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Round-off Effects in IIR Digital Filters 577 where ε 1 (n) and ε 2 (n) are the relative errors in the corresponding quantizers. The exact analysis even for the 1st-order case is tedious; hence we make a few practically reasonable approximations. If the absolute values of the errors are small, then we have ŷ(n − 1) ≈ y(n − 1) and ĝ(n) ≈ y(n); hence from (10.68a) we obtain e 1 (n) ≈ αε 1 (n) y(n − 1) e 2 (n) ≈ ε 2 (n) y(n) (10.69a) (10.69b) Furthermore, we make the following assumption about the noise sources: 1. ε 1 (n) and ε 2 (n) are white noise sources. 2. ε 1 (n) and ε 2 (n) are uncorrelated with each other. 3. ε 1 (n) and ε 2 (n) are uncorrelated with the input x(n). 4. ε 1 (n) and ε 2 (n) are uniformly distributed between −2 −B and 2 −B . Let x(n) beazero-mean, stationary random sequence. Then y(n) is also a zero-mean, stationary sequence. Hence from (10.69) σ 2 e 1 = |α| 2 σ 2 ε 1 σ 2 y (10.70a) σ 2 e 2 = σ 2 ε 2 σ 2 y (10.70b) Let the error in the output due e 1 (n) beq 1 (n) and that due to e 2 (n) be q 2 (n). Let h 1 (n) and h 2 (n) bethe corresponding noise impulse responses. Note that h 1 (n) =h 2 (n) =h(n) =α n u(n). Then the total error q(n) is with where q(n) =q 1 (n)+q 2 (n) (10.71) σ 2 q = σ 2 q 1 + σ 2 q 2 (10.72) ∑ ∞ ∑ ∞ σq 2 1 = σe 2 1 |h 1 (n)| 2 and σq 2 2 = σe 2 2 |h 2 (n)| 2 (10.73) 0 Hence using (10.72), (10.73), and (10.70), σq 2 = ( σe 2 1 + σe 2 ) ( ) ( ) 1 2 1 −|α| 2 = σy 2 1 (|α| 2 1 −|α| 2 σε 2 1 + σε 2 ) 2 0 (10.74) Using σε 2 1 = σε 2 2 =2 −2B /3, we obtain ( )( ) 2 σq 2 = σy 2 −2B 1+|α| 2 3 1 −|α| 2 Therefore SNR = σ2 y σ 2 q =3 ( 2 2B) ( 1 −|α| 2 1+|α| 2 ) (10.75) (10.76) 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.
578 Chapter 10 ROUND-OFF EFFECTS IN DIGITAL FILTERS or SNR dB =4.77+6.02B +10log 10 (1 −|α| 2 ) − 10 log 10 (1 + |α| 2 ) (10.77) which is also a very informative result. Some comments are in order. 1. The SNR in (10.76) was derived without assuming any input statistics, Hence the result is valid for a large class of inputs including white-noise, narrow-band, or wide-band signals. The floating-point arithmetic does not have to worry about the scaling or limiting input values since it can handle a large dynamic range. 2. Using 0
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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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IIR Filter Structures 223 Brow = r(
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IIR Filter Structures 225 □ EXAMP
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IIR Filter Structures 227 a1 = 1.00
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FIR Filter Structures 229 FIGURE 6.
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FIR Filter Structures 235 6.3.8 MAT
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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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Page 546 and 547: FIR Filter Structures for Sampling
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Page 620 and 621: Suppression of Narrowband Interfere
Page 622 and 623: Adaptive Channel Equalization 603 T
Page 624 and 625: Adaptive Channel Equalization 605 F
Page 626 and 627: CHAPTER 12 Applications in Communic
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Page 630 and 631: Differential PCM (DPCM) 611 the sig
Page 632 and 633: Differential PCM (DPCM) 613 digits
Page 634 and 635: Adaptive PCM and DPCM (ADPCM) 615 a
Page 636 and 637: Adaptive PCM and DPCM (ADPCM) 617 F
Page 638 and 639: Delta Modulation (DM) 619 FIGURE 12
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Page 642 and 643: 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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