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Quantization of Filter Coefficients 273 When Q[x] isdue to the rounding operation, then the error in the mantissa is between [− 1 2 2−B , 1 2 2−B ]. In this case we will denote the relative error by ε R . Then from (6.43), the absolute error, Q R [x] − x = ε R x, is between (− 1 2 2−B ) 2 E ≤ ε R x ≤ ( 1 2 2−B ) 2 E (6.62) Now for a given E, and since the mantissa is between 1 2 ≤ M
274 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS 6.8.2 EFFECT ON POLE-ZERO LOCATIONS One aspect can be reasonably analyzed, which is the movement of filter poles when a k is changed to â k . This can be used to check the stability of IIR filters. A similar movement of zeros to changes in numerator coefficients can also be analyzed. To evaluate this movement, consider the denominator polynomial of H(z) in(6.65) N∑ N∏ D(z) =1+ △ a k z −k ( = 1 − pl z −1) (6.67) k=1 where {p l }s are the poles of H(z). We will regard D(z) asafunction D(p 1 ,...,p N )ofpoles {p 1 ,...,p N } where each pole p l is a function of the filter coefficients {a 1 ,...,a N }—that is, p l = f(a 1 ,...,a N ), l =1,...N. Then the change in the denominator D(z) due to a change in the kth coefficient a k is given by ( ) ( )( ) ( )( ) ( )( ) ∂D(z) ∂D(z) ∂p1 ∂D(z) ∂p2 ∂D(z) ∂pN = + +···+ ∂a k ∂p 1 ∂a k ∂p 2 ∂a k ∂p N ∂a k (6.68) where from (6.67) ( ) [ ∂D(z) = ∂ ∏ N ( 1 − pl z −1)] = −z ∏ −1 ( 1 − pl z −1) (6.69) ∂p i ∂p i l=1 l≠i ( )∣ From (6.69), note that ∂D(z) ∣∣z=pl ∂p i =0for l ̸=i. Hence from (6.68) we obtain l=1 ( )∣ ( )∣ ( ) ∂D(z) ∣∣∣z=pl ∂D(z) ∣∣∣z=pl ∂pl = ∂a k ∂p l ∂a k or ( ) ∂pl = ∂a k ( )∣ ∂D(z) ∣∣z=pl ∂a k ( )∣ ∂D(z) ∣∣z=pl ∂p l Now ( ∂D(z) ∂a k ) ∣ ∣∣∣z=pl = ∂ ∂a k ( 1+ N∑ i=1 (6.70) a i z −i )∣ ∣∣∣∣z=pl = z −k∣ ∣ z=pl = p −k l (6.71) From (6.69), (6.70) and (6.71), we obtain ( ) ∂pl p −k l = ∂a k −z ∏ −1 i≠l (1 − p i z −1 ) ∣ z=pl p N−k l = −∏ i≠l (p l − p i ) (6.72) 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.
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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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Problems 135 4. x(n) =n 2 (2/3) n
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Problems 137 2. Using (4.32), deter
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Problems 139 It is also known that
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CHAPTER 5 The Discrete Fourier Tran
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The Discrete Fourier Series 143 den
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The Discrete Fourier Series 145 The
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The Discrete Fourier Series 147 DFS
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Sampling and Reconstruction in the
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The Discrete Fourier Transform 155
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Properties of the Discrete Fourier
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Linear Convolution Using the DFT 18
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The Fast Fourier Transform 187 5.6
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The Fast Fourier Transform 189 Usin
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The Fast Fourier Transform 191 and
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The Fast Fourier Transform 193 Fina
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The Fast Fourier Transform 195 Let
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The Fast Fourier Transform 197 N =2
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The Fast Fourier Transform 199 comp
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Problems 201 Clearly, ˜x 3(n) isdi
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Problems 203 determine the impulse
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Problems 205 and use this property.
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Problems 207 function x3 = circonvf
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Problems 209 where l is an integer.
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Problems 211 2. From the following
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Basic Elements 213 characteristics
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IIR Filter Structures 215 FIGURE 6.
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IIR Filter Structures 217 FIGURE 6.
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IIR Filter Structures 219 for i = 1
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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
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Page 270 and 271: Representation of Numbers 251 new f
Page 272 and 273: Representation of Numbers 253 rule
Page 274 and 275: Representation of Numbers 255 Solut
Page 276 and 277: Representation of Numbers 257 Ten
Page 278 and 279: Representation of Numbers 259 negat
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Page 282 and 283: Representation of Numbers 263 For t
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Page 286 and 287: The Process of Quantization and Err
Page 294 and 295: Quantization of Filter Coefficients
Page 308 and 309: Problems 289 z −1 x(n) K 2 2 0.5
Page 310 and 311: Problems 291 x(n) −1/2 1/2 −2/3
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Page 314 and 315: Problems 295 1. Direct form 2. Line
Page 316 and 317: Problems 297 P6.24 MATLAB provides
Page 318 and 319: Problems 299 4. Explain why the mag
Page 320 and 321: Problems 301 P6.38 An IIR bandstop
Page 322 and 323: CHAPTER 7 FIR Filter Design We now
Page 324 and 325: Preliminaries 305 1 + δ 1 1 1 −
Page 326 and 327: 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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