (LPC) of S
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Linear Predictive Coding (LPC) of S
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Dual-tone Multifrequency (DTMF) Sig
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Dual-tone Multifrequency (DTMF) Sig
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Binary Digital Communications 631 F
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Spread-Spectrum Communications 633
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BIBLIOGRAPHY [1] J. W. Cooley and J
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INDEX ’operator, 25 &operator, 26
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INDEX 639 adaptive delta modulation
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INDEX 641 finite-precision numerica
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INDEX 643 round-off effects, 580-59
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INDEX 645 DTFT formula, 152-153 err
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INDEX 647 optimal equiripple design
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INDEX 649 Real-valued exponential s
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INDEX 651 Sinusoidal signals, 83-84
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